Research ArticleMixing Dyadic and Deliberative Opinion Dynamics in anAgent-Based Model of Group Decision-Making

George Butler Gabriella Pigozzi and Juliette Rouchier

Universite Paris-Dauphine PSL Research University CNRS UMR 7243 LAMSADE Place du Marechal de Lattre de Tassigny75016 Paris France

Correspondence should be addressed to Juliette Rouchier julietterouchierdauphinefr

Received 15 December 2018 Revised 6 May 2019 Accepted 10 June 2019 Published 7 August 2019

Academic Editor Jose Manuel Galan

Copyright copy 2019 George Butler et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In this article we propose an agent-based model of opinion diffusion and voting where influence among individuals anddeliberation in a group are mixed The model is inspired from social modeling as it describes an iterative process of collectivedecision-making that repeats a series of interindividual influences and collective deliberation steps and studies the evolution ofopinions and decisions in a group It also aims at founding a comprehensive model to describe collective decision-making as acombination of two different paradigms argumentation theory and ABM-influence models which are not obvious to combineas a formal link between them is required In our model we find that deliberation through the exchange of arguments reducesthe variance of opinions and the proportion of extremists in a population as long as not too much deliberation takes place in thedecision processes Additionally if we define the correct collective decisions in the system in terms of the arguments that shouldbe accepted allowing for more deliberation favors convergence towards the correct decisions

1 Introduction

In a group opinions are formed over affinities and conflictsamong the individuals that compose it Axelrod [1] a pioneerin opinion dynamics cast light on two key factors requiredto model the processes of diffusion namely social influence(ie individuals become more similar when they interact)and homophily (ie individuals interact preferentially withsimilar others) He was the first to show that a radicaldifferentiation of culture in a group could emerge from simpleimitation through dyadic interactions His results suggestedthat interactions through homophily and social influencecould lead to collective states or collective opinions whoseexplanation in many situations went beyond the individualor micro level Further it was found that these collectivestates could be characterized by quantities like statisticaldistributions and averages which explains why in recentyears a growing body of research has endeavored to identifythe conditions under which social influence at the micro(dyadic)-level translates into macropatterns of diffusionthrough repeated iterations [2] In particular several modelshave been developed to reproduce the emergent properties of

opinion diffusion which may be classified in two groups onthe one hand the discrete opinion models where opinionsor other ontological equivalents take discrete values [3 4]on the other the continuous opinion models where opinionsare represented by real numbers [5ndash10]

In the context of continuous opinion dynamics intro-duced by Deffuant et al [6] and later extended by otherauthors to include network effects [10 11] trust [12] andmany other social phenomena [7] individuals meet in ran-dom pair-wise encounters and then converge to a commonopinion if and only if their respective opinions are suffi-ciently close to each other in a kind of bounded confidencemechanism based on confirmation bias After some transientevolution and social dynamics this leads to final states inwhich either full consensus is reached or the population splitsinto a finite number of clusters such that all individuals in onecluster share the same opinion So far thesemodels have beenmostly applied to political issues such as societal cleavagesand the emergence of extremism

However these representations of social interactions inopinion dynamics fail to take into account everyday com-munication settings that characterize democracies such as

HindawiComplexityVolume 2019 Article ID 3758159 31 pageshttpsdoiorg10115520193758159

2 Complexity

meetings debate arenas and the Media in which individualsexchange points of view and can influence one anotherin a collective manner In effect individual actions thattranslate into collective decisions such as voting are shapedby factors related to the structure and size of the channelsof communication and deliberation When a group engagesin a collective discussion group size what arguments areadvanced how discussion is organized over time and theacceptability criteria for proposals may lead to a transfor-mation of preferences [13] and play a crucial role in con-sensus formation [14ndash16] For instance work in [14] defendsthat deliberation polarizes individuals in the direction oftheir initial opinions due to social pressure and to limitedknowledge (biased or unbalanced argument pool) withinthe discussing group In contrast from several empiricalstudies of deliberation processes other authors infer thatdeliberation can have a stabilizing or moderating effect onopinions [16 17] which they interpret as opinions becomingmore informed [18] balanced andor confused [13] Authorsin [15] express that deliberation may encourage moderateopinion consensuses if it is procedural or be polarizing ifit ego-involves the participants All these phenomena mayintroduce some degree of discrepancy into the otherwisewell-known steady states of opinions (clusters) obtained inclassical opinion dynamics and should be modeled some-how

Opinion diffusion has also been used to track conver-gence towards ldquocorrectrdquo or accurate opinions in groupsAlthough authors in [19] study how interactions amongagents diffuse true information they focus their analyses onnetwork effects and noisy signaling and not on deliberationprotocols In [20] Rouchier and Tanimura explore the dif-fusion of information about an exogenous ldquotrue staterdquo of theworld (represented by bits) through social interactions andlearning but assume the interactions to be only dyadic andnotdeliberative In our context a correct decision correspondsto one derived from a dialectical situation in which allthe arguments for and against a proposed alternative aretaken into account [15] The existence of an ideal criteria ofcorrectness of collective decisions can be used to evaluate theoutcome of deliberative procedures and democratic decision-making Since deliberation imposes regulatory conditions todecision-making processes one may attribute a rational anddemocratic value to the collective decisions obtained from itDeliberation is a way of getting closer to the ideal state inwhich a group judges propositions as if it had all argumentsat its disposal In [18] Barabas shows empirical evidence thatdeliberation increases knowledge and is correlated to correctresponses to objective questions Up until now opiniondiffusion models have not taken into account this dimensionand models mixing deliberative and dyadic interactions maywell do so In the same manner collective truth-seekingmodels can benefit frommore insight on processes of peer-to-peer information diffusion of the like of continuous opiniondynamics models

From a social welfare perspective a claim in favorof deliberation is that promoting dialogue leads to betterdecision-making where ldquobetterrdquo goes as improving socialwelfare Preference structures that are rationally untenable

or unjustifiable are eliminated from the pool of admissiblepreferences outright [21] We argue that different ways ofpartaking deliberation eg the structure and protocol ofdecision-making processes may allow for more accuratecollective decision-making

In this direction agent-based modeling proves to bean interesting method to study and observe the effect ofdeliberation on opinions For one it helps infer knowledgefrom models in which a multiplicity of different modesof communication among heterogeneous agents are ana-lytically difficult to describe second given that empirical-based conclusions on the topic may be costly and difficultto ascertain due to confronting ideologies and theories onthe topic (see [15 16]) it provides an alternative way oftesting the effects of collective decisions and deliberationprotocols on public opinion and not an exhaustive third itfurnishes an interesting modeling environment for collectivechoice analysis by giving the possibility to account for dif-ferent levels of decision-making and a diversity of influenceloops

The aim of ourmodel is to breach a gap between delibera-tion and opinion diffusionWemodel dyadic or ego-involveddynamics using an opinion diffusion model based on socialjudgment theory [9 22] whereas formal or deliberativediscussion is modeled using abstract argumentation theory[23ndash25] We build a process of collective decision-makingwhere deliberation and voting are necessary conditions forcollective choice as we are inspired from results in thedeliberative democracy [16 17 26] and social psychology[14 15] literature We present the effects of deliberation onthe opinions of a group and on its ability to correctly judgepropositionsWe call the latter the grouprsquos judgment accuracyWe also observe how deliberation impacts a grouprsquos ability toeventually vote in favor of proposals that are discussed andaccepted during deliberation in other words on the grouprsquoscoherence in decision-making Furthermore since a collectivedecision in the model is an outcome of a structured groupdecision-making process a sequence of deliberative anddyadic interactions among agents we seek to study the impactof its structure on the grouprsquos opinions judgment accuracyand coherence The frequency of deliberative interactionswithin a decision-making process the size of deliberation(number of agents that deliberate) and the majority votingrules used determine the collective acceptance of proposalsare considered to this end Ultimately we strive to create atool that helps policy-makers analyze deliberation protocolsand make studied decisions about them

Our model allows us to explore a new paradigm in opin-ion diffusion and answer the following research questionsconcerning decision-making in groups

(1) What effect can deliberation have on a grouprsquos opin-ion distribution when the opinion dynamics aredescribed using bounded confidencemodels In whatway are opinion dynamics through deliberation aloneinteresting

(2) How do the structure and protocol of deliberativedecision-making processes affect a grouprsquos distribu-tion of opinions coherence and judgment accuracy

Complexity 3

(3) In what way can controlling for protocol relate tosocial situations in which collective decisions arecoherent and accurate

Simulations show that deliberation yields on averagequalitative loose consensus and group polarization whilereducing the number of different opinion clusters over thedistribution of opinions In effect our model shows thatdeliberation has a significant overall impact on the distribu-tion of opinions (on its variance) and on the shifts in individ-ual opinion In particular when specifying opinion dynamicsas only deliberative the proportion of extremists and thevariance of opinions are lower and the shifts of opinionsgreater than in a nondeliberative dyadic opinion dynamicsmodel However when considering a mix of deliberationand dyadic interactions parameters that promote biggeror more frequent deliberation during decision-making pro-cesses increase the variance of opinions and the proportionof extremists and limit the effect of deliberation on opinionsThe majority voting quota rule to accept a proposal plays apreponderant role as it determines if the consensus-drivingpower of deliberation outweighs the propensity to dissensusobserved in bounded-confidence models with rejection

Themodel sheds light on the fact that a grouprsquos coherenceand judgment accuracy may depend significantly on howdecision processes are structured The number of debatesallowed and the number of agents that may participate inthem increase judgment accuracy in a marginally decreasingfashion but have little or no effect on the grouprsquos coherencein decision-making Last we point out that results areconditioned on how many arguments agents have at theirdisposal and on how they advance them during deliberation

The remainder of this paper is organized as follows inSection 2 we present the model and provide some necessarybasics to understand its implementation in the subsequentsection we introduce the metrics of interest and the cali-bration of the model In Section 4 we report and discussthe results of the simulations and in Sections 5 and 6respectively we survey related work and we conclude thearticle

2 A Model for Collective Decision-Makingwith Deliberation

We propose a system of collective decision-making amongagents made of three objects arguments agents and tablesfor deliberation Agents have agency arguments representpieces of information and tables validate collective decisionswhile organizing all deliberative interactions among agents

21 Model Overview In this section we propose an overviewof themodelWe quickly introduce the agents and the objectsthat make collective decisions possible in the multiagentsystemWe also provide a basic interpretation of the collectivedecision-making procedure which we illustrate with anexample

211 Overview of Agents and Objects in the Model Argu-ments are objects that relate to each other by a defeat relation

They are characterized by their support of some value orprinciple Agents are characterized by their sensitivity todeliberated ideas by the arguments they possess whichreflect their opinions and their knowledge about argumentsThey communicate one-to-one or collectively by the dint ofarguments in a public arena Communication leads them toan eventual update of their opinions Tables are entities thatcontain both agents and argumentsThey are controlled by acentral authority (CA) [27] that fixes the rules in the delib-eration process the frequency of deliberative interactionsand the conditions for collective acceptability of deliberatedinformation The existence of a central authority namely aperson or a machine that can reveal the correct epistemicstatus of any set of arguments can be equated to Habermasrsquoclaim that the ldquounforced force of the better argumentrdquo willtriumph in an ideal speech situation1 A central authoritymayalso be associated with Rancierersquos notion of ldquopolicerdquo that hedefines as ldquoan order of bodies that defines the allocation ofways of doing ways of being and ways of saying and thatsees that those bodies are assigned by name to a particularplace and taskrdquo [28] (p29)

212 Overview of the Decision-Making Process Let 119873 be agroup of agents that is asked to deliberate and vote on aproposal P Given an argument 119868 in favor of P they haveto judge on whether P is desirable or not The argument119868 or proposal argument determines how much the proposalsupports a principle P or its opposite notP A proposalis a sentence that indicates a way of attaining a goal orsolving a problem A principle derives from the notion ofvaluemdashvalues seen as fundamental social or personal goodsthat are desirable in themselves [29] Environmentalism andpatriotism are examples of values to choose proposals thatminimize environmental impact or maximize welfare areexamples of two not mutually exclusive principles

Agents are assumed to decide on the acceptance of theproposal on the basis of their opinions or their adherenceto the said principle whether they argue formally or infor-mally about P When agents are not deliberating they aresubject to random pair-wise influence when the deliberativeexchange of arguments concludes they are influenced by theacceptance status of the proposal that is being discussedOnly a fraction of all agents deliberate in each deliberativeexchange all agents are prone to pair-wise discussion Theyvote for proposals according to their opinions A proposalP is accepted if and only if the argument 119868 that is givenwith it is accepted after deliberation and the proposal is votedfavorably by amajority of agents To vote in favor of a proposalis considered to be equivalent to accepting the argument thatcomes along with it

Example 1 P = ldquoProtect the environmentrdquo is a principle aproposal may be P = ldquoReduce carbon emissions by 2030using electric carsrdquo To justify the proposal one may advancethe argument 119868 = ldquoElectric cars will reduce societyrsquos depen-dency on fossil fuels and will result in a reduction of carbonemissions for 2030 while proctecting the environmentrdquo thatexpresses the degree to which the proposal is in line withenvironmental protection An argument that tackles the

4 Complexity

a b c

Figure 1 Argumentation framework 119860119865 = (AR) with A =119886 119887 119888 andR = (119886 119887) (119887 119888) The labeling 119887 119888 119886 is conflict-free 119886 119888 119887 0 is the only complete labeling

proposal argument 119868 and that opposes the principle P couldbe 119886 = ldquoElectric cars may protect the environment byreducing effective carbon emissions but the batteries theydepend on are very pollutantrdquo The proposal P may beaccepted in the deliberation arena if some other argumentthat defeats 119886 say 119887 = ldquoBattery recycling businesses arehatching everywhere By 2020 it is very likely that scientistsfind a viable solution to chemical pollution due to batteriesrdquo isadvanced It is not accepted otherwise If 119868 is accepted thenPis collectively accepted if a majority of agents vote favorablyfor it

22 Arguments as the Basic Units of Collective Discussionamong Agents In this subsection we introduce the sim-plest object in our system arguments We present abstractargumentation theory and Caminadarsquos labeling approach toargumentation [30] that allows us to track the epistemic statusof arguments during deliberation

221 Arguments Arguments are objects that represent piecesof information that agents can understand They are infor-mational cues that enable agents to discuss with one anotherin a public collective context and make decisions on theacceptability of other pieces of informationThey are assumedto be nonfallacious In our approach each argument 119886 ismodeled by a real number V119886 isin [minus1 1] that stands forhow much 119886 respects or supports the principle P V119886 = 1means that argument 119886 is totally coherent with the principleP whereas V119886 = minus1 reads that 119886 is totally incoherent withthe principle P Arguments relate to each other through anincompatibility relation that states that one cannot standbehind two conflicting arguments Arguments are also char-acterized by their acceptability status that indicates whetherthey are accepted refuted or undecided in a given discur-sive context Arguments have an epistemic reach (119890119903) or amaximum number of arguments they can attack This canbe interpreted as the argumentrsquos level of generality or as thepotential level of argumentative conflict within the argumentpool

222 Abstract Argumentation Theory Deliberation definedas an exchange of arguments is modeled by confronting rep-resentations of different eventually contending argumentsIn our model we use abstract argumentation theory [23] torepresent deliberation where an argument is just a node in agraph like arguments 119886 119887 and 119888 in Figure 1 Abstract argu-mentation theorymodels incompatibility between argumentsand abstracts away from their internal structureThe intuitionis that if an argument 119886 attacks an argument 119887 (representedby an arrow from node 119886 to node 119887 as in Figure 1) a rationalagent cannot accept both 119886 and 119887

Formally let A be a finite set of arguments and R asubset of A times A called attack relation The attack relation isintransitive and we note (119886 119887) isin R the fact that argument 119886attacks or is incompatible with argument 119887 One says that anargument 119886 defends an argument 119888 noted (119886 119888) isin D if thereexists an argument 119887 such that (119886 119887) isin R and (119887 119888) isin R

(see Figure 1) An argumentation framework (119860119865 = (AR))is a digraph in which the nodes represent the argumentsand the arcs represent the attacks among them Given anargumentation framework119860119865 the classic problem in abstractargumentation resides in finding which of the arguments in119860119865 can be accepted rejected or left undecided

In the labeling approach [30] a label L119886119887(119886) isin Λ =119894119899 119900119906119905 119906119899119889 of an argument 119886 isin A denotes the epistemicstatus of 119886 Intuitively an argument is labeled 119894119899 if it isjustifiable and 119900119906119905 if it is not If 119886 is labeled 119906119899119889 it isconsidered to be in abeyance due to for example insufficientgrounds for it to be labeled 119894119899 or 119900119906119905 Furthermore given anargumentation framework 119860119865 = (AR) one calls labeling acomplete function

L A 997888rarr Λ = 119894119899 119900119906119905 119906119899119889119886 997891997888rarr L119886119887 (119886) (1)

that assigns a label to each argument in 119860119865 A labeling iswritten as a triplet⨉120582isinΛ120582L(A) where 120582L(A) = 119886 isin A |L119886119887(119886) = 120582 For example 119894119899 L(A) stands for the set ofarguments in A that are labeled 119894119899 in the labelingL(A) Alabeling is said to be legal if all argument it labels is legallylabeled An argument 119886 isin A is said to be legally labeled

(i) L119886119887(119886) = 119894119899 if forall119887 isin A st (119887 119886) isin RL119886119887(119887) = 119900119906119905if 119886 is not attacked or is only attacked by argumentsthat are themselves labeled 119900119906119905

(ii) L119886119887(119886) = 119900119906119905 if exist119887 isin A st (119887 119886) isin RL119886119887(119887) = 119894119899that is if 119886 is attacked by at least one argument that islabeled 119894119899

(iii) L119886119887(119886) = 119906119899119889 if not(forall119887 isin A st (119887 119886) isin RL119886119887(119887) =119900119906119905) and (forall119888 isin A st (119888 119886) isin RLa119887(119888) = 119894119899)or equivalently if there exists at least one argumentlabeled 119906119899119889 that attacks 119886 and there is no argumentlabeled 119894119899 attacking 119886

Roughly speaking a semantics (denoted by 120590) is a rationalitycriteria to decide which arguments to accept given an argu-mentation framework The basic normative requirementsfor labeling-based semantics and argument acceptabilityin abstract argumentation repose on conflict-free labelingsor labelings in which no two 119894119899-labeled arguments attackeach other and admissible labelings which are conflict-free labelings that ask for arguments that are defended bythe 119894119899-labeled arguments to be themselves 119894119899-labeled Thefamily of admissibility-based labelings goes from completelabelings to preferred and grounded labelings which arecomplete labelings that capture properties such as credulityand skepticism in argumentation Formally a labelingL forthe arguments in an argumentation framework (AR) is saidto be

Complexity 5

(i) conflict-free if ∄119886 119887 isin A st (119886 119887) isin R and 119886 119887 isin119894119899L(A)

(ii) admissible if conflict-free and forall119886 isin AL119886119887(119886) =119906119899119889 997904rArr 119886 is legallyL119886119887(119886)

(iii) complete if admissible and forall119886 isin AL119886119887(119886) =119906119899119889 997904rArr 119886 is legally labeled 119906119899119889

(iv) preferred if complete and if it maximizes 119894119899L(A)(v) grounded if complete and if it maximizes 119906119899119889L(A)

We choose admissibility-based semantics because for onethey supply a comprehensive model of collective reasoningthey allow for a meaningful parameterization of credulousand skeptical collective reasoning and last they are relativelyeasy to interpret since the differences between them arewell documented in the literature (eg [18]) Additionallyand in contrast to other rank-based or graded semantics inabstract argumentation admissibility semantics assume thatall arguments have the same weight

223 Abstract Argumentation and Discursive Situations inCollective Decision-Making Abstract argumentation is aconvenient formalism for argument-based decision-makingsince it ignores difficulties relative to the nature generationand number of arguments and posits the possibility of usinggraph theoretic tools to model (collective) reasoning in aclear coherent and easy way [31] Example 2 below providesa simple overview of a debate between two agents over theacceptability of the proposal ldquoTax the richrdquo

Example 2 Let P = ldquoTax the richrdquo and P = ldquoLiberalismrdquoFigure 1 represents the abstract argumentation frameworkobtained from the following arguments

(i) 119868 = 119888 = ldquoOnly the rich should be taxed because theypossess most of the capital in the countryrdquo

(ii) 119887 = ldquoIf you only tax the rich the rich will leave andthen yoursquoll have no one to taxrdquo

(iii) 119886 = ldquoThe rich will not leave because they have theirlivelihoods here and it would cost themmore to leavethan to pay the taxesrdquo

Agent 1 advances the proposal argument 119888 agent 2 arguesthat given the justification 119888 for P P is not tenable Agent1 defends his proposal by advancing 119886 and defeating theargument 119887The conclusion is that holding 119888 and 119886 is a tenableposition so the rich should be taxed Notice that by simplyaccepting 119888 and taking no position on 119887 nor 119886 is conflict-freeand still corresponds to the position advocating that the richshould be taxed

Abstract argumentation comes as an immediate appli-cation to our model in the construction of an ideal argu-mentation framework that loosely models Habermasrsquo idealspeech situation The ideal speech situation is important inour work because it corresponds to a normative state that isin practice difficult to reach and that allows us to observehow different deliberation protocols affect deliberative out-comes

In the model a label given to an argument provideda semantics 120590 is said to be ideal if it is obtained from astate of affairs in which all arguments are presented duringdeliberation We call ideal or consensual argumentationframework the argumentation framework (AR) = 119860119865120576containing all arguments in the system and all consensualattacks among them In a multiagent approach a consensualattack between two arguments 119886 119887 isin A is a couple (119886 119887)such that (119886 119887) isin R if and only if a certain majority ofagents recognizes that such attack exists In our model allagents agree on the attack relation over 119860119865120576 It follows thatif two agents advance two distinct arguments 119886 119887 isin A suchthat (119886 119887) isin R then all agents will recognize that suchconflict exists An immediate consequence of this assumptionis that all deliberated results are consensual even if an opinionconsensus2 on the principle is not necessarily reached

23 Deliberative Social Agents In this section we presentthe agents in our model We define them on the basisof their opinions on the principle P the arguments theypossess their knowledge of the relationship among themtheir behavior during deliberation and their sensitivity todeliberated proposals

231 Agents in Dyadic Social Interactions Every agent 119894 hasan opinion a relative position or degree of adherence 119900119894 isin[minus1 1] to a principle P and a couple (119880119894 119879119894) isin [0 2] times [0 2](119880119894 le 119879119894) of latitudes of acceptance and rejection respectivelyof informational cues They live in [0 2] since 0 and 2 arerespectively the minimum and maximum distances of anytwo informational cues in the system The idea behind thecouple (119880119894 119879119894) is that there exist levels of relative tolerancefrom which informational cues have either an attractive ora repulsive effect on the individual [22 32] An 119900119894 close to 1implies that agent 119894 fully supports the principle P and closeto -1 that she rejects principleP or equivalently fully supportsnotP Moreover if an agent 119894rsquos opinion is such that |119900119894| ge 075119894 is considered to have an extreme opinion a moderate oneotherwise 119880119894 may be considered as 119894rsquos uncertainty about herown opinion [33] or as the limit below which the objectshe judges may attract her 119879119894 may be seen as 119894rsquos bound oftolerance from which informational cues disgust her andconfirmher initial positionDifferent combinations of (119880119894119879119894)can be associated with agent 119894rsquos ego or personal involvementin discussion processes as it is vividly explained in [32]Finally agents are assumed to be sincere and precise whencommunicating their positions to one another (no noise inthe interactions)

232 Agents in Deliberative Social Interactions Agents voteand participate in deliberation because they are aware ofthe potential changes an accepted proposal may induce inthe opinion of the group After all proposals promote aprinciple that potentially leads to a shift in other agentsrsquoopinions Agentsrsquo incentive to participate in deliberation isbased on the idea that every single one of them wants tomake her point across and at the same time reach a correctcollective decision Agents are endowed with a probability

6 Complexity

-1 0 1

Arguments

Adherence to principle

Figure 2 Representation of agents and their arguments on theopinion or adherence to the principles spectrum The rectangleson the agents are their argument sacks Arguments are depicted bydots and arrows from the arguments to the sacks depict both thearguments they hold in the opinion spectrum and the number ofarguments they hold 119896119894 = 4 and 120601119890119894 = 075 and 120601119898119894 = 05 for allagent 119894119909119894 = 119891(119889(119900119894 V119868) 119901119886) of being attracted to a deliberated cueand a probability of being repulsed by it 119910119894 = 119892(119889(119900119894 V119868) 119901119903)The two probabilities are assumed to be independent Theyare a function of the distance between the opinion andthe proposal argument and of the grouprsquos sensitivity3 todeliberation (119901119886 119901119903) The former is decreasing on 119889(119900119894 V119868)and increasing on 119901119886 while the latter is increasing on both119889(119900119894 V119868) and 119901119903 Agents are also assumed

(i) to be capable of assessing the degree of support for Pof all arguments

(ii) to trust4 one another when they utter informationalcues

233 Agents and Voting At the end of the decision-makingprocess agents vote on whether they agree or not on the pro-posal argument that has been discussed during deliberationVoting for an agent is the expression of her opinion in thefinal phase of the decision-making process An agent 119894 is saidto vote favorably for a proposal P of justification argument119868 if and only if V119868 times 119900119894 ge 0 or equivalently if the proposalargument does not adhere to the opposite of the principle 119894supports5

234 Agent Knowledge of Arguments Let A be a finite setof arguments Each agent 119894 has a sack of arguments A119894 subA of size |A119894| = 119896119894 whose content reflects her relativeposition 119900119894 on P (see Figure 2) For extreme-opinion agentsa proportion 120601119890 ge 05 of the 119896 arguments in their sacks areof the same adherence to the principle than their opinions

whereas for moderate agents a proportion 120601119898 le 120601119890 oftheir 119896 arguments are of the valence of their opinions Thehypothesis feeds from results found in [22] stating that ldquoanindividual places a verbal statement on an issue both in termsof the itemrsquos relative proximity to his own position and thelatitude which is acceptable to him around that focal point ofacceptancerdquo

The arguments in an agent 119894rsquos sack are those that sheknows how to use and advance in a deliberative interac-tion The size of the sack represents agent 119894rsquos ability tocommunicate in a deliberative context It follows that eachagent 119894 possesses partial knowledge (119870119894 sub A times A) of theattack relation between the arguments in A and of 119860119865120576which she derives from the arguments in her sack andwhat she observes in deliberation Knowledge of argumentsis assumed to be ldquoattack-orientedrdquomdashan agent 119894 knows anattack (119886 119887) isin A119894 times A if she knows upon observationthat neither she nor the group can rationally accept 119886alongside 119887 She may be aware of the existing defense relationamong arguments when concatenating the information sheaccumulates on their attack relation during deliberationShe may use this information strategically in deliberativecontexts

Let 119889119894119904119905 A times A 997888rarr N be the length of the shortestpath between two elements in the argumentation frameworkinduced by an agent 119894rsquos knowledge119870119894 119894 sees an argument 119886 asan attacker (defender) of an argument 119887 if119889119894119904119905(119886 119887) equiv 1mod 2(119889119894119904119905(119886 119887) equiv 0mod 2) An argument 119886 that is at an odddistance from another argument 119887 attacks it since either itdirectly attacks the argument or it attacks a defender of theargument Likewise if 119886 is at a pair distance of 119887 it attacksan attacker of 119887 and thus defends it If such path does notexist then the distance is undefined and the agent does notsee either argument as an attacker or defender of the otherFurther let AF119889 = (A119889R119889) denote the current state ofaffairs in deliberation An agent 119894rsquos knowledge ofR is the setof attacks and defenses she can infer from the arguments sheknows (119860119905119905119894119863119890119891119894) and the attacks and defenses she can inferfrom deliberation (119860119905119905119889119894 119863119890119891119889119894 )

119870119894 = 119860119905119905119894 cup 119860119905119905119889119894 cup 119863119890119891119894 cup 119863119890119891119889119894 (2)

where

(i) 119860119905119905119894 = (119886 119887) isin A119894 timesA119894 | 119889119894119904119905(119886 119887) equiv 1mod 2)(ii) 119863119890119891119894 = (119886 119887) isin A119894 timesA119894 | 119889119894119904119905(119886 119887) equiv 0mod 2(iii) 119860119905119905119889119894 = (119886 119887) isin (A119894 cup A119889) times A119889 | 119889119894119904119905(119886 119887) equiv1mod 2)(iv) 119863119890119891119889119894 = (119886 119887) isin (A119894 cup A119889) times A119889 | 119889119894119904119905(119886 119887) equiv0mod 2

Notice that agents have no restriction on the amount ofinformation they can carry and use during deliberation Themodel implies that agents have the ability to use and processall information on the attack relations if they could observethe attacks and that they all have equally high cognitivecapacity Agent knowledge resets at the end of each decision-making process but argument sacks stay untouched In otherwords argument sacks are static in the model

Complexity 7

235 Agent Behavior during Deliberation Agents maybehave in twodifferentways in deliberationTheymay behavenaively or focusedly Naive agents use deliberation to voicetheir opinions on the principle through arguments Focusedagents strategically argue in favor of proposal arguments thatsupport the principle they favor In both cases agents advancearguments in terms of the argumentsrsquo relative proximity totheir own positions [22]

Let 119886119894 isin A119894 denote the argument an agent advancesin a debate 119889 (of a deliberation process 119863119868) over a centralargument 119868 and V119886 the valence of an argument 119886 isin A Let

(i) 119891119894(119886) equiv (119900119894 times V119886 gt 0 or 119900119894 = V119886 = 0) the formula thatis satisfied if the argument 119886 is of the same valence asthe opinion of agent 119894

(ii) 119892119894(119886) equiv (119886 119887 isin A119894A119889 and forall119887(|119900119894 minus V119887| gt |119900119894 minus V119886|)) theformula that is satisfied only if the argument 119886 is in 119894rsquossack (A119894) and not in the debate and is the closest interms of adherence to the principle P to 119894rsquos opinion

(iii) ℎ119894(119886) equiv (119891119894(119886) and 119892119894(119886)) the conjunction of thepreceding formuli indicating that an argument 119886 is inagent 119894rsquos sack and not in the debate of the same signas 119900119894 and closest to 119900119894 in absolute value

Naive agents choose 119886119894 = 119886 such that ℎ119894(119886) is true given anyproposal argument 119868 Agents that behave strategically or tosay focusedly choose their moves in the debate as follows(119873119868119871 indicates no move or null argument)

(a) if agent 119894rsquos position with respect to P is of the samesign as the proposal argumentrsquos adherence to P (119891119894(119868)) sheadvances an argument 119886 such that

(i) ℎ119894(119886) and ((119886 119868) isin 119863119890119891119889119894 )) is true If 119886119894 = 119873119868119871(i1) notℎ119894(119886) and 119892119894(119886) and ((119886 119868) isin 119863119890119891119889119894 ) (chooses anargument not of the same sign and closest to heropinion and defending 119868) is true If 119886119894 = 119873119868119871(ii) ℎ119894(119886) and ((exist119888 isin A119894)((119888 119868) isin 119860119905119905119889119894 and (119886 119888) isin 119860119905119905119894))(anticipates an attack from an argument 119888 on 119868 andadvances an argument that attacks the argument 119888)6is true If 119886119894 = 119873119868119871(ii1) notℎ119894(119886)and119892119894(119886)and ((exist119888 isin A119894)((119888 119868) isin 119860119905119905119889119894 and(119886 119888) isin119860119905119905119894)) is true If 119886119894 = 119873119868119871(iii) ℎ119894(119886) and ((119886 119868) notin 119860119905119905119889119894 ) (simply does not attack 119868)is true If 119886119894 = 119873119868119871(iii1)notℎ119894(119886)and119892119894(119886)and((119886 119868) notin 119860119905119905119889119894 ) is true If 119886119894 = 119873119868119871(iv) 119886119894 = 119873119868119871 (the agent will advance no argument)

(b) if agent 119894rsquos position with respect toP is of the oppositesign of the proposal argumentrsquos adherence to P (not119891119894(119868)) sheplays the argument 119886 such that

(i) ℎ119894(119886) and ((119886 119868) isin 119860119905119905119889119894 ) 7 If 119886119894 = 119873119868119871(i1) notℎ119894(119886)and119892119894(119886)and ((119886 119868) isin 119860119905119905119889119894 ) is true If 119886119894 = 119873119868119871(ii) ℎ119894(119886) and ((exist119887 isin A119894)((119887 119868) isin 119863119890119891119889119894 and (119886 119887) isin 119860119905119905119894))(anticipates a defense from an argument 119888 to 119868 andadvances an argument that attacks 119888) is true If 119886119894 =119873119868119871

(ii1)notℎ119894(119886)and119892119894(119886)and((exist119887 isin A119894)((119887 119868) isin 119863119890119891119889119894 and(119886 119887) isin119860119905119905119894)) is true If 119886119894 = 119873119868119871(iii) ℎ119894(119886) and ((forall119887 isin A119889)((119887 119868) isin 119860119905119905119889119894 and (119886 119887) notin 119860119905119905119889119894 ))(avoids attacking any attacker of 119868) If 119886119894 = 119873119868119871(iii1)notℎ119894(119886)and119892119894(119886)and((forall119887 isin A119889)((119887 119868) isin 119860119905119905119889119894 and(119886 119887) notin119860119905119905119889119894 )) If 119886119894 = 119873119868119871(iv) 119886119894 = 119873119868119871

In English a focused agent that is opposed to the proposalattempts to either attack it directly or attack an argument thatis defending it whereas an agent that agrees with the proposalattempts to either defend it or avoid attacking it altogetherThis behavioral approach is similar to debate protocols inabstract argumentation but for the fact that agents mayadvance arguments that do not result in an ldquoadvancementrdquoof the debate In our case the debates stop for other reasons(see Section 24)

Two important points deserve to be highlighted Firstlyfocused agents with small argument sacks will regularly findthemselves applying rules (119894119894119894) or (119894V) In effect if they donot have enough arguments to infer attacks and defensestheir behavior in deliberation is likely to be similar to that ofthe naive agents Secondly the naive and focused behavioralassumptions presented here are very different in terms ofcomputational complexity The first assumes that the agentobserves the debate but what she sees has no effect on hercourse of action ie the argument she voices is uniquelydetermined by how she feels about the principle behindthe proposal argument In the second an agent debatesessentially to knock out any proposal argument she disagreeswith8

24 Tables for Deliberation Tables are the physical or virtualplaces where the exchange of arguments occurs Agentsdeliberate on these tables to determine whether the proposalargument is acceptable or not and are thus subdued bythe deliberation procedure imposed by the tablersquos centralauthority (CA) [27] The CA decides on the structure andlength of the collective decision-making process and thedeliberation procedure It controls the percentage of agentsfrom the population that actively participates in the debate(119899119863) and the labeling-based semantics used to extract accept-able arguments from the framework (120590) The percentage 119899119863denotes either the proportion of agents in the population thatgets to advance arguments in a population-scale deliberationor an independent sample of agents summoned to activelyparticipate in the debate 120590 stands for the procedure used toconclude on the epistemic status of the proposal argumentand the arguments advanced during deliberation

The CA has the ability to stop debates at will using a stoprule that inherently depends on the number of debates thatought to take place before a decision is deemed sufficientlydiscussed (119898) the maximum number of debates that cantake place before abandoning deliberation (119898) and the labelgiven to the proposal argument 119868 (119904119903(119898119898L119886119887(119868))) 119904119903is a Boolean function whose value ldquotruerdquo signals the callfor a vote andor the end of the decision-making process119898 is associated with a minimal dialectical or epistemic

8 Complexity

requirement to consider a proposal for voting and to a lowerbound of the length of the deliberation process 119898 providesan upper bound Moreover the CA controls the size of thetime interval between debates (119905119863) the collective decisionrule (eg if there is voting on the deliberated proposals) andthe majority quota rule for accepting proposal arguments(120572) in the decision-making process 119905119863 may represent thefrequency or density of pair-wise interactions in a decision-making process

241 The Construction of a Decision-Making Process Adeliberation process or debate in ourmodel is a tool to obtainlabels for proposal arguments that are as close as possible tothe ideal or consensual ones To define a deliberation processformally we introduce the notion of debate step as constituentof a deliberation process Informally a debate step is a timestep in which a debate occurs Formally and more event-oriented a debate step (1198891198731015840 1198611015840 1198781015840) of a debate 119863119868 on 119868 isa quadruplet composed of a set of agents 1198731015840 sub 119873 a set ofarguments 1198611015840 sub ⋃119894isin1198731015840 A119894 a set of attack relations 1198781015840 sub Rand a mapping

119889 (AF 2A times 2R) 997888rarr AF

(119860119865 (1198611015840 1198781015840)) 997891997888rarr 1198601198651015840 = (119861 cup 1198611015840 119878 ⊔ 1198781015840) (3)

that adds the arguments in 1198611015840 and some attacks in 1198781015840 to aframework 119860119865 = (119861 119878) and where 119878 ⊔ 1198781015840 = (119886 119887) isin 119878 cup 1198781015840 |119886 119887 isin 119861 cup 1198611015840 In the same spirit we define a deliberationprocess119863119868 on a proposal argument 119868 as a sequence of debatesteps (119889119873119904 119861119904 119878119904)119904gt0 such that1198730 = 0 and

119889((119904minus1⋃119895=0

119861119895 119904minus1⨆119895=0

119878119895) (119861119904 119878119904)) sub (AR) (4)

with

(i) 1198610 = 119868 (the initial argument is the proposal argu-ment)

(ii) 1198780 = 0 (no reflexive attack from 119868 to 119868 is allowed)(iii) (forall119904 gt 0)(⋃119895lt119904 119861119895cap119861119904 = 0) (any newly added argument

to the framework has yet to be added to it)(iv) (forall119904 gt 0)(⨆119895lt119904 119878119895 cap 119878119904 = 0) (any newly considered

attack among arguments cannot be declared amongarguments that are not in the framework)

(v) lim119904997888rarrinfin(⋃119904gt0 119861119904 ⋃119904gt0 119878119904) sube (AR) (the system isstable no arguments are created during deliberation)

Finally let L119886119887119889(119868) denote the label given to argument 119868 ata debate step 119889 during a deliberation process 119863119868 A decision-making process notedD119868 = (119899119863 119905119863 119904119903(119898119898L119886119887119889(119868)) 120572 120590)over a proposalwhose justification argument is 119868 is a sequenceof debate and nondebate steps (119911119905)119905isinN 119911119905 isin 119889 119889 such that

(i) 119889 are debate steps and 119889 nondebate steps that corre-spond to time steps at which there are no debates

(ii) If 119905(mod 119905119863) equiv 0 then 119911119905 = 119889 and 119889 otherwise

For Against

Table

CA

in out

out out in

out in

proparg

Figure 3 A representation of a table of deliberation betweentwo groups of 3 agents with 4 arguments each Each agent hasadvanced an argument Agents tacitly agree on the attacks betweenthe arguments The CA determines the labeling using the groundedlabeling-based semantics

d d d d d d d d d d d d dTime 0 1 2 3 4 5 6 7 8 9 10 11 12I

Figure 4 A decision-making processD119868 or collective decision stepstarting at 119905 = 0 composed of debate (119889) and nondebate (119889) substepsDebate steps are shaded The number of nondebate steps betweendebates is 119905119863 = 3 the number of debates in the decision-makingprocess is119898 = 4

(iii) The subsequence (119911119897119905)119897119905isinN of 119911119905 with 119897119905 = 119905 such that119905(mod 119905119863) equiv 0 is a deliberation process with |1198731015840| =119899119863 times |119873|(iv) 119904119903(sdot sdotL119886119887119889(119868)) pursues the deliberation process as

long asL119886119887119889(119868) = 119906119899119889 under the semantics 120590(v) The length of the sequence |119911119905| or equivalently the

duration of the process is somewhere in the interval⟦119898119905119863 119898119905119863⟧(vi) The final labeling for the proposal argument 119868

L119886119887(119868) is determined by a combination of its delib-erated labelL119886119887119889(119868) and a majority vote of majorityquota 120572

A decision-making process ends when a final decision hasbeen taken concerning the acceptability of the proposal Peg 119868 has been deemed unacceptable (L119886119887119889(119868) = 119900119906119905) or amajority of agents have voted againstP For a representationof a deliberative interaction on a table see Figure 3 for one ofa decision-making process see Figure 4

Please note that every collective decision can be seen asa ldquotime steprdquo in as much as it describes how agents updateopinions and make collective decisions In the model thedecision-making process and the parameters define the sub-steps or events that occur within the time step Henceforth

Complexity 9

Fix proceduralsm m α nD tD σ Init count m larr 0

Given proposal anargument I is randomly generated

nD times |N | agents arerandomly picked

m larr m + 1

Begin decision processon proposal at time t

t larr t + 1Deliberation step m

larr 0

t larr t + 1dyadic opinionupdate Eq (5)tm larr tm + 1

m ge m

tm gt tD

ℒd (I) = in

m ge m

yields ℒ(I)

ℒ(I) = in Deliberationupdate Eq (6)

End decision

yes

no

no

no

yes

yes

no

yes

yes

yes

no

no

I

Vote on ℒd(I)

ℒd (I) = und

ＪＬＩ＝ＭＭ I

Init count tmyields ℒd(I)

Figure 5 Process chart for a decision processD119868 on a proposalP 119905 corresponds to time and 119898 (119905119898) corresponds to the number of debate(non-debate) steps that have taken place in the deliberation process (in between two debate steps) over time for the mixed discussion modelWe useL119886119887(119868) andL(119868) interchangeably in the figure for presentation

comparing simulations on the basis of the different decision-making processes translates into comparing these collectivedecision steps

242 Deliberation Protocol To define the deliberation proto-col held in the table either as a consequence of the definitionprocess or as a statement we assume the following

(i) Agents may decide not to contribute to the debate(ii) All agents have the same probability conditional to

their opinions of being picked to participate in adebate or deliberation step

(iii) There is no restriction on the number of times eachagent can participate in a deliberation process

(iv) Each agent may only place one argument per debatestep

(v) Arguments that have already been advanced in thedebate may attack newly placed arguments9

The deliberation or debate protocol goes as follows

(1) The CA randomly generates and makes public acentral argument or proposal argument 119868 and informsall agents about the rules of the decision-makingprocess

(2) The CA randomly draws two sets of (119899119863 times |119873|)2agents with divergent views10 onP and P It mergesthem to create the set of debaters1198731015840

(3) Each agent 119894 advances an argument from her sackA119894 The CA makes sure that there are no repeatingarguments (agents already take argument repetitioninto account)

(4) The CA establishes the debate step 119889rsquos argumentationframework and computes its labelingL120590

119889(A119889) using120590(5) If the computed label for 119868 is L119886119887119889(119868) = 119906119899119889

or the number of debates steps is inferior or equalto 119898 at time 119905 then the CA stops the debate andresumes it at the (119905 + 119905119863 + 1)rsquoth time step by repeating(1) (2) (3) (4) and (5)

(6) Let L119886119887(119868) be the final label given to the proposalargument 119868 If voting is allowed then if more than120572 times |119873| agents agree with 119868 119868 is accepted (L119886119887(119868) =119894119899) it is rejected if strictly less than 120572times |119873| agree withit (L119886119887(119868) = 119900119906119905) When there is a tie ifL119886119887119889(119868) =119906119899119889 then L119886119887(119868) = 119900119906119905 if L119886119887119889(119868) = 119894119899 thenL119886119887(119868) = 119894119899 When voting is not part of the decision-making process (120572 = 0)L119886119887119889(119868) = L119886119887(119868)

One important point to notice about this protocol is that itinduces a stop rule 119904119903 (cf steps (5) and (6) of protocol) forthe decision process and thus it always ends Either agentsdebate and agree on the proposalrsquos acceptability throughargumentation or after 119898 debate steps they directly voteon it so that a decision concerning the proposal is alwaysreached Refer to Figure 5 for a sweeping description of thedeliberation protocol and the decision-making process

10 Complexity

-1

-1 1

1

Latitude of acceptance

Latitude of rejection Agent i

Agent j

Figure 6 A dyadic interaction between agents 119894 and 119895 as expressedin (5) Agent 119894rsquos opinion falls within agent 119895rsquos latitude of noncommit-ment and conversely Neither agent affects the opinion of the other

25 Opinion Dynamics for Social Interactions In this subsec-tion we describe the opinion diffusion model based on pair-wise interactions among agents and deliberation Sherif rsquosand Hovlandrsquos social judgment theory [22] motivates part ofour approach It describes how individualsrsquo opinions changeon the basis of their attitude structures Attitude structuresrefer to the relative scope width or latitude of categoriesused by individuals when evaluating information namelythe latitudes of acceptance rejection and noncommitment[32] The idea behind this theory is that individuals changetheir positions only in accordance to how far or close thecommunicative cues they receive are from (to) their anchorpositions It holds that if communicative cues are far (close)from (to) an agentrsquos position say over her latitude of rejection(acceptance) then the agent shifts her position away from(towards) the position defended by the cues In the casewherethe cues fall within the agentrsquos latitude of noncommitmenther position does not change (see Figure 6 and Equation (5))

251 Pair-Wise or Dyadic Opinion Dynamics As agentsmay communicate and deliberate collectively they may alsoengage in one-to-one conversations with other agents toponder on their positions We loosely associate this type ofcommunication with dyadic nonargumentative exchange ordiscussion based on fallacious arguments and persuasionIn the light of social judgment theory and the descriptionof agents in the system we model pair-wise symmetricinteractions following the opinion dynamicsmodel in [9] Anagent 119895rsquos influence on an agent 119894rsquos opinion at time 119905 is governedby the ensuing difference equation

119900119905+1119894 =

119900119905119894 + 120583119894 (119900119905119895 minus 119900119905119894) 119894119891 10038161003816100381610038161003816119900119905119894 minus 11990011990511989510038161003816100381610038161003816 lt 119880119894119900119905119894 minus 120583119894 (119900119905119895 minus 119900119905119894) 119894119891 1003816100381610038161003816119900119905119894 minus 119900119905119894 1003816100381610038161003816 gt 119879119894119900119905119894 119900119905ℎ119890119903119908119894119904119890(5)

where the parameter 120583119894 isin [0 05] controls for the strengthof the attraction and repulsion in social influence and 119879119894and 119880119894 are the latitudes of rejection and acceptance for agent119894 respectively The parameter 120583119894 may be thought of as therelative importance agent 119894 gives to the opinions of her peersor analogously the weight she gives to her own opinionwhen updating 120583 lt 05 means that agents will never givemore weight to the opinions of other agents than to their

own (egocentric bias) When two agents 119894 and 119895 discuss iffor instance 119895 advances an informational cue (argumentpersuasion tactic) and it happens to be close enough to119894rsquos opinion anchor then 119894 shifts her opinion towards thedirection of 119895rsquos informational cue The symmetric influencefrom agent 119894 to agent 119895 takes effect in the same way (seeFigure 6)

We add to the dyadic dynamics a rule of encounters ateach step of pair-wise interactions each agent meets exactlyone other agent at random This rule may be associated withrandom day-to-day encounters among agents Steps of socialinfluence correspond to nondebate steps (119889) in decision-making processes

252 Deliberative Opinion Dynamics We define an opinionupdate equation that links the proposal arguments that areadvanced during deliberation and opinions We combine theuncertain and probabilistic nature of the effect of delibera-tion have it a moderating [15 16] or polarizing [14] one witha mechanism similar to the one in social judgment theorybased on the distance between different informational cuesand opinion anchors [32] The probabilistic modeling of theopinion update can be related to deliberation encouragingopen-mindedness11 during collective discussion [18] Fromthis choice it follows that deliberated cues can affect even themost extreme of agents as opposed to some classic opiniondiffusion models (eg [5 8]) where once agents becomeextremists they may no longer become moderate

LetP be a proposal V119905119868 the proposal argument 119868rsquos level ofsupport for a principle P and 119900119905119894 an agent 119894rsquos opinion at time119905 Then given the difference 120575119905119894 = (V119905119868 minus 119900119905119894)2 we define 119894rsquosprobability of being attracted to a decisionrsquos informational cue119868 by 119909119905119894 = 119901|120575119905119894 |119886 where 119901119886 denotes a general probability param-eter that characterizes how important deliberated results arefor the group The parameter may also be interpreted asthe grouprsquos tendency to be swayed by a decisional majoritySimilarly we define 119894rsquos probability of being repulsed from adecision informational cue 119868 by 119910119905119894 = 1199011|120575119905119894 |119903 where 119901119903 denotesa general probability parameter that characterizes the grouprsquosdislike of deliberated results 119901119903 can also be thought of as thegrouprsquos distrust of the system symbolized by deliberation anddemocracy

Let L119886119887(119868) denote the epistemic status of the proposalargument 119868 at the end of a decision process over P everyagent 119894 updates her opinion as follows

(i) IfL119886119887(119868) = 119894119899

119900119905+1119894 =

119900119905119894 + 2120574119894120575119905119894 119908119894119905ℎ 119901119903119900119887119886119887119894119897119894119905119910 119909119905119894119900119905119894 minus 2120574119894120575t119894 119908119894119905ℎ 119901119903119900119887119886119887119894119897119894119905119910 119910119905119894119900119905119894 119908119894119905ℎ 119901119903119900119887119886119887119894119897119894119905119910 1 minus 119909119905119894 minus 119910119905119894

(6)

(ii) IfL119886119887(119868) = 119900119906119905 119900119905+1119894 = 119900119905119894where 120574119894 isin [0 05] is the strength of repulsion and attractionin the dynamicThemeaning of 120574119894 is analogous to the one of120583119894in the dyadic interactionsmodel but for the observation that 119894

Complexity 11

weights her opinion to an argument rather than to an opinionThe interpretation of this dynamics is straightforward Ifthe deliberated proposal argument is close to an agentrsquosopinion then it is very likely that the agent shifts its opiniontowards it Please note that the probability 119909119905119894 (119910119905119894 ) that anagent is attracted (repelled) to (from) an accepted deliberatedproposal argument is bounded below (above) by 119901119886 (119901119903)(equivalently 119909119905119894 isin [119901119886 1] 119910119905119894 isin [0 119901119903]) Steps of deliberativeopinion dynamics are associated with debate steps in which aconcluding collective decision is made

253 Mixed Opinion Dynamics Opinion dynamics in thecontext of decision-making processes can be best described asa combination of the two preceding dynamics In this opiniondynamics model agents are engaged in a democratic systemin which they deliberate vote for proposals and occasionallydiscuss with one another on the principle supported by theproposalsThe opinion dynamics for an agent 119894 can be writtenas a sequence of dyadic and deliberative opinion updates

119900119905+1119894 =

119864119902119906119886119905119894119900119899 (5) 119894119891 119905 equiv 0 (119898119900119889119905119863) and 119904119903 (sdot) = 119891119886119897119904119890119864119902119906119886119905119894119900119899 (6) 119894119891 119905 equiv 0 (119898119900119889119905119863) and 119904119903 (sdot) = 119905119903119906119890119900119905119894 119900119905ℎ119890119903119908119894119904119890

(7)

where 119904119903(sdot) is the decision processrsquos stop rule 119905119863 the numberof time steps between each deliberation step Equation (5)describes the dynamics for the dyadic interactions and Equa-tion (6) posits the changes in opinions due to the deliberatedcues Equation (5) applies when agents are not deliberatingand when each agent encounters another agent to exchangeinformation that may lead to local opinion updates Whenthere is deliberation a handful of agents deliberate and ifthey reach the stop condition imposed by the table all agentsvote for the proposal They update (see Equation (6)) theiropinions on the basis of the proposal argumentrsquos support forthe principle and the result of the scrutiny Otherwise thereis no change in the opinions of the agents A vote indicatesthe end of the decision-making process

3 Experiments and Calibration

In this section we introduce the metrics that enable us toobserve the simulations and characterize the calibration ofthe model

31 Observations and Initialization In this section wedescribe the simulations and the protocol used to test andobserve the results of our model We introduce the metricsof interest explain the calibration of the model describesome results obtained from the simulation data used forcalibration and conclude with a brief discussion on theexpected outcomes of the mixed opinion dynamics model

311 Metrics or Statistics of Interest Let 119878 denote the end ofa simulation We are interested in the effect of deliberationand of the modelrsquos procedural parameters on the followingmetrics or statistics

(i) Variance of opinions (119881119886119903(119900)) the variance ofopinions at time 119878 Since opinions live in the unionof the positive and negative unit intervals 119881119886119903(119900) isin[0 1] The higher the variance of the distribution isthe more ldquodiverserdquo the opinions are(ii) Proportion of extremists in the population(119901119903119900119901119890119909) the proportion of agents in the populationwith opinions such that |119900119894| ge 075 A high proportionof extremists makes ldquohealthyrdquo consensus12 difficult toreach and deliberation more or less informative(iii) Shifts of opinions (119878ℎ) [33] statistic that measuresthe aggregated change in individual opinion at time 119878with respect to aggregated individual opinion at thebeginning of the simulation

119878ℎ = 2sum119894isin119873 10038161003816100381610038161199000119894 minus 1199001198781198941003816100381610038161003816max119894isin1198731199000119894 minusmin119894isin1198731199000119894 (8)

119878ℎ is positive and bounded above by 2|119873| since 119900119894 isin[minus1 1] A low shift statistic implies that the processhas a small impact on opinions(iv) Judgment or consensual inaccuracy (119890119888) consen-sual accuracy of a group consists of an ad hoc statisticmeasuring a grouprsquos ability to infer correct labels forproposal arguments from a decision-making processgiven the ideal consensual labeling based on full-informationL120590

120576 (A) We use a Hamming-based dis-tance on labelings [34] to define the statistic

119890119888 = 1|I| sum119868isinI1198861198681003816100381610038161003816L119886119887120576 (119868) = L119886119887 (119868)1003816100381610038161003816 (9)

whereI is the set of all discussed proposal argumentsand 119886119868 = 12 ifL119886119887120576(119868) = 119906119899119889 or ifL119886119887(119868) = 119906119899119889119886119868 = 1 otherwise119890119888 lives in the interval [0 1] An inaccuracy statisticclose to 1 indicates that agents subdued to a particulardecision-making process make many mistakes injudging the labels of the proposal arguments Notethat when there is voting all proposal arguments arelabeled either 119894119899 or 119900119906119905 after deliberation(v) Coherence (119894119903) letL119886119887119863(119868) be the label obtainedfor 119868 from the deliberation process without votingThe coherence statistic measures how well votingresults adjust to results obtained during deliberationonly We use the proportion of arguments that havebeen labeled 119894119899 in the debate and that agents havevoted favorably for

119894119903 = 1003816100381610038161003816119868 isin I | L119886119887119863 (119868) = 119894119899 andL119886119887 (119868) = 11989411989910038161003816100381610038161003816100381610038161003816119868 isin I | L119886119887119863 (119868) = 1198941198991003816100381610038161003816 (10)

The coherence statisticrsquos domain is [0 1] If after 119872debates no deliberated central argument has beenlabeled 119894119899 or voting is not part of the procedure ofcollective decision-making then the statistic equals

12 Complexity

1 or said differently agents are perfectly coherentThis comes for the fact that if the central argumentis labeled 119900119906119905 then it is not even eligible for voting iflabeled 119906119899119889 then the debate will always be coherentwith the preferences of the agents since the result willbe a simple aggregation of their votes If the statisticequals 0 then none of the proposal arguments labeled119894119899 are voted in favorably by the agents It follows that ahigh coherence statistic implies that when agents votefor acceptable proposal arguments the results of thescrutiny reflect the consensual rationality expressedin the deliberation process

312 Parameters of Interest We recall the parameters ofinterest in our study that are linked to the structure of thedecision-making processes

(i) 119898 the minimum number of time steps in which adebate or a scrutiny occurs in a decision-making pro-cess before a final deliberated decision is submitted toa vote

(ii) 119899119863 the number of agents that deliberate as a propor-tion of the population

(iii) 119905119863 the number of time steps between debates inwhich pair-wise interactions among agents mayoccur

(iv) 120572 the proportional majority requirement for theacceptance of a proposal When 120572 = 0 it stands forno voting ldquoany proposal argument that is labeled 119894119899during deliberation is acceptedrdquo

Please recall that in terms of the definition of a decision-making process each parameter controls either the length orthe content of the sequence D119868 (119905119863 119898) or the rules that areapplied during the debate steps (120572 119899119863) (refer to Figure 4)313 Initialization All |119873| = 400 agents start off with anopinion 119900119894 drawn from a uniform distribution U(minus1 1) 13Given 119900119894 every agent 119894 randomly draws a setA119894 (|A119894| = 119896) ofarguments from a balanced14 argument pool A of119872 = 600nonneutral (V119886 = 0) arguments on the basis of 119900119894 if |119900119894| lt 025(the opinion is moderate) then agent 119894 randomly fills halfof her argument sack with arguments such that V119886 lt 0 andthe other half with arguments such that V119886 gt 0 (120601119898119894 = 05)Otherwise she randomly fills 120601119890119894 ge 05 of her sack witharguments such that V119886 and 119900119894 are of the same sign and theremaining (1 minus 120601119890119894 ) with arguments of the opposite sign15

Like for the opinion of agents every argument 119886rsquosadherence to the principle P V119886 is drawn from a uniformdistribution U(minus1 1) The attack relation that gives birth tothe ideal argumentation framework (119860119865120576) is established onthe basis of the V119886rsquos and the argumentsrsquo epistemic reach 119890119903

Let 120579(119886 119887 119903) = (|V119886 + V119887|2)119903 be an auxiliary positivereal-valued function that takes two arguments (119886 119887) and apositive real number (119903 gt 0) as inputThe probability that anyargument 119886 isin A creates a link to any other argument 119887 isin A

is given by the equation 119901 = 119901120579(1198861198872)119890 and thus argumentssupporting opposing views of the principle always attack each

other We fix 119901119890 the epistemic correlation parameter to 015If 119901119890 was greater than 015 then arguments of the same signwould attack each other too often (and focused agents wouldbe rarely incited to advance favorable arguments duringdeliberation) If 119901119890 was lower then the arguments of thesame type would induce an almost empty graph which isunrealistic The number of arguments that any argument 119886can attack is bounded by 119886rsquos epistemic reach that we fix to15 A higher value of epistemic reach makes the argumentlattice too conflicting and nearly bipartite A lower epistemicreachmakes the lattice not sufficiently conflicting in the sensethat too many arguments can attack the proposal argumentrelatively to the few that can defend it If any argument attacksthe proposal argument the chances that another argumentattacks the attacker are low Hence the proposal argumentis almost never accepted and deliberative opinion updatesbecome rare

The arguments in the resulting argumentation frame-work 119860119865120576 are given a permanent labelingL120590

120576 (A) (in shortL120576) using120590 = grounded semanticsWe choose the groundedlabeling-based semantics because it provides a unique admis-sible labeling respects minimal rationality constraints whilesimplifying the model and models a skeptic approach toaccepting arguments (see [23 25 31]) as it maximizes thecardinality of the set 119906119899119889 L120576 To some extent if we considerproposals as committing because they may guide collectiveaction choosing a skeptic semantics seems reasonable insofaras it labels an argument 119894119899 or 119900119906119905 only if it has no reasonto label it 119906119899119889 For instance if a proposal is a public policythat requires large amounts of resources and engages tofuture course of action then it may also determine policycycles and heavily burden a group and its future decisionsIn important situations like these it seems reasonable tolengthen deliberative cues and ask for more demanding andldquogroundedrdquo criteria for policy argument acceptability

On the proposalrsquos side we create an argument 119868 whoseadherence to P is also drawn from aU(minus1 1) and we label itL119886119887(119868) = 119906119899119889 119868 is interpreted as being the main argumentjustifying the discussed proposal P and V119905119868 as its supportfor P at time 119905 119868 cannot attack other arguments but otherarguments can attack it For each argument 119886 isin A a directedarc or attack from 119886 to 119868 is activated with probability 119901119901 =003120579(11988611986802) When 119901119901rsquos lower bound is too high the proposalargument is almost always defeated and thus deliberativeopinion updates do not happen When 119901119901rsquos lower boundis lower than 003 then the opposite occursmdashthe proposalargument is always accepted due to an absence of attackstowards it Finally we set the maximum number of debatesto119898 = 7 for the decision-making processes stop rules1632 Calibration and Simulation Protocol In this subsectionwe discuss the calibration of the models the terminationconditions for runs in the deliberative and mixed model andthe expected outcomes in terms of the observations

321 Calibrating Dyadic Opinion Dynamics Dyadic opiniondynamics correspond to a space of parameters in whichargumentation and deliberation spaces are not taken into

Complexity 13

Table 1 Parameters and parameter types and their domains used to compare dyadic collective and mixed opinion dynamics

Length ofD119868 Decisional in119863119868 Social influence Deliberation influence Cognitive119905119863 119898 119898 119899119863 120572 120583 119879 119880 119901119886 119901119903 120574 119896 119861119890ℎ119886V1198941199001199031 3 6 1 3 6 7 001 002 005 0 05 066 01 14 18 02 06 03 05 01 005 01 02 4 8 16 119873119886119894V119890 119865119900119888119906119904119890119889account Deliberated arguments and informational cues haveno effect on agent opinion but agents still vote for theproposal argument17 We use and calibrate this model forcomparability in terms of all our metrics since they arenot explicitly observed in [9] a reference opinion dynamicsmodel Furthermore for simplicity and comparability againwe suppose that agents are homogeneous in terms of theirattitude structures and opinion weightings We fix 120583119894 =120583 119879119894 = 119879 and 119880119894 = 119880 for all 119894 isin 119873 and for some triplet(120583 119880 119879) isin [0 05] times [0 2] times [119880 2]

Experiments suggest that the strength of the dyadic inter-actions is an explanatory factor of the time of convergence(120583 lt 05) and of the dyamicsrsquo steady states Bigger values of 120583speed up convergence towards a stable set of opinion clustersand affect the size and the relative position of these clustersin the opinion distribution 120583 however does not play a majorrole in the number of opinion clusters observed at the steadystate of the dynamics

We set 120583 to the value given to it in Jagger et alrsquos opiniondynamics model (120583 = 01) [9] For one it is the smallestreal number for which most results found in [9] hold and thedynamics are smooth secondly it seems to be a reasonableldquostrengthrdquo of influence considering that we would not wantpair-wise social interactions to completely shadow or maskan eventual effect of deliberation in the opinion dynamics

For different values of attitude structures 119879 and 119880 weget exactly the same results as in [9] in terms of the numberand density of the opinion clusters The different couplesof values of 119880 and 119879 model the size of the latitude ofnoncommitment (119879 minus 119880) and thus the agentsrsquo propensityto update opinions Whenever 119879 minus 119880 le 02 regularitiesin convergence and opinion clustering appear If 119880 gt 1there is always central convergence if 119879 lt 1 there is alwaysbipolar convergence with clusters of similar sizes For 119879 minus119880 gt 02 relatively low values of 119879 (119879 lt 1) and relativelyhigh values of 119880 always yield extreme bipolar convergencewith a relatively small cluster of moderate agents (bipolar-central convergence) For 119879 sufficiently high (119879 ge 12) and119880 sufficiently low (119880 le 08) in other words not very ego-involved agents there is a bigger spectrum of steady statesFor this reason we studied the metrics of interest for thesevalue differences and kept the couples of (119880 119879) depicted inTable 1 For these couples central bipolar bipolar-centralconvergence (3 clusters) and multicluster convergence (4 or5 clusters) are possible and result frommeaningfully differentattitude structures Judgment accuracy varies significantlyacross the four different attitude structures we retain and soit happens with the variance of opinions These scenariosprovides us with reference results from which to build upon

322 Calibrating Deliberative Social Interactions Delibera-tive social interactions correspond to social situations in

which individuals are not influenced by pair-wise discussionwith peers but are sensitive to collectively deliberated infor-mational cues On these terms parameters such as the sizeof agentsrsquo argument sacks (119896119894) the distribution of argumentsin them (120601119894) the minimum number of debates (119898) andthe proportion of debaters from the population (119899119863) are nolonger mute We assume for simplicity that the formulaexist120601119890exist119896forall119894(119896119894 = 119896 and 120601119890119894 = 120601119890 and 120601119898119894 = 05) is true with 119894 isin 119873for some value 120601119890 isin [05 1] and for 119896 isin N

The chosen domains for these parameters (see Table 1)are justified by the size of the population and the size of theargument pool (119872 = 600) which we set from the start If 119896is too big (119896 ge 32) andor too many agents are allowed inthe debate (119899119863 ge 02) then individuals always disprove thecentral argument and deliberation is never or rarely takeninto account in the update of opinions Similar effects occurwhen 119898 is high yet we only calibrate it with respect to themean running time of a simulation and the values of 119905119863Different distributions of 120601119890119894 seem to significantly or directlyexplain none of the statistics we analyze so we convenientlyset 120601119890119894 = 075 We chose the parameter 119896 to be always divisibleby 4 for it is convenient given the initial conditions regardingthe argument distribution in the agentsrsquo argument sacks

We fix the value space for the parameters 120574 119901119886 119901119903 as inTable 1 to account for different intensities of the effect ofdeliberative voting retroaction on the system For examplethe scenarios where the three parameters are at their lowestvalues correspond to the scenarios with the smallest fixedprior effect of deliberation in the model Increasing any ofthese parameters should mechanically be associated witha world in which agents are more sensitive to collectivedecisions The values for 119901119886 are taken from [13] wherethe author states that by means of deliberation 7 to28 of individuals changed their opinion from agreeingto disagreeing or vice versa on a referendum question forDenmarkrsquos participation in the EuroWe choose two differentvalues for 119901119886 that account for high (03) and very high (05)sensitivity to deliberation in a group 119901119903 on the other handis taken to be small (equal to 01) since we believe that thereexist agents that will always go against the reached consensusTheir numbers are meager

For simplicity we assume that for all 119894 isin 119873 120574119894 = 120574 forsome 120574 isin [0 05] In effect we posit that the heterogeneityof agents in the deliberative context only derives from theirarguments and their opinions We decide to calibrate 120574 inallusion to the strength of the social dynamics in pair-wiseinteractions In doing so we limit ourselves to consider threedifferent scenarios in which deliberation has respectivelyhalf equal and twice as much opinion-shifting power thanone-to-one social influence Finally we include the accept-ability voting quota rule 120572 whose domain is inspired fromclassical majority rules [35] observed empirically (Table 1)

14 Complexity

Table 2 Parameters and their domains used to study the effect of the procedural parameters on the metrics

Procedural parameters Other parameters119905119863 119898 119899119863 120572 119898 120583 119879 119880 119901119886 119901119903 120574 119896 119861119890ℎ119886V1198941199001199031 2 3 4 5 6 1 2 3 4 5 001 002 003 004 005 0 05 066 7 01 16 02 03 005 02 12 119873119886119894V119890 119865119900119888119906119904119890119889

323 Calibrating the Mixed Social Interactions Model Themixed interaction model ascribes to a parameter space inwhich the effect of collective choices on our metrics isnontrivial and where deliberation and voting on proposalsdetermine their acceptability Since the mixed model isequivalent to periodic iterations of pair-wise and collectivesocial interactions the calibration of the parameters in bothpreceding models holds (see Table 1 for the calibration)The first reason for this is comparability the second is thatthe previous calibration takes into account the fact thatboth dynamics are going to be combined We include thefrequency of debates by adding the parameter 119905119863 whichcontrols for the amount of pair-wise interaction amongagents between two distinct deliberation steps

324 Termination Conditions for Runs Simulations stoponce 100 proposals are deliberated on andor voted for inother words after 100 collective decision steps have occuredThe number of proposals discussed seems arbitrary yet it ishigh enough to observe the effects of deliberation on opiniondistributions and on other metrics related to coherence andjudgment accuracy We choose the number of runs as afunction of our research questions which give relevance tothe procedural parameters in the mixed model rather than tothe parameters set to describe the population and its behavior

325 Expected Outcomes of the Simulations We expect moredeliberation in terms of higher 119898 and 119899119863 to increasejudgment accuracy and at the same time to reduce thevariance of opinions In turn a smaller variance of opinionsimplies that the argument pool for deliberation is smaller andtherefore judgment accuracy should be lower Moreovervariance increasing dyadic interactions (or rejection) shouldalso increase judgment accuracy and coherence since bipo-larization and dissensus foster argument diversity in delibera-tionThe coherence statistic should be stronger in simulationsin which central convergence appears quickly and agentsare naive Attitude structures sensitivity to informationalcues and weights given to deliberated cues that makes pathsto bipolar convergence smaller or to central convergencelonger should be associated with higher judgment accuracyShifts of opinion should be more visible in scenarios inwhich extreme agents are pulled away from the extremesHence the sensitivity to deliberated cues (119901119886 119901119903) and 120574should explain the shifts When crossed with high values of119880 and 119879 the shifting power of deliberation should be at itshighest

In the end we do not know how these mechanismswill play out The results of the subsequent experimentsgive us insight on the interplay between the aforementionedeffects

4 Results

Before pointing at any result obtained from the mixedinteractions model we describe and compare the dyadicand deliberative interactions models in the parameter spaceobtained from the calibration in Section 3 (see Table 1)Primarily we require our results to describe two orthogonaltypes of opinion dynamics the pair-wise dyadic and thedeliberative The latter comprises scenarios in which indi-viduals do not influence each other by means of pair-wisediscussion and only update their opinions from delibera-tion the former scenarios where only pair-wise interactionsamong agents determine the dynamics We describe both onthe parameter space given in Table 1 by performing at least20 simulations per scenario Quantitative results from thepair-wise interaction model alone are described during thecalibration and are not treated heremdashdeliberative parametershave no effect on it Moreover we show how these differentmodel specifications yield qualitatively different opiniondistributions and compare them on the basis of the metricsof interest To this end we perform independent two-sampleStudentrsquos t-tests18 or Welch t-tests19 and compute confidenceintervals at a 095 level of confidence Last but not least wecomment ordinary least squares (OLS) regression estimatesto account for the direction and magnitude20 of the effectsof the parameters on the metrics Estimates are declaredsignificant at a 5 level of risk

We perform the same analysis on the mixed interactionsmodel as we compare it to the pair-wise dyadic and deliber-ative interactions models and discuss the marginal effects ofthe parameters on the metrics More precisely we obtain twodifferent types of results for the mixed interactions modelthe first compares the mixed scenarios to their monolithiccounterparts with respect to each metric and with respect tothe regression estimates the second gives a clear idea of themarginal effect of each governance parameter (or parameterof interest) on our observations In the first we allow controland procedural parameters to vary on a parameter spacesimilar to those of the dyadic and deliberative opiniondynamics (see Table 1) In the second we fix all of ourcontrol parameters execute 36000 balanced21 runs andfocus only on the one-way pooled effects of our proceduraland behavioral (all agents are focused or naive) parametersWe generate comprehensive graphs and tables to account forthe obtained results

Please notice the slight change in the size of the parameterspace for the procedural parameters and in the values ofthe nonprocedural ones for the first and second type ofresults (refer to Tables 1 and 2 respectively) Since we wantto have a finer idea of the marginal effects of each of theprocedural parameters on how well a group decides onproposals we make the value jumps for each parameter (1)

Complexity 15

() (U T) = (02 18) (b) ( pa m nD) = (005 05 00 2 02) (＝) ０ＬＧＮＬＭ ＣＨ (＜) (＝) Ｈ＞ tD = 3

(d) (５ ４) = (02 14) (e) ( pa m nD) = (01 03 05 3 02) () ０ＬＧＮＬＭ ＣＨ (＞) () Ｈ＞ tD = 6

Figure 7 From left to right common opinion trajectories and distributions for dyadic deliberative and mixed opinion dynamics within thestructure of different decision-making processes

small enough to detect significant differences in the estimatesand (2) meaningful for interpretation For the fixed one-valued parameters in Table 2 we use Table 1 to set them totheir minimum mean or median values

41 Comments on Dyadic and Deliberative Opinions Dynam-ics In this subsection we present our first results regardingthe deliberative opinion dynamics described in Equation (6)and its differences with the dyadic dynamics (Equation (5))in terms of our observations

411 Qualitative Analysis of Pair-Wise Opinion DynamicsPair-wise opinion dynamics alone produces multiclusterstable convergencewith variable cluster sizes (see Figures 7(a)and 7(d)) Insight on the complexity of these dynamics wasdiscussed in Section 321

In Figures 7(a) and 7(d) agents discuss randomly withone another in pairs Clusters in the extremes form quicklysince either agents that are close to the extremes attractone another or are convinced to stay close to the extremesby interacting with other agents they disagree with Otheragents with near-extreme positions may be simply attractedto one of the several moderate foci of agents in the opiniondistribution On the other hand moderate agents are eitherattracted to the closest opinion focus or pushed towards theextremes of the distribution Over time agents in the centralfocus either attract other agents into it or ignore the opinions

of the extreme agents that interact with them The result ofthese dynamics yields amulticlustered convergencewhere theopinion foci are at least at distance 119880 from one another andonly the extreme foci (clusters of agents with |119900119894| ge 075) areat a distance greater than or equal to 119879 from one anotherThefoci at the bounds of the opinion distribution are very stableand in analogy to the (12119889)-rule in assimilation bounded-confidence models [33] the number of clusters that form isroughly equal to 119880minus1412 Qualitative Analysis of Deliberative Opinion DynamicsQualitatively deliberative opinion dynamics yield a looseunipolar convergence of opinions near the center (centralconvergence see Figure 7(b)) near the center of either the leftor right portions of the distribution (group polarization seeFigure 7(e)) or to a sparse bipolar opinion distribution withclusters at the center and at the extremes (see Figure 7(c)) ifmixed with pair-wise interactions This is probably a reasonto expect the variance of opinions and the proportion ofextremists (almost perfectly correlated 120588 = 089 119901 lt 0001)to be low for these scenarios The side towards which theopinion cluster skews depends essentially on the valence ofthe first argument that is collectively accepted

In Figures 7(e) and 7(b) agents that update their opin-ions do so at the same time and only when deliberationis successful In these scenarios every time a decision iscollectively accepted agents update their opinions towards

16 Complexity

Table 3 Ordinary least squares estimates for pair-wise interactions scenarios for most of the observed metrics

Dependent variable119881119886119903(119900) 119878ℎ 119890119888 119894119903(1) (2) (3) (4)Constant 0610lowastlowastlowast 4429lowastlowastlowast 0472lowastlowastlowast 0000

(0002) (0206) (0003) (0000)

119880 = 06 (119880 = 02) minus0012lowastlowastlowast 2981lowastlowastlowast 0061lowastlowastlowast 0000(0002) (0260) (0004) (0000)

119879 = 18 (ref 119879 = 14) minus0264lowastlowastlowast minus3261lowastlowastlowast 0004 0000(0002) (0260) (0004) (0000)

120572 = 066 (ref 120572 = 05) 0002 minus0059 minus0100lowastlowastlowast 0000(0002) (0184) (0003) (0000)

(119880 119879) = (06 18) (ref (119880 119879) = (02 14)) minus0324lowastlowastlowast minus3728lowastlowastlowast minus0033lowastlowastlowast 0000(0003) (0368) (0006) (0000)

Observations 1440 1440 1440 1440R2 0983 0389 0529Adjusted R2 0983 0388 0527Residual Std Error (df = 1435) 0032 3492 0053 0000F Statistic (df = 4 1435) 21 272740lowastlowastlowast 228686lowastlowastlowast 402326lowastlowastlowastNote lowastplt01 lowastlowastplt005 lowastlowastlowastplt001

the argumentrsquos defended position Hence if there is votingand if agents happen to obtain 119894119899 proposal arguments theyinexorably cluster towards one or the other side of the opiniondistribution resulting in opinion convergence remeniscent ofgroup polarization (see Figure 7(e)) Furthermore as agentscluster in the same side of the opinion spectrum the argu-ments that may be advanced in deliberation are fewer andmore skewed towards that one side of the opinion spectrumAgents become more easily persuaded in deliberation andonly by arguments supporting one side of the spectrumConvergence of opinions towards one loose opinion focusbecomes faster and more certain If there is no vote (seeFigure 7(b)) the dynamics are similar but for the fact thatit is the uniform randomness of the proposal argument thatguarantees a central convergence of opinions

413 Quantitative Analysis of Pair-Wise Opinion DynamicsThe three parameters that allow for quantitative analysis are119880 and119879 the agentsrsquo attitude structure and themajority quotavoting rule 120572 Table 3 supports the claim that going from 120572 =05 to 120572 = 066 does not affect the variance nor the shifts ofopinions However judgment accuracy is significantly higherwhen accepting proposals becomes more difficult This israther surprising but also intuitive since decisions that aretaken using more restrictive methods of scrutiny should bemore accurate

Concerning the attitude structures the interaction termbetween 119880 and 119879 shows that 119880 has a small effect on thevariance of opinions given that 119879 is low This shows thateven when the assimilation threshold is high a sufficientlysmall rejection threshold can make its one-way effect vanishHowever high levels of both 119880 and 119879 reduce the variance ofopinions dramatically In contrast when it comes to overallchanges in opinion the magnitude of 119880rsquos one-way marginal

effect increases the shifts of opinions whereas 119879 lowers itThis can be interpreted as having the rejection dynamics pushmoderate-to-extreme agents to positions that are close tothe extremes and very moderate agents to the center of thedistribution Not too many quantitative changes (on average)can therefore be observedThe fact that higher119880 is associatedwith faster convergence of opinions seems to imply lowerjudgment accuracy At some point voting for the same type ofproposals for too longmay result inmany collectivemistakesThis is an interesting conclusion to consider in the analysis ofthe mixed model Finally the interaction term shows that 119879has no strong impact on the metric

414 Quantitative Analysis of Deliberative OpinionDynamicsSeven parameters can be considered for quantitative anal-ysis in this dynamics the parameters of interest recalledin Section 312 the size of the argument sacks (119896) agentbehavior during deliberation the strength of the deliberationdynamics (120574) and the measure sensitivity to deliberatedresults (119901119886) The most ldquounexpectedrdquo result obtained fromthe deliberation dynamics is that it shows how deliberationcontributes to the proportion of extremists and the varianceof opinions It happens that having more agents discussproposals and having more of these discussions increasesthe variance of opinions and the proportion of extremistswhile lowering judgment inaccuracy and coherence For theformer the fact that far too many arguments are advancedduring deliberation plummets the chances that the centralargument is accepted hence deliberation does not moveopinions too often For the latter we observe that only ldquobigrdquodifferences (going from the lowest value of a parameter to thehighest) in protocol actually prove to have a preponderantimpact on the metrics These observations hint at a possibletrade-off between judgment accuracy and the variance of

Complexity 17

Table 4 Ordinary least squares estimates for the deliberative interactions scenarios for most of the observed metrics

Dependent variable119881119886119903(119900) 119878ℎ 119890119888 119894119903(1) (2) (3) (4)Constant 0032lowastlowastlowast 13092lowastlowastlowast 0370lowastlowastlowast 1007lowastlowastlowast

(0001) (0406) (0001) (0001)

119898 = 3 (ref119898 = 1) 0012lowastlowastlowast minus2062lowastlowastlowast minus0012lowastlowastlowast minus0001(0001) (0276) (0001) (0001)

119898 = 6 (ref119898 = 1) 0023lowastlowastlowast minus3285lowastlowastlowast minus0018lowastlowastlowast minus0004lowastlowastlowast(0001) (0276) (0001) (0001)

119899119863 = 002 (ref 119899119863 = 001) 0007lowastlowastlowast minus0965lowastlowastlowast minus0006lowastlowastlowast minus0002lowastlowast(0001) (0276) (0001) (0001)

119899119863 = 005 (ref 119899119863 = 001) 0020lowastlowastlowast minus3316lowastlowastlowast minus0015lowastlowastlowast minus0004lowastlowastlowast(0001) (0276) (0001) (0001)

120572 = 05 (ref 120572 = 0) 0146lowastlowastlowast 68932lowastlowastlowast 0068lowastlowastlowast minus0507lowastlowastlowast(0001) (0276) (0001) (0001)

120572 = 066 (ref 120572 = 0) 0291lowastlowastlowast minus13842lowastlowastlowast 00004 minus0000(0001) (0276) (0001) (0001)

119896 = 8 (ref 119896 = 4) 0004lowastlowastlowast minus0815lowastlowastlowast minus0002lowastlowastlowast minus0001(0001) (0276) (0001) (0001)

119896 = 16 (ref 119896 = 4) 0008lowastlowastlowast minus1118lowastlowastlowast minus0004lowastlowastlowast minus0003lowastlowastlowast(0001) (0276) (0001) (0001)

119861119890ℎ119886V119894119900119903 = ldquoFocusedrdquo (ref ldquoNaiverdquo) 0015lowastlowastlowast minus1980lowastlowastlowast minus0012lowastlowastlowast minus0004lowastlowastlowast(00005) (0225) (00004) (0001)

120574 = 01 (ref 120574 = 005) minus0020lowastlowastlowast 5762lowastlowastlowast minus00003 minus0001(0001) (0276) (0001) (0001)

120574 = 02 (ref 120574 = 005) minus0025lowastlowastlowast 8648lowastlowastlowast minus00003 minus0001(0001) (0276) (0001) (0001)

119901119886 = 05 (ref 119901119886 = 03) minus0013lowastlowastlowast 1581lowastlowastlowast 00004 00003(00005) (0225) (00004) (0001)

Observations 19440 19440 19440 19440R2 0931 0844 0562 0958Adjusted R2 0931 0844 0562 0958Residual Std Error (df = 19427) 0033 15701 0030 0050F Statistic (df = 12 19427) 21 844790lowastlowastlowast 8 733563lowastlowastlowast 2 079585lowastlowastlowast 36 761250lowastlowastlowastNote lowastplt01 lowastlowastplt005 lowastlowastlowastplt001

opinions that may be of interest in the mixed discussionmodel

Group coherence shifts of opinions and the varianceof opinions are highly affected by the voting thresholdthe coherence metric is reduced by half when allowing forclassic majority voting shifts explode and the variance ofopinions increases dramatically This is not surprising sincethe majority quota rule works as a buffer to the success ofdecision-making processes and in consequence it hampersthe deliberative opinion updates For the shifts of opinions itis clear that the two-third majority voting rule results in littleopinion change and that the simple majority rule inducesgroup polarization As expected there are less extremists inscenarios where there is no voting and the shifts of opinionare more likely to happen when agents are more sensitiveto deliberation When we include voting in the system

the grouprsquos judgment accuracy decreases due to the correctdeliberated decisions that are vetoed by a salient opinion-agent majority Strangely the effect only holds for the simplemajority decision rule Lastly the proportion of individualsthat participate in the debate and the minimal amount ofdebates have similar marginal effects on all the metrics (seeTable 4)

415 Comparing Deliberative and Pair-Wise Opinion Dynam-ics in a Nuthshell Qualitatively both models yield similaroutcomes but for the fact that deliberation never producesopinion polarization with both types of extremists (seeFigure 7) Furthermore clusters in the deliberative interac-tions model are less homogeneous in size and multiclusterconvergence occurs only when deliberation is rarely suc-cessful which happens sporadically given the calibration of

18 Complexity

Table 5 Mean difference 095 confidence intervals for metrics by model comparison

Model Metric119881119886119903(119900) 119901119903119900119901119890119909 119878ℎ 119890119888 119894119903Dyadic vs Deliberative [0191 0216] [0170 0197] [-2948 -2828] [0074 0082] [0165 0172]Mixed vs Deliberative [0142 0147] [0150 0155] [-2085 -1976] [0003 0002] [-0038 -0030]Mixed vs Dyadic [-0072 -0046] [-0045 -0018] [8332 8859] [-0079 -0071] [-0205 -0201]the model On the other hand pair-wise opinion dynamicsproduces multicluster convergence most of the time andcentral convergence only in the infrequent cases where thelatitude of acceptance is large

The comparison of these two scenarios in terms ofthe metrics is straightforward Whilst dyadic interactionsallow for more variability in opinions through pair-wiseinteractions and more mistakes in judging proposals delib-erative updates simply act as an antagonist force On aver-age the deliberative interaction model produces opiniondistributions with smaller variance and yields steady stateswith higher judgment accuracy a higher shift statistic andnaturally lower group coherence since only proposals that aredeliberated 119894119899 are eligible for voting (see Table 5)

42 Articulating Dyadic and Deliberative Social Interactionsin Opinion Dynamics So far we have considered scenarioswhere argumentation had no place and where individualsmade decisions according to opinions that were purelyconstructed from pair-wise discussions We also observedcases in which agents formed their opinions only by inte-grating deliberated proposals into their opinion updates Thescenarios we present in this section combine the two pre-ceding scenarios to account for both the effects of pair-wisediscussion and the effects of deliberation on opinions judg-ment accuracy and group coherence From the precedingobservations we conclude the following deliberative opiniondynamics reduces the variance and the proportion of extrem-ists because it shifts opinions greatly and polarizes groupsby eliminating at least one type of extremist the one whoseopinions are initially opposed to the deliberated resultsor it brings opinions close to neutrality (see Figure 7(b))This may not be desired in a group since the diversity andstability of opinions are necessary for deliberation to makesense and lead to the collective acceptance of controversialproposals Dyadic opinion dynamics on the other handtend to increase the variance of opinions and the numberof extremists while having a lesser aggregated effect on theshifts of opinion This is an immediate result of the pair-wiseinteraction dynamics inasmuch as it polarizes individualsvery quickly

To sum up in the mixed interactions model deliberationhinders the effect of voting on opinions by making someproposals that would normally be submitted for voting noteligible for voting It may also undo opinion changes dueto pair-wise interactions The mixed model can be seen asthe deliberation opinion dynamics model in which pair-wiseinteractions occur in between collective discussion and mayaffect which arguments are advanced during deliberationEquivalently it can be seen as the dyadic opinion dynamics

model in which deliberation accounts for endogenousldquoshocksrdquo on the opinion distribution

421 Qualitative Analysis of the Mixed Dynamics In Figures7(c) and 7(f) agents update their opinions through delibera-tion and pair-wise interactions Opinion trajectories visualsshow a combination of both previously described opiniontrajectories Over time as agents discuss with one anotheropinion clusters form as in the pair-wise or dyadic opiniondynamics Deliberation may disrupt the formation of theseclusters and three different situations may occur

(i) Deliberation disrupts the formation of the opinionclusters but not sufficiently to jeopardize pair-wiseopinion cluster formation (eg for high values 119905119863)Two phenomena may be responsible for it (1) thevariability of the proposal argumentrsquos support for theprinciple P and (2) too few (successive) deliberationsteps that result in opinion updates Once the opinionclusters are formed successful deliberation foreshad-ows the merging of the previously formed clustersmay bring extremists closer to moderate agents andthanks to the assimilation component of the pair-wisedynamics allow bigger and more moderate opinionclusters to form (refer to Figure 7(c))

(ii) Deliberation significantly disrupts the pair-wisedynamic formation of clusters This may happenwhen the requirements for accepting proposals areweak in the decision-making process (eg low 120572) Inthis case one has either central convergence sinceall agents follow the deliberated results in the samemanner or 3-cluster convergence with two extremeopinion foci In the latter situation the extremegroups form quickly (like in the pair-wise model) anddeliberation cannot pull extreme agents away fromthe extreme foci they are in The main reason for thisis that immediately after an extreme agent attempts toleave her foci through deliberation updates she getspulled or pushed back into it by means of a dyadicinteraction with another extreme agent

(iii) Deliberation does not disrupt the formation of theopinions cluster This is the rare case scenario forvery demanding decision-making processes in termsof inclusion (eg 119899119863) and argument acceptability (eg120572) Opinions converge as in the pair-wise opiniondynamics model (refer to Figures 7(a) and 7(d))

Last when there is voting (120572 = 0) the position ofthe moderate opinion cluster is determined by the first

Complexity 19

minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

Var(o)

sh

ec

ir

Var(o)

propex

sh

ec

ir

1

098

minus02

minus004

001

1

minus014

minus001

minus003

1

026

minus028

1

minus057 1

ＪＬＩＪ

Ｒ

Figure 8 Significant correlation estimates for themetrics of interestin the mixed opinion dyamics scenarios

accepted proposal argument andor the opinionmajority thatis formed from previous dyadic interactions (see Figure 7(f))

422 Relating the Metrics of Interest in Mixed OpinionDynamics The first thing to notice is that for all of thesescenarios all the metrics are significantly correlated (seeFigure 8) The variance of opinions and the proportion ofextremists are almost perfectly correlated To some extentthis means that a high variance of opinions known thatthe mean of opinions is statistically null equates to havingmany extremists in the population be towards one or theother side of the opinion distribution (eg extreme bipolar-ization) Similarly we find a negative and strong correlationbetween the judgment inaccuracymeasure and the coherencecoefficient (see Figure 8)mdashwhen judgment inaccuracy is high(low) coherence to deliberation is low (high) Cases likethese are telling since they may be related to sequences ofcollective decisions in which agents do not make too manymistakes when judging proposal arguments and still makedeliberated decisions that are on average in line with theirprinciples

The shift statistic interestingly is negatively correlatedwith the variance of opinions with judgment accuracy andwith the coherence coefficient In a way this means thathaving more moderate agents conduces on average to moremistakes in judging arguments and to less coherent collectivedecisions An explanation for this phenomenon could be thatas agents reach consensus in opinion important (possiblyextreme) arguments for correctly judging argument propos-als are not advanced in the deliberation arenawhich results inmore mistakes and in possibly accepting proposal argumentsthat after voting are not meant to be accepted Indeed the

negative correlation between the variance of opinions andjudgment inaccuracy points towards this direction since onaverage the more shifts there are the smaller the variance ofopinions is (see Figure 8)

423 Comparing Mixed Dynamics with Other MonolithicDynamics of Interactions Qualitatively opinion trajectoriesin the mixed model are more chaotic than in the pair-wise and deliberative interaction models (see Figure 7) Asanalyzed previously deliberation dynamics disrupt the pair-wise dynamics and create situations in which central andmulticlustered convergence (with two clusters at the extremesof the distribution) are possible at the expense of more(pair-wise dynamics) or less (deliberative dynamics) clustersIt tolerates the survival of very small isolated group ofagents that the previous models rarely tolerate in a hundredcollective decision-making steps (see Figure 7(c))

One of the most interesting observations one can makeof this family of models regards how groups change (in termsof our metrics) when dyadic social interactions are addedto deliberative interactions and vice-versa When comparingthe mixed interactions with only deliberative ones shiftsagentsrsquo aggregated change in opinions are on average lesspronounced The variance of opinions and the proportionof extremists are significantly higher for the former than forthe latter group coherence in contrast is lower with respectto the scenarios where opinions are formed only throughdeliberation Judgment accuracy follows that same trendA reason for this might be that mixed dynamics are fasterto yield semistable clusters andor opinion convergence Inconsequence the potential argument pool for deliberativeexchange shrinks faster and deliberation becomes less effec-tive over time which results in more collective mistakes Theoutcomeon coherence is predictable inasmuch as deliberativeinteractions produce group polarization If a group agrees inopinion debates are sterile because themembers of the groupwill most likely refrain from attacking the proposal argu-ments they all support and will work together to disparagethe proposal arguments they do not support (see Table 5)

In the same manner we report that the proportion ofextremists and the variance of opinions are barely lower inthe pair-wise interaction model when including deliberationin the decision-making process Shifts of opinions are alsobarely lower when there is no deliberation This can indicatethat deliberation does not undo the changes in opiniondue to pair-wise interactions nor does it amplifies them Incontrast when we consider the labeling-based metrics themixed interaction model ensures lower judgment inaccuracybecause of deliberation and lower group coherence (seeTable 5)

In sum the mixed discussion model is a compromisebetween the pair-wise interaction and deliberative opiniondynamic models in that it offers some guarantees in termsof judgment accuracy and coherence while allowing forldquoreasonablerdquo variance in the opinion pool It establishes somekind of trade-off between labeling-based metrics and thevariance of opinions that may be of interest for decision-making process design

20 Complexity

Table 6 Summary of sign or relative marginal effects for all varying parameters on dyadic deliberative and mixed interactions models inand for all experiments

Metrics Parameters119905119863 119898 119899119863 120572 119879 119880 119901119886 120574 119896 119861119890ℎ119886V119894119900119903119881119886119903(119900) XX X X O X X X X X X119901119903119900119901119890119909 XX X X O X X X X X X119878ℎ XX X X O X X X X X119890119888 XX X X XO X XOO XOO X X119894119903 XX X X O XO O X OOX XOO X XNota gray symbols correspond to dyadic dynamics light gray to deliberative dynamics and bold black to mixed dynamics Given a scenario the symbol 119883stands for constant parameter and 119874 for insignificant marginal effect and as parameters grow for positive (negative) marginal effect for the parameterFrom left to right at the parameter level the leftmost arrow corresponds to the scenario with the strongest (in absolute value) marginal effect of the parameter(with respect to to its factors) Example Let 119901 be a parameter and 119874 stands for ldquo119901 has a significant positive effect in dyadic discussion scenario negativein the deliberative one and insignificant in mixed discussion The positive effect in dyadic discussion is greater than the absolute value of the negative one inthe purely deliberative scenariordquo

424 Comments on the Differences between the Three Modelsfor OLS Regression Estimates The significance of most of theregression parameters estimated for the mixed interactionsscenarios are similar to those estimated for the pure dyadicanddeliberative scenariosNevertheless themagnitude of theeffects is different insomuch that the dynamics themselvesare different So instead of commenting the linear regressionestimates of the mixed model we compare when possiblethe standardized significant coefficients of the regressors ofthe mixed model with those of the pair-wise and deliberativeinteractionmodels (see Table 6 for a summary of the results)

In mixed interactions for instance the influence of119898 onthe opinion distribution is much more preponderant than inthe deliberative dynamics scenarios Going from 119898 = 1 to119898 = 3 increases the standard variation units of the variance ofopinions by twice as much as the one in the deliberative sce-nariosThis in particular is due to the design of the decision-making process If 119898 is higher then the number of socialinfluence steps (119905119863) in the decision process is also higher(compared to 119905119863 = 0 in the deliberative case) If we decideto consider these steps as contributing to the variance ofopinions then the effect of119898 on the variance of opinions hasto be stronger in the mixed interactions model In contrastthe number of agents participating in the debate has a weakereffect on the variance of opinions in the mixed interactionsmodel than in the deliberative dynamics specially for 119899119863 =005 (more than twice as much) The difference may comefrom the fact that in mixed discussions the part of theargument pool used in deliberation is more restrictive thanin the deliberative case We can argue that since pair-wisedynamics may get individuals closer to one another aftersome deliberation the same happens to the arguments thatindividuals will most likely use in deliberation Taking thisinto account means that deliberation will at many instancesbe unfruitful and have no effect on opinions thus failing tomoderate extreme opinions andor polarize the group In thesame direction a high number of participants in deliberationcan make the deliberation dynamics stiff In this case thedistribution of opinion is close to the uniform distributionwhich has a relatively high-to-moderate variancewith respectto the reference results

The strength of the deliberative dynamics has a positivestronger effect on the variance of opinions in themixedmodelthan in the deliberative model when 120574 = 02 and a negativeweaker one when 120574 = 01 A sound interpretation of thisresult is that when deliberation is successful and its effect onopinions does not get agents to move sufficiently away fromtheir current positions agents are pulled or pushed back totheir previously held opinions the variance of opinions doesnot changemuchOn the other hand if the effect of successfuldeliberation is strong opinions can change sufficiently tomake agents much more prone to assimilation into moderateopinion foci the variance of opinions falls

The deliberative opinion dynamics model is much moresensitive to 120572 than the mixed discussion model Differ-ences in marginal effects can double The probability ofattraction to deliberated results (119901119886) for instance has astrongmoderating effect (negative) on agentsrsquo opinions in thedeliberative opinion dynamics but a meager one (two-fold)in the mixed and dyadic dynamic models This is becauseopinion updates in the deliberative model depend only onthat parameter whereas in the mixed model more numeroustypes of interactions make opinion updates possible

In terms of judgment accuracy all parameters except thesize of the argument sacks (119896) have a significant slightlystronger effect on the deliberative dynamics than in themixedmodel In effect the difference may be attributed to theupshot of pair-wise interactions on the distribution of opin-ions and on the arguments that would be advanced duringdeliberation The result respecting parameter 119896 however iscounterintuitive Since argument sacks are static scenarioswhere shifts are higher should make 119896 more important fordeliberation to succeed and it is precisely in the scenarios ofpure deliberative dynamics that we find the biggest opinionshifts This may point to the hypothesis that in deliberativeinteractions agents change opinions substantially but do notoften change their adherence to the principle (go from anegative opinion to a positive one or vice versa) when thereis no group polarization

43 Sensitivity Analysis for the Procedural Parameters ofInterest Weperform a sensitivity analysis of the observationson the parameters of interest in themixed interactionsmodel

Complexity 21

We extend the parameter domains of the pure dyadic anddeliberative scenarios as we make it explicit in Table 2

431 Minimum Number of Debates (119898) We observe that theminimum number of debates has a significant well-observedeffect on all of our metrics excluding group coherence Forthe variance of opinions we can observe that the moredebates there are the bigger the value of the metric isalthough the higher 119899119863 and 119905119863 are the weaker the overalleffect of119898 becomes (Figures 9(d) and 9(a)) Furthermore themarginal effect of increasing minimal debates is decreasingan additional deliberation step requirement increases thevariance of opinions but less and less as it grows 119898 coupledwith 119899119863 has a shy S-shaped effect (see Figure 9(d)) on thevariance of opinions meaning that it has a stronger effectwhen 119899119863 is low which suggests a possible trade-off betweenthe number of arguments accepted in the debate and thenumber of debates required before accepting a proposal

In contrast the shifts in opinion are less and less likelyas 119898 grows and this is independent of the variations of theother parameters Again the effect is marginally decreasingand is only truly significant when 120572 = 05 An explanationto these effects may be linked to the design of the systemFirst the variance of opinions is higher when deliberation isasked for because the more deliberation steps there are thehigher the chances are that the central proposal argumentis deemed unacceptable specially when agents are focusedFurthermoremechanically speaking increasing theminimalamount of debates implies that whenever a decision is to betaken at least 119898 times 119905119863 time steps have to take place and if atany moment a proposal argument is considered undecided119905119863 time steps are added to the process So unless the debatedoes not yield 119906119899119889 labels for proposal arguments (highlyunlikely considering that 120590 = grounded semantics) the morenondeliberation steps there are in the decision process thehigher the variance of opinions is Concerning the shiftswhen 120572 = 05 either the system is too stiff to accept anyproposal argument and opinions do not changemuch or theeffects of pair-wise discussion and deliberation cancel out insuch a way that individual shifts are minimal (Figure 9(b))

On the side of labeling-based metrics the more debatesare asked for the more accurate a group is in its judgmentthe effect being smaller as 119898 grows When agents are naivethe effect is quasilinear while when they are focused thestrongest effects of acquiring more deliberation are foundwhen levels of deliberation are already low (Figure 9(f))Thiscan be explained by the fact that the more debates thereare in the decision process the closer one gets to the idealargumentation framework

432 Proportion of the Population in Deliberation Steps (119899119863)Like for 119898 119899119863 has a significant effect on the proportionof extremists and on the shifts and variance of opinions(Figure 9(e)) This may result from the fact that being ableto put more arguments in play at the same debate step mayincrease the odds of revealing cycles around the proposalargument in119860119865120576 Given that we use grounded semantics thearguments in the cycles are labeled 119906119899119889 and in consequence

debates are more often postponed when 119899119863 is higher Post-poning debates in turn increases the number of nondebatessteps in the decision process which increases the varianceof opinions and limits over time the moderating effect ofdeliberation Moreover the effect of this parameter is verydependent on the value of 120572 (Figure 9(e)) For shifts forinstance 120572 = 05 makes the effect of 119899119863 negative while120572 = 0 makes it positive but to a lesser extent Furthermorefor higher requirements of deliberation (119898) adding moreindividuals to the deliberation process has weaker effectson the variance of opinions and on the other metrics It isalso quite interesting to notice that it has the same effect onagents independently of agent behavior during deliberationThis is a surprising result as one would have expected morefocused agents in a deliberation arena to cause a raise inthe proportion of extremists as they play to knock outopposing proposal arguments and thereof reducing the effectof deliberation (lowers variance) on the whole population

Finally similar to119898 addingmore people into the deliber-ation process increases judgment accuracy (see Figure 9(c))and has no clear effect if any on group coherence (seeFigure 9(i))

433 Steps between Deliberation Steps (119905119863) In all configu-rations 119905119863 increases the proportion of extremists (variance)and decreases the shifts of opinions among agents Theshifts and the effect on the variance of opinions are presentfor 120572 = 05 insignificant otherwise (Figure 9(b)) 119905119863 ishighly linked to 119898 by construction When 119898 = 1 thecurve linking the variance of opinions and 119905119863 is convex As119898 increases the curve becomes more and more concavewhich means that 119905119863 has a more important effect on theopinion distribution as collective decisions take longer to beachieved This seems counterintuitive but in reality it reflectsthemultiplicative relationship between deliberation and pair-wise interactions and the semantics in the model If 119898 is lowand 119905119863 is high the effective number of pair-wise interactionsis on average smaller in the deliberation process Hencethe structure of the decision-making process hinders anyincrease in the variance of opinions If119898 is high the oppositeeffect is observed Additionally since the grounded semanticsyields few 119894119899 arguments with respect to other admissibility-based semantics getting closer to the ideal argumentationframeworkmaymake it difficult to obtain the expected effectsof deliberative discussion on agents Lower 119898 constrains thevariance of opinions by making deliberation more influentialon opinions

In terms of judgment accuracy and coherence we observethat 119905119863 does not explain coherence yet it tends to decreasejudgment inaccuracy This may be due to a more variedargument pool in the debate which in turn may be aconsequence of a higher variance of opinions or vice versa

434 Acceptability Voting Quota (120572) By far it is the mostinfluential parameter in our study It changes the directionand the intensity of the effect of all the other proceduralparameters and by construction heavily constrains the roadto accepting a proposal (see Figure 9)

22 Complexity

015

020

025

030

035

040

1 2 3 4 5 6

12345

Values of Ｇ

(a) 119905119863 times 119901119903119900119901119890119909 by119898

0

10

20

30

40

50

1 2 3 4 5 6

005066

Values of

(b) 119905119863 times 119878ℎ by 120572

040

045

050

055

060

1 2 3 4 5

001002003004005

Values of Ｈ＄

(c) 119898 times 119890119888 by 119899119863

015

020

025

030

035

040

1 2 3 4 5

001002003004005

Values of Ｈ＄

(d) 119898 times 119901119903119900119901119890119909 by 119899119863

0

10

20

30

40

50

001 002 003 004 005

00005066

Values of

(e) 119899119863 times 119878ℎ by 120572

040

045

050

055

060

1 2 3 4 5

BehaviorNaiveFocused

(f) 119898 times 119890119888 by 119887119890ℎ119886V119894119900119903

015

020

025

030

035

040

0 05 066Behavior

NaiveFocused

(g) 120572 times 119901119903119900119901119890119909 by 119887119890ℎ119886V119894119900119903

05

06

07

08

09

10

1 2 3 4 5 6

00005066

Values of

(h) 119905119863 times 119894119903 by 120572

08700

08725

08750

08775

08800

001 002 003 004 005

BehaviorNaiveFocused

(i) 119899119863 times 119894119903 by 119887119890ℎ119886V119894119900119903

Figure 9 Curves of mean observations for different parameters and metrics for 36000 runs Bars are 095 confidence intervals Plots withthe same observables (ordinate) except for 119894119903 are on the same scale

Complexity 23

The higher the requirement for accepting the proposal isthe higher the proportion of extremists and the variance ofopinions are and the lower the shift statistic is In few words120572 constrains the world to dyadic discussion or throws it intoa process in which deliberation is much more importantthan dyadic interactions This is why for any 120572 = 05 oneeither gives too much weight to deliberated results whichshadows pair-wise interactions or too little weight in thesense that no deliberated result is ever accepted and thusnever integrated into the opinions of the agents that havevoted for it The reason for the latter is that the latitudeacceptance is too low and thereafter pair-wise interactionsare not enough to unevenly polarize the population in such away that deliberated proposal arguments are voted favorablyby a non-negligeable majority Shifts for 120572 = 0 are stable fordifferent levels of 119905119863 which implies that deliberation keeps incheck all the shifts of opinions related to an increase in thenumber of pair-wise interactions For 120572 = 05 deliberationis less likely to be successful and shifts tend towards the shiftlevels for 120572 = 0 (see Figure 9(b))

Concerning labeling-basedmetrics120572 entirely determinesthe coherence statistic For 120572 = 05 there is no difference incoherence because of how coherence is defined it is maximalwhen 120572 = 0 and strangely maximal when 120572 = 066(see Figure 9(h)) Surprisingly 120572 has no effect on judgmentaccuracy One would have expected a higher 120572 to increasejudgment accuracy since agents would hardly collectivelyaccept a proposal argument discussed during deliberationProposal arguments have given how the argument lattice isgenerated a higher probability of being rejected than of beingaccepted in the ideal argumentation framework

435 A Word on Focused and Naive Agents Focused andnaive specifications for agent behavior are an importantparameter in the decision process since they model howagents choose arguments in the deliberation arena In the sce-narios where agents are focused agents show equal or highervariance of opinions and are therefore more extreme (seeFigure 9(g)) independently of the voting requirements Shiftsof opinion are less likely as well We can see this happeningbecause focused agents knock out proposal arguments moreoften than naive agents provided that 119899119863 is low

One could have expected naive agents to yield morecoherent decisions since they argue sincerely However thisseems not to be the case (see Figure 9(i)) Agents are unable todeliberate in away thatmakes deliberation reflect their votingintentions either because they do not have the necessaryarguments to do so or because such arguments do not exist

In terms of judgment accuracy focused agents do betterthan naive agents in finding the ideal labels for the proposalarguments This is likely because when reconstructing theframework focused agents take into account the deliberatedproposal argument and choose themost pertinent argumentsin their sacks that relate to the state of affairs at the deliber-ation arena In fine debates result in a better approximationofL119886119887120576(119868) even though one could have believed the oppositethat is focused agents are ready to sacrifice a correct collectivelabel of a proposal argument in the goal of defending theirpositions

44 Points of Discussion In this subsection we discuss theresults obtained from the simulations We attempt to makesense of them from a social sciences perspective

441 Opinion Consensus and Dissensus The deliberativeopinion dynamics corresponds to the simulated scenarios inwhich the variance of opinions is at its lowest and judgmentaccuracy at its highest Dyadic opinion dynamics are associ-ated with the scenarios in which the variance of opinions is atits highest judgment accuracy at its lowest and coherence atitsmaximumpossible levelThemixed interactions dynamicsis a combination of these aspects for all the metrics exceptfor coherence at which it is the worsemdashdeliberation leadsto some kind of loose opinion consensus that is reinforcedby pair-wise interactions Likewise the mixed discussionmodel confirms some results exposed in [16] expressingthat opinions and voting intentions in a deliberative contextldquooften changerdquo That being said and more particularly indeliberative opinion dynamics outputs align well with twointeresting results on consensus formation given by aninterpretation of Moscovici and Doisersquos [15] work Sunsteinrsquos[14] account on deliberation and group polarization andsocial impact theory [4] The first states that mild consensusis reached in deliberation as procedure disengages agentsthe second is that within discussion groups majorities tendto become larger and opinions more extrememdashnot on thebasis of competing arguments but of the initial balance ofopinions After several decision processes we find opiniondistributions that reflect these observations even though weare not able to precisely coin the set of parameters that yieldssuch distributions (see Figure 7)

Although differences between mixed and dyadic opiniondynamics in terms of the variance of opinions and shiftsare slim the former does much better in terms of judg-ment accuracy In this sense deliberation does moderateopinions but not as much when non-deliberation steps arerelatively frequent throughout the decision-making processIn terms of Moscovici and Doise theory of consensus [15]this makes sense since pair-wise discussion engages or ego-involve agents in the discussion and pushes to convergenceof opinion at the extremes of the opinion distributionHowever what remains unclear is the reason why proceduralparameters that constrain deliberation do not disengageagents from discussion and thus result in a mild consensusas also expressed by Moscovici and Doise [15] A possibleexplanation of this phenomenon may stem from the factthat deliberation in the model is always accompanied by aseries of ego-involving discussions that shadowor counter themoderating effect of deliberation Not only this deliberationcould also be responsible for crystallizing the opinions ofextremists for it favors one or the other part of the distributionof opinions as explained in [14] and to some extent in [13]Furthermore the mixed discussion model gives an exampleof polarization that results from a minority of dissenterson one side of an issue acceding to views of the majorityrsquosside [4] as opposed to the strong polarization observed indyadic interactions Situations in which the status quo orthe undecidability of the proposal argument lingers cannotpossibly yield situations of moderation In cognitive science

24 Complexity

one can be inclined to believe that the longer a deliberationprocess is the more likely individuals will take an immutablestance on the situation In view of the assumption that longdecision-making processes can be more costly for a group orsociety as a whole agents may feel forced to take a stancejust for the sake of ending the process The stance that isadopted would therefore depend on pair-wise interactionsamong individuals and not on deliberation since deliberationtakes too long to be conclusive

442 For Hypothetical Recommendations in Decision-MakingProcess Elaboration On another tune results from themodelsuggest that procedural parameters have marginally decreas-ing positive or negative effects on the metrics of interest Ina hypothetical policy analysis perspective this may indicatethat after choosing a correct number of individuals andimposing the right amount of debates one can attain rea-sonable levels of judgment accuracy coherence and varianceof opinions As accounted in [18] deliberation increasesknowledge (collective knowledge as well) and therefore itheightens the chances of making correct decisions If onebelieves that social welfare is linked to deliberation andeventually to the metrics studied previously then proceduralparameters can both be used as an instrument to attaindesired levels of social welfare and reinforce the notionof legitimacy of collective decisions It can also provide ajustification for choosing a deliberation regime rather thananother on the basis of how important (or urgent) a topic ofdiscussion is Indeed if we adopt the claim that deliberationis a forerunner of welfare then any state in which deliberativecues are at their maximum (highest number of participantsmany deliberation steps before making a decision and asfrequent as possible) has to be mapped to the highest attain-able social welfare In this case is a situation where judgmentaccuracy is at its highest yet the variance of opinions or theproportion of extremists at its lowest an ideal situation Wecannot say for sure since strong opinion consensus and longdecision-making processes in many situations may result ina loss of welfare rather than in a gain of it

Another interesting result in procedure for deliberationis the very similar estimates we find for the effects of theminimum number (119898) of debates necessary to make adecision on a proposal and the proportion (119899119863) of individualsparticipating in deliberation on judgment accuracy In thescope of social welfare this raises the question of whetheradvocating for longer decision processes by demandingmoredeliberation steps (higher 119898) rather than bigger debates(higher 119899119863) may increase welfare Depending on the objec-tives of the decider of the deliberation procedure he or shemay choose to concentrate on one or the other A recom-mendation we can assert from the mixed social interactionsmodel is that if one prefers low to high variance of opinionsandor high to low coherence the decider is better off ifshe focuses on increasing the size of the debate rather thanorganizing debates very frequently One of the reasons forthis might be that when making bigger debates all agents aretruly considered as equals in the decision-making processIn organizing many debates individuals that are chosen toparticipate in many of these will be more influential and will

somehow bias the set of arguments in the deliberation arenaThemajority rulemodeled through120572 also determines the sizeof the shifts and the distribution of opinions Because a bigger120572 is associated with higher levels of extremism and opinionvariance deliberated cues are harder to accept and thereforeit becomesmore difficult for agents to reach concensus andormoderate their opinions

On the other hand and further into the idea of legitimacythe goal of deliberative processes is the agreement amongall participants This agreement stands for the right or bestdecision from a formal or procedural perspective and it is alsothe best from a substantive point of view In other words thelegitimacy of decisions not only derives fromwell-establishedprocedural requirements (respect of the protocol) but alsobegs for the fulfillment of two essential conditions (1) satisfy-ing the procedural requirements for a correct procedure (for-mal legitimacy) and (2) the rational acceptability of the resultsof this procedure (substantive acceptability) Legitimacy canbe then obtained from the coherence statistic scrutinizedin the model insofar as it measures how well agents acceptthe results of a correct procedure (eg deliberation) So fora proposal to be legitimate one has to choose parametersthat would maximize group coherence in decision-makingprocesses

As deliberation may give rise to consensus in the delib-eration arena it may also sow the seeds of dissensus inthe group Our model clearly illustrates this phenomenonthrough the procedural parameters of deliberation and themetrics observed From another perspective in deliberativedemocracy dissensus in collective decision-making may bedesirable According to Landermore and Pagersquos accounts [36]lack of consensus can be perceived as having agents withalternative ideas that combined with other ideas can providea better approximation of the ideal status of an argumentTheauthors in [36] call this dissensus ldquopositive dissensusrdquo andit makes normative sense when solving complex collectivedecision-making problems Our model captures the ideaof having normative requirements to define a good andlegitimate collective decision requirements of high consen-sual deliberative accuracy (judgment accuracy) of legitimacy(coherence) and of diversity (variance of opinions) are alltaken into account and for such we can conclude that ourmodel is successful in describing a sound process of collectivedecision-making with interpretable outcomes

Of course the present simulation results and analysis donot prove that all opinion dynamics can be accounted forby such simple processes like the one presented here Theydo however shed light on the expressiveness of combiningdifferent paradigms to account formore interpretablemodelsand pave the way to creating original models of the sortThe simulations suggest the desirability of discovering theconsequences of relatively simple laws of communication atdifferent levels (micro meso and macro) to determine whatstill needs to be explained

5 Related Work

We see our model as a contribution to the influence andopinion dynamics field in ABM and a pragmatic application

Complexity 25

of abstract argumentation theory To our knowledge we areunaware of existing literature on agent-based modeling thatexplicitly relates collective choice and the notions of deliber-ation and opinion diffusion through abstract argumentationas we have done it

Most models in the literature of opinion diffusion areinterested in opinions because these have an influence incollective decision and in questions of social order Forinstance in [3] the authors are interested in consensus andin how a group collectively chooses among two alternativesIn other models authors are interested in the emergenceof extremism [8] and on the distribution of opinions whenextremists are introduced in the population [5 33 37] whileother authors coin the notion of opinion polarization as anemergent property of the system [8 12] They show usingmodels of ldquobounded confidencerdquo and opinion diffusion withtrust that three different kinds of steady states (unipolarbipolar and central) are possible depending on whetheragents are sufficiently uncertain about their opinions andsufficiently connected andor a certain proportion of indi-viduals are already extreme Recent articles on informationand opinion dynamics stress the importance of governanceand government intervention in the spread of emotions(opinions) in the frangibility of social consensus systemIn [10] the authors show that government intervention inthe spread of negative emotions can lead to an even fasterspread of negative emotions (single-peeked convergence) andin a faster collapse of the social consensus system Opinionas a function of trust is studied in [12] and relates to howindividuals form their opinions on the basis of how muchthey trust agents in their networks

Another stream of discrete opinion dynamics modelsthat explain the emergence of opinion can be found in theliterature Computer multiagent models of attitude changebased on Latanersquos social impact theory are presented in [4] Inthe theory an agent changes her opinion on the basis of theinformational impact she is subject to which depends on thepersuasiveness (strength) supportiveness and immediacy(group structure) of the environment and information shereceives The simulations predict two emergent groups ofphenomena the shifting of attitudes towards incompletelypolarized equilibria and the formation of coherent clusteringof subgroups with deviant attitudes Similarly but in acontinuous more argumentative fashion Mas and Flache[11] present a model of opinion diffusion using argumentsArguments are considered to be for or against a proposal andare given an agent-dependant relevance in dyadic persuasiveinteractions Arguments also determine the agentsrsquo opinionsand their dissimilarities which posits the rules of pair-wiseencounters in theirmodel as agents are assumed to form theiropinions according to the arguments they own They showthat in an argumentative opinion diffusion model the onlyobservable steady states are bipolarization and consensusand that bipolarization can emerge from interactions amongsimilar agents In other words bipolarization of opinionsis possible with homophily and without negative influenceif there is argumentation They subsequently compare theirresults to experimental data In [38] a similar approach toargumentation and opinion formation is taken but for the fact

that they introduce different types of arguments explicitlyapply social judgment theory [22] and use survey data toestablish model-to-real-world comparisons

Closer to opinion formation and abstract argumentationGabbriellini and Torroni [39] are the first to have origi-nally and soundly merged opinion diffusion and abstractargumentation They define an agentrsquos opinion as a functionof the arguments she holds and the attack relation amongthem They devised a focused peer-to-peer dialogue systemof persuasion (NetArg) inspired from Mercier and Sperberrsquosargumentative theory of reasoning [40] which used onlyabstract argumentation to study opinion polarization andopinion dynamics Moreover they considered networks andthe notions of trust and epistemic vigilance to define adynamics for knowledge revision and trust itself whichwe clearly lack in our model They showed that if a con-servative belief operator in argumentation was applied byagents when they reasoned then their dialogue protocoldid not increase polarization among agents That said theyused the model to study the effect of Groenvetterrsquos weaklink theory in the spreading of arguments and were notparticularly interested in notions like judgment accuracy orcoherence nor in tackling questions related to collectivedecision-making procedures Their system is very expressiveand helps position our work in the litterature our work is atthe frontier of pure argumentative opinion diffusion modelsopinion diffusion with arguments and continuous bounded-confidence models of opinion diffusion a la Deffuant

On another note work on collective cognitive conver-gence [41] and opinion sharing [19] shows that consensustowards a certain ldquocorrectrdquo opinion or cognitive state is alwayspossible yet dependent on noise variability and awarenessof agents In [20] the authors show that learning about anexogenous correct state of the world (represented by bits)under confidence was possible but only if the agents werenot too confident In a homogeneous population they showthat the higher the confidence the worse the learningmdashthevery confident agents do not learn the properties of the truestate of the world and disrupt the learning process of the lessconfident ones Collective cognitive convergence can be seenas a result of deliberation in truth-seeking models

When it comes to abstract argumentation theory we takean approach that wires two types of dialogues that are well-studied in the literature persuasion dialogues [42] and delib-eration dialogues [43] Another interesting line of work is theone on mechanism design [44] or the problem of devisingan argumentation protocol where strategic argumentation isnot a liability for success in debates We tackle mechanismdesign in a different way though Instead of consideringstrategy-proofness we are interested in how differences inprotocol can result in epistemologically ldquobetterrdquo collectivechoices and guarantee that opinion distributions are favorablefor deliberation For a survey on persuasion dialogue see[42]

Work on agent-based argumentation usually assumes thatthe semantic relationship between arguments is fixed Inother words if two individuals were to put two argumentsin the public arena such that one logically attacked the otherthen everyone would agree that such attack exists [45 46]

26 Complexity

Other models which do not make this restrictive assumptioncan also be found in the literature [27 47] and derive fromthe class or family of opponent models [27 48] in which twoopposing sides attempt to win a dialogue Ourmodel is in theintersection of these two but the frame combining opiniondiffusion of the kind and argumentation remains new

The idea of mixing interpersonal influence and verticalcommunication however is not original It is described andimplemented in innovation diffusion models such as [5 49]In both vertical communication is modeled as exogenoustransparent informationmdashagents are aware of the existenceof an innovation thus triggering several processes of choiceand stabilization of opinions Also in [50] an Eulerian modelis implemented to show the effect of media and exogenousinformation on opinion distributions With respect to thispoint the originality of our work is in that the informationemitted as vertical communication is endogenous It is issuedfrom a deliberation model that agents shape on the basisof their opinions arguments and behavior In the spirit of[51] where the authors control for the design of verticalcommunication (to whom the vertical communication isaddressed its timing how it influences agents opinions) wecontrol for the process generating the information for ourown set of variables length of debates majority quotas andthe frequency at which discussions take place

6 Conclusion and Perspectives

The main objective of this article was to build a bridgewhere decision-making argumentation and opinion dif-fusion could come together We proposed a model thatcombined abstract argumentation theory and a boundedconfidence opinion diffusion model and showed to whatextent it could explain variance of opinions extremismcoherence in collective decisions and judgment accuracyon arguments given an ideal state of full information Thesecond objective was to show in what way governance pro-viding agents with arenas of discussion and deliberation wasimportant in addressing the success of collective decision-making processes and the quality of their outcomes

The model revealed that allowing for more deliberationtime though in lower frequency and allowing for widerparticipation in deliberation increased the variance of opin-ions and the proportion of extremists in a group Theseobservations are consistent with the combination of resultsfound in [9] and in [13 14] which stress that deliberationmay polarize groups and may have a meager effect on theshifts of opinions and inconsistent with [16] where it isargued that opinions tend to moderate after deliberationIt happens that deliberation alone did moderate opinionyet when integrated into a more complex system in whichindividuals were allowed to interact with one another andnot all that was deliberated was accepted its influence wasshadowed by other more individualistic dynamics

Undeniably the grounded semantics played an importantrole in the weak effect of deliberation since it models askeptical way of reasoning over arguments For agentsaccepting arguments thus updating their opinions accord-ingly happened not too often Nevertheless asking for more

deliberation did increase judgment accuracy as observed in[18] yet in a marginally decreasing fashion We showed thatvoting within the deliberation protocol not only increasedthe proportion of extremists and the variance of opinions ina group but also determined how coherent deliberation andvoting procedures were with each other Lastly we showedthat agents that are focused judged arguments better hadmore stable opinions and constituted a group with higherproportion of extremists than their naive counterparts

In terms of governance the model says that there is per-haps no trade-off between extremismand judgment accuracyInstead it asserts that the higher the variance of opinions thecloser the group gets to the correct decisions This situationmay be interpreted as Landermore and Pagersquos notion of ldquoposi-tive dissensusrdquo (in complex collective decision-making tasks)[36] as opposed to consensus as a normative requirementfor correct collective decision-making If a decider has toorganize deliberation to legitimize a public policy dependingon what kind of world he or she wants different set ofparameters may be chosen to account for different levels oflegitimacy and correctness For instance one may ask for adecision to be legitimate to respect a certain level of coherenceor of judgment accuracy and to be discussed by a sundryof different-opinionated peoples If so deliberation has tobe made more often more participants have to be includedin debates or in deliberation instances and not too manytime steps should be introduced in between two deliberationinstances So the model may capture normative ideas todefine correct and legitimate collective decisions and tomake recommendations on those grounds Requirements ofhigh consensual deliberative accuracy (judgment accuracy)legitimacy (coherence) and diversity (variance of opinions)are all taken into account and for such a reason the model israther successful in describing a sound process of collectivedecision-making with interpretable outcomes

61 Extensions of the Model and Other Ideas Our model canbe extended in many ways Value-based abstract argumen-tation frameworks (VAFs) for example as an extension ofDungrsquos argumentation framework provide a formal descrip-tion of the process of decision-making in which argumentsare given values and audiences (set of agents) ignore attacksbetween arguments on the basis of their preferences over val-ues Social abstract argumentation frameworks (SAFs) alsoprovide an interesting and seemingly convenient formalismfor our model In SAFs agents vote on the acceptability ofarguments and the resulting labeling is a combination of thearguments that are voted for and labeling-based semantics Inour model we apply the same rule yet only for one argumentand given precise agent-based votingmechanisms and votingrules which are eventually parametrized

To consider our model as a specification of a mixed VAFand SAF model may be of great interest for future workThis formalism can provide solid foundations to describeexplain interpret and eventually extend the results foundhere from an argument-based perspective On those groundsfuture work will focus on creating a hybrid argumentationframeworkmdashwith its semanticsmdashthat takes heed of bothvalues or principles and voting

Complexity 27

Concerning other possible extensions of the model anatural reflex would be to argue on the validity and therobustness of the mix between the bounded confidenceopinion diffusion models and deliberation Consequentiallyadding deliberation to different bounded confidence modelsof opinion diffusion such as the Hegelsmann-Krause model[37] or to models of social impact theory like in [4] canbe of great interest for future work We acknowledge that afiner understanding of argumentation frameworks randomlattices and their respective applications for this modelare necessary Testing for different labeling-based semanticsor even trading labeling-based semantics for scoring-basedsemantics (semantics that defines acceptability on scoresgiven to arguments) such as debate semantics and ranking-based semantics (semantics that yield a preorder) may beinteresting to consider in future work

More into modeling deliberation it is not inadequateto believe that agents may also advance proposals duringdeliberation and replace to some extent the proposal forwhich the deliberation is taking place Endogenizing theprocess that produces the type of argument to be deliberatedon may be interesting to explore either by having agentsstrategically replace proposals at certain moments duringthe debate or having the central authority attempt to chooseproposals such that the probability of having them acceptedis high Agents may also be thought to be autoorganizing(independent of a central authority) and concede on the sizeof the instances of deliberation To consider differences insemantics across agents or even endogenizing the semanticson how urgent or important the proposal that is beingdiscussed is can also be an interesting direction to take

Last but not least many nontrivial modifications of ourmodel are possible and most should include better-thoughtdeliberation protocols It may be interesting to design andobserve protocols in which deliberation only affects agentsthat are actually debating for example Trust network effectsmultidimensionality of opinions new processes of argumentexchange or learning among agents are notions to furtherdevelop in order to make the model more realistic and relaxsome unreliable assumptions In sum the model is to berefined and extended with the objective of either studyingconcrete cases of deliberative polling and opinion dynamicsor implementing more intuitive thought experiments

Appendix

We present below some of the pseudocode we used toimplement the model

Algorithms

(a) Algorithm 1 implements the dyadic influence in themodel It describes how two agents interact andupdate their opinions

(b) Algorithm 2 implements the deliberative influence inthe model which describes how agents update theiropinions in sight of the accepted deliberated proposalarguments

(c) Algorithm 3 is used to obtain the grounded labelingof an argumentation framework

(d) Algorithm 4 provides an overview of a run in thesimulations119872 is the number of proposal argumentsto discuss which we set to 100 in the model

Each object in the system is indexed by a119908ℎ119900 number Hencea set of agents is annotated by a set of natural numbers Weuse this fact in the implementation of the algorithm 119874119887119895119890119888119905119894denotes the object of 119908ℎ119900 number 119894 Agentsrsquo opinions (119900) arecalibrated so that 119900 isin [minus1 1]Data Availability

The program and data used to support the findings of thisstudy are available from the corresponding author uponrequest

Disclosure

A preliminary version of this work was presented anddiscussed at the conference Bridging the Gap between FormalArgumentation and Actual Human Reasoning on October 4-5 2018 at BochumGermanyThe authors thank the audiencefor their valuable comments and suggestions

Conflicts of Interest

The authors declare that they have no conflicts of interestregarding the publication of this article

Acknowledgments

The contribution of Gabriella Pigozzi was supported by theDeutsche Forschungsgemeinschaft (DFG) and the CzechScience Foundation (GACR) as part of the joint project FromShared Evidence to Group Attitudes (RO 45486-1)

Endnotes

1 In other words the superiority of the ldquoright answerrdquo orlabeling in our case will appear as such to all

2 All agents hold the same opinion3 By sensitivity we mean how much an agent is ready to

change her opinion on the sole fact that an informationalcue has been debated on

4 The assumption is strong but this trust among agentsmay be rooted from a genuine individual motivation toreach a democratic consensus

5 We adopted a rule of thumbs that says that if an agentis neutral with respect to the principle then she willvote for any proposal that is presented to her duringdeliberation

6 If the argument is not unique she chooses one atrandom

7 If there are more than one candidate the agent will favorthe argument at minimal distance from 119868

28 Complexity

Require Vector of values for parameters (119881 = (120583 119880 119879))Ensure Successful interactions among agents119896 larr997888 0119873119890 larr997888 0

while 119896 le |119873|2 do ⊳Fill119873119890 with |119873|2 agents119896 larr997888 119896 + 1119873119890 larr997888 119877119886119899119889119900119898(|119873| + 1) ⊳Pseudo-random generator Generates integer between 0 and |119873|119873119903 larr997888 0 120575119894119895 larr997888 0 ⊳Variable to store differences in opinionfor all 119894 isin 119873119890 doRandomly pick a number 119895 isin 119873(119873119890 cup 119873119903)119873119903 larr997888 119873119903 cup 119895 120575119894119895 larr997888 |119900119894 minus 119900119895|procedure dyadic-update(119894 119895) ⊳Two agents 119894 and 119895 discuss and update opinions1199001015840 larr997888 119900119894 (120583119895 119880119895 119879119895) larr997888 (120583119894 119880119894 119879119894) larr997888 (120583 119880 119879) ⊳Save agent irsquos opinion before updating and homogenize agents

if 120575119894119895 lt 119880119894 then119900119894 larr997888 120583119894119900119895 + (1 minus 120583119894)119900119894if 120575119894119895 gt 119879119894 then119900119894 larr997888 (1 + 120583119894)119900119894 minus 120583119894119900119895if 120575119894119895 lt 119880119895 then119900119895 larr997888 1205831198951199001015840 + (1 minus 120583119895)119900119895if 120575119894119895 gt 119879119895 then119900119895 larr997888 (1 + 120583119895)119900119895 minus 1205831198951199001015840

Algorithm 1 Dyadic social influence

Require Vector of values for parameters (119881 = (120574 119901119886 119901119903))Ensure Successful interaction between proposal arguments and agents

for all 119894 isin 119873 doprocedure Probability-Update(V119886 119901119886 119901119903) ⊳Update individual probabilities of change due to deliberated cues119909119894 larr997888 119901|V119886minus119900119894 |2119886 119910119894 larr997888 1199012|V119886minus119900119894 |119903119902 larr997888 0if L119886119887(119868) = 119894119899 then

procedure deliberative-update(119894 119886) ⊳Agent i argument 119886120574119894 larr997888 120574 119902 larr997888 119906 sim U(0 1) ⊳ 119906 is a realization of a uniform random variableif 119902 lt 119909119894 then119900119894 larr997888 120574119894119900119894 + (1 minus 120574119894)V119886break

if 119902 lt 119910119894(1 minus 119909119894) then119900119894 larr997888 (1 + 120574119894)119900119894 minus 120574119894V119886Algorithm 2 Deliberation influence

8 In terms of computational complexity naive agentsmaketheirmoves in119874(|A119894| log(|A119894|)) while focused agents doat least as bad

9 Tables have memory Therefore debates are not neces-sarily trees

10 If there are less than (119899119863 times |119873|)2 agents for any of bothgroups the CA summons all of them to the deliberationtable

11 Or careful consideration of the viewpoints of otherswhich implies that citizens keep an open mind and donot reject arguments outright

12 A ldquohealthyrdquo consensus refers to the case in which manyconsensual decisions are taken but opinion-wise agentsdo not agree with each other Quotes are used on theword healthy because some political science theorists

(eg [36]) believe that in certain situations agreement inopinions for deliberation is epistemologically damaging

13 Simulations on groups of |119873| = 800 and |119873| = 400agents shows that scaling the population has an effect onall themetrics but it is confounding depending onwhichcouples of attitudes structures are considered (119880 119879) Westick to |119873| = 400 as in [9] for comparability andsimulation running time

14 By balanced wemean with as many arguments with V119886 lt0 as with V119886 gt 0

15 Experiments on the system suggest that a bigger(smaller) argument pool 119872 = 1000 (119872 = 200) isassociated with a higher (lower) variance of opinions(and extremism) and higher (lower) judgment accuracyFor the rest of the observations 119872 seem to have noeffect We choose a middle value for 119872 Similarly theargument pool chosen to be balanced or uniformly

Complexity 29

Require Argumentation framework 119860119865 = (AR)Ensure Assignment of labels to arguments from the grounded labeling of 119860119865L(A)

L0(A) larr997888 (0 0 0)119896 larr997888 0do119894119899 L119896+1(A) larr997888 119894119899 L119896(A) cup 119886 | 119886 is not labeled 119894119899 L119896(A) 119886119899119889 forall119887 isin A 119904119905 119887 R 119886 then 119887 isin 119900119906119905 L119896(A)119900119906119905 L119896+1(A) larr997888 119900119906119905 L119896(A) cup 119886 | 119886 is not labeled 119894119899 L119896(A) 119886119899119889 exist119887 isin A 119904119905 119887 R 119886 and 119887 isin 119894119899 L119896+1(A)

while L119896+1(A) = L119896(A) = 0Algorithm 3 Solve argumentation frameworks (with grounded semantics)

Require Vector of values for parameters (119881)Ensure Vector of statistics (119880)

procedure Set-parameters(119881) ⊳Initialize parameters for scenarioprocedure Init-objs(119899119898 119896 ) ⊳Create119898 arguments 119899 agents and 119860119865120576procedure Solve-Argumentation-Framework(119860119865) ⊳Assigns the epistemic labelingL120576 toA119894 larr997888 1119905 larr997888 1while 119894 le 119872 do ⊳ 119872 the number of arguments to discuss119868119894 larr997888 119866119890119899 119886119903119892(1) ⊳Generates proposal argument 119868119860 119894119905 larr997888 (119868119894 0)L119886119887(119868119894) larr997888 119906119899119889 ⊳Initialize label for 119868119895 larr997888 1 ⊳Initialize debate counter 119895while ((L119886119887(119868) = 119906119899119889 or 119898 le 119895) and 119895 le 119898) doif 119905 equiv 0(119898119900119889 119905119863) then

procedure build-argumentation-framework(119899119863 119887119890ℎ119886V119894119900119903) ⊳Build the argument graph agents deliberate(119861119894119905 119878119894119905) larr997888 (119883 119884) sub (⋃119897isin119873119899119863 A119897119894119905 ⋃119897isin119873119899119863 R119897119894119905) ⊳ 119873119899119863 sub 119873 are agents sampled by the CA119860 119894119905 larr997888 119889(119860 119894119905 (119861119894119905 119878119894119905))procedure Solve-Argumentation-Framework(A119889) ⊳FindL(A119889)L119886119887(119868119894) larr997888 L119886119887119863(119868119894)119895 larr997888 119895 + 1

else119896 larr997888 1while 119896 le 119905119863 doprocedure dyadic-social-influence(120583 119879 119880) ⊳Agents discuss one-to-one119896 larr997888 119896 + 1119905 larr997888 119905 + 1

if 120572 gt 0 andL119886119887(119868119894) = 119900119906119905 then ⊳Agents vote for the proposalif (|119897 isin 119873|119900119897 times V119868119894 ge 0| ge 120572 times 119899) then

L119886119887(119868119894) larr997888 119894119899else

L119886119887(119868119894) larr997888 119900119906119905if L119886119887(119868119894) = 119894119899 thenprocedure Deliberation-influence(120574 119901119886 119901119903) ⊳Agents are influenced by the result of the decision process119894 larr997888 119894 + 1119905 larr997888 119905 + 1119880 larr997888 119862119900119898119901119906119905119890 119878119905119886119905119894119904119905119894119888119904(119881)

return 119880 ⊳Get the statisticsAlgorithm 4 Run for one scenario

generated does not seem to play a significant explanatoryrole on any of the metrics

16 Experiments show that themaximumnumber of debatesis not an important parameter even if it is an intuitiveone when considering decision-making processes Inpractice debates end after at most 119898 + 1 deliberation

steps and even then it happens rarely However to avoidunpleasant surprises like infinite debates on a completeargument graph containing the proposal argument weadded the constraint 119898 Its value is precisely 7 becausethe biggest minimum number of debates we studyis 6

30 Complexity

17 To ensure comparability with the deliberative andmixedopinion dynamics models on judgment accuracy andcoherence agents vote on proposal arguments every 119905119863times(119898+1) 119905119863 and119898does not explain variability in judgmentaccuracy nor in coherence

18 Student t-tests for independent samples are robust tothe violation of two hypothesis on the distribution ofthe samples normality and homogeneity of variancesas long as the sample size is big enough (central limittheorem) and if the difference in size of the comparedsamples is smallrespectively

19 Studentrsquos test version forwhen variances are not assumedequal and the difference in size of the compared samplesis big

20 We do not correct for homoskedasticity nor normality oferrors in the estimates

21 Balanced as the same number of runs per scenario orsuch that no significant correlations are found betweenparameters

References

[1] R Axelrod ldquoThe dissemination of culture a model withlocal convergence and global polarizationrdquo Journal of ConflictResolution vol 41 no 2 pp 203ndash226 1997

[2] D M Rousseau Reinforcing the MicroMacro Bridge Organiza-tionalThinking and Pluralistic Vehicles SAGEPublications SageLos Angeles Calif USA 2011

[3] S Galam and SMoscovici ldquoTowards a theory of collective phe-nomena consensus and attitude changes in groupsrdquo EuropeanJournal of Social Psychology vol 21 no 1 pp 49ndash74 1991

[4] A Nowak J Szamrej and B Latane ldquoFrom private attitude topublic opinion a dynamic theory of social impactrdquo Psychologi-cal Review vol 97 no 3 pp 362ndash376 1990

[5] G Deffuant S Huet and F Amblard ldquoAn individual-basedmodel of innovation diffusion mixing social value and individ-ual benefitrdquo American Journal of Sociology vol 110 no 4 pp1041ndash1069 2005

[6] G Deffuant D Neau F Amblard and G Weisbuch ldquoMixingbeliefs among interacting agentsrdquoAdvances in Complex Systems(ACS) vol 3 no 01n04 pp 87ndash98 2000

[7] J Rouchier P Tubaro and C Emery ldquoOpinion transmission inorganizations an agent-based modeling approachrdquo Computa-tional and Mathematical OrganizationTheory vol 20 no 3 pp252ndash277 2014

[8] M Meadows and D Cliff ldquoThe relative disagreement model ofopinion dynamics where do extremists come fromrdquo in Pro-ceedings of the International Workshop on Self-Organizing Sys-tems Lecture Notes in Computer Science pp 66ndash77 Springer2013

[9] W Jager and F Amblard ldquoUniformity bipolarization and plu-riformity captured as generic stylized behavior with an agent-based simulation model of attitude changerdquo Computational ampMathematical Organization Theory vol 10 no 4 pp 295ndash3032005

[10] Y-F Zhang H-Y Duan and Z-L Geng ldquoEvolutionary mech-anism of frangibility in social consensus system based onnegative emotions spreadrdquo Complexity vol 2017 Article ID4037049 8 pages 2017

[11] A Flache andMWMacy ldquoSmall worlds and cultural polariza-tionrdquoThe Journal of Mathematical Sociology vol 35 no 1-3 pp146ndash176 2011

[12] A Tsang and K Larson ldquoOpinion dynamics of skepticalagentsrdquo in Proceedings of the 13th International Conference onAutonomous Agents and Multiagent Systems AAMAS 2014 pp277ndash284 International Foundation forAutonomousAgents andMultiagent Systems France May 2014

[13] K M Hansen Deliberative democracy and opinion formationUniversity Press of Southern Denmark Odense 2004

[14] C R Sunstein ldquoThe law of group polarizationrdquo Journal ofPolitical Philosophy vol 10 no 2 pp 175ndash195 2002 WileyOnline Library

[15] S Moscovici and W Doise Dissensions et Consensus unetheorie generale des decisions collectives Presses Universitairesde France-PUF 1992

[16] J S Fishkin and R C Luskin ldquoExperimenting with a demo-cratic ideal deliberative polling and public opinionrdquo ActaPolitica vol 40 no 3 pp 284ndash298 2005

[17] R C Luskin J S Fishkin and S Iyengar Considered Opinionson US Foreign Policy Face-To-Face versus Online Delibera-tive Polling International Communication Association NewOrleans La USA 2004

[18] J Barabas ldquoHowdeliberation affects policy opinionsrdquoAmericanPolitical Science Review vol 98 no 4 pp 687ndash701 2004

[19] O Pryymak A Rogers and N R Jennings ldquoEfficient opinionsharing in large decentralised teamsrdquo in Proceedings of the 11thInternational Conference on Autonomous Agents andMultiagentSystems vol 1 pp 543ndash550 Spain June 2012

[20] J Rouchier and E Tanimura ldquoWhen overconfident agents slowdown collective learningrdquo Simulation vol 88 no 1 pp 33ndash492012

[21] D Miller ldquoDeliberative democracy and social choicerdquo PoliticalStudies vol 40 S1 pp 54ndash67 1992

[22] M Sherif and C I Hovland Social Judgment Assimilation andContrast Effects in Communication and Attitude Change YaleUniversity Press 1961

[23] P M Dung ldquoOn the acceptability of arguments and its funda-mental role in nonmonotonic reasoning logic programmingand n-person gamesrdquo Artificial Intelligence vol 77 no 2 pp321ndash357 1995

[24] M Caminada and G Pigozzi ldquoOn judgment aggregation inabstract argumentationrdquo Autonomous Agents and Multi-AgentSystems vol 22 no 1 pp 64ndash102 2011

[25] P Baroni M Caminada andM Giacomin ldquoAn introduction toargumentation semanticsrdquoThe Knowledge Engineering Reviewvol 26 no 4 pp 365ndash410 2011

[26] S Chambers ldquoDeliberative democratic theoryrdquo Annual Reviewof Political Science vol 6 no 1 pp 307ndash326 2003

[27] E Bonzon andNMaudet ldquoOn the outcomes of multiparty per-suasionrdquo in Proceedings of the 10th International Conference onAutonomous Agents and Multiagent Systems 2011 AAMAS 2011vol 1 pp 47ndash54 International Foundation for AutonomousAgents and Multiagent Systems Taiwan May 2011

[28] J Ranciere Disagreement Politics and Philosophy University ofMinnesota Press 1999

[29] T Bench-Capon and K Atkinson ldquoAbstract argumentation andvaluesrdquo in Argumentation in Artificial Intelligence pp 45ndash64Springer 2009

[30] M Caminada ldquoOn the issue of reinstatement in argumenta-tionrdquo in Proceedings of the European Workshop on Logics in

Complexity 31

Artificial Intelligence vol 4160 of Lecture Notes in ComputerScience pp 111ndash123 Springer Berlin Heidelberg

[31] GAVreeswijk ldquoAn algorithm to computeminimally groundedand admissible defence sets in argument systemsrdquo in Pro-ceedings of the 2006 conference on Computational Models ofArgument Proceedings of COMMA 2006 vol 144 pp 109ndash120IOS Press Amsterdam The Netherlands 2006

[32] C W Sherif M Kelly H L Rodgers et al ldquoPersonal involve-ment social judgment and actionrdquo Journal of Personality andSocial Psychology vol 27 no 3 pp 311ndash328 1973

[33] G Deffuant F Amblard G Weisbuch and T Faure ldquoHow canextremism prevail A study based on the relative agreementinteraction modelrdquo Journal of Artificial Societies and SocialSimulation vol 5 no 4 p 1 2002

[34] E Awad M W Caminada G Pigozzi M Podlaszewski andI Rahwan ldquoPareto optimality and strategy-proofness in groupargument evaluationrdquo Journal of Logic and Computation vol 27no 8 pp 2581ndash2609 2017

[35] D Grossi and G Pigozzi ldquoJudgment aggregation a primerrdquoSynthesis Lectures on Artificial Intelligence and Machine Learn-ing vol 8 no 2 pp 1ndash151 2014

[36] H Landemore and S E Page ldquoDeliberation and disagreementproblem solving prediction and positive dissensusrdquo PoliticsPhilosophy and Economics vol 14 no 3 pp 229ndash254 2015

[37] R Hegselmann and U Krause ldquoOpinion dynamics andbounded confidence models analysis and simulationrdquo Journalof Artificial Societies and Social Simulation vol 5 no 3 2002

[38] A Stefanelli and R Seidl ldquoModerate and polarized opinionsUsing empirical data for an agent-based simulationrdquo inProceed-ings of the Social Simulation Conference 2014

[39] S Gabbriellini and P Torroni ldquoA new framework for ABMsbased on argumentative reasoningrdquo in Advances in SocialSimulation vol 229 pp 25ndash36 Springer Berlin Germany 2014

[40] H Mercier and D Sperber ldquoWhy do humans reason Argu-ments for an argumentative theoryrdquo Behavioral and BrainSciences vol 34 no 2 pp 57ndash74 2011

[41] H V Parunak T C Belding R Hilscher and S BruecknerldquoModeling and managing collective cognitive convergencerdquoin In Proceedings of the 7th iNternational Joint Conference onAutonomous Agents and Multiagent Systems vol 3 pp 1505ndash1508 International Foundation for Autonomous Agents andMultiagent Systems 2008

[42] H Prakken ldquoFormal systems for persuasion dialoguerdquo TheKnowledge Engineering Review vol 21 no 2 pp 163ndash188 2006

[43] K Atkinson T Bench-Capon and P Mcburney ldquoA dialoguegame protocol for multi-agent argument over proposals foractionrdquoAutonomousAgents andMulti-Agent Systems vol 11 no2 pp 153ndash171 2005

[44] M Thimm and A J Garcıa ldquoClassification and strategicalissues of argumentation games on structured argumentationframeworksrdquo in Proceedings of the 9th International Conferenceon Autonomous Agents and Multiagent Systems vol 1 pp 1247ndash1254 International Foundation for Autonomous Agents andMultiagent Systems Canada 2010

[45] I Rahwan and K Larson ldquoMechanism design for abstractargumentationrdquo in Proceedings of the 7th International JointConference on Autonomous Agents and Multiagent SystemsAAMAS 2008 pp 1031ndash1038 International Foundation forAutonomous Agents and Multiagent Systems Portugal May2008

[46] H Prakken R Riveret A Rotolo and G Sartor ldquoHeuristics inArgumentation A Game-Theoretical Investigationrdquo EuropeanUniversity Institute 2008

[47] N Hadidi Y Dimopoulos and P Moraitis ldquoTactics and con-cessions for argumentation-based negotiationrdquo in Proceedingsof the COMMA vol 245 pp 285ndash296 2012

[48] N Oren and T J Norman ldquoArguing using opponent modelsrdquoin Proceedings of the International Workshop on Argumentationin Multi-Agent Systems pp 160ndash174 Springer 2009

[49] S Delre W Jager T Bijmolt and M Janssen ldquoTargeting andtiming promotional activities An agent-based model for thetakeoff of new productsrdquo Journal of Business Research vol 60no 8 pp 826ndash835 2007

[50] A Mirtabatabaei P Jia and F Bullo ldquoEulerian opinion dynam-ics with bounded confidence and exogenous inputsrdquo SIAMJournal on Applied Dynamical Systems vol 13 no 1 pp 425ndash446 2014

[51] V Barbet N Guiraud V Laperriere and J Rouchier Hagglingon Values Towards Consensus or Trouble 2019

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Complexity 3

(3) In what way can controlling for protocol relate tosocial situations in which collective decisions arecoherent and accurate

Simulations show that deliberation yields on averagequalitative loose consensus and group polarization whilereducing the number of different opinion clusters over thedistribution of opinions In effect our model shows thatdeliberation has a significant overall impact on the distribu-tion of opinions (on its variance) and on the shifts in individ-ual opinion In particular when specifying opinion dynamicsas only deliberative the proportion of extremists and thevariance of opinions are lower and the shifts of opinionsgreater than in a nondeliberative dyadic opinion dynamicsmodel However when considering a mix of deliberationand dyadic interactions parameters that promote biggeror more frequent deliberation during decision-making pro-cesses increase the variance of opinions and the proportionof extremists and limit the effect of deliberation on opinionsThe majority voting quota rule to accept a proposal plays apreponderant role as it determines if the consensus-drivingpower of deliberation outweighs the propensity to dissensusobserved in bounded-confidence models with rejection

Themodel sheds light on the fact that a grouprsquos coherenceand judgment accuracy may depend significantly on howdecision processes are structured The number of debatesallowed and the number of agents that may participate inthem increase judgment accuracy in a marginally decreasingfashion but have little or no effect on the grouprsquos coherencein decision-making Last we point out that results areconditioned on how many arguments agents have at theirdisposal and on how they advance them during deliberation

The remainder of this paper is organized as follows inSection 2 we present the model and provide some necessarybasics to understand its implementation in the subsequentsection we introduce the metrics of interest and the cali-bration of the model In Section 4 we report and discussthe results of the simulations and in Sections 5 and 6respectively we survey related work and we conclude thearticle

2 A Model for Collective Decision-Makingwith Deliberation

We propose a system of collective decision-making amongagents made of three objects arguments agents and tablesfor deliberation Agents have agency arguments representpieces of information and tables validate collective decisionswhile organizing all deliberative interactions among agents

21 Model Overview In this section we propose an overviewof themodelWe quickly introduce the agents and the objectsthat make collective decisions possible in the multiagentsystemWe also provide a basic interpretation of the collectivedecision-making procedure which we illustrate with anexample

211 Overview of Agents and Objects in the Model Argu-ments are objects that relate to each other by a defeat relation

They are characterized by their support of some value orprinciple Agents are characterized by their sensitivity todeliberated ideas by the arguments they possess whichreflect their opinions and their knowledge about argumentsThey communicate one-to-one or collectively by the dint ofarguments in a public arena Communication leads them toan eventual update of their opinions Tables are entities thatcontain both agents and argumentsThey are controlled by acentral authority (CA) [27] that fixes the rules in the delib-eration process the frequency of deliberative interactionsand the conditions for collective acceptability of deliberatedinformation The existence of a central authority namely aperson or a machine that can reveal the correct epistemicstatus of any set of arguments can be equated to Habermasrsquoclaim that the ldquounforced force of the better argumentrdquo willtriumph in an ideal speech situation1 A central authoritymayalso be associated with Rancierersquos notion of ldquopolicerdquo that hedefines as ldquoan order of bodies that defines the allocation ofways of doing ways of being and ways of saying and thatsees that those bodies are assigned by name to a particularplace and taskrdquo [28] (p29)

212 Overview of the Decision-Making Process Let 119873 be agroup of agents that is asked to deliberate and vote on aproposal P Given an argument 119868 in favor of P they haveto judge on whether P is desirable or not The argument119868 or proposal argument determines how much the proposalsupports a principle P or its opposite notP A proposalis a sentence that indicates a way of attaining a goal orsolving a problem A principle derives from the notion ofvaluemdashvalues seen as fundamental social or personal goodsthat are desirable in themselves [29] Environmentalism andpatriotism are examples of values to choose proposals thatminimize environmental impact or maximize welfare areexamples of two not mutually exclusive principles

Agents are assumed to decide on the acceptance of theproposal on the basis of their opinions or their adherenceto the said principle whether they argue formally or infor-mally about P When agents are not deliberating they aresubject to random pair-wise influence when the deliberativeexchange of arguments concludes they are influenced by theacceptance status of the proposal that is being discussedOnly a fraction of all agents deliberate in each deliberativeexchange all agents are prone to pair-wise discussion Theyvote for proposals according to their opinions A proposalP is accepted if and only if the argument 119868 that is givenwith it is accepted after deliberation and the proposal is votedfavorably by amajority of agents To vote in favor of a proposalis considered to be equivalent to accepting the argument thatcomes along with it

Example 1 P = ldquoProtect the environmentrdquo is a principle aproposal may be P = ldquoReduce carbon emissions by 2030using electric carsrdquo To justify the proposal one may advancethe argument 119868 = ldquoElectric cars will reduce societyrsquos depen-dency on fossil fuels and will result in a reduction of carbonemissions for 2030 while proctecting the environmentrdquo thatexpresses the degree to which the proposal is in line withenvironmental protection An argument that tackles the

4 Complexity

a b c

Figure 1 Argumentation framework 119860119865 = (AR) with A =119886 119887 119888 andR = (119886 119887) (119887 119888) The labeling 119887 119888 119886 is conflict-free 119886 119888 119887 0 is the only complete labeling

proposal argument 119868 and that opposes the principle P couldbe 119886 = ldquoElectric cars may protect the environment byreducing effective carbon emissions but the batteries theydepend on are very pollutantrdquo The proposal P may beaccepted in the deliberation arena if some other argumentthat defeats 119886 say 119887 = ldquoBattery recycling businesses arehatching everywhere By 2020 it is very likely that scientistsfind a viable solution to chemical pollution due to batteriesrdquo isadvanced It is not accepted otherwise If 119868 is accepted thenPis collectively accepted if a majority of agents vote favorablyfor it

22 Arguments as the Basic Units of Collective Discussionamong Agents In this subsection we introduce the sim-plest object in our system arguments We present abstractargumentation theory and Caminadarsquos labeling approach toargumentation [30] that allows us to track the epistemic statusof arguments during deliberation

221 Arguments Arguments are objects that represent piecesof information that agents can understand They are infor-mational cues that enable agents to discuss with one anotherin a public collective context and make decisions on theacceptability of other pieces of informationThey are assumedto be nonfallacious In our approach each argument 119886 ismodeled by a real number V119886 isin [minus1 1] that stands forhow much 119886 respects or supports the principle P V119886 = 1means that argument 119886 is totally coherent with the principleP whereas V119886 = minus1 reads that 119886 is totally incoherent withthe principle P Arguments relate to each other through anincompatibility relation that states that one cannot standbehind two conflicting arguments Arguments are also char-acterized by their acceptability status that indicates whetherthey are accepted refuted or undecided in a given discur-sive context Arguments have an epistemic reach (119890119903) or amaximum number of arguments they can attack This canbe interpreted as the argumentrsquos level of generality or as thepotential level of argumentative conflict within the argumentpool

222 Abstract Argumentation Theory Deliberation definedas an exchange of arguments is modeled by confronting rep-resentations of different eventually contending argumentsIn our model we use abstract argumentation theory [23] torepresent deliberation where an argument is just a node in agraph like arguments 119886 119887 and 119888 in Figure 1 Abstract argu-mentation theorymodels incompatibility between argumentsand abstracts away from their internal structureThe intuitionis that if an argument 119886 attacks an argument 119887 (representedby an arrow from node 119886 to node 119887 as in Figure 1) a rationalagent cannot accept both 119886 and 119887

Formally let A be a finite set of arguments and R asubset of A times A called attack relation The attack relation isintransitive and we note (119886 119887) isin R the fact that argument 119886attacks or is incompatible with argument 119887 One says that anargument 119886 defends an argument 119888 noted (119886 119888) isin D if thereexists an argument 119887 such that (119886 119887) isin R and (119887 119888) isin R

(see Figure 1) An argumentation framework (119860119865 = (AR))is a digraph in which the nodes represent the argumentsand the arcs represent the attacks among them Given anargumentation framework119860119865 the classic problem in abstractargumentation resides in finding which of the arguments in119860119865 can be accepted rejected or left undecided

In the labeling approach [30] a label L119886119887(119886) isin Λ =119894119899 119900119906119905 119906119899119889 of an argument 119886 isin A denotes the epistemicstatus of 119886 Intuitively an argument is labeled 119894119899 if it isjustifiable and 119900119906119905 if it is not If 119886 is labeled 119906119899119889 it isconsidered to be in abeyance due to for example insufficientgrounds for it to be labeled 119894119899 or 119900119906119905 Furthermore given anargumentation framework 119860119865 = (AR) one calls labeling acomplete function

L A 997888rarr Λ = 119894119899 119900119906119905 119906119899119889119886 997891997888rarr L119886119887 (119886) (1)

that assigns a label to each argument in 119860119865 A labeling iswritten as a triplet⨉120582isinΛ120582L(A) where 120582L(A) = 119886 isin A |L119886119887(119886) = 120582 For example 119894119899 L(A) stands for the set ofarguments in A that are labeled 119894119899 in the labelingL(A) Alabeling is said to be legal if all argument it labels is legallylabeled An argument 119886 isin A is said to be legally labeled

(i) L119886119887(119886) = 119894119899 if forall119887 isin A st (119887 119886) isin RL119886119887(119887) = 119900119906119905if 119886 is not attacked or is only attacked by argumentsthat are themselves labeled 119900119906119905

(ii) L119886119887(119886) = 119900119906119905 if exist119887 isin A st (119887 119886) isin RL119886119887(119887) = 119894119899that is if 119886 is attacked by at least one argument that islabeled 119894119899

(iii) L119886119887(119886) = 119906119899119889 if not(forall119887 isin A st (119887 119886) isin RL119886119887(119887) =119900119906119905) and (forall119888 isin A st (119888 119886) isin RLa119887(119888) = 119894119899)or equivalently if there exists at least one argumentlabeled 119906119899119889 that attacks 119886 and there is no argumentlabeled 119894119899 attacking 119886

Roughly speaking a semantics (denoted by 120590) is a rationalitycriteria to decide which arguments to accept given an argu-mentation framework The basic normative requirementsfor labeling-based semantics and argument acceptabilityin abstract argumentation repose on conflict-free labelingsor labelings in which no two 119894119899-labeled arguments attackeach other and admissible labelings which are conflict-free labelings that ask for arguments that are defended bythe 119894119899-labeled arguments to be themselves 119894119899-labeled Thefamily of admissibility-based labelings goes from completelabelings to preferred and grounded labelings which arecomplete labelings that capture properties such as credulityand skepticism in argumentation Formally a labelingL forthe arguments in an argumentation framework (AR) is saidto be

Complexity 5

(i) conflict-free if ∄119886 119887 isin A st (119886 119887) isin R and 119886 119887 isin119894119899L(A)

(ii) admissible if conflict-free and forall119886 isin AL119886119887(119886) =119906119899119889 997904rArr 119886 is legallyL119886119887(119886)

(iii) complete if admissible and forall119886 isin AL119886119887(119886) =119906119899119889 997904rArr 119886 is legally labeled 119906119899119889

(iv) preferred if complete and if it maximizes 119894119899L(A)(v) grounded if complete and if it maximizes 119906119899119889L(A)

We choose admissibility-based semantics because for onethey supply a comprehensive model of collective reasoningthey allow for a meaningful parameterization of credulousand skeptical collective reasoning and last they are relativelyeasy to interpret since the differences between them arewell documented in the literature (eg [18]) Additionallyand in contrast to other rank-based or graded semantics inabstract argumentation admissibility semantics assume thatall arguments have the same weight

223 Abstract Argumentation and Discursive Situations inCollective Decision-Making Abstract argumentation is aconvenient formalism for argument-based decision-makingsince it ignores difficulties relative to the nature generationand number of arguments and posits the possibility of usinggraph theoretic tools to model (collective) reasoning in aclear coherent and easy way [31] Example 2 below providesa simple overview of a debate between two agents over theacceptability of the proposal ldquoTax the richrdquo

Example 2 Let P = ldquoTax the richrdquo and P = ldquoLiberalismrdquoFigure 1 represents the abstract argumentation frameworkobtained from the following arguments

(i) 119868 = 119888 = ldquoOnly the rich should be taxed because theypossess most of the capital in the countryrdquo

(ii) 119887 = ldquoIf you only tax the rich the rich will leave andthen yoursquoll have no one to taxrdquo

(iii) 119886 = ldquoThe rich will not leave because they have theirlivelihoods here and it would cost themmore to leavethan to pay the taxesrdquo

Agent 1 advances the proposal argument 119888 agent 2 arguesthat given the justification 119888 for P P is not tenable Agent1 defends his proposal by advancing 119886 and defeating theargument 119887The conclusion is that holding 119888 and 119886 is a tenableposition so the rich should be taxed Notice that by simplyaccepting 119888 and taking no position on 119887 nor 119886 is conflict-freeand still corresponds to the position advocating that the richshould be taxed

Abstract argumentation comes as an immediate appli-cation to our model in the construction of an ideal argu-mentation framework that loosely models Habermasrsquo idealspeech situation The ideal speech situation is important inour work because it corresponds to a normative state that isin practice difficult to reach and that allows us to observehow different deliberation protocols affect deliberative out-comes

In the model a label given to an argument provideda semantics 120590 is said to be ideal if it is obtained from astate of affairs in which all arguments are presented duringdeliberation We call ideal or consensual argumentationframework the argumentation framework (AR) = 119860119865120576containing all arguments in the system and all consensualattacks among them In a multiagent approach a consensualattack between two arguments 119886 119887 isin A is a couple (119886 119887)such that (119886 119887) isin R if and only if a certain majority ofagents recognizes that such attack exists In our model allagents agree on the attack relation over 119860119865120576 It follows thatif two agents advance two distinct arguments 119886 119887 isin A suchthat (119886 119887) isin R then all agents will recognize that suchconflict exists An immediate consequence of this assumptionis that all deliberated results are consensual even if an opinionconsensus2 on the principle is not necessarily reached

23 Deliberative Social Agents In this section we presentthe agents in our model We define them on the basisof their opinions on the principle P the arguments theypossess their knowledge of the relationship among themtheir behavior during deliberation and their sensitivity todeliberated proposals

231 Agents in Dyadic Social Interactions Every agent 119894 hasan opinion a relative position or degree of adherence 119900119894 isin[minus1 1] to a principle P and a couple (119880119894 119879119894) isin [0 2] times [0 2](119880119894 le 119879119894) of latitudes of acceptance and rejection respectivelyof informational cues They live in [0 2] since 0 and 2 arerespectively the minimum and maximum distances of anytwo informational cues in the system The idea behind thecouple (119880119894 119879119894) is that there exist levels of relative tolerancefrom which informational cues have either an attractive ora repulsive effect on the individual [22 32] An 119900119894 close to 1implies that agent 119894 fully supports the principle P and closeto -1 that she rejects principleP or equivalently fully supportsnotP Moreover if an agent 119894rsquos opinion is such that |119900119894| ge 075119894 is considered to have an extreme opinion a moderate oneotherwise 119880119894 may be considered as 119894rsquos uncertainty about herown opinion [33] or as the limit below which the objectshe judges may attract her 119879119894 may be seen as 119894rsquos bound oftolerance from which informational cues disgust her andconfirmher initial positionDifferent combinations of (119880119894119879119894)can be associated with agent 119894rsquos ego or personal involvementin discussion processes as it is vividly explained in [32]Finally agents are assumed to be sincere and precise whencommunicating their positions to one another (no noise inthe interactions)

232 Agents in Deliberative Social Interactions Agents voteand participate in deliberation because they are aware ofthe potential changes an accepted proposal may induce inthe opinion of the group After all proposals promote aprinciple that potentially leads to a shift in other agentsrsquoopinions Agentsrsquo incentive to participate in deliberation isbased on the idea that every single one of them wants tomake her point across and at the same time reach a correctcollective decision Agents are endowed with a probability

6 Complexity

-1 0 1

Arguments

Adherence to principle

Figure 2 Representation of agents and their arguments on theopinion or adherence to the principles spectrum The rectangleson the agents are their argument sacks Arguments are depicted bydots and arrows from the arguments to the sacks depict both thearguments they hold in the opinion spectrum and the number ofarguments they hold 119896119894 = 4 and 120601119890119894 = 075 and 120601119898119894 = 05 for allagent 119894119909119894 = 119891(119889(119900119894 V119868) 119901119886) of being attracted to a deliberated cueand a probability of being repulsed by it 119910119894 = 119892(119889(119900119894 V119868) 119901119903)The two probabilities are assumed to be independent Theyare a function of the distance between the opinion andthe proposal argument and of the grouprsquos sensitivity3 todeliberation (119901119886 119901119903) The former is decreasing on 119889(119900119894 V119868)and increasing on 119901119886 while the latter is increasing on both119889(119900119894 V119868) and 119901119903 Agents are also assumed

(i) to be capable of assessing the degree of support for Pof all arguments

(ii) to trust4 one another when they utter informationalcues

233 Agents and Voting At the end of the decision-makingprocess agents vote on whether they agree or not on the pro-posal argument that has been discussed during deliberationVoting for an agent is the expression of her opinion in thefinal phase of the decision-making process An agent 119894 is saidto vote favorably for a proposal P of justification argument119868 if and only if V119868 times 119900119894 ge 0 or equivalently if the proposalargument does not adhere to the opposite of the principle 119894supports5

234 Agent Knowledge of Arguments Let A be a finite setof arguments Each agent 119894 has a sack of arguments A119894 subA of size |A119894| = 119896119894 whose content reflects her relativeposition 119900119894 on P (see Figure 2) For extreme-opinion agentsa proportion 120601119890 ge 05 of the 119896 arguments in their sacks areof the same adherence to the principle than their opinions

whereas for moderate agents a proportion 120601119898 le 120601119890 oftheir 119896 arguments are of the valence of their opinions Thehypothesis feeds from results found in [22] stating that ldquoanindividual places a verbal statement on an issue both in termsof the itemrsquos relative proximity to his own position and thelatitude which is acceptable to him around that focal point ofacceptancerdquo

The arguments in an agent 119894rsquos sack are those that sheknows how to use and advance in a deliberative interac-tion The size of the sack represents agent 119894rsquos ability tocommunicate in a deliberative context It follows that eachagent 119894 possesses partial knowledge (119870119894 sub A times A) of theattack relation between the arguments in A and of 119860119865120576which she derives from the arguments in her sack andwhat she observes in deliberation Knowledge of argumentsis assumed to be ldquoattack-orientedrdquomdashan agent 119894 knows anattack (119886 119887) isin A119894 times A if she knows upon observationthat neither she nor the group can rationally accept 119886alongside 119887 She may be aware of the existing defense relationamong arguments when concatenating the information sheaccumulates on their attack relation during deliberationShe may use this information strategically in deliberativecontexts

Let 119889119894119904119905 A times A 997888rarr N be the length of the shortestpath between two elements in the argumentation frameworkinduced by an agent 119894rsquos knowledge119870119894 119894 sees an argument 119886 asan attacker (defender) of an argument 119887 if119889119894119904119905(119886 119887) equiv 1mod 2(119889119894119904119905(119886 119887) equiv 0mod 2) An argument 119886 that is at an odddistance from another argument 119887 attacks it since either itdirectly attacks the argument or it attacks a defender of theargument Likewise if 119886 is at a pair distance of 119887 it attacksan attacker of 119887 and thus defends it If such path does notexist then the distance is undefined and the agent does notsee either argument as an attacker or defender of the otherFurther let AF119889 = (A119889R119889) denote the current state ofaffairs in deliberation An agent 119894rsquos knowledge ofR is the setof attacks and defenses she can infer from the arguments sheknows (119860119905119905119894119863119890119891119894) and the attacks and defenses she can inferfrom deliberation (119860119905119905119889119894 119863119890119891119889119894 )

119870119894 = 119860119905119905119894 cup 119860119905119905119889119894 cup 119863119890119891119894 cup 119863119890119891119889119894 (2)

where

(i) 119860119905119905119894 = (119886 119887) isin A119894 timesA119894 | 119889119894119904119905(119886 119887) equiv 1mod 2)(ii) 119863119890119891119894 = (119886 119887) isin A119894 timesA119894 | 119889119894119904119905(119886 119887) equiv 0mod 2(iii) 119860119905119905119889119894 = (119886 119887) isin (A119894 cup A119889) times A119889 | 119889119894119904119905(119886 119887) equiv1mod 2)(iv) 119863119890119891119889119894 = (119886 119887) isin (A119894 cup A119889) times A119889 | 119889119894119904119905(119886 119887) equiv0mod 2

Notice that agents have no restriction on the amount ofinformation they can carry and use during deliberation Themodel implies that agents have the ability to use and processall information on the attack relations if they could observethe attacks and that they all have equally high cognitivecapacity Agent knowledge resets at the end of each decision-making process but argument sacks stay untouched In otherwords argument sacks are static in the model

Complexity 7

235 Agent Behavior during Deliberation Agents maybehave in twodifferentways in deliberationTheymay behavenaively or focusedly Naive agents use deliberation to voicetheir opinions on the principle through arguments Focusedagents strategically argue in favor of proposal arguments thatsupport the principle they favor In both cases agents advancearguments in terms of the argumentsrsquo relative proximity totheir own positions [22]

Let 119886119894 isin A119894 denote the argument an agent advancesin a debate 119889 (of a deliberation process 119863119868) over a centralargument 119868 and V119886 the valence of an argument 119886 isin A Let

(i) 119891119894(119886) equiv (119900119894 times V119886 gt 0 or 119900119894 = V119886 = 0) the formula thatis satisfied if the argument 119886 is of the same valence asthe opinion of agent 119894

(ii) 119892119894(119886) equiv (119886 119887 isin A119894A119889 and forall119887(|119900119894 minus V119887| gt |119900119894 minus V119886|)) theformula that is satisfied only if the argument 119886 is in 119894rsquossack (A119894) and not in the debate and is the closest interms of adherence to the principle P to 119894rsquos opinion

(iii) ℎ119894(119886) equiv (119891119894(119886) and 119892119894(119886)) the conjunction of thepreceding formuli indicating that an argument 119886 is inagent 119894rsquos sack and not in the debate of the same signas 119900119894 and closest to 119900119894 in absolute value

Naive agents choose 119886119894 = 119886 such that ℎ119894(119886) is true given anyproposal argument 119868 Agents that behave strategically or tosay focusedly choose their moves in the debate as follows(119873119868119871 indicates no move or null argument)

(a) if agent 119894rsquos position with respect to P is of the samesign as the proposal argumentrsquos adherence to P (119891119894(119868)) sheadvances an argument 119886 such that

(i) ℎ119894(119886) and ((119886 119868) isin 119863119890119891119889119894 )) is true If 119886119894 = 119873119868119871(i1) notℎ119894(119886) and 119892119894(119886) and ((119886 119868) isin 119863119890119891119889119894 ) (chooses anargument not of the same sign and closest to heropinion and defending 119868) is true If 119886119894 = 119873119868119871(ii) ℎ119894(119886) and ((exist119888 isin A119894)((119888 119868) isin 119860119905119905119889119894 and (119886 119888) isin 119860119905119905119894))(anticipates an attack from an argument 119888 on 119868 andadvances an argument that attacks the argument 119888)6is true If 119886119894 = 119873119868119871(ii1) notℎ119894(119886)and119892119894(119886)and ((exist119888 isin A119894)((119888 119868) isin 119860119905119905119889119894 and(119886 119888) isin119860119905119905119894)) is true If 119886119894 = 119873119868119871(iii) ℎ119894(119886) and ((119886 119868) notin 119860119905119905119889119894 ) (simply does not attack 119868)is true If 119886119894 = 119873119868119871(iii1)notℎ119894(119886)and119892119894(119886)and((119886 119868) notin 119860119905119905119889119894 ) is true If 119886119894 = 119873119868119871(iv) 119886119894 = 119873119868119871 (the agent will advance no argument)

(b) if agent 119894rsquos position with respect toP is of the oppositesign of the proposal argumentrsquos adherence to P (not119891119894(119868)) sheplays the argument 119886 such that

(i) ℎ119894(119886) and ((119886 119868) isin 119860119905119905119889119894 ) 7 If 119886119894 = 119873119868119871(i1) notℎ119894(119886)and119892119894(119886)and ((119886 119868) isin 119860119905119905119889119894 ) is true If 119886119894 = 119873119868119871(ii) ℎ119894(119886) and ((exist119887 isin A119894)((119887 119868) isin 119863119890119891119889119894 and (119886 119887) isin 119860119905119905119894))(anticipates a defense from an argument 119888 to 119868 andadvances an argument that attacks 119888) is true If 119886119894 =119873119868119871

(ii1)notℎ119894(119886)and119892119894(119886)and((exist119887 isin A119894)((119887 119868) isin 119863119890119891119889119894 and(119886 119887) isin119860119905119905119894)) is true If 119886119894 = 119873119868119871(iii) ℎ119894(119886) and ((forall119887 isin A119889)((119887 119868) isin 119860119905119905119889119894 and (119886 119887) notin 119860119905119905119889119894 ))(avoids attacking any attacker of 119868) If 119886119894 = 119873119868119871(iii1)notℎ119894(119886)and119892119894(119886)and((forall119887 isin A119889)((119887 119868) isin 119860119905119905119889119894 and(119886 119887) notin119860119905119905119889119894 )) If 119886119894 = 119873119868119871(iv) 119886119894 = 119873119868119871

In English a focused agent that is opposed to the proposalattempts to either attack it directly or attack an argument thatis defending it whereas an agent that agrees with the proposalattempts to either defend it or avoid attacking it altogetherThis behavioral approach is similar to debate protocols inabstract argumentation but for the fact that agents mayadvance arguments that do not result in an ldquoadvancementrdquoof the debate In our case the debates stop for other reasons(see Section 24)

Two important points deserve to be highlighted Firstlyfocused agents with small argument sacks will regularly findthemselves applying rules (119894119894119894) or (119894V) In effect if they donot have enough arguments to infer attacks and defensestheir behavior in deliberation is likely to be similar to that ofthe naive agents Secondly the naive and focused behavioralassumptions presented here are very different in terms ofcomputational complexity The first assumes that the agentobserves the debate but what she sees has no effect on hercourse of action ie the argument she voices is uniquelydetermined by how she feels about the principle behindthe proposal argument In the second an agent debatesessentially to knock out any proposal argument she disagreeswith8

24 Tables for Deliberation Tables are the physical or virtualplaces where the exchange of arguments occurs Agentsdeliberate on these tables to determine whether the proposalargument is acceptable or not and are thus subdued bythe deliberation procedure imposed by the tablersquos centralauthority (CA) [27] The CA decides on the structure andlength of the collective decision-making process and thedeliberation procedure It controls the percentage of agentsfrom the population that actively participates in the debate(119899119863) and the labeling-based semantics used to extract accept-able arguments from the framework (120590) The percentage 119899119863denotes either the proportion of agents in the population thatgets to advance arguments in a population-scale deliberationor an independent sample of agents summoned to activelyparticipate in the debate 120590 stands for the procedure used toconclude on the epistemic status of the proposal argumentand the arguments advanced during deliberation

The CA has the ability to stop debates at will using a stoprule that inherently depends on the number of debates thatought to take place before a decision is deemed sufficientlydiscussed (119898) the maximum number of debates that cantake place before abandoning deliberation (119898) and the labelgiven to the proposal argument 119868 (119904119903(119898119898L119886119887(119868))) 119904119903is a Boolean function whose value ldquotruerdquo signals the callfor a vote andor the end of the decision-making process119898 is associated with a minimal dialectical or epistemic

8 Complexity

requirement to consider a proposal for voting and to a lowerbound of the length of the deliberation process 119898 providesan upper bound Moreover the CA controls the size of thetime interval between debates (119905119863) the collective decisionrule (eg if there is voting on the deliberated proposals) andthe majority quota rule for accepting proposal arguments(120572) in the decision-making process 119905119863 may represent thefrequency or density of pair-wise interactions in a decision-making process

241 The Construction of a Decision-Making Process Adeliberation process or debate in ourmodel is a tool to obtainlabels for proposal arguments that are as close as possible tothe ideal or consensual ones To define a deliberation processformally we introduce the notion of debate step as constituentof a deliberation process Informally a debate step is a timestep in which a debate occurs Formally and more event-oriented a debate step (1198891198731015840 1198611015840 1198781015840) of a debate 119863119868 on 119868 isa quadruplet composed of a set of agents 1198731015840 sub 119873 a set ofarguments 1198611015840 sub ⋃119894isin1198731015840 A119894 a set of attack relations 1198781015840 sub Rand a mapping

119889 (AF 2A times 2R) 997888rarr AF

(119860119865 (1198611015840 1198781015840)) 997891997888rarr 1198601198651015840 = (119861 cup 1198611015840 119878 ⊔ 1198781015840) (3)

that adds the arguments in 1198611015840 and some attacks in 1198781015840 to aframework 119860119865 = (119861 119878) and where 119878 ⊔ 1198781015840 = (119886 119887) isin 119878 cup 1198781015840 |119886 119887 isin 119861 cup 1198611015840 In the same spirit we define a deliberationprocess119863119868 on a proposal argument 119868 as a sequence of debatesteps (119889119873119904 119861119904 119878119904)119904gt0 such that1198730 = 0 and

119889((119904minus1⋃119895=0

119861119895 119904minus1⨆119895=0

119878119895) (119861119904 119878119904)) sub (AR) (4)

with

(i) 1198610 = 119868 (the initial argument is the proposal argu-ment)

(ii) 1198780 = 0 (no reflexive attack from 119868 to 119868 is allowed)(iii) (forall119904 gt 0)(⋃119895lt119904 119861119895cap119861119904 = 0) (any newly added argument

to the framework has yet to be added to it)(iv) (forall119904 gt 0)(⨆119895lt119904 119878119895 cap 119878119904 = 0) (any newly considered

attack among arguments cannot be declared amongarguments that are not in the framework)

(v) lim119904997888rarrinfin(⋃119904gt0 119861119904 ⋃119904gt0 119878119904) sube (AR) (the system isstable no arguments are created during deliberation)

Finally let L119886119887119889(119868) denote the label given to argument 119868 ata debate step 119889 during a deliberation process 119863119868 A decision-making process notedD119868 = (119899119863 119905119863 119904119903(119898119898L119886119887119889(119868)) 120572 120590)over a proposalwhose justification argument is 119868 is a sequenceof debate and nondebate steps (119911119905)119905isinN 119911119905 isin 119889 119889 such that

(i) 119889 are debate steps and 119889 nondebate steps that corre-spond to time steps at which there are no debates

(ii) If 119905(mod 119905119863) equiv 0 then 119911119905 = 119889 and 119889 otherwise

For Against

Table

CA

in out

out out in

out in

proparg

Figure 3 A representation of a table of deliberation betweentwo groups of 3 agents with 4 arguments each Each agent hasadvanced an argument Agents tacitly agree on the attacks betweenthe arguments The CA determines the labeling using the groundedlabeling-based semantics

d d d d d d d d d d d d dTime 0 1 2 3 4 5 6 7 8 9 10 11 12I

Figure 4 A decision-making processD119868 or collective decision stepstarting at 119905 = 0 composed of debate (119889) and nondebate (119889) substepsDebate steps are shaded The number of nondebate steps betweendebates is 119905119863 = 3 the number of debates in the decision-makingprocess is119898 = 4

(iii) The subsequence (119911119897119905)119897119905isinN of 119911119905 with 119897119905 = 119905 such that119905(mod 119905119863) equiv 0 is a deliberation process with |1198731015840| =119899119863 times |119873|(iv) 119904119903(sdot sdotL119886119887119889(119868)) pursues the deliberation process as

long asL119886119887119889(119868) = 119906119899119889 under the semantics 120590(v) The length of the sequence |119911119905| or equivalently the

duration of the process is somewhere in the interval⟦119898119905119863 119898119905119863⟧(vi) The final labeling for the proposal argument 119868

L119886119887(119868) is determined by a combination of its delib-erated labelL119886119887119889(119868) and a majority vote of majorityquota 120572

A decision-making process ends when a final decision hasbeen taken concerning the acceptability of the proposal Peg 119868 has been deemed unacceptable (L119886119887119889(119868) = 119900119906119905) or amajority of agents have voted againstP For a representationof a deliberative interaction on a table see Figure 3 for one ofa decision-making process see Figure 4

Please note that every collective decision can be seen asa ldquotime steprdquo in as much as it describes how agents updateopinions and make collective decisions In the model thedecision-making process and the parameters define the sub-steps or events that occur within the time step Henceforth

Complexity 9

Fix proceduralsm m α nD tD σ Init count m larr 0

Given proposal anargument I is randomly generated

nD times |N | agents arerandomly picked

m larr m + 1

Begin decision processon proposal at time t

t larr t + 1Deliberation step m

larr 0

t larr t + 1dyadic opinionupdate Eq (5)tm larr tm + 1

m ge m

tm gt tD

ℒd (I) = in

m ge m

yields ℒ(I)

ℒ(I) = in Deliberationupdate Eq (6)

End decision

yes

no

no

no

yes

yes

no

yes

yes

yes

no

no

I

Vote on ℒd(I)

ℒd (I) = und

ＪＬＩ＝ＭＭ I

Init count tmyields ℒd(I)

Figure 5 Process chart for a decision processD119868 on a proposalP 119905 corresponds to time and 119898 (119905119898) corresponds to the number of debate(non-debate) steps that have taken place in the deliberation process (in between two debate steps) over time for the mixed discussion modelWe useL119886119887(119868) andL(119868) interchangeably in the figure for presentation

comparing simulations on the basis of the different decision-making processes translates into comparing these collectivedecision steps

242 Deliberation Protocol To define the deliberation proto-col held in the table either as a consequence of the definitionprocess or as a statement we assume the following

(i) Agents may decide not to contribute to the debate(ii) All agents have the same probability conditional to

their opinions of being picked to participate in adebate or deliberation step

(iii) There is no restriction on the number of times eachagent can participate in a deliberation process

(iv) Each agent may only place one argument per debatestep

(v) Arguments that have already been advanced in thedebate may attack newly placed arguments9

The deliberation or debate protocol goes as follows

(1) The CA randomly generates and makes public acentral argument or proposal argument 119868 and informsall agents about the rules of the decision-makingprocess

(2) The CA randomly draws two sets of (119899119863 times |119873|)2agents with divergent views10 onP and P It mergesthem to create the set of debaters1198731015840

(3) Each agent 119894 advances an argument from her sackA119894 The CA makes sure that there are no repeatingarguments (agents already take argument repetitioninto account)

(4) The CA establishes the debate step 119889rsquos argumentationframework and computes its labelingL120590

119889(A119889) using120590(5) If the computed label for 119868 is L119886119887119889(119868) = 119906119899119889

or the number of debates steps is inferior or equalto 119898 at time 119905 then the CA stops the debate andresumes it at the (119905 + 119905119863 + 1)rsquoth time step by repeating(1) (2) (3) (4) and (5)

(6) Let L119886119887(119868) be the final label given to the proposalargument 119868 If voting is allowed then if more than120572 times |119873| agents agree with 119868 119868 is accepted (L119886119887(119868) =119894119899) it is rejected if strictly less than 120572times |119873| agree withit (L119886119887(119868) = 119900119906119905) When there is a tie ifL119886119887119889(119868) =119906119899119889 then L119886119887(119868) = 119900119906119905 if L119886119887119889(119868) = 119894119899 thenL119886119887(119868) = 119894119899 When voting is not part of the decision-making process (120572 = 0)L119886119887119889(119868) = L119886119887(119868)

One important point to notice about this protocol is that itinduces a stop rule 119904119903 (cf steps (5) and (6) of protocol) forthe decision process and thus it always ends Either agentsdebate and agree on the proposalrsquos acceptability throughargumentation or after 119898 debate steps they directly voteon it so that a decision concerning the proposal is alwaysreached Refer to Figure 5 for a sweeping description of thedeliberation protocol and the decision-making process

10 Complexity

-1

-1 1

1

Latitude of acceptance

Latitude of rejection Agent i

Agent j

Figure 6 A dyadic interaction between agents 119894 and 119895 as expressedin (5) Agent 119894rsquos opinion falls within agent 119895rsquos latitude of noncommit-ment and conversely Neither agent affects the opinion of the other

25 Opinion Dynamics for Social Interactions In this subsec-tion we describe the opinion diffusion model based on pair-wise interactions among agents and deliberation Sherif rsquosand Hovlandrsquos social judgment theory [22] motivates part ofour approach It describes how individualsrsquo opinions changeon the basis of their attitude structures Attitude structuresrefer to the relative scope width or latitude of categoriesused by individuals when evaluating information namelythe latitudes of acceptance rejection and noncommitment[32] The idea behind this theory is that individuals changetheir positions only in accordance to how far or close thecommunicative cues they receive are from (to) their anchorpositions It holds that if communicative cues are far (close)from (to) an agentrsquos position say over her latitude of rejection(acceptance) then the agent shifts her position away from(towards) the position defended by the cues In the casewherethe cues fall within the agentrsquos latitude of noncommitmenther position does not change (see Figure 6 and Equation (5))

251 Pair-Wise or Dyadic Opinion Dynamics As agentsmay communicate and deliberate collectively they may alsoengage in one-to-one conversations with other agents toponder on their positions We loosely associate this type ofcommunication with dyadic nonargumentative exchange ordiscussion based on fallacious arguments and persuasionIn the light of social judgment theory and the descriptionof agents in the system we model pair-wise symmetricinteractions following the opinion dynamicsmodel in [9] Anagent 119895rsquos influence on an agent 119894rsquos opinion at time 119905 is governedby the ensuing difference equation

119900119905+1119894 =

119900119905119894 + 120583119894 (119900119905119895 minus 119900119905119894) 119894119891 10038161003816100381610038161003816119900119905119894 minus 11990011990511989510038161003816100381610038161003816 lt 119880119894119900119905119894 minus 120583119894 (119900119905119895 minus 119900119905119894) 119894119891 1003816100381610038161003816119900119905119894 minus 119900119905119894 1003816100381610038161003816 gt 119879119894119900119905119894 119900119905ℎ119890119903119908119894119904119890(5)

where the parameter 120583119894 isin [0 05] controls for the strengthof the attraction and repulsion in social influence and 119879119894and 119880119894 are the latitudes of rejection and acceptance for agent119894 respectively The parameter 120583119894 may be thought of as therelative importance agent 119894 gives to the opinions of her peersor analogously the weight she gives to her own opinionwhen updating 120583 lt 05 means that agents will never givemore weight to the opinions of other agents than to their

own (egocentric bias) When two agents 119894 and 119895 discuss iffor instance 119895 advances an informational cue (argumentpersuasion tactic) and it happens to be close enough to119894rsquos opinion anchor then 119894 shifts her opinion towards thedirection of 119895rsquos informational cue The symmetric influencefrom agent 119894 to agent 119895 takes effect in the same way (seeFigure 6)

We add to the dyadic dynamics a rule of encounters ateach step of pair-wise interactions each agent meets exactlyone other agent at random This rule may be associated withrandom day-to-day encounters among agents Steps of socialinfluence correspond to nondebate steps (119889) in decision-making processes

252 Deliberative Opinion Dynamics We define an opinionupdate equation that links the proposal arguments that areadvanced during deliberation and opinions We combine theuncertain and probabilistic nature of the effect of delibera-tion have it a moderating [15 16] or polarizing [14] one witha mechanism similar to the one in social judgment theorybased on the distance between different informational cuesand opinion anchors [32] The probabilistic modeling of theopinion update can be related to deliberation encouragingopen-mindedness11 during collective discussion [18] Fromthis choice it follows that deliberated cues can affect even themost extreme of agents as opposed to some classic opiniondiffusion models (eg [5 8]) where once agents becomeextremists they may no longer become moderate

LetP be a proposal V119905119868 the proposal argument 119868rsquos level ofsupport for a principle P and 119900119905119894 an agent 119894rsquos opinion at time119905 Then given the difference 120575119905119894 = (V119905119868 minus 119900119905119894)2 we define 119894rsquosprobability of being attracted to a decisionrsquos informational cue119868 by 119909119905119894 = 119901|120575119905119894 |119886 where 119901119886 denotes a general probability param-eter that characterizes how important deliberated results arefor the group The parameter may also be interpreted asthe grouprsquos tendency to be swayed by a decisional majoritySimilarly we define 119894rsquos probability of being repulsed from adecision informational cue 119868 by 119910119905119894 = 1199011|120575119905119894 |119903 where 119901119903 denotesa general probability parameter that characterizes the grouprsquosdislike of deliberated results 119901119903 can also be thought of as thegrouprsquos distrust of the system symbolized by deliberation anddemocracy

Let L119886119887(119868) denote the epistemic status of the proposalargument 119868 at the end of a decision process over P everyagent 119894 updates her opinion as follows

(i) IfL119886119887(119868) = 119894119899

119900119905+1119894 =

119900119905119894 + 2120574119894120575119905119894 119908119894119905ℎ 119901119903119900119887119886119887119894119897119894119905119910 119909119905119894119900119905119894 minus 2120574119894120575t119894 119908119894119905ℎ 119901119903119900119887119886119887119894119897119894119905119910 119910119905119894119900119905119894 119908119894119905ℎ 119901119903119900119887119886119887119894119897119894119905119910 1 minus 119909119905119894 minus 119910119905119894

(6)

(ii) IfL119886119887(119868) = 119900119906119905 119900119905+1119894 = 119900119905119894where 120574119894 isin [0 05] is the strength of repulsion and attractionin the dynamicThemeaning of 120574119894 is analogous to the one of120583119894in the dyadic interactionsmodel but for the observation that 119894

Complexity 11

weights her opinion to an argument rather than to an opinionThe interpretation of this dynamics is straightforward Ifthe deliberated proposal argument is close to an agentrsquosopinion then it is very likely that the agent shifts its opiniontowards it Please note that the probability 119909119905119894 (119910119905119894 ) that anagent is attracted (repelled) to (from) an accepted deliberatedproposal argument is bounded below (above) by 119901119886 (119901119903)(equivalently 119909119905119894 isin [119901119886 1] 119910119905119894 isin [0 119901119903]) Steps of deliberativeopinion dynamics are associated with debate steps in which aconcluding collective decision is made

253 Mixed Opinion Dynamics Opinion dynamics in thecontext of decision-making processes can be best described asa combination of the two preceding dynamics In this opiniondynamics model agents are engaged in a democratic systemin which they deliberate vote for proposals and occasionallydiscuss with one another on the principle supported by theproposalsThe opinion dynamics for an agent 119894 can be writtenas a sequence of dyadic and deliberative opinion updates

119900119905+1119894 =

119864119902119906119886119905119894119900119899 (5) 119894119891 119905 equiv 0 (119898119900119889119905119863) and 119904119903 (sdot) = 119891119886119897119904119890119864119902119906119886119905119894119900119899 (6) 119894119891 119905 equiv 0 (119898119900119889119905119863) and 119904119903 (sdot) = 119905119903119906119890119900119905119894 119900119905ℎ119890119903119908119894119904119890

(7)

where 119904119903(sdot) is the decision processrsquos stop rule 119905119863 the numberof time steps between each deliberation step Equation (5)describes the dynamics for the dyadic interactions and Equa-tion (6) posits the changes in opinions due to the deliberatedcues Equation (5) applies when agents are not deliberatingand when each agent encounters another agent to exchangeinformation that may lead to local opinion updates Whenthere is deliberation a handful of agents deliberate and ifthey reach the stop condition imposed by the table all agentsvote for the proposal They update (see Equation (6)) theiropinions on the basis of the proposal argumentrsquos support forthe principle and the result of the scrutiny Otherwise thereis no change in the opinions of the agents A vote indicatesthe end of the decision-making process

3 Experiments and Calibration

In this section we introduce the metrics that enable us toobserve the simulations and characterize the calibration ofthe model

31 Observations and Initialization In this section wedescribe the simulations and the protocol used to test andobserve the results of our model We introduce the metricsof interest explain the calibration of the model describesome results obtained from the simulation data used forcalibration and conclude with a brief discussion on theexpected outcomes of the mixed opinion dynamics model

311 Metrics or Statistics of Interest Let 119878 denote the end ofa simulation We are interested in the effect of deliberationand of the modelrsquos procedural parameters on the followingmetrics or statistics

(i) Variance of opinions (119881119886119903(119900)) the variance ofopinions at time 119878 Since opinions live in the unionof the positive and negative unit intervals 119881119886119903(119900) isin[0 1] The higher the variance of the distribution isthe more ldquodiverserdquo the opinions are(ii) Proportion of extremists in the population(119901119903119900119901119890119909) the proportion of agents in the populationwith opinions such that |119900119894| ge 075 A high proportionof extremists makes ldquohealthyrdquo consensus12 difficult toreach and deliberation more or less informative(iii) Shifts of opinions (119878ℎ) [33] statistic that measuresthe aggregated change in individual opinion at time 119878with respect to aggregated individual opinion at thebeginning of the simulation

119878ℎ = 2sum119894isin119873 10038161003816100381610038161199000119894 minus 1199001198781198941003816100381610038161003816max119894isin1198731199000119894 minusmin119894isin1198731199000119894 (8)

119878ℎ is positive and bounded above by 2|119873| since 119900119894 isin[minus1 1] A low shift statistic implies that the processhas a small impact on opinions(iv) Judgment or consensual inaccuracy (119890119888) consen-sual accuracy of a group consists of an ad hoc statisticmeasuring a grouprsquos ability to infer correct labels forproposal arguments from a decision-making processgiven the ideal consensual labeling based on full-informationL120590

120576 (A) We use a Hamming-based dis-tance on labelings [34] to define the statistic

119890119888 = 1|I| sum119868isinI1198861198681003816100381610038161003816L119886119887120576 (119868) = L119886119887 (119868)1003816100381610038161003816 (9)

whereI is the set of all discussed proposal argumentsand 119886119868 = 12 ifL119886119887120576(119868) = 119906119899119889 or ifL119886119887(119868) = 119906119899119889119886119868 = 1 otherwise119890119888 lives in the interval [0 1] An inaccuracy statisticclose to 1 indicates that agents subdued to a particulardecision-making process make many mistakes injudging the labels of the proposal arguments Notethat when there is voting all proposal arguments arelabeled either 119894119899 or 119900119906119905 after deliberation(v) Coherence (119894119903) letL119886119887119863(119868) be the label obtainedfor 119868 from the deliberation process without votingThe coherence statistic measures how well votingresults adjust to results obtained during deliberationonly We use the proportion of arguments that havebeen labeled 119894119899 in the debate and that agents havevoted favorably for

119894119903 = 1003816100381610038161003816119868 isin I | L119886119887119863 (119868) = 119894119899 andL119886119887 (119868) = 11989411989910038161003816100381610038161003816100381610038161003816119868 isin I | L119886119887119863 (119868) = 1198941198991003816100381610038161003816 (10)

The coherence statisticrsquos domain is [0 1] If after 119872debates no deliberated central argument has beenlabeled 119894119899 or voting is not part of the procedure ofcollective decision-making then the statistic equals

12 Complexity

1 or said differently agents are perfectly coherentThis comes for the fact that if the central argumentis labeled 119900119906119905 then it is not even eligible for voting iflabeled 119906119899119889 then the debate will always be coherentwith the preferences of the agents since the result willbe a simple aggregation of their votes If the statisticequals 0 then none of the proposal arguments labeled119894119899 are voted in favorably by the agents It follows that ahigh coherence statistic implies that when agents votefor acceptable proposal arguments the results of thescrutiny reflect the consensual rationality expressedin the deliberation process

312 Parameters of Interest We recall the parameters ofinterest in our study that are linked to the structure of thedecision-making processes

(i) 119898 the minimum number of time steps in which adebate or a scrutiny occurs in a decision-making pro-cess before a final deliberated decision is submitted toa vote

(ii) 119899119863 the number of agents that deliberate as a propor-tion of the population

(iii) 119905119863 the number of time steps between debates inwhich pair-wise interactions among agents mayoccur

(iv) 120572 the proportional majority requirement for theacceptance of a proposal When 120572 = 0 it stands forno voting ldquoany proposal argument that is labeled 119894119899during deliberation is acceptedrdquo

Please recall that in terms of the definition of a decision-making process each parameter controls either the length orthe content of the sequence D119868 (119905119863 119898) or the rules that areapplied during the debate steps (120572 119899119863) (refer to Figure 4)313 Initialization All |119873| = 400 agents start off with anopinion 119900119894 drawn from a uniform distribution U(minus1 1) 13Given 119900119894 every agent 119894 randomly draws a setA119894 (|A119894| = 119896) ofarguments from a balanced14 argument pool A of119872 = 600nonneutral (V119886 = 0) arguments on the basis of 119900119894 if |119900119894| lt 025(the opinion is moderate) then agent 119894 randomly fills halfof her argument sack with arguments such that V119886 lt 0 andthe other half with arguments such that V119886 gt 0 (120601119898119894 = 05)Otherwise she randomly fills 120601119890119894 ge 05 of her sack witharguments such that V119886 and 119900119894 are of the same sign and theremaining (1 minus 120601119890119894 ) with arguments of the opposite sign15

Like for the opinion of agents every argument 119886rsquosadherence to the principle P V119886 is drawn from a uniformdistribution U(minus1 1) The attack relation that gives birth tothe ideal argumentation framework (119860119865120576) is established onthe basis of the V119886rsquos and the argumentsrsquo epistemic reach 119890119903

Let 120579(119886 119887 119903) = (|V119886 + V119887|2)119903 be an auxiliary positivereal-valued function that takes two arguments (119886 119887) and apositive real number (119903 gt 0) as inputThe probability that anyargument 119886 isin A creates a link to any other argument 119887 isin A

is given by the equation 119901 = 119901120579(1198861198872)119890 and thus argumentssupporting opposing views of the principle always attack each

other We fix 119901119890 the epistemic correlation parameter to 015If 119901119890 was greater than 015 then arguments of the same signwould attack each other too often (and focused agents wouldbe rarely incited to advance favorable arguments duringdeliberation) If 119901119890 was lower then the arguments of thesame type would induce an almost empty graph which isunrealistic The number of arguments that any argument 119886can attack is bounded by 119886rsquos epistemic reach that we fix to15 A higher value of epistemic reach makes the argumentlattice too conflicting and nearly bipartite A lower epistemicreachmakes the lattice not sufficiently conflicting in the sensethat too many arguments can attack the proposal argumentrelatively to the few that can defend it If any argument attacksthe proposal argument the chances that another argumentattacks the attacker are low Hence the proposal argumentis almost never accepted and deliberative opinion updatesbecome rare

The arguments in the resulting argumentation frame-work 119860119865120576 are given a permanent labelingL120590

120576 (A) (in shortL120576) using120590 = grounded semanticsWe choose the groundedlabeling-based semantics because it provides a unique admis-sible labeling respects minimal rationality constraints whilesimplifying the model and models a skeptic approach toaccepting arguments (see [23 25 31]) as it maximizes thecardinality of the set 119906119899119889 L120576 To some extent if we considerproposals as committing because they may guide collectiveaction choosing a skeptic semantics seems reasonable insofaras it labels an argument 119894119899 or 119900119906119905 only if it has no reasonto label it 119906119899119889 For instance if a proposal is a public policythat requires large amounts of resources and engages tofuture course of action then it may also determine policycycles and heavily burden a group and its future decisionsIn important situations like these it seems reasonable tolengthen deliberative cues and ask for more demanding andldquogroundedrdquo criteria for policy argument acceptability

On the proposalrsquos side we create an argument 119868 whoseadherence to P is also drawn from aU(minus1 1) and we label itL119886119887(119868) = 119906119899119889 119868 is interpreted as being the main argumentjustifying the discussed proposal P and V119905119868 as its supportfor P at time 119905 119868 cannot attack other arguments but otherarguments can attack it For each argument 119886 isin A a directedarc or attack from 119886 to 119868 is activated with probability 119901119901 =003120579(11988611986802) When 119901119901rsquos lower bound is too high the proposalargument is almost always defeated and thus deliberativeopinion updates do not happen When 119901119901rsquos lower boundis lower than 003 then the opposite occursmdashthe proposalargument is always accepted due to an absence of attackstowards it Finally we set the maximum number of debatesto119898 = 7 for the decision-making processes stop rules1632 Calibration and Simulation Protocol In this subsectionwe discuss the calibration of the models the terminationconditions for runs in the deliberative and mixed model andthe expected outcomes in terms of the observations

321 Calibrating Dyadic Opinion Dynamics Dyadic opiniondynamics correspond to a space of parameters in whichargumentation and deliberation spaces are not taken into

Complexity 13

Table 1 Parameters and parameter types and their domains used to compare dyadic collective and mixed opinion dynamics

Length ofD119868 Decisional in119863119868 Social influence Deliberation influence Cognitive119905119863 119898 119898 119899119863 120572 120583 119879 119880 119901119886 119901119903 120574 119896 119861119890ℎ119886V1198941199001199031 3 6 1 3 6 7 001 002 005 0 05 066 01 14 18 02 06 03 05 01 005 01 02 4 8 16 119873119886119894V119890 119865119900119888119906119904119890119889account Deliberated arguments and informational cues haveno effect on agent opinion but agents still vote for theproposal argument17 We use and calibrate this model forcomparability in terms of all our metrics since they arenot explicitly observed in [9] a reference opinion dynamicsmodel Furthermore for simplicity and comparability againwe suppose that agents are homogeneous in terms of theirattitude structures and opinion weightings We fix 120583119894 =120583 119879119894 = 119879 and 119880119894 = 119880 for all 119894 isin 119873 and for some triplet(120583 119880 119879) isin [0 05] times [0 2] times [119880 2]

Experiments suggest that the strength of the dyadic inter-actions is an explanatory factor of the time of convergence(120583 lt 05) and of the dyamicsrsquo steady states Bigger values of 120583speed up convergence towards a stable set of opinion clustersand affect the size and the relative position of these clustersin the opinion distribution 120583 however does not play a majorrole in the number of opinion clusters observed at the steadystate of the dynamics

We set 120583 to the value given to it in Jagger et alrsquos opiniondynamics model (120583 = 01) [9] For one it is the smallestreal number for which most results found in [9] hold and thedynamics are smooth secondly it seems to be a reasonableldquostrengthrdquo of influence considering that we would not wantpair-wise social interactions to completely shadow or maskan eventual effect of deliberation in the opinion dynamics

For different values of attitude structures 119879 and 119880 weget exactly the same results as in [9] in terms of the numberand density of the opinion clusters The different couplesof values of 119880 and 119879 model the size of the latitude ofnoncommitment (119879 minus 119880) and thus the agentsrsquo propensityto update opinions Whenever 119879 minus 119880 le 02 regularitiesin convergence and opinion clustering appear If 119880 gt 1there is always central convergence if 119879 lt 1 there is alwaysbipolar convergence with clusters of similar sizes For 119879 minus119880 gt 02 relatively low values of 119879 (119879 lt 1) and relativelyhigh values of 119880 always yield extreme bipolar convergencewith a relatively small cluster of moderate agents (bipolar-central convergence) For 119879 sufficiently high (119879 ge 12) and119880 sufficiently low (119880 le 08) in other words not very ego-involved agents there is a bigger spectrum of steady statesFor this reason we studied the metrics of interest for thesevalue differences and kept the couples of (119880 119879) depicted inTable 1 For these couples central bipolar bipolar-centralconvergence (3 clusters) and multicluster convergence (4 or5 clusters) are possible and result frommeaningfully differentattitude structures Judgment accuracy varies significantlyacross the four different attitude structures we retain and soit happens with the variance of opinions These scenariosprovides us with reference results from which to build upon

322 Calibrating Deliberative Social Interactions Delibera-tive social interactions correspond to social situations in

which individuals are not influenced by pair-wise discussionwith peers but are sensitive to collectively deliberated infor-mational cues On these terms parameters such as the sizeof agentsrsquo argument sacks (119896119894) the distribution of argumentsin them (120601119894) the minimum number of debates (119898) andthe proportion of debaters from the population (119899119863) are nolonger mute We assume for simplicity that the formulaexist120601119890exist119896forall119894(119896119894 = 119896 and 120601119890119894 = 120601119890 and 120601119898119894 = 05) is true with 119894 isin 119873for some value 120601119890 isin [05 1] and for 119896 isin N

The chosen domains for these parameters (see Table 1)are justified by the size of the population and the size of theargument pool (119872 = 600) which we set from the start If 119896is too big (119896 ge 32) andor too many agents are allowed inthe debate (119899119863 ge 02) then individuals always disprove thecentral argument and deliberation is never or rarely takeninto account in the update of opinions Similar effects occurwhen 119898 is high yet we only calibrate it with respect to themean running time of a simulation and the values of 119905119863Different distributions of 120601119890119894 seem to significantly or directlyexplain none of the statistics we analyze so we convenientlyset 120601119890119894 = 075 We chose the parameter 119896 to be always divisibleby 4 for it is convenient given the initial conditions regardingthe argument distribution in the agentsrsquo argument sacks

We fix the value space for the parameters 120574 119901119886 119901119903 as inTable 1 to account for different intensities of the effect ofdeliberative voting retroaction on the system For examplethe scenarios where the three parameters are at their lowestvalues correspond to the scenarios with the smallest fixedprior effect of deliberation in the model Increasing any ofthese parameters should mechanically be associated witha world in which agents are more sensitive to collectivedecisions The values for 119901119886 are taken from [13] wherethe author states that by means of deliberation 7 to28 of individuals changed their opinion from agreeingto disagreeing or vice versa on a referendum question forDenmarkrsquos participation in the EuroWe choose two differentvalues for 119901119886 that account for high (03) and very high (05)sensitivity to deliberation in a group 119901119903 on the other handis taken to be small (equal to 01) since we believe that thereexist agents that will always go against the reached consensusTheir numbers are meager

For simplicity we assume that for all 119894 isin 119873 120574119894 = 120574 forsome 120574 isin [0 05] In effect we posit that the heterogeneityof agents in the deliberative context only derives from theirarguments and their opinions We decide to calibrate 120574 inallusion to the strength of the social dynamics in pair-wiseinteractions In doing so we limit ourselves to consider threedifferent scenarios in which deliberation has respectivelyhalf equal and twice as much opinion-shifting power thanone-to-one social influence Finally we include the accept-ability voting quota rule 120572 whose domain is inspired fromclassical majority rules [35] observed empirically (Table 1)

14 Complexity

Table 2 Parameters and their domains used to study the effect of the procedural parameters on the metrics

Procedural parameters Other parameters119905119863 119898 119899119863 120572 119898 120583 119879 119880 119901119886 119901119903 120574 119896 119861119890ℎ119886V1198941199001199031 2 3 4 5 6 1 2 3 4 5 001 002 003 004 005 0 05 066 7 01 16 02 03 005 02 12 119873119886119894V119890 119865119900119888119906119904119890119889

323 Calibrating the Mixed Social Interactions Model Themixed interaction model ascribes to a parameter space inwhich the effect of collective choices on our metrics isnontrivial and where deliberation and voting on proposalsdetermine their acceptability Since the mixed model isequivalent to periodic iterations of pair-wise and collectivesocial interactions the calibration of the parameters in bothpreceding models holds (see Table 1 for the calibration)The first reason for this is comparability the second is thatthe previous calibration takes into account the fact thatboth dynamics are going to be combined We include thefrequency of debates by adding the parameter 119905119863 whichcontrols for the amount of pair-wise interaction amongagents between two distinct deliberation steps

324 Termination Conditions for Runs Simulations stoponce 100 proposals are deliberated on andor voted for inother words after 100 collective decision steps have occuredThe number of proposals discussed seems arbitrary yet it ishigh enough to observe the effects of deliberation on opiniondistributions and on other metrics related to coherence andjudgment accuracy We choose the number of runs as afunction of our research questions which give relevance tothe procedural parameters in the mixed model rather than tothe parameters set to describe the population and its behavior

325 Expected Outcomes of the Simulations We expect moredeliberation in terms of higher 119898 and 119899119863 to increasejudgment accuracy and at the same time to reduce thevariance of opinions In turn a smaller variance of opinionsimplies that the argument pool for deliberation is smaller andtherefore judgment accuracy should be lower Moreovervariance increasing dyadic interactions (or rejection) shouldalso increase judgment accuracy and coherence since bipo-larization and dissensus foster argument diversity in delibera-tionThe coherence statistic should be stronger in simulationsin which central convergence appears quickly and agentsare naive Attitude structures sensitivity to informationalcues and weights given to deliberated cues that makes pathsto bipolar convergence smaller or to central convergencelonger should be associated with higher judgment accuracyShifts of opinion should be more visible in scenarios inwhich extreme agents are pulled away from the extremesHence the sensitivity to deliberated cues (119901119886 119901119903) and 120574should explain the shifts When crossed with high values of119880 and 119879 the shifting power of deliberation should be at itshighest

In the end we do not know how these mechanismswill play out The results of the subsequent experimentsgive us insight on the interplay between the aforementionedeffects

4 Results

Before pointing at any result obtained from the mixedinteractions model we describe and compare the dyadicand deliberative interactions models in the parameter spaceobtained from the calibration in Section 3 (see Table 1)Primarily we require our results to describe two orthogonaltypes of opinion dynamics the pair-wise dyadic and thedeliberative The latter comprises scenarios in which indi-viduals do not influence each other by means of pair-wisediscussion and only update their opinions from delibera-tion the former scenarios where only pair-wise interactionsamong agents determine the dynamics We describe both onthe parameter space given in Table 1 by performing at least20 simulations per scenario Quantitative results from thepair-wise interaction model alone are described during thecalibration and are not treated heremdashdeliberative parametershave no effect on it Moreover we show how these differentmodel specifications yield qualitatively different opiniondistributions and compare them on the basis of the metricsof interest To this end we perform independent two-sampleStudentrsquos t-tests18 or Welch t-tests19 and compute confidenceintervals at a 095 level of confidence Last but not least wecomment ordinary least squares (OLS) regression estimatesto account for the direction and magnitude20 of the effectsof the parameters on the metrics Estimates are declaredsignificant at a 5 level of risk

We perform the same analysis on the mixed interactionsmodel as we compare it to the pair-wise dyadic and deliber-ative interactions models and discuss the marginal effects ofthe parameters on the metrics More precisely we obtain twodifferent types of results for the mixed interactions modelthe first compares the mixed scenarios to their monolithiccounterparts with respect to each metric and with respect tothe regression estimates the second gives a clear idea of themarginal effect of each governance parameter (or parameterof interest) on our observations In the first we allow controland procedural parameters to vary on a parameter spacesimilar to those of the dyadic and deliberative opiniondynamics (see Table 1) In the second we fix all of ourcontrol parameters execute 36000 balanced21 runs andfocus only on the one-way pooled effects of our proceduraland behavioral (all agents are focused or naive) parametersWe generate comprehensive graphs and tables to account forthe obtained results

Please notice the slight change in the size of the parameterspace for the procedural parameters and in the values ofthe nonprocedural ones for the first and second type ofresults (refer to Tables 1 and 2 respectively) Since we wantto have a finer idea of the marginal effects of each of theprocedural parameters on how well a group decides onproposals we make the value jumps for each parameter (1)

Complexity 15

() (U T) = (02 18) (b) ( pa m nD) = (005 05 00 2 02) (＝) ０ＬＧＮＬＭ ＣＨ (＜) (＝) Ｈ＞ tD = 3

(d) (５ ４) = (02 14) (e) ( pa m nD) = (01 03 05 3 02) () ０ＬＧＮＬＭ ＣＨ (＞) () Ｈ＞ tD = 6

Figure 7 From left to right common opinion trajectories and distributions for dyadic deliberative and mixed opinion dynamics within thestructure of different decision-making processes

small enough to detect significant differences in the estimatesand (2) meaningful for interpretation For the fixed one-valued parameters in Table 2 we use Table 1 to set them totheir minimum mean or median values

41 Comments on Dyadic and Deliberative Opinions Dynam-ics In this subsection we present our first results regardingthe deliberative opinion dynamics described in Equation (6)and its differences with the dyadic dynamics (Equation (5))in terms of our observations

411 Qualitative Analysis of Pair-Wise Opinion DynamicsPair-wise opinion dynamics alone produces multiclusterstable convergencewith variable cluster sizes (see Figures 7(a)and 7(d)) Insight on the complexity of these dynamics wasdiscussed in Section 321

In Figures 7(a) and 7(d) agents discuss randomly withone another in pairs Clusters in the extremes form quicklysince either agents that are close to the extremes attractone another or are convinced to stay close to the extremesby interacting with other agents they disagree with Otheragents with near-extreme positions may be simply attractedto one of the several moderate foci of agents in the opiniondistribution On the other hand moderate agents are eitherattracted to the closest opinion focus or pushed towards theextremes of the distribution Over time agents in the centralfocus either attract other agents into it or ignore the opinions

of the extreme agents that interact with them The result ofthese dynamics yields amulticlustered convergencewhere theopinion foci are at least at distance 119880 from one another andonly the extreme foci (clusters of agents with |119900119894| ge 075) areat a distance greater than or equal to 119879 from one anotherThefoci at the bounds of the opinion distribution are very stableand in analogy to the (12119889)-rule in assimilation bounded-confidence models [33] the number of clusters that form isroughly equal to 119880minus1412 Qualitative Analysis of Deliberative Opinion DynamicsQualitatively deliberative opinion dynamics yield a looseunipolar convergence of opinions near the center (centralconvergence see Figure 7(b)) near the center of either the leftor right portions of the distribution (group polarization seeFigure 7(e)) or to a sparse bipolar opinion distribution withclusters at the center and at the extremes (see Figure 7(c)) ifmixed with pair-wise interactions This is probably a reasonto expect the variance of opinions and the proportion ofextremists (almost perfectly correlated 120588 = 089 119901 lt 0001)to be low for these scenarios The side towards which theopinion cluster skews depends essentially on the valence ofthe first argument that is collectively accepted

In Figures 7(e) and 7(b) agents that update their opin-ions do so at the same time and only when deliberationis successful In these scenarios every time a decision iscollectively accepted agents update their opinions towards

16 Complexity

Table 3 Ordinary least squares estimates for pair-wise interactions scenarios for most of the observed metrics

Dependent variable119881119886119903(119900) 119878ℎ 119890119888 119894119903(1) (2) (3) (4)Constant 0610lowastlowastlowast 4429lowastlowastlowast 0472lowastlowastlowast 0000

(0002) (0206) (0003) (0000)

119880 = 06 (119880 = 02) minus0012lowastlowastlowast 2981lowastlowastlowast 0061lowastlowastlowast 0000(0002) (0260) (0004) (0000)

119879 = 18 (ref 119879 = 14) minus0264lowastlowastlowast minus3261lowastlowastlowast 0004 0000(0002) (0260) (0004) (0000)

120572 = 066 (ref 120572 = 05) 0002 minus0059 minus0100lowastlowastlowast 0000(0002) (0184) (0003) (0000)

(119880 119879) = (06 18) (ref (119880 119879) = (02 14)) minus0324lowastlowastlowast minus3728lowastlowastlowast minus0033lowastlowastlowast 0000(0003) (0368) (0006) (0000)

Observations 1440 1440 1440 1440R2 0983 0389 0529Adjusted R2 0983 0388 0527Residual Std Error (df = 1435) 0032 3492 0053 0000F Statistic (df = 4 1435) 21 272740lowastlowastlowast 228686lowastlowastlowast 402326lowastlowastlowastNote lowastplt01 lowastlowastplt005 lowastlowastlowastplt001

the argumentrsquos defended position Hence if there is votingand if agents happen to obtain 119894119899 proposal arguments theyinexorably cluster towards one or the other side of the opiniondistribution resulting in opinion convergence remeniscent ofgroup polarization (see Figure 7(e)) Furthermore as agentscluster in the same side of the opinion spectrum the argu-ments that may be advanced in deliberation are fewer andmore skewed towards that one side of the opinion spectrumAgents become more easily persuaded in deliberation andonly by arguments supporting one side of the spectrumConvergence of opinions towards one loose opinion focusbecomes faster and more certain If there is no vote (seeFigure 7(b)) the dynamics are similar but for the fact thatit is the uniform randomness of the proposal argument thatguarantees a central convergence of opinions

413 Quantitative Analysis of Pair-Wise Opinion DynamicsThe three parameters that allow for quantitative analysis are119880 and119879 the agentsrsquo attitude structure and themajority quotavoting rule 120572 Table 3 supports the claim that going from 120572 =05 to 120572 = 066 does not affect the variance nor the shifts ofopinions However judgment accuracy is significantly higherwhen accepting proposals becomes more difficult This israther surprising but also intuitive since decisions that aretaken using more restrictive methods of scrutiny should bemore accurate

Concerning the attitude structures the interaction termbetween 119880 and 119879 shows that 119880 has a small effect on thevariance of opinions given that 119879 is low This shows thateven when the assimilation threshold is high a sufficientlysmall rejection threshold can make its one-way effect vanishHowever high levels of both 119880 and 119879 reduce the variance ofopinions dramatically In contrast when it comes to overallchanges in opinion the magnitude of 119880rsquos one-way marginal

effect increases the shifts of opinions whereas 119879 lowers itThis can be interpreted as having the rejection dynamics pushmoderate-to-extreme agents to positions that are close tothe extremes and very moderate agents to the center of thedistribution Not too many quantitative changes (on average)can therefore be observedThe fact that higher119880 is associatedwith faster convergence of opinions seems to imply lowerjudgment accuracy At some point voting for the same type ofproposals for too longmay result inmany collectivemistakesThis is an interesting conclusion to consider in the analysis ofthe mixed model Finally the interaction term shows that 119879has no strong impact on the metric

414 Quantitative Analysis of Deliberative OpinionDynamicsSeven parameters can be considered for quantitative anal-ysis in this dynamics the parameters of interest recalledin Section 312 the size of the argument sacks (119896) agentbehavior during deliberation the strength of the deliberationdynamics (120574) and the measure sensitivity to deliberatedresults (119901119886) The most ldquounexpectedrdquo result obtained fromthe deliberation dynamics is that it shows how deliberationcontributes to the proportion of extremists and the varianceof opinions It happens that having more agents discussproposals and having more of these discussions increasesthe variance of opinions and the proportion of extremistswhile lowering judgment inaccuracy and coherence For theformer the fact that far too many arguments are advancedduring deliberation plummets the chances that the centralargument is accepted hence deliberation does not moveopinions too often For the latter we observe that only ldquobigrdquodifferences (going from the lowest value of a parameter to thehighest) in protocol actually prove to have a preponderantimpact on the metrics These observations hint at a possibletrade-off between judgment accuracy and the variance of

Complexity 17

Table 4 Ordinary least squares estimates for the deliberative interactions scenarios for most of the observed metrics

Dependent variable119881119886119903(119900) 119878ℎ 119890119888 119894119903(1) (2) (3) (4)Constant 0032lowastlowastlowast 13092lowastlowastlowast 0370lowastlowastlowast 1007lowastlowastlowast

(0001) (0406) (0001) (0001)

119898 = 3 (ref119898 = 1) 0012lowastlowastlowast minus2062lowastlowastlowast minus0012lowastlowastlowast minus0001(0001) (0276) (0001) (0001)

119898 = 6 (ref119898 = 1) 0023lowastlowastlowast minus3285lowastlowastlowast minus0018lowastlowastlowast minus0004lowastlowastlowast(0001) (0276) (0001) (0001)

119899119863 = 002 (ref 119899119863 = 001) 0007lowastlowastlowast minus0965lowastlowastlowast minus0006lowastlowastlowast minus0002lowastlowast(0001) (0276) (0001) (0001)

119899119863 = 005 (ref 119899119863 = 001) 0020lowastlowastlowast minus3316lowastlowastlowast minus0015lowastlowastlowast minus0004lowastlowastlowast(0001) (0276) (0001) (0001)

120572 = 05 (ref 120572 = 0) 0146lowastlowastlowast 68932lowastlowastlowast 0068lowastlowastlowast minus0507lowastlowastlowast(0001) (0276) (0001) (0001)

120572 = 066 (ref 120572 = 0) 0291lowastlowastlowast minus13842lowastlowastlowast 00004 minus0000(0001) (0276) (0001) (0001)

119896 = 8 (ref 119896 = 4) 0004lowastlowastlowast minus0815lowastlowastlowast minus0002lowastlowastlowast minus0001(0001) (0276) (0001) (0001)

119896 = 16 (ref 119896 = 4) 0008lowastlowastlowast minus1118lowastlowastlowast minus0004lowastlowastlowast minus0003lowastlowastlowast(0001) (0276) (0001) (0001)

119861119890ℎ119886V119894119900119903 = ldquoFocusedrdquo (ref ldquoNaiverdquo) 0015lowastlowastlowast minus1980lowastlowastlowast minus0012lowastlowastlowast minus0004lowastlowastlowast(00005) (0225) (00004) (0001)

120574 = 01 (ref 120574 = 005) minus0020lowastlowastlowast 5762lowastlowastlowast minus00003 minus0001(0001) (0276) (0001) (0001)

120574 = 02 (ref 120574 = 005) minus0025lowastlowastlowast 8648lowastlowastlowast minus00003 minus0001(0001) (0276) (0001) (0001)

119901119886 = 05 (ref 119901119886 = 03) minus0013lowastlowastlowast 1581lowastlowastlowast 00004 00003(00005) (0225) (00004) (0001)

Observations 19440 19440 19440 19440R2 0931 0844 0562 0958Adjusted R2 0931 0844 0562 0958Residual Std Error (df = 19427) 0033 15701 0030 0050F Statistic (df = 12 19427) 21 844790lowastlowastlowast 8 733563lowastlowastlowast 2 079585lowastlowastlowast 36 761250lowastlowastlowastNote lowastplt01 lowastlowastplt005 lowastlowastlowastplt001

opinions that may be of interest in the mixed discussionmodel

Group coherence shifts of opinions and the varianceof opinions are highly affected by the voting thresholdthe coherence metric is reduced by half when allowing forclassic majority voting shifts explode and the variance ofopinions increases dramatically This is not surprising sincethe majority quota rule works as a buffer to the success ofdecision-making processes and in consequence it hampersthe deliberative opinion updates For the shifts of opinions itis clear that the two-third majority voting rule results in littleopinion change and that the simple majority rule inducesgroup polarization As expected there are less extremists inscenarios where there is no voting and the shifts of opinionare more likely to happen when agents are more sensitiveto deliberation When we include voting in the system

the grouprsquos judgment accuracy decreases due to the correctdeliberated decisions that are vetoed by a salient opinion-agent majority Strangely the effect only holds for the simplemajority decision rule Lastly the proportion of individualsthat participate in the debate and the minimal amount ofdebates have similar marginal effects on all the metrics (seeTable 4)

415 Comparing Deliberative and Pair-Wise Opinion Dynam-ics in a Nuthshell Qualitatively both models yield similaroutcomes but for the fact that deliberation never producesopinion polarization with both types of extremists (seeFigure 7) Furthermore clusters in the deliberative interac-tions model are less homogeneous in size and multiclusterconvergence occurs only when deliberation is rarely suc-cessful which happens sporadically given the calibration of

18 Complexity

Table 5 Mean difference 095 confidence intervals for metrics by model comparison

Model Metric119881119886119903(119900) 119901119903119900119901119890119909 119878ℎ 119890119888 119894119903Dyadic vs Deliberative [0191 0216] [0170 0197] [-2948 -2828] [0074 0082] [0165 0172]Mixed vs Deliberative [0142 0147] [0150 0155] [-2085 -1976] [0003 0002] [-0038 -0030]Mixed vs Dyadic [-0072 -0046] [-0045 -0018] [8332 8859] [-0079 -0071] [-0205 -0201]the model On the other hand pair-wise opinion dynamicsproduces multicluster convergence most of the time andcentral convergence only in the infrequent cases where thelatitude of acceptance is large

The comparison of these two scenarios in terms ofthe metrics is straightforward Whilst dyadic interactionsallow for more variability in opinions through pair-wiseinteractions and more mistakes in judging proposals delib-erative updates simply act as an antagonist force On aver-age the deliberative interaction model produces opiniondistributions with smaller variance and yields steady stateswith higher judgment accuracy a higher shift statistic andnaturally lower group coherence since only proposals that aredeliberated 119894119899 are eligible for voting (see Table 5)

42 Articulating Dyadic and Deliberative Social Interactionsin Opinion Dynamics So far we have considered scenarioswhere argumentation had no place and where individualsmade decisions according to opinions that were purelyconstructed from pair-wise discussions We also observedcases in which agents formed their opinions only by inte-grating deliberated proposals into their opinion updates Thescenarios we present in this section combine the two pre-ceding scenarios to account for both the effects of pair-wisediscussion and the effects of deliberation on opinions judg-ment accuracy and group coherence From the precedingobservations we conclude the following deliberative opiniondynamics reduces the variance and the proportion of extrem-ists because it shifts opinions greatly and polarizes groupsby eliminating at least one type of extremist the one whoseopinions are initially opposed to the deliberated resultsor it brings opinions close to neutrality (see Figure 7(b))This may not be desired in a group since the diversity andstability of opinions are necessary for deliberation to makesense and lead to the collective acceptance of controversialproposals Dyadic opinion dynamics on the other handtend to increase the variance of opinions and the numberof extremists while having a lesser aggregated effect on theshifts of opinion This is an immediate result of the pair-wiseinteraction dynamics inasmuch as it polarizes individualsvery quickly

To sum up in the mixed interactions model deliberationhinders the effect of voting on opinions by making someproposals that would normally be submitted for voting noteligible for voting It may also undo opinion changes dueto pair-wise interactions The mixed model can be seen asthe deliberation opinion dynamics model in which pair-wiseinteractions occur in between collective discussion and mayaffect which arguments are advanced during deliberationEquivalently it can be seen as the dyadic opinion dynamics

model in which deliberation accounts for endogenousldquoshocksrdquo on the opinion distribution

421 Qualitative Analysis of the Mixed Dynamics In Figures7(c) and 7(f) agents update their opinions through delibera-tion and pair-wise interactions Opinion trajectories visualsshow a combination of both previously described opiniontrajectories Over time as agents discuss with one anotheropinion clusters form as in the pair-wise or dyadic opiniondynamics Deliberation may disrupt the formation of theseclusters and three different situations may occur

(i) Deliberation disrupts the formation of the opinionclusters but not sufficiently to jeopardize pair-wiseopinion cluster formation (eg for high values 119905119863)Two phenomena may be responsible for it (1) thevariability of the proposal argumentrsquos support for theprinciple P and (2) too few (successive) deliberationsteps that result in opinion updates Once the opinionclusters are formed successful deliberation foreshad-ows the merging of the previously formed clustersmay bring extremists closer to moderate agents andthanks to the assimilation component of the pair-wisedynamics allow bigger and more moderate opinionclusters to form (refer to Figure 7(c))

(ii) Deliberation significantly disrupts the pair-wisedynamic formation of clusters This may happenwhen the requirements for accepting proposals areweak in the decision-making process (eg low 120572) Inthis case one has either central convergence sinceall agents follow the deliberated results in the samemanner or 3-cluster convergence with two extremeopinion foci In the latter situation the extremegroups form quickly (like in the pair-wise model) anddeliberation cannot pull extreme agents away fromthe extreme foci they are in The main reason for thisis that immediately after an extreme agent attempts toleave her foci through deliberation updates she getspulled or pushed back into it by means of a dyadicinteraction with another extreme agent

(iii) Deliberation does not disrupt the formation of theopinions cluster This is the rare case scenario forvery demanding decision-making processes in termsof inclusion (eg 119899119863) and argument acceptability (eg120572) Opinions converge as in the pair-wise opiniondynamics model (refer to Figures 7(a) and 7(d))

Last when there is voting (120572 = 0) the position ofthe moderate opinion cluster is determined by the first

Complexity 19

minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

Var(o)

sh

ec

ir

Var(o)

propex

sh

ec

ir

1

098

minus02

minus004

001

1

minus014

minus001

minus003

1

026

minus028

1

minus057 1

ＪＬＩＪ

Ｒ

Figure 8 Significant correlation estimates for themetrics of interestin the mixed opinion dyamics scenarios

accepted proposal argument andor the opinionmajority thatis formed from previous dyadic interactions (see Figure 7(f))

422 Relating the Metrics of Interest in Mixed OpinionDynamics The first thing to notice is that for all of thesescenarios all the metrics are significantly correlated (seeFigure 8) The variance of opinions and the proportion ofextremists are almost perfectly correlated To some extentthis means that a high variance of opinions known thatthe mean of opinions is statistically null equates to havingmany extremists in the population be towards one or theother side of the opinion distribution (eg extreme bipolar-ization) Similarly we find a negative and strong correlationbetween the judgment inaccuracymeasure and the coherencecoefficient (see Figure 8)mdashwhen judgment inaccuracy is high(low) coherence to deliberation is low (high) Cases likethese are telling since they may be related to sequences ofcollective decisions in which agents do not make too manymistakes when judging proposal arguments and still makedeliberated decisions that are on average in line with theirprinciples

The shift statistic interestingly is negatively correlatedwith the variance of opinions with judgment accuracy andwith the coherence coefficient In a way this means thathaving more moderate agents conduces on average to moremistakes in judging arguments and to less coherent collectivedecisions An explanation for this phenomenon could be thatas agents reach consensus in opinion important (possiblyextreme) arguments for correctly judging argument propos-als are not advanced in the deliberation arenawhich results inmore mistakes and in possibly accepting proposal argumentsthat after voting are not meant to be accepted Indeed the

negative correlation between the variance of opinions andjudgment inaccuracy points towards this direction since onaverage the more shifts there are the smaller the variance ofopinions is (see Figure 8)

423 Comparing Mixed Dynamics with Other MonolithicDynamics of Interactions Qualitatively opinion trajectoriesin the mixed model are more chaotic than in the pair-wise and deliberative interaction models (see Figure 7) Asanalyzed previously deliberation dynamics disrupt the pair-wise dynamics and create situations in which central andmulticlustered convergence (with two clusters at the extremesof the distribution) are possible at the expense of more(pair-wise dynamics) or less (deliberative dynamics) clustersIt tolerates the survival of very small isolated group ofagents that the previous models rarely tolerate in a hundredcollective decision-making steps (see Figure 7(c))

One of the most interesting observations one can makeof this family of models regards how groups change (in termsof our metrics) when dyadic social interactions are addedto deliberative interactions and vice-versa When comparingthe mixed interactions with only deliberative ones shiftsagentsrsquo aggregated change in opinions are on average lesspronounced The variance of opinions and the proportionof extremists are significantly higher for the former than forthe latter group coherence in contrast is lower with respectto the scenarios where opinions are formed only throughdeliberation Judgment accuracy follows that same trendA reason for this might be that mixed dynamics are fasterto yield semistable clusters andor opinion convergence Inconsequence the potential argument pool for deliberativeexchange shrinks faster and deliberation becomes less effec-tive over time which results in more collective mistakes Theoutcomeon coherence is predictable inasmuch as deliberativeinteractions produce group polarization If a group agrees inopinion debates are sterile because themembers of the groupwill most likely refrain from attacking the proposal argu-ments they all support and will work together to disparagethe proposal arguments they do not support (see Table 5)

In the same manner we report that the proportion ofextremists and the variance of opinions are barely lower inthe pair-wise interaction model when including deliberationin the decision-making process Shifts of opinions are alsobarely lower when there is no deliberation This can indicatethat deliberation does not undo the changes in opiniondue to pair-wise interactions nor does it amplifies them Incontrast when we consider the labeling-based metrics themixed interaction model ensures lower judgment inaccuracybecause of deliberation and lower group coherence (seeTable 5)

In sum the mixed discussion model is a compromisebetween the pair-wise interaction and deliberative opiniondynamic models in that it offers some guarantees in termsof judgment accuracy and coherence while allowing forldquoreasonablerdquo variance in the opinion pool It establishes somekind of trade-off between labeling-based metrics and thevariance of opinions that may be of interest for decision-making process design

20 Complexity

Table 6 Summary of sign or relative marginal effects for all varying parameters on dyadic deliberative and mixed interactions models inand for all experiments

Metrics Parameters119905119863 119898 119899119863 120572 119879 119880 119901119886 120574 119896 119861119890ℎ119886V119894119900119903119881119886119903(119900) XX X X O X X X X X X119901119903119900119901119890119909 XX X X O X X X X X X119878ℎ XX X X O X X X X X119890119888 XX X X XO X XOO XOO X X119894119903 XX X X O XO O X OOX XOO X XNota gray symbols correspond to dyadic dynamics light gray to deliberative dynamics and bold black to mixed dynamics Given a scenario the symbol 119883stands for constant parameter and 119874 for insignificant marginal effect and as parameters grow for positive (negative) marginal effect for the parameterFrom left to right at the parameter level the leftmost arrow corresponds to the scenario with the strongest (in absolute value) marginal effect of the parameter(with respect to to its factors) Example Let 119901 be a parameter and 119874 stands for ldquo119901 has a significant positive effect in dyadic discussion scenario negativein the deliberative one and insignificant in mixed discussion The positive effect in dyadic discussion is greater than the absolute value of the negative one inthe purely deliberative scenariordquo

424 Comments on the Differences between the Three Modelsfor OLS Regression Estimates The significance of most of theregression parameters estimated for the mixed interactionsscenarios are similar to those estimated for the pure dyadicanddeliberative scenariosNevertheless themagnitude of theeffects is different insomuch that the dynamics themselvesare different So instead of commenting the linear regressionestimates of the mixed model we compare when possiblethe standardized significant coefficients of the regressors ofthe mixed model with those of the pair-wise and deliberativeinteractionmodels (see Table 6 for a summary of the results)

In mixed interactions for instance the influence of119898 onthe opinion distribution is much more preponderant than inthe deliberative dynamics scenarios Going from 119898 = 1 to119898 = 3 increases the standard variation units of the variance ofopinions by twice as much as the one in the deliberative sce-nariosThis in particular is due to the design of the decision-making process If 119898 is higher then the number of socialinfluence steps (119905119863) in the decision process is also higher(compared to 119905119863 = 0 in the deliberative case) If we decideto consider these steps as contributing to the variance ofopinions then the effect of119898 on the variance of opinions hasto be stronger in the mixed interactions model In contrastthe number of agents participating in the debate has a weakereffect on the variance of opinions in the mixed interactionsmodel than in the deliberative dynamics specially for 119899119863 =005 (more than twice as much) The difference may comefrom the fact that in mixed discussions the part of theargument pool used in deliberation is more restrictive thanin the deliberative case We can argue that since pair-wisedynamics may get individuals closer to one another aftersome deliberation the same happens to the arguments thatindividuals will most likely use in deliberation Taking thisinto account means that deliberation will at many instancesbe unfruitful and have no effect on opinions thus failing tomoderate extreme opinions andor polarize the group In thesame direction a high number of participants in deliberationcan make the deliberation dynamics stiff In this case thedistribution of opinion is close to the uniform distributionwhich has a relatively high-to-moderate variancewith respectto the reference results

The strength of the deliberative dynamics has a positivestronger effect on the variance of opinions in themixedmodelthan in the deliberative model when 120574 = 02 and a negativeweaker one when 120574 = 01 A sound interpretation of thisresult is that when deliberation is successful and its effect onopinions does not get agents to move sufficiently away fromtheir current positions agents are pulled or pushed back totheir previously held opinions the variance of opinions doesnot changemuchOn the other hand if the effect of successfuldeliberation is strong opinions can change sufficiently tomake agents much more prone to assimilation into moderateopinion foci the variance of opinions falls

The deliberative opinion dynamics model is much moresensitive to 120572 than the mixed discussion model Differ-ences in marginal effects can double The probability ofattraction to deliberated results (119901119886) for instance has astrongmoderating effect (negative) on agentsrsquo opinions in thedeliberative opinion dynamics but a meager one (two-fold)in the mixed and dyadic dynamic models This is becauseopinion updates in the deliberative model depend only onthat parameter whereas in the mixed model more numeroustypes of interactions make opinion updates possible

In terms of judgment accuracy all parameters except thesize of the argument sacks (119896) have a significant slightlystronger effect on the deliberative dynamics than in themixedmodel In effect the difference may be attributed to theupshot of pair-wise interactions on the distribution of opin-ions and on the arguments that would be advanced duringdeliberation The result respecting parameter 119896 however iscounterintuitive Since argument sacks are static scenarioswhere shifts are higher should make 119896 more important fordeliberation to succeed and it is precisely in the scenarios ofpure deliberative dynamics that we find the biggest opinionshifts This may point to the hypothesis that in deliberativeinteractions agents change opinions substantially but do notoften change their adherence to the principle (go from anegative opinion to a positive one or vice versa) when thereis no group polarization

43 Sensitivity Analysis for the Procedural Parameters ofInterest Weperform a sensitivity analysis of the observationson the parameters of interest in themixed interactionsmodel

Complexity 21

We extend the parameter domains of the pure dyadic anddeliberative scenarios as we make it explicit in Table 2

431 Minimum Number of Debates (119898) We observe that theminimum number of debates has a significant well-observedeffect on all of our metrics excluding group coherence Forthe variance of opinions we can observe that the moredebates there are the bigger the value of the metric isalthough the higher 119899119863 and 119905119863 are the weaker the overalleffect of119898 becomes (Figures 9(d) and 9(a)) Furthermore themarginal effect of increasing minimal debates is decreasingan additional deliberation step requirement increases thevariance of opinions but less and less as it grows 119898 coupledwith 119899119863 has a shy S-shaped effect (see Figure 9(d)) on thevariance of opinions meaning that it has a stronger effectwhen 119899119863 is low which suggests a possible trade-off betweenthe number of arguments accepted in the debate and thenumber of debates required before accepting a proposal

In contrast the shifts in opinion are less and less likelyas 119898 grows and this is independent of the variations of theother parameters Again the effect is marginally decreasingand is only truly significant when 120572 = 05 An explanationto these effects may be linked to the design of the systemFirst the variance of opinions is higher when deliberation isasked for because the more deliberation steps there are thehigher the chances are that the central proposal argumentis deemed unacceptable specially when agents are focusedFurthermoremechanically speaking increasing theminimalamount of debates implies that whenever a decision is to betaken at least 119898 times 119905119863 time steps have to take place and if atany moment a proposal argument is considered undecided119905119863 time steps are added to the process So unless the debatedoes not yield 119906119899119889 labels for proposal arguments (highlyunlikely considering that 120590 = grounded semantics) the morenondeliberation steps there are in the decision process thehigher the variance of opinions is Concerning the shiftswhen 120572 = 05 either the system is too stiff to accept anyproposal argument and opinions do not changemuch or theeffects of pair-wise discussion and deliberation cancel out insuch a way that individual shifts are minimal (Figure 9(b))

On the side of labeling-based metrics the more debatesare asked for the more accurate a group is in its judgmentthe effect being smaller as 119898 grows When agents are naivethe effect is quasilinear while when they are focused thestrongest effects of acquiring more deliberation are foundwhen levels of deliberation are already low (Figure 9(f))Thiscan be explained by the fact that the more debates thereare in the decision process the closer one gets to the idealargumentation framework

432 Proportion of the Population in Deliberation Steps (119899119863)Like for 119898 119899119863 has a significant effect on the proportionof extremists and on the shifts and variance of opinions(Figure 9(e)) This may result from the fact that being ableto put more arguments in play at the same debate step mayincrease the odds of revealing cycles around the proposalargument in119860119865120576 Given that we use grounded semantics thearguments in the cycles are labeled 119906119899119889 and in consequence

debates are more often postponed when 119899119863 is higher Post-poning debates in turn increases the number of nondebatessteps in the decision process which increases the varianceof opinions and limits over time the moderating effect ofdeliberation Moreover the effect of this parameter is verydependent on the value of 120572 (Figure 9(e)) For shifts forinstance 120572 = 05 makes the effect of 119899119863 negative while120572 = 0 makes it positive but to a lesser extent Furthermorefor higher requirements of deliberation (119898) adding moreindividuals to the deliberation process has weaker effectson the variance of opinions and on the other metrics It isalso quite interesting to notice that it has the same effect onagents independently of agent behavior during deliberationThis is a surprising result as one would have expected morefocused agents in a deliberation arena to cause a raise inthe proportion of extremists as they play to knock outopposing proposal arguments and thereof reducing the effectof deliberation (lowers variance) on the whole population

Finally similar to119898 addingmore people into the deliber-ation process increases judgment accuracy (see Figure 9(c))and has no clear effect if any on group coherence (seeFigure 9(i))

433 Steps between Deliberation Steps (119905119863) In all configu-rations 119905119863 increases the proportion of extremists (variance)and decreases the shifts of opinions among agents Theshifts and the effect on the variance of opinions are presentfor 120572 = 05 insignificant otherwise (Figure 9(b)) 119905119863 ishighly linked to 119898 by construction When 119898 = 1 thecurve linking the variance of opinions and 119905119863 is convex As119898 increases the curve becomes more and more concavewhich means that 119905119863 has a more important effect on theopinion distribution as collective decisions take longer to beachieved This seems counterintuitive but in reality it reflectsthemultiplicative relationship between deliberation and pair-wise interactions and the semantics in the model If 119898 is lowand 119905119863 is high the effective number of pair-wise interactionsis on average smaller in the deliberation process Hencethe structure of the decision-making process hinders anyincrease in the variance of opinions If119898 is high the oppositeeffect is observed Additionally since the grounded semanticsyields few 119894119899 arguments with respect to other admissibility-based semantics getting closer to the ideal argumentationframeworkmaymake it difficult to obtain the expected effectsof deliberative discussion on agents Lower 119898 constrains thevariance of opinions by making deliberation more influentialon opinions

In terms of judgment accuracy and coherence we observethat 119905119863 does not explain coherence yet it tends to decreasejudgment inaccuracy This may be due to a more variedargument pool in the debate which in turn may be aconsequence of a higher variance of opinions or vice versa

434 Acceptability Voting Quota (120572) By far it is the mostinfluential parameter in our study It changes the directionand the intensity of the effect of all the other proceduralparameters and by construction heavily constrains the roadto accepting a proposal (see Figure 9)

22 Complexity

015

020

025

030

035

040

1 2 3 4 5 6

12345

Values of Ｇ

(a) 119905119863 times 119901119903119900119901119890119909 by119898

0

10

20

30

40

50

1 2 3 4 5 6

005066

Values of

(b) 119905119863 times 119878ℎ by 120572

040

045

050

055

060

1 2 3 4 5

001002003004005

Values of Ｈ＄

(c) 119898 times 119890119888 by 119899119863

015

020

025

030

035

040

1 2 3 4 5

001002003004005

Values of Ｈ＄

(d) 119898 times 119901119903119900119901119890119909 by 119899119863

0

10

20

30

40

50

001 002 003 004 005

00005066

Values of

(e) 119899119863 times 119878ℎ by 120572

040

045

050

055

060

1 2 3 4 5

BehaviorNaiveFocused

(f) 119898 times 119890119888 by 119887119890ℎ119886V119894119900119903

015

020

025

030

035

040

0 05 066Behavior

NaiveFocused

(g) 120572 times 119901119903119900119901119890119909 by 119887119890ℎ119886V119894119900119903

05

06

07

08

09

10

1 2 3 4 5 6

00005066

Values of

(h) 119905119863 times 119894119903 by 120572

08700

08725

08750

08775

08800

001 002 003 004 005

BehaviorNaiveFocused

(i) 119899119863 times 119894119903 by 119887119890ℎ119886V119894119900119903

Figure 9 Curves of mean observations for different parameters and metrics for 36000 runs Bars are 095 confidence intervals Plots withthe same observables (ordinate) except for 119894119903 are on the same scale

Complexity 23

The higher the requirement for accepting the proposal isthe higher the proportion of extremists and the variance ofopinions are and the lower the shift statistic is In few words120572 constrains the world to dyadic discussion or throws it intoa process in which deliberation is much more importantthan dyadic interactions This is why for any 120572 = 05 oneeither gives too much weight to deliberated results whichshadows pair-wise interactions or too little weight in thesense that no deliberated result is ever accepted and thusnever integrated into the opinions of the agents that havevoted for it The reason for the latter is that the latitudeacceptance is too low and thereafter pair-wise interactionsare not enough to unevenly polarize the population in such away that deliberated proposal arguments are voted favorablyby a non-negligeable majority Shifts for 120572 = 0 are stable fordifferent levels of 119905119863 which implies that deliberation keeps incheck all the shifts of opinions related to an increase in thenumber of pair-wise interactions For 120572 = 05 deliberationis less likely to be successful and shifts tend towards the shiftlevels for 120572 = 0 (see Figure 9(b))

Concerning labeling-basedmetrics120572 entirely determinesthe coherence statistic For 120572 = 05 there is no difference incoherence because of how coherence is defined it is maximalwhen 120572 = 0 and strangely maximal when 120572 = 066(see Figure 9(h)) Surprisingly 120572 has no effect on judgmentaccuracy One would have expected a higher 120572 to increasejudgment accuracy since agents would hardly collectivelyaccept a proposal argument discussed during deliberationProposal arguments have given how the argument lattice isgenerated a higher probability of being rejected than of beingaccepted in the ideal argumentation framework

435 A Word on Focused and Naive Agents Focused andnaive specifications for agent behavior are an importantparameter in the decision process since they model howagents choose arguments in the deliberation arena In the sce-narios where agents are focused agents show equal or highervariance of opinions and are therefore more extreme (seeFigure 9(g)) independently of the voting requirements Shiftsof opinion are less likely as well We can see this happeningbecause focused agents knock out proposal arguments moreoften than naive agents provided that 119899119863 is low

One could have expected naive agents to yield morecoherent decisions since they argue sincerely However thisseems not to be the case (see Figure 9(i)) Agents are unable todeliberate in away thatmakes deliberation reflect their votingintentions either because they do not have the necessaryarguments to do so or because such arguments do not exist

In terms of judgment accuracy focused agents do betterthan naive agents in finding the ideal labels for the proposalarguments This is likely because when reconstructing theframework focused agents take into account the deliberatedproposal argument and choose themost pertinent argumentsin their sacks that relate to the state of affairs at the deliber-ation arena In fine debates result in a better approximationofL119886119887120576(119868) even though one could have believed the oppositethat is focused agents are ready to sacrifice a correct collectivelabel of a proposal argument in the goal of defending theirpositions

44 Points of Discussion In this subsection we discuss theresults obtained from the simulations We attempt to makesense of them from a social sciences perspective

441 Opinion Consensus and Dissensus The deliberativeopinion dynamics corresponds to the simulated scenarios inwhich the variance of opinions is at its lowest and judgmentaccuracy at its highest Dyadic opinion dynamics are associ-ated with the scenarios in which the variance of opinions is atits highest judgment accuracy at its lowest and coherence atitsmaximumpossible levelThemixed interactions dynamicsis a combination of these aspects for all the metrics exceptfor coherence at which it is the worsemdashdeliberation leadsto some kind of loose opinion consensus that is reinforcedby pair-wise interactions Likewise the mixed discussionmodel confirms some results exposed in [16] expressingthat opinions and voting intentions in a deliberative contextldquooften changerdquo That being said and more particularly indeliberative opinion dynamics outputs align well with twointeresting results on consensus formation given by aninterpretation of Moscovici and Doisersquos [15] work Sunsteinrsquos[14] account on deliberation and group polarization andsocial impact theory [4] The first states that mild consensusis reached in deliberation as procedure disengages agentsthe second is that within discussion groups majorities tendto become larger and opinions more extrememdashnot on thebasis of competing arguments but of the initial balance ofopinions After several decision processes we find opiniondistributions that reflect these observations even though weare not able to precisely coin the set of parameters that yieldssuch distributions (see Figure 7)

Although differences between mixed and dyadic opiniondynamics in terms of the variance of opinions and shiftsare slim the former does much better in terms of judg-ment accuracy In this sense deliberation does moderateopinions but not as much when non-deliberation steps arerelatively frequent throughout the decision-making processIn terms of Moscovici and Doise theory of consensus [15]this makes sense since pair-wise discussion engages or ego-involve agents in the discussion and pushes to convergenceof opinion at the extremes of the opinion distributionHowever what remains unclear is the reason why proceduralparameters that constrain deliberation do not disengageagents from discussion and thus result in a mild consensusas also expressed by Moscovici and Doise [15] A possibleexplanation of this phenomenon may stem from the factthat deliberation in the model is always accompanied by aseries of ego-involving discussions that shadowor counter themoderating effect of deliberation Not only this deliberationcould also be responsible for crystallizing the opinions ofextremists for it favors one or the other part of the distributionof opinions as explained in [14] and to some extent in [13]Furthermore the mixed discussion model gives an exampleof polarization that results from a minority of dissenterson one side of an issue acceding to views of the majorityrsquosside [4] as opposed to the strong polarization observed indyadic interactions Situations in which the status quo orthe undecidability of the proposal argument lingers cannotpossibly yield situations of moderation In cognitive science

24 Complexity

one can be inclined to believe that the longer a deliberationprocess is the more likely individuals will take an immutablestance on the situation In view of the assumption that longdecision-making processes can be more costly for a group orsociety as a whole agents may feel forced to take a stancejust for the sake of ending the process The stance that isadopted would therefore depend on pair-wise interactionsamong individuals and not on deliberation since deliberationtakes too long to be conclusive

442 For Hypothetical Recommendations in Decision-MakingProcess Elaboration On another tune results from themodelsuggest that procedural parameters have marginally decreas-ing positive or negative effects on the metrics of interest Ina hypothetical policy analysis perspective this may indicatethat after choosing a correct number of individuals andimposing the right amount of debates one can attain rea-sonable levels of judgment accuracy coherence and varianceof opinions As accounted in [18] deliberation increasesknowledge (collective knowledge as well) and therefore itheightens the chances of making correct decisions If onebelieves that social welfare is linked to deliberation andeventually to the metrics studied previously then proceduralparameters can both be used as an instrument to attaindesired levels of social welfare and reinforce the notionof legitimacy of collective decisions It can also provide ajustification for choosing a deliberation regime rather thananother on the basis of how important (or urgent) a topic ofdiscussion is Indeed if we adopt the claim that deliberationis a forerunner of welfare then any state in which deliberativecues are at their maximum (highest number of participantsmany deliberation steps before making a decision and asfrequent as possible) has to be mapped to the highest attain-able social welfare In this case is a situation where judgmentaccuracy is at its highest yet the variance of opinions or theproportion of extremists at its lowest an ideal situation Wecannot say for sure since strong opinion consensus and longdecision-making processes in many situations may result ina loss of welfare rather than in a gain of it

Another interesting result in procedure for deliberationis the very similar estimates we find for the effects of theminimum number (119898) of debates necessary to make adecision on a proposal and the proportion (119899119863) of individualsparticipating in deliberation on judgment accuracy In thescope of social welfare this raises the question of whetheradvocating for longer decision processes by demandingmoredeliberation steps (higher 119898) rather than bigger debates(higher 119899119863) may increase welfare Depending on the objec-tives of the decider of the deliberation procedure he or shemay choose to concentrate on one or the other A recom-mendation we can assert from the mixed social interactionsmodel is that if one prefers low to high variance of opinionsandor high to low coherence the decider is better off ifshe focuses on increasing the size of the debate rather thanorganizing debates very frequently One of the reasons forthis might be that when making bigger debates all agents aretruly considered as equals in the decision-making processIn organizing many debates individuals that are chosen toparticipate in many of these will be more influential and will

somehow bias the set of arguments in the deliberation arenaThemajority rulemodeled through120572 also determines the sizeof the shifts and the distribution of opinions Because a bigger120572 is associated with higher levels of extremism and opinionvariance deliberated cues are harder to accept and thereforeit becomesmore difficult for agents to reach concensus andormoderate their opinions

On the other hand and further into the idea of legitimacythe goal of deliberative processes is the agreement amongall participants This agreement stands for the right or bestdecision from a formal or procedural perspective and it is alsothe best from a substantive point of view In other words thelegitimacy of decisions not only derives fromwell-establishedprocedural requirements (respect of the protocol) but alsobegs for the fulfillment of two essential conditions (1) satisfy-ing the procedural requirements for a correct procedure (for-mal legitimacy) and (2) the rational acceptability of the resultsof this procedure (substantive acceptability) Legitimacy canbe then obtained from the coherence statistic scrutinizedin the model insofar as it measures how well agents acceptthe results of a correct procedure (eg deliberation) So fora proposal to be legitimate one has to choose parametersthat would maximize group coherence in decision-makingprocesses

As deliberation may give rise to consensus in the delib-eration arena it may also sow the seeds of dissensus inthe group Our model clearly illustrates this phenomenonthrough the procedural parameters of deliberation and themetrics observed From another perspective in deliberativedemocracy dissensus in collective decision-making may bedesirable According to Landermore and Pagersquos accounts [36]lack of consensus can be perceived as having agents withalternative ideas that combined with other ideas can providea better approximation of the ideal status of an argumentTheauthors in [36] call this dissensus ldquopositive dissensusrdquo andit makes normative sense when solving complex collectivedecision-making problems Our model captures the ideaof having normative requirements to define a good andlegitimate collective decision requirements of high consen-sual deliberative accuracy (judgment accuracy) of legitimacy(coherence) and of diversity (variance of opinions) are alltaken into account and for such we can conclude that ourmodel is successful in describing a sound process of collectivedecision-making with interpretable outcomes

Of course the present simulation results and analysis donot prove that all opinion dynamics can be accounted forby such simple processes like the one presented here Theydo however shed light on the expressiveness of combiningdifferent paradigms to account formore interpretablemodelsand pave the way to creating original models of the sortThe simulations suggest the desirability of discovering theconsequences of relatively simple laws of communication atdifferent levels (micro meso and macro) to determine whatstill needs to be explained

5 Related Work

We see our model as a contribution to the influence andopinion dynamics field in ABM and a pragmatic application

Complexity 25

of abstract argumentation theory To our knowledge we areunaware of existing literature on agent-based modeling thatexplicitly relates collective choice and the notions of deliber-ation and opinion diffusion through abstract argumentationas we have done it

Most models in the literature of opinion diffusion areinterested in opinions because these have an influence incollective decision and in questions of social order Forinstance in [3] the authors are interested in consensus andin how a group collectively chooses among two alternativesIn other models authors are interested in the emergenceof extremism [8] and on the distribution of opinions whenextremists are introduced in the population [5 33 37] whileother authors coin the notion of opinion polarization as anemergent property of the system [8 12] They show usingmodels of ldquobounded confidencerdquo and opinion diffusion withtrust that three different kinds of steady states (unipolarbipolar and central) are possible depending on whetheragents are sufficiently uncertain about their opinions andsufficiently connected andor a certain proportion of indi-viduals are already extreme Recent articles on informationand opinion dynamics stress the importance of governanceand government intervention in the spread of emotions(opinions) in the frangibility of social consensus systemIn [10] the authors show that government intervention inthe spread of negative emotions can lead to an even fasterspread of negative emotions (single-peeked convergence) andin a faster collapse of the social consensus system Opinionas a function of trust is studied in [12] and relates to howindividuals form their opinions on the basis of how muchthey trust agents in their networks

Another stream of discrete opinion dynamics modelsthat explain the emergence of opinion can be found in theliterature Computer multiagent models of attitude changebased on Latanersquos social impact theory are presented in [4] Inthe theory an agent changes her opinion on the basis of theinformational impact she is subject to which depends on thepersuasiveness (strength) supportiveness and immediacy(group structure) of the environment and information shereceives The simulations predict two emergent groups ofphenomena the shifting of attitudes towards incompletelypolarized equilibria and the formation of coherent clusteringof subgroups with deviant attitudes Similarly but in acontinuous more argumentative fashion Mas and Flache[11] present a model of opinion diffusion using argumentsArguments are considered to be for or against a proposal andare given an agent-dependant relevance in dyadic persuasiveinteractions Arguments also determine the agentsrsquo opinionsand their dissimilarities which posits the rules of pair-wiseencounters in theirmodel as agents are assumed to form theiropinions according to the arguments they own They showthat in an argumentative opinion diffusion model the onlyobservable steady states are bipolarization and consensusand that bipolarization can emerge from interactions amongsimilar agents In other words bipolarization of opinionsis possible with homophily and without negative influenceif there is argumentation They subsequently compare theirresults to experimental data In [38] a similar approach toargumentation and opinion formation is taken but for the fact

that they introduce different types of arguments explicitlyapply social judgment theory [22] and use survey data toestablish model-to-real-world comparisons

Closer to opinion formation and abstract argumentationGabbriellini and Torroni [39] are the first to have origi-nally and soundly merged opinion diffusion and abstractargumentation They define an agentrsquos opinion as a functionof the arguments she holds and the attack relation amongthem They devised a focused peer-to-peer dialogue systemof persuasion (NetArg) inspired from Mercier and Sperberrsquosargumentative theory of reasoning [40] which used onlyabstract argumentation to study opinion polarization andopinion dynamics Moreover they considered networks andthe notions of trust and epistemic vigilance to define adynamics for knowledge revision and trust itself whichwe clearly lack in our model They showed that if a con-servative belief operator in argumentation was applied byagents when they reasoned then their dialogue protocoldid not increase polarization among agents That said theyused the model to study the effect of Groenvetterrsquos weaklink theory in the spreading of arguments and were notparticularly interested in notions like judgment accuracy orcoherence nor in tackling questions related to collectivedecision-making procedures Their system is very expressiveand helps position our work in the litterature our work is atthe frontier of pure argumentative opinion diffusion modelsopinion diffusion with arguments and continuous bounded-confidence models of opinion diffusion a la Deffuant

On another note work on collective cognitive conver-gence [41] and opinion sharing [19] shows that consensustowards a certain ldquocorrectrdquo opinion or cognitive state is alwayspossible yet dependent on noise variability and awarenessof agents In [20] the authors show that learning about anexogenous correct state of the world (represented by bits)under confidence was possible but only if the agents werenot too confident In a homogeneous population they showthat the higher the confidence the worse the learningmdashthevery confident agents do not learn the properties of the truestate of the world and disrupt the learning process of the lessconfident ones Collective cognitive convergence can be seenas a result of deliberation in truth-seeking models

When it comes to abstract argumentation theory we takean approach that wires two types of dialogues that are well-studied in the literature persuasion dialogues [42] and delib-eration dialogues [43] Another interesting line of work is theone on mechanism design [44] or the problem of devisingan argumentation protocol where strategic argumentation isnot a liability for success in debates We tackle mechanismdesign in a different way though Instead of consideringstrategy-proofness we are interested in how differences inprotocol can result in epistemologically ldquobetterrdquo collectivechoices and guarantee that opinion distributions are favorablefor deliberation For a survey on persuasion dialogue see[42]

Work on agent-based argumentation usually assumes thatthe semantic relationship between arguments is fixed Inother words if two individuals were to put two argumentsin the public arena such that one logically attacked the otherthen everyone would agree that such attack exists [45 46]

26 Complexity

Other models which do not make this restrictive assumptioncan also be found in the literature [27 47] and derive fromthe class or family of opponent models [27 48] in which twoopposing sides attempt to win a dialogue Ourmodel is in theintersection of these two but the frame combining opiniondiffusion of the kind and argumentation remains new

The idea of mixing interpersonal influence and verticalcommunication however is not original It is described andimplemented in innovation diffusion models such as [5 49]In both vertical communication is modeled as exogenoustransparent informationmdashagents are aware of the existenceof an innovation thus triggering several processes of choiceand stabilization of opinions Also in [50] an Eulerian modelis implemented to show the effect of media and exogenousinformation on opinion distributions With respect to thispoint the originality of our work is in that the informationemitted as vertical communication is endogenous It is issuedfrom a deliberation model that agents shape on the basisof their opinions arguments and behavior In the spirit of[51] where the authors control for the design of verticalcommunication (to whom the vertical communication isaddressed its timing how it influences agents opinions) wecontrol for the process generating the information for ourown set of variables length of debates majority quotas andthe frequency at which discussions take place

6 Conclusion and Perspectives

The main objective of this article was to build a bridgewhere decision-making argumentation and opinion dif-fusion could come together We proposed a model thatcombined abstract argumentation theory and a boundedconfidence opinion diffusion model and showed to whatextent it could explain variance of opinions extremismcoherence in collective decisions and judgment accuracyon arguments given an ideal state of full information Thesecond objective was to show in what way governance pro-viding agents with arenas of discussion and deliberation wasimportant in addressing the success of collective decision-making processes and the quality of their outcomes

The model revealed that allowing for more deliberationtime though in lower frequency and allowing for widerparticipation in deliberation increased the variance of opin-ions and the proportion of extremists in a group Theseobservations are consistent with the combination of resultsfound in [9] and in [13 14] which stress that deliberationmay polarize groups and may have a meager effect on theshifts of opinions and inconsistent with [16] where it isargued that opinions tend to moderate after deliberationIt happens that deliberation alone did moderate opinionyet when integrated into a more complex system in whichindividuals were allowed to interact with one another andnot all that was deliberated was accepted its influence wasshadowed by other more individualistic dynamics

Undeniably the grounded semantics played an importantrole in the weak effect of deliberation since it models askeptical way of reasoning over arguments For agentsaccepting arguments thus updating their opinions accord-ingly happened not too often Nevertheless asking for more

deliberation did increase judgment accuracy as observed in[18] yet in a marginally decreasing fashion We showed thatvoting within the deliberation protocol not only increasedthe proportion of extremists and the variance of opinions ina group but also determined how coherent deliberation andvoting procedures were with each other Lastly we showedthat agents that are focused judged arguments better hadmore stable opinions and constituted a group with higherproportion of extremists than their naive counterparts

In terms of governance the model says that there is per-haps no trade-off between extremismand judgment accuracyInstead it asserts that the higher the variance of opinions thecloser the group gets to the correct decisions This situationmay be interpreted as Landermore and Pagersquos notion of ldquoposi-tive dissensusrdquo (in complex collective decision-making tasks)[36] as opposed to consensus as a normative requirementfor correct collective decision-making If a decider has toorganize deliberation to legitimize a public policy dependingon what kind of world he or she wants different set ofparameters may be chosen to account for different levels oflegitimacy and correctness For instance one may ask for adecision to be legitimate to respect a certain level of coherenceor of judgment accuracy and to be discussed by a sundryof different-opinionated peoples If so deliberation has tobe made more often more participants have to be includedin debates or in deliberation instances and not too manytime steps should be introduced in between two deliberationinstances So the model may capture normative ideas todefine correct and legitimate collective decisions and tomake recommendations on those grounds Requirements ofhigh consensual deliberative accuracy (judgment accuracy)legitimacy (coherence) and diversity (variance of opinions)are all taken into account and for such a reason the model israther successful in describing a sound process of collectivedecision-making with interpretable outcomes

61 Extensions of the Model and Other Ideas Our model canbe extended in many ways Value-based abstract argumen-tation frameworks (VAFs) for example as an extension ofDungrsquos argumentation framework provide a formal descrip-tion of the process of decision-making in which argumentsare given values and audiences (set of agents) ignore attacksbetween arguments on the basis of their preferences over val-ues Social abstract argumentation frameworks (SAFs) alsoprovide an interesting and seemingly convenient formalismfor our model In SAFs agents vote on the acceptability ofarguments and the resulting labeling is a combination of thearguments that are voted for and labeling-based semantics Inour model we apply the same rule yet only for one argumentand given precise agent-based votingmechanisms and votingrules which are eventually parametrized

To consider our model as a specification of a mixed VAFand SAF model may be of great interest for future workThis formalism can provide solid foundations to describeexplain interpret and eventually extend the results foundhere from an argument-based perspective On those groundsfuture work will focus on creating a hybrid argumentationframeworkmdashwith its semanticsmdashthat takes heed of bothvalues or principles and voting

Complexity 27

Concerning other possible extensions of the model anatural reflex would be to argue on the validity and therobustness of the mix between the bounded confidenceopinion diffusion models and deliberation Consequentiallyadding deliberation to different bounded confidence modelsof opinion diffusion such as the Hegelsmann-Krause model[37] or to models of social impact theory like in [4] canbe of great interest for future work We acknowledge that afiner understanding of argumentation frameworks randomlattices and their respective applications for this modelare necessary Testing for different labeling-based semanticsor even trading labeling-based semantics for scoring-basedsemantics (semantics that defines acceptability on scoresgiven to arguments) such as debate semantics and ranking-based semantics (semantics that yield a preorder) may beinteresting to consider in future work

More into modeling deliberation it is not inadequateto believe that agents may also advance proposals duringdeliberation and replace to some extent the proposal forwhich the deliberation is taking place Endogenizing theprocess that produces the type of argument to be deliberatedon may be interesting to explore either by having agentsstrategically replace proposals at certain moments duringthe debate or having the central authority attempt to chooseproposals such that the probability of having them acceptedis high Agents may also be thought to be autoorganizing(independent of a central authority) and concede on the sizeof the instances of deliberation To consider differences insemantics across agents or even endogenizing the semanticson how urgent or important the proposal that is beingdiscussed is can also be an interesting direction to take

Last but not least many nontrivial modifications of ourmodel are possible and most should include better-thoughtdeliberation protocols It may be interesting to design andobserve protocols in which deliberation only affects agentsthat are actually debating for example Trust network effectsmultidimensionality of opinions new processes of argumentexchange or learning among agents are notions to furtherdevelop in order to make the model more realistic and relaxsome unreliable assumptions In sum the model is to berefined and extended with the objective of either studyingconcrete cases of deliberative polling and opinion dynamicsor implementing more intuitive thought experiments

Appendix

We present below some of the pseudocode we used toimplement the model

Algorithms

(a) Algorithm 1 implements the dyadic influence in themodel It describes how two agents interact andupdate their opinions

(b) Algorithm 2 implements the deliberative influence inthe model which describes how agents update theiropinions in sight of the accepted deliberated proposalarguments

(c) Algorithm 3 is used to obtain the grounded labelingof an argumentation framework

(d) Algorithm 4 provides an overview of a run in thesimulations119872 is the number of proposal argumentsto discuss which we set to 100 in the model

Each object in the system is indexed by a119908ℎ119900 number Hencea set of agents is annotated by a set of natural numbers Weuse this fact in the implementation of the algorithm 119874119887119895119890119888119905119894denotes the object of 119908ℎ119900 number 119894 Agentsrsquo opinions (119900) arecalibrated so that 119900 isin [minus1 1]Data Availability

The program and data used to support the findings of thisstudy are available from the corresponding author uponrequest

Disclosure

A preliminary version of this work was presented anddiscussed at the conference Bridging the Gap between FormalArgumentation and Actual Human Reasoning on October 4-5 2018 at BochumGermanyThe authors thank the audiencefor their valuable comments and suggestions

Conflicts of Interest

The authors declare that they have no conflicts of interestregarding the publication of this article

Acknowledgments

The contribution of Gabriella Pigozzi was supported by theDeutsche Forschungsgemeinschaft (DFG) and the CzechScience Foundation (GACR) as part of the joint project FromShared Evidence to Group Attitudes (RO 45486-1)

Endnotes

1 In other words the superiority of the ldquoright answerrdquo orlabeling in our case will appear as such to all

2 All agents hold the same opinion3 By sensitivity we mean how much an agent is ready to

change her opinion on the sole fact that an informationalcue has been debated on

4 The assumption is strong but this trust among agentsmay be rooted from a genuine individual motivation toreach a democratic consensus

5 We adopted a rule of thumbs that says that if an agentis neutral with respect to the principle then she willvote for any proposal that is presented to her duringdeliberation

6 If the argument is not unique she chooses one atrandom

7 If there are more than one candidate the agent will favorthe argument at minimal distance from 119868

28 Complexity

Require Vector of values for parameters (119881 = (120583 119880 119879))Ensure Successful interactions among agents119896 larr997888 0119873119890 larr997888 0

while 119896 le |119873|2 do ⊳Fill119873119890 with |119873|2 agents119896 larr997888 119896 + 1119873119890 larr997888 119877119886119899119889119900119898(|119873| + 1) ⊳Pseudo-random generator Generates integer between 0 and |119873|119873119903 larr997888 0 120575119894119895 larr997888 0 ⊳Variable to store differences in opinionfor all 119894 isin 119873119890 doRandomly pick a number 119895 isin 119873(119873119890 cup 119873119903)119873119903 larr997888 119873119903 cup 119895 120575119894119895 larr997888 |119900119894 minus 119900119895|procedure dyadic-update(119894 119895) ⊳Two agents 119894 and 119895 discuss and update opinions1199001015840 larr997888 119900119894 (120583119895 119880119895 119879119895) larr997888 (120583119894 119880119894 119879119894) larr997888 (120583 119880 119879) ⊳Save agent irsquos opinion before updating and homogenize agents

if 120575119894119895 lt 119880119894 then119900119894 larr997888 120583119894119900119895 + (1 minus 120583119894)119900119894if 120575119894119895 gt 119879119894 then119900119894 larr997888 (1 + 120583119894)119900119894 minus 120583119894119900119895if 120575119894119895 lt 119880119895 then119900119895 larr997888 1205831198951199001015840 + (1 minus 120583119895)119900119895if 120575119894119895 gt 119879119895 then119900119895 larr997888 (1 + 120583119895)119900119895 minus 1205831198951199001015840

Algorithm 1 Dyadic social influence

Require Vector of values for parameters (119881 = (120574 119901119886 119901119903))Ensure Successful interaction between proposal arguments and agents

for all 119894 isin 119873 doprocedure Probability-Update(V119886 119901119886 119901119903) ⊳Update individual probabilities of change due to deliberated cues119909119894 larr997888 119901|V119886minus119900119894 |2119886 119910119894 larr997888 1199012|V119886minus119900119894 |119903119902 larr997888 0if L119886119887(119868) = 119894119899 then

procedure deliberative-update(119894 119886) ⊳Agent i argument 119886120574119894 larr997888 120574 119902 larr997888 119906 sim U(0 1) ⊳ 119906 is a realization of a uniform random variableif 119902 lt 119909119894 then119900119894 larr997888 120574119894119900119894 + (1 minus 120574119894)V119886break

if 119902 lt 119910119894(1 minus 119909119894) then119900119894 larr997888 (1 + 120574119894)119900119894 minus 120574119894V119886Algorithm 2 Deliberation influence

8 In terms of computational complexity naive agentsmaketheirmoves in119874(|A119894| log(|A119894|)) while focused agents doat least as bad

9 Tables have memory Therefore debates are not neces-sarily trees

10 If there are less than (119899119863 times |119873|)2 agents for any of bothgroups the CA summons all of them to the deliberationtable

11 Or careful consideration of the viewpoints of otherswhich implies that citizens keep an open mind and donot reject arguments outright

12 A ldquohealthyrdquo consensus refers to the case in which manyconsensual decisions are taken but opinion-wise agentsdo not agree with each other Quotes are used on theword healthy because some political science theorists

(eg [36]) believe that in certain situations agreement inopinions for deliberation is epistemologically damaging

13 Simulations on groups of |119873| = 800 and |119873| = 400agents shows that scaling the population has an effect onall themetrics but it is confounding depending onwhichcouples of attitudes structures are considered (119880 119879) Westick to |119873| = 400 as in [9] for comparability andsimulation running time

14 By balanced wemean with as many arguments with V119886 lt0 as with V119886 gt 0

15 Experiments on the system suggest that a bigger(smaller) argument pool 119872 = 1000 (119872 = 200) isassociated with a higher (lower) variance of opinions(and extremism) and higher (lower) judgment accuracyFor the rest of the observations 119872 seem to have noeffect We choose a middle value for 119872 Similarly theargument pool chosen to be balanced or uniformly

Complexity 29

Require Argumentation framework 119860119865 = (AR)Ensure Assignment of labels to arguments from the grounded labeling of 119860119865L(A)

L0(A) larr997888 (0 0 0)119896 larr997888 0do119894119899 L119896+1(A) larr997888 119894119899 L119896(A) cup 119886 | 119886 is not labeled 119894119899 L119896(A) 119886119899119889 forall119887 isin A 119904119905 119887 R 119886 then 119887 isin 119900119906119905 L119896(A)119900119906119905 L119896+1(A) larr997888 119900119906119905 L119896(A) cup 119886 | 119886 is not labeled 119894119899 L119896(A) 119886119899119889 exist119887 isin A 119904119905 119887 R 119886 and 119887 isin 119894119899 L119896+1(A)

while L119896+1(A) = L119896(A) = 0Algorithm 3 Solve argumentation frameworks (with grounded semantics)

Require Vector of values for parameters (119881)Ensure Vector of statistics (119880)

procedure Set-parameters(119881) ⊳Initialize parameters for scenarioprocedure Init-objs(119899119898 119896 ) ⊳Create119898 arguments 119899 agents and 119860119865120576procedure Solve-Argumentation-Framework(119860119865) ⊳Assigns the epistemic labelingL120576 toA119894 larr997888 1119905 larr997888 1while 119894 le 119872 do ⊳ 119872 the number of arguments to discuss119868119894 larr997888 119866119890119899 119886119903119892(1) ⊳Generates proposal argument 119868119860 119894119905 larr997888 (119868119894 0)L119886119887(119868119894) larr997888 119906119899119889 ⊳Initialize label for 119868119895 larr997888 1 ⊳Initialize debate counter 119895while ((L119886119887(119868) = 119906119899119889 or 119898 le 119895) and 119895 le 119898) doif 119905 equiv 0(119898119900119889 119905119863) then

procedure build-argumentation-framework(119899119863 119887119890ℎ119886V119894119900119903) ⊳Build the argument graph agents deliberate(119861119894119905 119878119894119905) larr997888 (119883 119884) sub (⋃119897isin119873119899119863 A119897119894119905 ⋃119897isin119873119899119863 R119897119894119905) ⊳ 119873119899119863 sub 119873 are agents sampled by the CA119860 119894119905 larr997888 119889(119860 119894119905 (119861119894119905 119878119894119905))procedure Solve-Argumentation-Framework(A119889) ⊳FindL(A119889)L119886119887(119868119894) larr997888 L119886119887119863(119868119894)119895 larr997888 119895 + 1

else119896 larr997888 1while 119896 le 119905119863 doprocedure dyadic-social-influence(120583 119879 119880) ⊳Agents discuss one-to-one119896 larr997888 119896 + 1119905 larr997888 119905 + 1

if 120572 gt 0 andL119886119887(119868119894) = 119900119906119905 then ⊳Agents vote for the proposalif (|119897 isin 119873|119900119897 times V119868119894 ge 0| ge 120572 times 119899) then

L119886119887(119868119894) larr997888 119894119899else

L119886119887(119868119894) larr997888 119900119906119905if L119886119887(119868119894) = 119894119899 thenprocedure Deliberation-influence(120574 119901119886 119901119903) ⊳Agents are influenced by the result of the decision process119894 larr997888 119894 + 1119905 larr997888 119905 + 1119880 larr997888 119862119900119898119901119906119905119890 119878119905119886119905119894119904119905119894119888119904(119881)

return 119880 ⊳Get the statisticsAlgorithm 4 Run for one scenario

generated does not seem to play a significant explanatoryrole on any of the metrics

16 Experiments show that themaximumnumber of debatesis not an important parameter even if it is an intuitiveone when considering decision-making processes Inpractice debates end after at most 119898 + 1 deliberation

steps and even then it happens rarely However to avoidunpleasant surprises like infinite debates on a completeargument graph containing the proposal argument weadded the constraint 119898 Its value is precisely 7 becausethe biggest minimum number of debates we studyis 6

30 Complexity

17 To ensure comparability with the deliberative andmixedopinion dynamics models on judgment accuracy andcoherence agents vote on proposal arguments every 119905119863times(119898+1) 119905119863 and119898does not explain variability in judgmentaccuracy nor in coherence

18 Student t-tests for independent samples are robust tothe violation of two hypothesis on the distribution ofthe samples normality and homogeneity of variancesas long as the sample size is big enough (central limittheorem) and if the difference in size of the comparedsamples is smallrespectively

19 Studentrsquos test version forwhen variances are not assumedequal and the difference in size of the compared samplesis big

20 We do not correct for homoskedasticity nor normality oferrors in the estimates

21 Balanced as the same number of runs per scenario orsuch that no significant correlations are found betweenparameters

References

[1] R Axelrod ldquoThe dissemination of culture a model withlocal convergence and global polarizationrdquo Journal of ConflictResolution vol 41 no 2 pp 203ndash226 1997

[2] D M Rousseau Reinforcing the MicroMacro Bridge Organiza-tionalThinking and Pluralistic Vehicles SAGEPublications SageLos Angeles Calif USA 2011

[3] S Galam and SMoscovici ldquoTowards a theory of collective phe-nomena consensus and attitude changes in groupsrdquo EuropeanJournal of Social Psychology vol 21 no 1 pp 49ndash74 1991

[4] A Nowak J Szamrej and B Latane ldquoFrom private attitude topublic opinion a dynamic theory of social impactrdquo Psychologi-cal Review vol 97 no 3 pp 362ndash376 1990

[5] G Deffuant S Huet and F Amblard ldquoAn individual-basedmodel of innovation diffusion mixing social value and individ-ual benefitrdquo American Journal of Sociology vol 110 no 4 pp1041ndash1069 2005

[6] G Deffuant D Neau F Amblard and G Weisbuch ldquoMixingbeliefs among interacting agentsrdquoAdvances in Complex Systems(ACS) vol 3 no 01n04 pp 87ndash98 2000

[7] J Rouchier P Tubaro and C Emery ldquoOpinion transmission inorganizations an agent-based modeling approachrdquo Computa-tional and Mathematical OrganizationTheory vol 20 no 3 pp252ndash277 2014

[8] M Meadows and D Cliff ldquoThe relative disagreement model ofopinion dynamics where do extremists come fromrdquo in Pro-ceedings of the International Workshop on Self-Organizing Sys-tems Lecture Notes in Computer Science pp 66ndash77 Springer2013

[9] W Jager and F Amblard ldquoUniformity bipolarization and plu-riformity captured as generic stylized behavior with an agent-based simulation model of attitude changerdquo Computational ampMathematical Organization Theory vol 10 no 4 pp 295ndash3032005

[10] Y-F Zhang H-Y Duan and Z-L Geng ldquoEvolutionary mech-anism of frangibility in social consensus system based onnegative emotions spreadrdquo Complexity vol 2017 Article ID4037049 8 pages 2017

[11] A Flache andMWMacy ldquoSmall worlds and cultural polariza-tionrdquoThe Journal of Mathematical Sociology vol 35 no 1-3 pp146ndash176 2011

[12] A Tsang and K Larson ldquoOpinion dynamics of skepticalagentsrdquo in Proceedings of the 13th International Conference onAutonomous Agents and Multiagent Systems AAMAS 2014 pp277ndash284 International Foundation forAutonomousAgents andMultiagent Systems France May 2014

[13] K M Hansen Deliberative democracy and opinion formationUniversity Press of Southern Denmark Odense 2004

[14] C R Sunstein ldquoThe law of group polarizationrdquo Journal ofPolitical Philosophy vol 10 no 2 pp 175ndash195 2002 WileyOnline Library

[15] S Moscovici and W Doise Dissensions et Consensus unetheorie generale des decisions collectives Presses Universitairesde France-PUF 1992

[16] J S Fishkin and R C Luskin ldquoExperimenting with a demo-cratic ideal deliberative polling and public opinionrdquo ActaPolitica vol 40 no 3 pp 284ndash298 2005

[17] R C Luskin J S Fishkin and S Iyengar Considered Opinionson US Foreign Policy Face-To-Face versus Online Delibera-tive Polling International Communication Association NewOrleans La USA 2004

[18] J Barabas ldquoHowdeliberation affects policy opinionsrdquoAmericanPolitical Science Review vol 98 no 4 pp 687ndash701 2004

[19] O Pryymak A Rogers and N R Jennings ldquoEfficient opinionsharing in large decentralised teamsrdquo in Proceedings of the 11thInternational Conference on Autonomous Agents andMultiagentSystems vol 1 pp 543ndash550 Spain June 2012

[20] J Rouchier and E Tanimura ldquoWhen overconfident agents slowdown collective learningrdquo Simulation vol 88 no 1 pp 33ndash492012

[21] D Miller ldquoDeliberative democracy and social choicerdquo PoliticalStudies vol 40 S1 pp 54ndash67 1992

[22] M Sherif and C I Hovland Social Judgment Assimilation andContrast Effects in Communication and Attitude Change YaleUniversity Press 1961

[23] P M Dung ldquoOn the acceptability of arguments and its funda-mental role in nonmonotonic reasoning logic programmingand n-person gamesrdquo Artificial Intelligence vol 77 no 2 pp321ndash357 1995

[24] M Caminada and G Pigozzi ldquoOn judgment aggregation inabstract argumentationrdquo Autonomous Agents and Multi-AgentSystems vol 22 no 1 pp 64ndash102 2011

[25] P Baroni M Caminada andM Giacomin ldquoAn introduction toargumentation semanticsrdquoThe Knowledge Engineering Reviewvol 26 no 4 pp 365ndash410 2011

[26] S Chambers ldquoDeliberative democratic theoryrdquo Annual Reviewof Political Science vol 6 no 1 pp 307ndash326 2003

[27] E Bonzon andNMaudet ldquoOn the outcomes of multiparty per-suasionrdquo in Proceedings of the 10th International Conference onAutonomous Agents and Multiagent Systems 2011 AAMAS 2011vol 1 pp 47ndash54 International Foundation for AutonomousAgents and Multiagent Systems Taiwan May 2011

[28] J Ranciere Disagreement Politics and Philosophy University ofMinnesota Press 1999

[29] T Bench-Capon and K Atkinson ldquoAbstract argumentation andvaluesrdquo in Argumentation in Artificial Intelligence pp 45ndash64Springer 2009

[30] M Caminada ldquoOn the issue of reinstatement in argumenta-tionrdquo in Proceedings of the European Workshop on Logics in

Complexity 31

Artificial Intelligence vol 4160 of Lecture Notes in ComputerScience pp 111ndash123 Springer Berlin Heidelberg

[31] GAVreeswijk ldquoAn algorithm to computeminimally groundedand admissible defence sets in argument systemsrdquo in Pro-ceedings of the 2006 conference on Computational Models ofArgument Proceedings of COMMA 2006 vol 144 pp 109ndash120IOS Press Amsterdam The Netherlands 2006

[32] C W Sherif M Kelly H L Rodgers et al ldquoPersonal involve-ment social judgment and actionrdquo Journal of Personality andSocial Psychology vol 27 no 3 pp 311ndash328 1973

[33] G Deffuant F Amblard G Weisbuch and T Faure ldquoHow canextremism prevail A study based on the relative agreementinteraction modelrdquo Journal of Artificial Societies and SocialSimulation vol 5 no 4 p 1 2002

[34] E Awad M W Caminada G Pigozzi M Podlaszewski andI Rahwan ldquoPareto optimality and strategy-proofness in groupargument evaluationrdquo Journal of Logic and Computation vol 27no 8 pp 2581ndash2609 2017

[35] D Grossi and G Pigozzi ldquoJudgment aggregation a primerrdquoSynthesis Lectures on Artificial Intelligence and Machine Learn-ing vol 8 no 2 pp 1ndash151 2014

[36] H Landemore and S E Page ldquoDeliberation and disagreementproblem solving prediction and positive dissensusrdquo PoliticsPhilosophy and Economics vol 14 no 3 pp 229ndash254 2015

[37] R Hegselmann and U Krause ldquoOpinion dynamics andbounded confidence models analysis and simulationrdquo Journalof Artificial Societies and Social Simulation vol 5 no 3 2002

[38] A Stefanelli and R Seidl ldquoModerate and polarized opinionsUsing empirical data for an agent-based simulationrdquo inProceed-ings of the Social Simulation Conference 2014

[39] S Gabbriellini and P Torroni ldquoA new framework for ABMsbased on argumentative reasoningrdquo in Advances in SocialSimulation vol 229 pp 25ndash36 Springer Berlin Germany 2014

[40] H Mercier and D Sperber ldquoWhy do humans reason Argu-ments for an argumentative theoryrdquo Behavioral and BrainSciences vol 34 no 2 pp 57ndash74 2011

[41] H V Parunak T C Belding R Hilscher and S BruecknerldquoModeling and managing collective cognitive convergencerdquoin In Proceedings of the 7th iNternational Joint Conference onAutonomous Agents and Multiagent Systems vol 3 pp 1505ndash1508 International Foundation for Autonomous Agents andMultiagent Systems 2008

[42] H Prakken ldquoFormal systems for persuasion dialoguerdquo TheKnowledge Engineering Review vol 21 no 2 pp 163ndash188 2006

[43] K Atkinson T Bench-Capon and P Mcburney ldquoA dialoguegame protocol for multi-agent argument over proposals foractionrdquoAutonomousAgents andMulti-Agent Systems vol 11 no2 pp 153ndash171 2005

[44] M Thimm and A J Garcıa ldquoClassification and strategicalissues of argumentation games on structured argumentationframeworksrdquo in Proceedings of the 9th International Conferenceon Autonomous Agents and Multiagent Systems vol 1 pp 1247ndash1254 International Foundation for Autonomous Agents andMultiagent Systems Canada 2010

[45] I Rahwan and K Larson ldquoMechanism design for abstractargumentationrdquo in Proceedings of the 7th International JointConference on Autonomous Agents and Multiagent SystemsAAMAS 2008 pp 1031ndash1038 International Foundation forAutonomous Agents and Multiagent Systems Portugal May2008

[46] H Prakken R Riveret A Rotolo and G Sartor ldquoHeuristics inArgumentation A Game-Theoretical Investigationrdquo EuropeanUniversity Institute 2008

[47] N Hadidi Y Dimopoulos and P Moraitis ldquoTactics and con-cessions for argumentation-based negotiationrdquo in Proceedingsof the COMMA vol 245 pp 285ndash296 2012

[48] N Oren and T J Norman ldquoArguing using opponent modelsrdquoin Proceedings of the International Workshop on Argumentationin Multi-Agent Systems pp 160ndash174 Springer 2009

[49] S Delre W Jager T Bijmolt and M Janssen ldquoTargeting andtiming promotional activities An agent-based model for thetakeoff of new productsrdquo Journal of Business Research vol 60no 8 pp 826ndash835 2007

[50] A Mirtabatabaei P Jia and F Bullo ldquoEulerian opinion dynam-ics with bounded confidence and exogenous inputsrdquo SIAMJournal on Applied Dynamical Systems vol 13 no 1 pp 425ndash446 2014

[51] V Barbet N Guiraud V Laperriere and J Rouchier Hagglingon Values Towards Consensus or Trouble 2019

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4 Complexity

a b c

Figure 1 Argumentation framework 119860119865 = (AR) with A =119886 119887 119888 andR = (119886 119887) (119887 119888) The labeling 119887 119888 119886 is conflict-free 119886 119888 119887 0 is the only complete labeling

proposal argument 119868 and that opposes the principle P couldbe 119886 = ldquoElectric cars may protect the environment byreducing effective carbon emissions but the batteries theydepend on are very pollutantrdquo The proposal P may beaccepted in the deliberation arena if some other argumentthat defeats 119886 say 119887 = ldquoBattery recycling businesses arehatching everywhere By 2020 it is very likely that scientistsfind a viable solution to chemical pollution due to batteriesrdquo isadvanced It is not accepted otherwise If 119868 is accepted thenPis collectively accepted if a majority of agents vote favorablyfor it

22 Arguments as the Basic Units of Collective Discussionamong Agents In this subsection we introduce the sim-plest object in our system arguments We present abstractargumentation theory and Caminadarsquos labeling approach toargumentation [30] that allows us to track the epistemic statusof arguments during deliberation

221 Arguments Arguments are objects that represent piecesof information that agents can understand They are infor-mational cues that enable agents to discuss with one anotherin a public collective context and make decisions on theacceptability of other pieces of informationThey are assumedto be nonfallacious In our approach each argument 119886 ismodeled by a real number V119886 isin [minus1 1] that stands forhow much 119886 respects or supports the principle P V119886 = 1means that argument 119886 is totally coherent with the principleP whereas V119886 = minus1 reads that 119886 is totally incoherent withthe principle P Arguments relate to each other through anincompatibility relation that states that one cannot standbehind two conflicting arguments Arguments are also char-acterized by their acceptability status that indicates whetherthey are accepted refuted or undecided in a given discur-sive context Arguments have an epistemic reach (119890119903) or amaximum number of arguments they can attack This canbe interpreted as the argumentrsquos level of generality or as thepotential level of argumentative conflict within the argumentpool

222 Abstract Argumentation Theory Deliberation definedas an exchange of arguments is modeled by confronting rep-resentations of different eventually contending argumentsIn our model we use abstract argumentation theory [23] torepresent deliberation where an argument is just a node in agraph like arguments 119886 119887 and 119888 in Figure 1 Abstract argu-mentation theorymodels incompatibility between argumentsand abstracts away from their internal structureThe intuitionis that if an argument 119886 attacks an argument 119887 (representedby an arrow from node 119886 to node 119887 as in Figure 1) a rationalagent cannot accept both 119886 and 119887

Formally let A be a finite set of arguments and R asubset of A times A called attack relation The attack relation isintransitive and we note (119886 119887) isin R the fact that argument 119886attacks or is incompatible with argument 119887 One says that anargument 119886 defends an argument 119888 noted (119886 119888) isin D if thereexists an argument 119887 such that (119886 119887) isin R and (119887 119888) isin R

(see Figure 1) An argumentation framework (119860119865 = (AR))is a digraph in which the nodes represent the argumentsand the arcs represent the attacks among them Given anargumentation framework119860119865 the classic problem in abstractargumentation resides in finding which of the arguments in119860119865 can be accepted rejected or left undecided

In the labeling approach [30] a label L119886119887(119886) isin Λ =119894119899 119900119906119905 119906119899119889 of an argument 119886 isin A denotes the epistemicstatus of 119886 Intuitively an argument is labeled 119894119899 if it isjustifiable and 119900119906119905 if it is not If 119886 is labeled 119906119899119889 it isconsidered to be in abeyance due to for example insufficientgrounds for it to be labeled 119894119899 or 119900119906119905 Furthermore given anargumentation framework 119860119865 = (AR) one calls labeling acomplete function

L A 997888rarr Λ = 119894119899 119900119906119905 119906119899119889119886 997891997888rarr L119886119887 (119886) (1)

that assigns a label to each argument in 119860119865 A labeling iswritten as a triplet⨉120582isinΛ120582L(A) where 120582L(A) = 119886 isin A |L119886119887(119886) = 120582 For example 119894119899 L(A) stands for the set ofarguments in A that are labeled 119894119899 in the labelingL(A) Alabeling is said to be legal if all argument it labels is legallylabeled An argument 119886 isin A is said to be legally labeled

(i) L119886119887(119886) = 119894119899 if forall119887 isin A st (119887 119886) isin RL119886119887(119887) = 119900119906119905if 119886 is not attacked or is only attacked by argumentsthat are themselves labeled 119900119906119905

(ii) L119886119887(119886) = 119900119906119905 if exist119887 isin A st (119887 119886) isin RL119886119887(119887) = 119894119899that is if 119886 is attacked by at least one argument that islabeled 119894119899

(iii) L119886119887(119886) = 119906119899119889 if not(forall119887 isin A st (119887 119886) isin RL119886119887(119887) =119900119906119905) and (forall119888 isin A st (119888 119886) isin RLa119887(119888) = 119894119899)or equivalently if there exists at least one argumentlabeled 119906119899119889 that attacks 119886 and there is no argumentlabeled 119894119899 attacking 119886

Roughly speaking a semantics (denoted by 120590) is a rationalitycriteria to decide which arguments to accept given an argu-mentation framework The basic normative requirementsfor labeling-based semantics and argument acceptabilityin abstract argumentation repose on conflict-free labelingsor labelings in which no two 119894119899-labeled arguments attackeach other and admissible labelings which are conflict-free labelings that ask for arguments that are defended bythe 119894119899-labeled arguments to be themselves 119894119899-labeled Thefamily of admissibility-based labelings goes from completelabelings to preferred and grounded labelings which arecomplete labelings that capture properties such as credulityand skepticism in argumentation Formally a labelingL forthe arguments in an argumentation framework (AR) is saidto be

Complexity 5

(i) conflict-free if ∄119886 119887 isin A st (119886 119887) isin R and 119886 119887 isin119894119899L(A)

(ii) admissible if conflict-free and forall119886 isin AL119886119887(119886) =119906119899119889 997904rArr 119886 is legallyL119886119887(119886)

(iii) complete if admissible and forall119886 isin AL119886119887(119886) =119906119899119889 997904rArr 119886 is legally labeled 119906119899119889

(iv) preferred if complete and if it maximizes 119894119899L(A)(v) grounded if complete and if it maximizes 119906119899119889L(A)

We choose admissibility-based semantics because for onethey supply a comprehensive model of collective reasoningthey allow for a meaningful parameterization of credulousand skeptical collective reasoning and last they are relativelyeasy to interpret since the differences between them arewell documented in the literature (eg [18]) Additionallyand in contrast to other rank-based or graded semantics inabstract argumentation admissibility semantics assume thatall arguments have the same weight

223 Abstract Argumentation and Discursive Situations inCollective Decision-Making Abstract argumentation is aconvenient formalism for argument-based decision-makingsince it ignores difficulties relative to the nature generationand number of arguments and posits the possibility of usinggraph theoretic tools to model (collective) reasoning in aclear coherent and easy way [31] Example 2 below providesa simple overview of a debate between two agents over theacceptability of the proposal ldquoTax the richrdquo

Example 2 Let P = ldquoTax the richrdquo and P = ldquoLiberalismrdquoFigure 1 represents the abstract argumentation frameworkobtained from the following arguments

(i) 119868 = 119888 = ldquoOnly the rich should be taxed because theypossess most of the capital in the countryrdquo

(ii) 119887 = ldquoIf you only tax the rich the rich will leave andthen yoursquoll have no one to taxrdquo

(iii) 119886 = ldquoThe rich will not leave because they have theirlivelihoods here and it would cost themmore to leavethan to pay the taxesrdquo

Agent 1 advances the proposal argument 119888 agent 2 arguesthat given the justification 119888 for P P is not tenable Agent1 defends his proposal by advancing 119886 and defeating theargument 119887The conclusion is that holding 119888 and 119886 is a tenableposition so the rich should be taxed Notice that by simplyaccepting 119888 and taking no position on 119887 nor 119886 is conflict-freeand still corresponds to the position advocating that the richshould be taxed

Abstract argumentation comes as an immediate appli-cation to our model in the construction of an ideal argu-mentation framework that loosely models Habermasrsquo idealspeech situation The ideal speech situation is important inour work because it corresponds to a normative state that isin practice difficult to reach and that allows us to observehow different deliberation protocols affect deliberative out-comes

In the model a label given to an argument provideda semantics 120590 is said to be ideal if it is obtained from astate of affairs in which all arguments are presented duringdeliberation We call ideal or consensual argumentationframework the argumentation framework (AR) = 119860119865120576containing all arguments in the system and all consensualattacks among them In a multiagent approach a consensualattack between two arguments 119886 119887 isin A is a couple (119886 119887)such that (119886 119887) isin R if and only if a certain majority ofagents recognizes that such attack exists In our model allagents agree on the attack relation over 119860119865120576 It follows thatif two agents advance two distinct arguments 119886 119887 isin A suchthat (119886 119887) isin R then all agents will recognize that suchconflict exists An immediate consequence of this assumptionis that all deliberated results are consensual even if an opinionconsensus2 on the principle is not necessarily reached

23 Deliberative Social Agents In this section we presentthe agents in our model We define them on the basisof their opinions on the principle P the arguments theypossess their knowledge of the relationship among themtheir behavior during deliberation and their sensitivity todeliberated proposals

231 Agents in Dyadic Social Interactions Every agent 119894 hasan opinion a relative position or degree of adherence 119900119894 isin[minus1 1] to a principle P and a couple (119880119894 119879119894) isin [0 2] times [0 2](119880119894 le 119879119894) of latitudes of acceptance and rejection respectivelyof informational cues They live in [0 2] since 0 and 2 arerespectively the minimum and maximum distances of anytwo informational cues in the system The idea behind thecouple (119880119894 119879119894) is that there exist levels of relative tolerancefrom which informational cues have either an attractive ora repulsive effect on the individual [22 32] An 119900119894 close to 1implies that agent 119894 fully supports the principle P and closeto -1 that she rejects principleP or equivalently fully supportsnotP Moreover if an agent 119894rsquos opinion is such that |119900119894| ge 075119894 is considered to have an extreme opinion a moderate oneotherwise 119880119894 may be considered as 119894rsquos uncertainty about herown opinion [33] or as the limit below which the objectshe judges may attract her 119879119894 may be seen as 119894rsquos bound oftolerance from which informational cues disgust her andconfirmher initial positionDifferent combinations of (119880119894119879119894)can be associated with agent 119894rsquos ego or personal involvementin discussion processes as it is vividly explained in [32]Finally agents are assumed to be sincere and precise whencommunicating their positions to one another (no noise inthe interactions)

232 Agents in Deliberative Social Interactions Agents voteand participate in deliberation because they are aware ofthe potential changes an accepted proposal may induce inthe opinion of the group After all proposals promote aprinciple that potentially leads to a shift in other agentsrsquoopinions Agentsrsquo incentive to participate in deliberation isbased on the idea that every single one of them wants tomake her point across and at the same time reach a correctcollective decision Agents are endowed with a probability

6 Complexity

-1 0 1

Arguments

Adherence to principle

Figure 2 Representation of agents and their arguments on theopinion or adherence to the principles spectrum The rectangleson the agents are their argument sacks Arguments are depicted bydots and arrows from the arguments to the sacks depict both thearguments they hold in the opinion spectrum and the number ofarguments they hold 119896119894 = 4 and 120601119890119894 = 075 and 120601119898119894 = 05 for allagent 119894119909119894 = 119891(119889(119900119894 V119868) 119901119886) of being attracted to a deliberated cueand a probability of being repulsed by it 119910119894 = 119892(119889(119900119894 V119868) 119901119903)The two probabilities are assumed to be independent Theyare a function of the distance between the opinion andthe proposal argument and of the grouprsquos sensitivity3 todeliberation (119901119886 119901119903) The former is decreasing on 119889(119900119894 V119868)and increasing on 119901119886 while the latter is increasing on both119889(119900119894 V119868) and 119901119903 Agents are also assumed

(i) to be capable of assessing the degree of support for Pof all arguments

(ii) to trust4 one another when they utter informationalcues

233 Agents and Voting At the end of the decision-makingprocess agents vote on whether they agree or not on the pro-posal argument that has been discussed during deliberationVoting for an agent is the expression of her opinion in thefinal phase of the decision-making process An agent 119894 is saidto vote favorably for a proposal P of justification argument119868 if and only if V119868 times 119900119894 ge 0 or equivalently if the proposalargument does not adhere to the opposite of the principle 119894supports5

234 Agent Knowledge of Arguments Let A be a finite setof arguments Each agent 119894 has a sack of arguments A119894 subA of size |A119894| = 119896119894 whose content reflects her relativeposition 119900119894 on P (see Figure 2) For extreme-opinion agentsa proportion 120601119890 ge 05 of the 119896 arguments in their sacks areof the same adherence to the principle than their opinions

whereas for moderate agents a proportion 120601119898 le 120601119890 oftheir 119896 arguments are of the valence of their opinions Thehypothesis feeds from results found in [22] stating that ldquoanindividual places a verbal statement on an issue both in termsof the itemrsquos relative proximity to his own position and thelatitude which is acceptable to him around that focal point ofacceptancerdquo

The arguments in an agent 119894rsquos sack are those that sheknows how to use and advance in a deliberative interac-tion The size of the sack represents agent 119894rsquos ability tocommunicate in a deliberative context It follows that eachagent 119894 possesses partial knowledge (119870119894 sub A times A) of theattack relation between the arguments in A and of 119860119865120576which she derives from the arguments in her sack andwhat she observes in deliberation Knowledge of argumentsis assumed to be ldquoattack-orientedrdquomdashan agent 119894 knows anattack (119886 119887) isin A119894 times A if she knows upon observationthat neither she nor the group can rationally accept 119886alongside 119887 She may be aware of the existing defense relationamong arguments when concatenating the information sheaccumulates on their attack relation during deliberationShe may use this information strategically in deliberativecontexts

Let 119889119894119904119905 A times A 997888rarr N be the length of the shortestpath between two elements in the argumentation frameworkinduced by an agent 119894rsquos knowledge119870119894 119894 sees an argument 119886 asan attacker (defender) of an argument 119887 if119889119894119904119905(119886 119887) equiv 1mod 2(119889119894119904119905(119886 119887) equiv 0mod 2) An argument 119886 that is at an odddistance from another argument 119887 attacks it since either itdirectly attacks the argument or it attacks a defender of theargument Likewise if 119886 is at a pair distance of 119887 it attacksan attacker of 119887 and thus defends it If such path does notexist then the distance is undefined and the agent does notsee either argument as an attacker or defender of the otherFurther let AF119889 = (A119889R119889) denote the current state ofaffairs in deliberation An agent 119894rsquos knowledge ofR is the setof attacks and defenses she can infer from the arguments sheknows (119860119905119905119894119863119890119891119894) and the attacks and defenses she can inferfrom deliberation (119860119905119905119889119894 119863119890119891119889119894 )

119870119894 = 119860119905119905119894 cup 119860119905119905119889119894 cup 119863119890119891119894 cup 119863119890119891119889119894 (2)

where

(i) 119860119905119905119894 = (119886 119887) isin A119894 timesA119894 | 119889119894119904119905(119886 119887) equiv 1mod 2)(ii) 119863119890119891119894 = (119886 119887) isin A119894 timesA119894 | 119889119894119904119905(119886 119887) equiv 0mod 2(iii) 119860119905119905119889119894 = (119886 119887) isin (A119894 cup A119889) times A119889 | 119889119894119904119905(119886 119887) equiv1mod 2)(iv) 119863119890119891119889119894 = (119886 119887) isin (A119894 cup A119889) times A119889 | 119889119894119904119905(119886 119887) equiv0mod 2

Notice that agents have no restriction on the amount ofinformation they can carry and use during deliberation Themodel implies that agents have the ability to use and processall information on the attack relations if they could observethe attacks and that they all have equally high cognitivecapacity Agent knowledge resets at the end of each decision-making process but argument sacks stay untouched In otherwords argument sacks are static in the model

Complexity 7

235 Agent Behavior during Deliberation Agents maybehave in twodifferentways in deliberationTheymay behavenaively or focusedly Naive agents use deliberation to voicetheir opinions on the principle through arguments Focusedagents strategically argue in favor of proposal arguments thatsupport the principle they favor In both cases agents advancearguments in terms of the argumentsrsquo relative proximity totheir own positions [22]

Let 119886119894 isin A119894 denote the argument an agent advancesin a debate 119889 (of a deliberation process 119863119868) over a centralargument 119868 and V119886 the valence of an argument 119886 isin A Let

(i) 119891119894(119886) equiv (119900119894 times V119886 gt 0 or 119900119894 = V119886 = 0) the formula thatis satisfied if the argument 119886 is of the same valence asthe opinion of agent 119894

(ii) 119892119894(119886) equiv (119886 119887 isin A119894A119889 and forall119887(|119900119894 minus V119887| gt |119900119894 minus V119886|)) theformula that is satisfied only if the argument 119886 is in 119894rsquossack (A119894) and not in the debate and is the closest interms of adherence to the principle P to 119894rsquos opinion

(iii) ℎ119894(119886) equiv (119891119894(119886) and 119892119894(119886)) the conjunction of thepreceding formuli indicating that an argument 119886 is inagent 119894rsquos sack and not in the debate of the same signas 119900119894 and closest to 119900119894 in absolute value

Naive agents choose 119886119894 = 119886 such that ℎ119894(119886) is true given anyproposal argument 119868 Agents that behave strategically or tosay focusedly choose their moves in the debate as follows(119873119868119871 indicates no move or null argument)

(a) if agent 119894rsquos position with respect to P is of the samesign as the proposal argumentrsquos adherence to P (119891119894(119868)) sheadvances an argument 119886 such that

(i) ℎ119894(119886) and ((119886 119868) isin 119863119890119891119889119894 )) is true If 119886119894 = 119873119868119871(i1) notℎ119894(119886) and 119892119894(119886) and ((119886 119868) isin 119863119890119891119889119894 ) (chooses anargument not of the same sign and closest to heropinion and defending 119868) is true If 119886119894 = 119873119868119871(ii) ℎ119894(119886) and ((exist119888 isin A119894)((119888 119868) isin 119860119905119905119889119894 and (119886 119888) isin 119860119905119905119894))(anticipates an attack from an argument 119888 on 119868 andadvances an argument that attacks the argument 119888)6is true If 119886119894 = 119873119868119871(ii1) notℎ119894(119886)and119892119894(119886)and ((exist119888 isin A119894)((119888 119868) isin 119860119905119905119889119894 and(119886 119888) isin119860119905119905119894)) is true If 119886119894 = 119873119868119871(iii) ℎ119894(119886) and ((119886 119868) notin 119860119905119905119889119894 ) (simply does not attack 119868)is true If 119886119894 = 119873119868119871(iii1)notℎ119894(119886)and119892119894(119886)and((119886 119868) notin 119860119905119905119889119894 ) is true If 119886119894 = 119873119868119871(iv) 119886119894 = 119873119868119871 (the agent will advance no argument)

(b) if agent 119894rsquos position with respect toP is of the oppositesign of the proposal argumentrsquos adherence to P (not119891119894(119868)) sheplays the argument 119886 such that

(i) ℎ119894(119886) and ((119886 119868) isin 119860119905119905119889119894 ) 7 If 119886119894 = 119873119868119871(i1) notℎ119894(119886)and119892119894(119886)and ((119886 119868) isin 119860119905119905119889119894 ) is true If 119886119894 = 119873119868119871(ii) ℎ119894(119886) and ((exist119887 isin A119894)((119887 119868) isin 119863119890119891119889119894 and (119886 119887) isin 119860119905119905119894))(anticipates a defense from an argument 119888 to 119868 andadvances an argument that attacks 119888) is true If 119886119894 =119873119868119871

(ii1)notℎ119894(119886)and119892119894(119886)and((exist119887 isin A119894)((119887 119868) isin 119863119890119891119889119894 and(119886 119887) isin119860119905119905119894)) is true If 119886119894 = 119873119868119871(iii) ℎ119894(119886) and ((forall119887 isin A119889)((119887 119868) isin 119860119905119905119889119894 and (119886 119887) notin 119860119905119905119889119894 ))(avoids attacking any attacker of 119868) If 119886119894 = 119873119868119871(iii1)notℎ119894(119886)and119892119894(119886)and((forall119887 isin A119889)((119887 119868) isin 119860119905119905119889119894 and(119886 119887) notin119860119905119905119889119894 )) If 119886119894 = 119873119868119871(iv) 119886119894 = 119873119868119871

In English a focused agent that is opposed to the proposalattempts to either attack it directly or attack an argument thatis defending it whereas an agent that agrees with the proposalattempts to either defend it or avoid attacking it altogetherThis behavioral approach is similar to debate protocols inabstract argumentation but for the fact that agents mayadvance arguments that do not result in an ldquoadvancementrdquoof the debate In our case the debates stop for other reasons(see Section 24)

Two important points deserve to be highlighted Firstlyfocused agents with small argument sacks will regularly findthemselves applying rules (119894119894119894) or (119894V) In effect if they donot have enough arguments to infer attacks and defensestheir behavior in deliberation is likely to be similar to that ofthe naive agents Secondly the naive and focused behavioralassumptions presented here are very different in terms ofcomputational complexity The first assumes that the agentobserves the debate but what she sees has no effect on hercourse of action ie the argument she voices is uniquelydetermined by how she feels about the principle behindthe proposal argument In the second an agent debatesessentially to knock out any proposal argument she disagreeswith8

24 Tables for Deliberation Tables are the physical or virtualplaces where the exchange of arguments occurs Agentsdeliberate on these tables to determine whether the proposalargument is acceptable or not and are thus subdued bythe deliberation procedure imposed by the tablersquos centralauthority (CA) [27] The CA decides on the structure andlength of the collective decision-making process and thedeliberation procedure It controls the percentage of agentsfrom the population that actively participates in the debate(119899119863) and the labeling-based semantics used to extract accept-able arguments from the framework (120590) The percentage 119899119863denotes either the proportion of agents in the population thatgets to advance arguments in a population-scale deliberationor an independent sample of agents summoned to activelyparticipate in the debate 120590 stands for the procedure used toconclude on the epistemic status of the proposal argumentand the arguments advanced during deliberation

The CA has the ability to stop debates at will using a stoprule that inherently depends on the number of debates thatought to take place before a decision is deemed sufficientlydiscussed (119898) the maximum number of debates that cantake place before abandoning deliberation (119898) and the labelgiven to the proposal argument 119868 (119904119903(119898119898L119886119887(119868))) 119904119903is a Boolean function whose value ldquotruerdquo signals the callfor a vote andor the end of the decision-making process119898 is associated with a minimal dialectical or epistemic

8 Complexity

requirement to consider a proposal for voting and to a lowerbound of the length of the deliberation process 119898 providesan upper bound Moreover the CA controls the size of thetime interval between debates (119905119863) the collective decisionrule (eg if there is voting on the deliberated proposals) andthe majority quota rule for accepting proposal arguments(120572) in the decision-making process 119905119863 may represent thefrequency or density of pair-wise interactions in a decision-making process

241 The Construction of a Decision-Making Process Adeliberation process or debate in ourmodel is a tool to obtainlabels for proposal arguments that are as close as possible tothe ideal or consensual ones To define a deliberation processformally we introduce the notion of debate step as constituentof a deliberation process Informally a debate step is a timestep in which a debate occurs Formally and more event-oriented a debate step (1198891198731015840 1198611015840 1198781015840) of a debate 119863119868 on 119868 isa quadruplet composed of a set of agents 1198731015840 sub 119873 a set ofarguments 1198611015840 sub ⋃119894isin1198731015840 A119894 a set of attack relations 1198781015840 sub Rand a mapping

119889 (AF 2A times 2R) 997888rarr AF

(119860119865 (1198611015840 1198781015840)) 997891997888rarr 1198601198651015840 = (119861 cup 1198611015840 119878 ⊔ 1198781015840) (3)

that adds the arguments in 1198611015840 and some attacks in 1198781015840 to aframework 119860119865 = (119861 119878) and where 119878 ⊔ 1198781015840 = (119886 119887) isin 119878 cup 1198781015840 |119886 119887 isin 119861 cup 1198611015840 In the same spirit we define a deliberationprocess119863119868 on a proposal argument 119868 as a sequence of debatesteps (119889119873119904 119861119904 119878119904)119904gt0 such that1198730 = 0 and

119889((119904minus1⋃119895=0

119861119895 119904minus1⨆119895=0

119878119895) (119861119904 119878119904)) sub (AR) (4)

with

(i) 1198610 = 119868 (the initial argument is the proposal argu-ment)

(ii) 1198780 = 0 (no reflexive attack from 119868 to 119868 is allowed)(iii) (forall119904 gt 0)(⋃119895lt119904 119861119895cap119861119904 = 0) (any newly added argument

to the framework has yet to be added to it)(iv) (forall119904 gt 0)(⨆119895lt119904 119878119895 cap 119878119904 = 0) (any newly considered

attack among arguments cannot be declared amongarguments that are not in the framework)

(v) lim119904997888rarrinfin(⋃119904gt0 119861119904 ⋃119904gt0 119878119904) sube (AR) (the system isstable no arguments are created during deliberation)

Finally let L119886119887119889(119868) denote the label given to argument 119868 ata debate step 119889 during a deliberation process 119863119868 A decision-making process notedD119868 = (119899119863 119905119863 119904119903(119898119898L119886119887119889(119868)) 120572 120590)over a proposalwhose justification argument is 119868 is a sequenceof debate and nondebate steps (119911119905)119905isinN 119911119905 isin 119889 119889 such that

(i) 119889 are debate steps and 119889 nondebate steps that corre-spond to time steps at which there are no debates

(ii) If 119905(mod 119905119863) equiv 0 then 119911119905 = 119889 and 119889 otherwise

For Against

Table

CA

in out

out out in

out in

proparg

Figure 3 A representation of a table of deliberation betweentwo groups of 3 agents with 4 arguments each Each agent hasadvanced an argument Agents tacitly agree on the attacks betweenthe arguments The CA determines the labeling using the groundedlabeling-based semantics

d d d d d d d d d d d d dTime 0 1 2 3 4 5 6 7 8 9 10 11 12I

Figure 4 A decision-making processD119868 or collective decision stepstarting at 119905 = 0 composed of debate (119889) and nondebate (119889) substepsDebate steps are shaded The number of nondebate steps betweendebates is 119905119863 = 3 the number of debates in the decision-makingprocess is119898 = 4

(iii) The subsequence (119911119897119905)119897119905isinN of 119911119905 with 119897119905 = 119905 such that119905(mod 119905119863) equiv 0 is a deliberation process with |1198731015840| =119899119863 times |119873|(iv) 119904119903(sdot sdotL119886119887119889(119868)) pursues the deliberation process as

long asL119886119887119889(119868) = 119906119899119889 under the semantics 120590(v) The length of the sequence |119911119905| or equivalently the

duration of the process is somewhere in the interval⟦119898119905119863 119898119905119863⟧(vi) The final labeling for the proposal argument 119868

L119886119887(119868) is determined by a combination of its delib-erated labelL119886119887119889(119868) and a majority vote of majorityquota 120572

A decision-making process ends when a final decision hasbeen taken concerning the acceptability of the proposal Peg 119868 has been deemed unacceptable (L119886119887119889(119868) = 119900119906119905) or amajority of agents have voted againstP For a representationof a deliberative interaction on a table see Figure 3 for one ofa decision-making process see Figure 4

Please note that every collective decision can be seen asa ldquotime steprdquo in as much as it describes how agents updateopinions and make collective decisions In the model thedecision-making process and the parameters define the sub-steps or events that occur within the time step Henceforth

Complexity 9

Fix proceduralsm m α nD tD σ Init count m larr 0

Given proposal anargument I is randomly generated

nD times |N | agents arerandomly picked

m larr m + 1

Begin decision processon proposal at time t

t larr t + 1Deliberation step m

larr 0

t larr t + 1dyadic opinionupdate Eq (5)tm larr tm + 1

m ge m

tm gt tD

ℒd (I) = in

m ge m

yields ℒ(I)

ℒ(I) = in Deliberationupdate Eq (6)

End decision

yes

no

no

no

yes

yes

no

yes

yes

yes

no

no

I

Vote on ℒd(I)

ℒd (I) = und

ＪＬＩ＝ＭＭ I

Init count tmyields ℒd(I)

Figure 5 Process chart for a decision processD119868 on a proposalP 119905 corresponds to time and 119898 (119905119898) corresponds to the number of debate(non-debate) steps that have taken place in the deliberation process (in between two debate steps) over time for the mixed discussion modelWe useL119886119887(119868) andL(119868) interchangeably in the figure for presentation

comparing simulations on the basis of the different decision-making processes translates into comparing these collectivedecision steps

242 Deliberation Protocol To define the deliberation proto-col held in the table either as a consequence of the definitionprocess or as a statement we assume the following

(i) Agents may decide not to contribute to the debate(ii) All agents have the same probability conditional to

their opinions of being picked to participate in adebate or deliberation step

(iii) There is no restriction on the number of times eachagent can participate in a deliberation process

(iv) Each agent may only place one argument per debatestep

(v) Arguments that have already been advanced in thedebate may attack newly placed arguments9

The deliberation or debate protocol goes as follows

(1) The CA randomly generates and makes public acentral argument or proposal argument 119868 and informsall agents about the rules of the decision-makingprocess

(2) The CA randomly draws two sets of (119899119863 times |119873|)2agents with divergent views10 onP and P It mergesthem to create the set of debaters1198731015840

(3) Each agent 119894 advances an argument from her sackA119894 The CA makes sure that there are no repeatingarguments (agents already take argument repetitioninto account)

(4) The CA establishes the debate step 119889rsquos argumentationframework and computes its labelingL120590

119889(A119889) using120590(5) If the computed label for 119868 is L119886119887119889(119868) = 119906119899119889

or the number of debates steps is inferior or equalto 119898 at time 119905 then the CA stops the debate andresumes it at the (119905 + 119905119863 + 1)rsquoth time step by repeating(1) (2) (3) (4) and (5)

(6) Let L119886119887(119868) be the final label given to the proposalargument 119868 If voting is allowed then if more than120572 times |119873| agents agree with 119868 119868 is accepted (L119886119887(119868) =119894119899) it is rejected if strictly less than 120572times |119873| agree withit (L119886119887(119868) = 119900119906119905) When there is a tie ifL119886119887119889(119868) =119906119899119889 then L119886119887(119868) = 119900119906119905 if L119886119887119889(119868) = 119894119899 thenL119886119887(119868) = 119894119899 When voting is not part of the decision-making process (120572 = 0)L119886119887119889(119868) = L119886119887(119868)

One important point to notice about this protocol is that itinduces a stop rule 119904119903 (cf steps (5) and (6) of protocol) forthe decision process and thus it always ends Either agentsdebate and agree on the proposalrsquos acceptability throughargumentation or after 119898 debate steps they directly voteon it so that a decision concerning the proposal is alwaysreached Refer to Figure 5 for a sweeping description of thedeliberation protocol and the decision-making process

10 Complexity

-1

-1 1

1

Latitude of acceptance

Latitude of rejection Agent i

Agent j

Figure 6 A dyadic interaction between agents 119894 and 119895 as expressedin (5) Agent 119894rsquos opinion falls within agent 119895rsquos latitude of noncommit-ment and conversely Neither agent affects the opinion of the other

25 Opinion Dynamics for Social Interactions In this subsec-tion we describe the opinion diffusion model based on pair-wise interactions among agents and deliberation Sherif rsquosand Hovlandrsquos social judgment theory [22] motivates part ofour approach It describes how individualsrsquo opinions changeon the basis of their attitude structures Attitude structuresrefer to the relative scope width or latitude of categoriesused by individuals when evaluating information namelythe latitudes of acceptance rejection and noncommitment[32] The idea behind this theory is that individuals changetheir positions only in accordance to how far or close thecommunicative cues they receive are from (to) their anchorpositions It holds that if communicative cues are far (close)from (to) an agentrsquos position say over her latitude of rejection(acceptance) then the agent shifts her position away from(towards) the position defended by the cues In the casewherethe cues fall within the agentrsquos latitude of noncommitmenther position does not change (see Figure 6 and Equation (5))

251 Pair-Wise or Dyadic Opinion Dynamics As agentsmay communicate and deliberate collectively they may alsoengage in one-to-one conversations with other agents toponder on their positions We loosely associate this type ofcommunication with dyadic nonargumentative exchange ordiscussion based on fallacious arguments and persuasionIn the light of social judgment theory and the descriptionof agents in the system we model pair-wise symmetricinteractions following the opinion dynamicsmodel in [9] Anagent 119895rsquos influence on an agent 119894rsquos opinion at time 119905 is governedby the ensuing difference equation

119900119905+1119894 =

119900119905119894 + 120583119894 (119900119905119895 minus 119900119905119894) 119894119891 10038161003816100381610038161003816119900119905119894 minus 11990011990511989510038161003816100381610038161003816 lt 119880119894119900119905119894 minus 120583119894 (119900119905119895 minus 119900119905119894) 119894119891 1003816100381610038161003816119900119905119894 minus 119900119905119894 1003816100381610038161003816 gt 119879119894119900119905119894 119900119905ℎ119890119903119908119894119904119890(5)

where the parameter 120583119894 isin [0 05] controls for the strengthof the attraction and repulsion in social influence and 119879119894and 119880119894 are the latitudes of rejection and acceptance for agent119894 respectively The parameter 120583119894 may be thought of as therelative importance agent 119894 gives to the opinions of her peersor analogously the weight she gives to her own opinionwhen updating 120583 lt 05 means that agents will never givemore weight to the opinions of other agents than to their

own (egocentric bias) When two agents 119894 and 119895 discuss iffor instance 119895 advances an informational cue (argumentpersuasion tactic) and it happens to be close enough to119894rsquos opinion anchor then 119894 shifts her opinion towards thedirection of 119895rsquos informational cue The symmetric influencefrom agent 119894 to agent 119895 takes effect in the same way (seeFigure 6)

We add to the dyadic dynamics a rule of encounters ateach step of pair-wise interactions each agent meets exactlyone other agent at random This rule may be associated withrandom day-to-day encounters among agents Steps of socialinfluence correspond to nondebate steps (119889) in decision-making processes

252 Deliberative Opinion Dynamics We define an opinionupdate equation that links the proposal arguments that areadvanced during deliberation and opinions We combine theuncertain and probabilistic nature of the effect of delibera-tion have it a moderating [15 16] or polarizing [14] one witha mechanism similar to the one in social judgment theorybased on the distance between different informational cuesand opinion anchors [32] The probabilistic modeling of theopinion update can be related to deliberation encouragingopen-mindedness11 during collective discussion [18] Fromthis choice it follows that deliberated cues can affect even themost extreme of agents as opposed to some classic opiniondiffusion models (eg [5 8]) where once agents becomeextremists they may no longer become moderate

LetP be a proposal V119905119868 the proposal argument 119868rsquos level ofsupport for a principle P and 119900119905119894 an agent 119894rsquos opinion at time119905 Then given the difference 120575119905119894 = (V119905119868 minus 119900119905119894)2 we define 119894rsquosprobability of being attracted to a decisionrsquos informational cue119868 by 119909119905119894 = 119901|120575119905119894 |119886 where 119901119886 denotes a general probability param-eter that characterizes how important deliberated results arefor the group The parameter may also be interpreted asthe grouprsquos tendency to be swayed by a decisional majoritySimilarly we define 119894rsquos probability of being repulsed from adecision informational cue 119868 by 119910119905119894 = 1199011|120575119905119894 |119903 where 119901119903 denotesa general probability parameter that characterizes the grouprsquosdislike of deliberated results 119901119903 can also be thought of as thegrouprsquos distrust of the system symbolized by deliberation anddemocracy

Let L119886119887(119868) denote the epistemic status of the proposalargument 119868 at the end of a decision process over P everyagent 119894 updates her opinion as follows

(i) IfL119886119887(119868) = 119894119899

119900119905+1119894 =

119900119905119894 + 2120574119894120575119905119894 119908119894119905ℎ 119901119903119900119887119886119887119894119897119894119905119910 119909119905119894119900119905119894 minus 2120574119894120575t119894 119908119894119905ℎ 119901119903119900119887119886119887119894119897119894119905119910 119910119905119894119900119905119894 119908119894119905ℎ 119901119903119900119887119886119887119894119897119894119905119910 1 minus 119909119905119894 minus 119910119905119894

(6)

(ii) IfL119886119887(119868) = 119900119906119905 119900119905+1119894 = 119900119905119894where 120574119894 isin [0 05] is the strength of repulsion and attractionin the dynamicThemeaning of 120574119894 is analogous to the one of120583119894in the dyadic interactionsmodel but for the observation that 119894

Complexity 11

weights her opinion to an argument rather than to an opinionThe interpretation of this dynamics is straightforward Ifthe deliberated proposal argument is close to an agentrsquosopinion then it is very likely that the agent shifts its opiniontowards it Please note that the probability 119909119905119894 (119910119905119894 ) that anagent is attracted (repelled) to (from) an accepted deliberatedproposal argument is bounded below (above) by 119901119886 (119901119903)(equivalently 119909119905119894 isin [119901119886 1] 119910119905119894 isin [0 119901119903]) Steps of deliberativeopinion dynamics are associated with debate steps in which aconcluding collective decision is made

253 Mixed Opinion Dynamics Opinion dynamics in thecontext of decision-making processes can be best described asa combination of the two preceding dynamics In this opiniondynamics model agents are engaged in a democratic systemin which they deliberate vote for proposals and occasionallydiscuss with one another on the principle supported by theproposalsThe opinion dynamics for an agent 119894 can be writtenas a sequence of dyadic and deliberative opinion updates

119900119905+1119894 =

119864119902119906119886119905119894119900119899 (5) 119894119891 119905 equiv 0 (119898119900119889119905119863) and 119904119903 (sdot) = 119891119886119897119904119890119864119902119906119886119905119894119900119899 (6) 119894119891 119905 equiv 0 (119898119900119889119905119863) and 119904119903 (sdot) = 119905119903119906119890119900119905119894 119900119905ℎ119890119903119908119894119904119890

(7)

where 119904119903(sdot) is the decision processrsquos stop rule 119905119863 the numberof time steps between each deliberation step Equation (5)describes the dynamics for the dyadic interactions and Equa-tion (6) posits the changes in opinions due to the deliberatedcues Equation (5) applies when agents are not deliberatingand when each agent encounters another agent to exchangeinformation that may lead to local opinion updates Whenthere is deliberation a handful of agents deliberate and ifthey reach the stop condition imposed by the table all agentsvote for the proposal They update (see Equation (6)) theiropinions on the basis of the proposal argumentrsquos support forthe principle and the result of the scrutiny Otherwise thereis no change in the opinions of the agents A vote indicatesthe end of the decision-making process

3 Experiments and Calibration

In this section we introduce the metrics that enable us toobserve the simulations and characterize the calibration ofthe model

31 Observations and Initialization In this section wedescribe the simulations and the protocol used to test andobserve the results of our model We introduce the metricsof interest explain the calibration of the model describesome results obtained from the simulation data used forcalibration and conclude with a brief discussion on theexpected outcomes of the mixed opinion dynamics model

311 Metrics or Statistics of Interest Let 119878 denote the end ofa simulation We are interested in the effect of deliberationand of the modelrsquos procedural parameters on the followingmetrics or statistics

(i) Variance of opinions (119881119886119903(119900)) the variance ofopinions at time 119878 Since opinions live in the unionof the positive and negative unit intervals 119881119886119903(119900) isin[0 1] The higher the variance of the distribution isthe more ldquodiverserdquo the opinions are(ii) Proportion of extremists in the population(119901119903119900119901119890119909) the proportion of agents in the populationwith opinions such that |119900119894| ge 075 A high proportionof extremists makes ldquohealthyrdquo consensus12 difficult toreach and deliberation more or less informative(iii) Shifts of opinions (119878ℎ) [33] statistic that measuresthe aggregated change in individual opinion at time 119878with respect to aggregated individual opinion at thebeginning of the simulation

119878ℎ = 2sum119894isin119873 10038161003816100381610038161199000119894 minus 1199001198781198941003816100381610038161003816max119894isin1198731199000119894 minusmin119894isin1198731199000119894 (8)

119878ℎ is positive and bounded above by 2|119873| since 119900119894 isin[minus1 1] A low shift statistic implies that the processhas a small impact on opinions(iv) Judgment or consensual inaccuracy (119890119888) consen-sual accuracy of a group consists of an ad hoc statisticmeasuring a grouprsquos ability to infer correct labels forproposal arguments from a decision-making processgiven the ideal consensual labeling based on full-informationL120590

120576 (A) We use a Hamming-based dis-tance on labelings [34] to define the statistic

119890119888 = 1|I| sum119868isinI1198861198681003816100381610038161003816L119886119887120576 (119868) = L119886119887 (119868)1003816100381610038161003816 (9)

whereI is the set of all discussed proposal argumentsand 119886119868 = 12 ifL119886119887120576(119868) = 119906119899119889 or ifL119886119887(119868) = 119906119899119889119886119868 = 1 otherwise119890119888 lives in the interval [0 1] An inaccuracy statisticclose to 1 indicates that agents subdued to a particulardecision-making process make many mistakes injudging the labels of the proposal arguments Notethat when there is voting all proposal arguments arelabeled either 119894119899 or 119900119906119905 after deliberation(v) Coherence (119894119903) letL119886119887119863(119868) be the label obtainedfor 119868 from the deliberation process without votingThe coherence statistic measures how well votingresults adjust to results obtained during deliberationonly We use the proportion of arguments that havebeen labeled 119894119899 in the debate and that agents havevoted favorably for

119894119903 = 1003816100381610038161003816119868 isin I | L119886119887119863 (119868) = 119894119899 andL119886119887 (119868) = 11989411989910038161003816100381610038161003816100381610038161003816119868 isin I | L119886119887119863 (119868) = 1198941198991003816100381610038161003816 (10)

The coherence statisticrsquos domain is [0 1] If after 119872debates no deliberated central argument has beenlabeled 119894119899 or voting is not part of the procedure ofcollective decision-making then the statistic equals

12 Complexity

1 or said differently agents are perfectly coherentThis comes for the fact that if the central argumentis labeled 119900119906119905 then it is not even eligible for voting iflabeled 119906119899119889 then the debate will always be coherentwith the preferences of the agents since the result willbe a simple aggregation of their votes If the statisticequals 0 then none of the proposal arguments labeled119894119899 are voted in favorably by the agents It follows that ahigh coherence statistic implies that when agents votefor acceptable proposal arguments the results of thescrutiny reflect the consensual rationality expressedin the deliberation process

312 Parameters of Interest We recall the parameters ofinterest in our study that are linked to the structure of thedecision-making processes

(i) 119898 the minimum number of time steps in which adebate or a scrutiny occurs in a decision-making pro-cess before a final deliberated decision is submitted toa vote

(ii) 119899119863 the number of agents that deliberate as a propor-tion of the population

(iii) 119905119863 the number of time steps between debates inwhich pair-wise interactions among agents mayoccur

(iv) 120572 the proportional majority requirement for theacceptance of a proposal When 120572 = 0 it stands forno voting ldquoany proposal argument that is labeled 119894119899during deliberation is acceptedrdquo

Please recall that in terms of the definition of a decision-making process each parameter controls either the length orthe content of the sequence D119868 (119905119863 119898) or the rules that areapplied during the debate steps (120572 119899119863) (refer to Figure 4)313 Initialization All |119873| = 400 agents start off with anopinion 119900119894 drawn from a uniform distribution U(minus1 1) 13Given 119900119894 every agent 119894 randomly draws a setA119894 (|A119894| = 119896) ofarguments from a balanced14 argument pool A of119872 = 600nonneutral (V119886 = 0) arguments on the basis of 119900119894 if |119900119894| lt 025(the opinion is moderate) then agent 119894 randomly fills halfof her argument sack with arguments such that V119886 lt 0 andthe other half with arguments such that V119886 gt 0 (120601119898119894 = 05)Otherwise she randomly fills 120601119890119894 ge 05 of her sack witharguments such that V119886 and 119900119894 are of the same sign and theremaining (1 minus 120601119890119894 ) with arguments of the opposite sign15

Like for the opinion of agents every argument 119886rsquosadherence to the principle P V119886 is drawn from a uniformdistribution U(minus1 1) The attack relation that gives birth tothe ideal argumentation framework (119860119865120576) is established onthe basis of the V119886rsquos and the argumentsrsquo epistemic reach 119890119903

Let 120579(119886 119887 119903) = (|V119886 + V119887|2)119903 be an auxiliary positivereal-valued function that takes two arguments (119886 119887) and apositive real number (119903 gt 0) as inputThe probability that anyargument 119886 isin A creates a link to any other argument 119887 isin A

is given by the equation 119901 = 119901120579(1198861198872)119890 and thus argumentssupporting opposing views of the principle always attack each

other We fix 119901119890 the epistemic correlation parameter to 015If 119901119890 was greater than 015 then arguments of the same signwould attack each other too often (and focused agents wouldbe rarely incited to advance favorable arguments duringdeliberation) If 119901119890 was lower then the arguments of thesame type would induce an almost empty graph which isunrealistic The number of arguments that any argument 119886can attack is bounded by 119886rsquos epistemic reach that we fix to15 A higher value of epistemic reach makes the argumentlattice too conflicting and nearly bipartite A lower epistemicreachmakes the lattice not sufficiently conflicting in the sensethat too many arguments can attack the proposal argumentrelatively to the few that can defend it If any argument attacksthe proposal argument the chances that another argumentattacks the attacker are low Hence the proposal argumentis almost never accepted and deliberative opinion updatesbecome rare

The arguments in the resulting argumentation frame-work 119860119865120576 are given a permanent labelingL120590

120576 (A) (in shortL120576) using120590 = grounded semanticsWe choose the groundedlabeling-based semantics because it provides a unique admis-sible labeling respects minimal rationality constraints whilesimplifying the model and models a skeptic approach toaccepting arguments (see [23 25 31]) as it maximizes thecardinality of the set 119906119899119889 L120576 To some extent if we considerproposals as committing because they may guide collectiveaction choosing a skeptic semantics seems reasonable insofaras it labels an argument 119894119899 or 119900119906119905 only if it has no reasonto label it 119906119899119889 For instance if a proposal is a public policythat requires large amounts of resources and engages tofuture course of action then it may also determine policycycles and heavily burden a group and its future decisionsIn important situations like these it seems reasonable tolengthen deliberative cues and ask for more demanding andldquogroundedrdquo criteria for policy argument acceptability

On the proposalrsquos side we create an argument 119868 whoseadherence to P is also drawn from aU(minus1 1) and we label itL119886119887(119868) = 119906119899119889 119868 is interpreted as being the main argumentjustifying the discussed proposal P and V119905119868 as its supportfor P at time 119905 119868 cannot attack other arguments but otherarguments can attack it For each argument 119886 isin A a directedarc or attack from 119886 to 119868 is activated with probability 119901119901 =003120579(11988611986802) When 119901119901rsquos lower bound is too high the proposalargument is almost always defeated and thus deliberativeopinion updates do not happen When 119901119901rsquos lower boundis lower than 003 then the opposite occursmdashthe proposalargument is always accepted due to an absence of attackstowards it Finally we set the maximum number of debatesto119898 = 7 for the decision-making processes stop rules1632 Calibration and Simulation Protocol In this subsectionwe discuss the calibration of the models the terminationconditions for runs in the deliberative and mixed model andthe expected outcomes in terms of the observations

321 Calibrating Dyadic Opinion Dynamics Dyadic opiniondynamics correspond to a space of parameters in whichargumentation and deliberation spaces are not taken into

Complexity 13

Table 1 Parameters and parameter types and their domains used to compare dyadic collective and mixed opinion dynamics

Length ofD119868 Decisional in119863119868 Social influence Deliberation influence Cognitive119905119863 119898 119898 119899119863 120572 120583 119879 119880 119901119886 119901119903 120574 119896 119861119890ℎ119886V1198941199001199031 3 6 1 3 6 7 001 002 005 0 05 066 01 14 18 02 06 03 05 01 005 01 02 4 8 16 119873119886119894V119890 119865119900119888119906119904119890119889account Deliberated arguments and informational cues haveno effect on agent opinion but agents still vote for theproposal argument17 We use and calibrate this model forcomparability in terms of all our metrics since they arenot explicitly observed in [9] a reference opinion dynamicsmodel Furthermore for simplicity and comparability againwe suppose that agents are homogeneous in terms of theirattitude structures and opinion weightings We fix 120583119894 =120583 119879119894 = 119879 and 119880119894 = 119880 for all 119894 isin 119873 and for some triplet(120583 119880 119879) isin [0 05] times [0 2] times [119880 2]

Experiments suggest that the strength of the dyadic inter-actions is an explanatory factor of the time of convergence(120583 lt 05) and of the dyamicsrsquo steady states Bigger values of 120583speed up convergence towards a stable set of opinion clustersand affect the size and the relative position of these clustersin the opinion distribution 120583 however does not play a majorrole in the number of opinion clusters observed at the steadystate of the dynamics

We set 120583 to the value given to it in Jagger et alrsquos opiniondynamics model (120583 = 01) [9] For one it is the smallestreal number for which most results found in [9] hold and thedynamics are smooth secondly it seems to be a reasonableldquostrengthrdquo of influence considering that we would not wantpair-wise social interactions to completely shadow or maskan eventual effect of deliberation in the opinion dynamics

For different values of attitude structures 119879 and 119880 weget exactly the same results as in [9] in terms of the numberand density of the opinion clusters The different couplesof values of 119880 and 119879 model the size of the latitude ofnoncommitment (119879 minus 119880) and thus the agentsrsquo propensityto update opinions Whenever 119879 minus 119880 le 02 regularitiesin convergence and opinion clustering appear If 119880 gt 1there is always central convergence if 119879 lt 1 there is alwaysbipolar convergence with clusters of similar sizes For 119879 minus119880 gt 02 relatively low values of 119879 (119879 lt 1) and relativelyhigh values of 119880 always yield extreme bipolar convergencewith a relatively small cluster of moderate agents (bipolar-central convergence) For 119879 sufficiently high (119879 ge 12) and119880 sufficiently low (119880 le 08) in other words not very ego-involved agents there is a bigger spectrum of steady statesFor this reason we studied the metrics of interest for thesevalue differences and kept the couples of (119880 119879) depicted inTable 1 For these couples central bipolar bipolar-centralconvergence (3 clusters) and multicluster convergence (4 or5 clusters) are possible and result frommeaningfully differentattitude structures Judgment accuracy varies significantlyacross the four different attitude structures we retain and soit happens with the variance of opinions These scenariosprovides us with reference results from which to build upon

322 Calibrating Deliberative Social Interactions Delibera-tive social interactions correspond to social situations in

which individuals are not influenced by pair-wise discussionwith peers but are sensitive to collectively deliberated infor-mational cues On these terms parameters such as the sizeof agentsrsquo argument sacks (119896119894) the distribution of argumentsin them (120601119894) the minimum number of debates (119898) andthe proportion of debaters from the population (119899119863) are nolonger mute We assume for simplicity that the formulaexist120601119890exist119896forall119894(119896119894 = 119896 and 120601119890119894 = 120601119890 and 120601119898119894 = 05) is true with 119894 isin 119873for some value 120601119890 isin [05 1] and for 119896 isin N

The chosen domains for these parameters (see Table 1)are justified by the size of the population and the size of theargument pool (119872 = 600) which we set from the start If 119896is too big (119896 ge 32) andor too many agents are allowed inthe debate (119899119863 ge 02) then individuals always disprove thecentral argument and deliberation is never or rarely takeninto account in the update of opinions Similar effects occurwhen 119898 is high yet we only calibrate it with respect to themean running time of a simulation and the values of 119905119863Different distributions of 120601119890119894 seem to significantly or directlyexplain none of the statistics we analyze so we convenientlyset 120601119890119894 = 075 We chose the parameter 119896 to be always divisibleby 4 for it is convenient given the initial conditions regardingthe argument distribution in the agentsrsquo argument sacks

We fix the value space for the parameters 120574 119901119886 119901119903 as inTable 1 to account for different intensities of the effect ofdeliberative voting retroaction on the system For examplethe scenarios where the three parameters are at their lowestvalues correspond to the scenarios with the smallest fixedprior effect of deliberation in the model Increasing any ofthese parameters should mechanically be associated witha world in which agents are more sensitive to collectivedecisions The values for 119901119886 are taken from [13] wherethe author states that by means of deliberation 7 to28 of individuals changed their opinion from agreeingto disagreeing or vice versa on a referendum question forDenmarkrsquos participation in the EuroWe choose two differentvalues for 119901119886 that account for high (03) and very high (05)sensitivity to deliberation in a group 119901119903 on the other handis taken to be small (equal to 01) since we believe that thereexist agents that will always go against the reached consensusTheir numbers are meager

For simplicity we assume that for all 119894 isin 119873 120574119894 = 120574 forsome 120574 isin [0 05] In effect we posit that the heterogeneityof agents in the deliberative context only derives from theirarguments and their opinions We decide to calibrate 120574 inallusion to the strength of the social dynamics in pair-wiseinteractions In doing so we limit ourselves to consider threedifferent scenarios in which deliberation has respectivelyhalf equal and twice as much opinion-shifting power thanone-to-one social influence Finally we include the accept-ability voting quota rule 120572 whose domain is inspired fromclassical majority rules [35] observed empirically (Table 1)

14 Complexity

Table 2 Parameters and their domains used to study the effect of the procedural parameters on the metrics

Procedural parameters Other parameters119905119863 119898 119899119863 120572 119898 120583 119879 119880 119901119886 119901119903 120574 119896 119861119890ℎ119886V1198941199001199031 2 3 4 5 6 1 2 3 4 5 001 002 003 004 005 0 05 066 7 01 16 02 03 005 02 12 119873119886119894V119890 119865119900119888119906119904119890119889

323 Calibrating the Mixed Social Interactions Model Themixed interaction model ascribes to a parameter space inwhich the effect of collective choices on our metrics isnontrivial and where deliberation and voting on proposalsdetermine their acceptability Since the mixed model isequivalent to periodic iterations of pair-wise and collectivesocial interactions the calibration of the parameters in bothpreceding models holds (see Table 1 for the calibration)The first reason for this is comparability the second is thatthe previous calibration takes into account the fact thatboth dynamics are going to be combined We include thefrequency of debates by adding the parameter 119905119863 whichcontrols for the amount of pair-wise interaction amongagents between two distinct deliberation steps

324 Termination Conditions for Runs Simulations stoponce 100 proposals are deliberated on andor voted for inother words after 100 collective decision steps have occuredThe number of proposals discussed seems arbitrary yet it ishigh enough to observe the effects of deliberation on opiniondistributions and on other metrics related to coherence andjudgment accuracy We choose the number of runs as afunction of our research questions which give relevance tothe procedural parameters in the mixed model rather than tothe parameters set to describe the population and its behavior

325 Expected Outcomes of the Simulations We expect moredeliberation in terms of higher 119898 and 119899119863 to increasejudgment accuracy and at the same time to reduce thevariance of opinions In turn a smaller variance of opinionsimplies that the argument pool for deliberation is smaller andtherefore judgment accuracy should be lower Moreovervariance increasing dyadic interactions (or rejection) shouldalso increase judgment accuracy and coherence since bipo-larization and dissensus foster argument diversity in delibera-tionThe coherence statistic should be stronger in simulationsin which central convergence appears quickly and agentsare naive Attitude structures sensitivity to informationalcues and weights given to deliberated cues that makes pathsto bipolar convergence smaller or to central convergencelonger should be associated with higher judgment accuracyShifts of opinion should be more visible in scenarios inwhich extreme agents are pulled away from the extremesHence the sensitivity to deliberated cues (119901119886 119901119903) and 120574should explain the shifts When crossed with high values of119880 and 119879 the shifting power of deliberation should be at itshighest

In the end we do not know how these mechanismswill play out The results of the subsequent experimentsgive us insight on the interplay between the aforementionedeffects

4 Results

Before pointing at any result obtained from the mixedinteractions model we describe and compare the dyadicand deliberative interactions models in the parameter spaceobtained from the calibration in Section 3 (see Table 1)Primarily we require our results to describe two orthogonaltypes of opinion dynamics the pair-wise dyadic and thedeliberative The latter comprises scenarios in which indi-viduals do not influence each other by means of pair-wisediscussion and only update their opinions from delibera-tion the former scenarios where only pair-wise interactionsamong agents determine the dynamics We describe both onthe parameter space given in Table 1 by performing at least20 simulations per scenario Quantitative results from thepair-wise interaction model alone are described during thecalibration and are not treated heremdashdeliberative parametershave no effect on it Moreover we show how these differentmodel specifications yield qualitatively different opiniondistributions and compare them on the basis of the metricsof interest To this end we perform independent two-sampleStudentrsquos t-tests18 or Welch t-tests19 and compute confidenceintervals at a 095 level of confidence Last but not least wecomment ordinary least squares (OLS) regression estimatesto account for the direction and magnitude20 of the effectsof the parameters on the metrics Estimates are declaredsignificant at a 5 level of risk

We perform the same analysis on the mixed interactionsmodel as we compare it to the pair-wise dyadic and deliber-ative interactions models and discuss the marginal effects ofthe parameters on the metrics More precisely we obtain twodifferent types of results for the mixed interactions modelthe first compares the mixed scenarios to their monolithiccounterparts with respect to each metric and with respect tothe regression estimates the second gives a clear idea of themarginal effect of each governance parameter (or parameterof interest) on our observations In the first we allow controland procedural parameters to vary on a parameter spacesimilar to those of the dyadic and deliberative opiniondynamics (see Table 1) In the second we fix all of ourcontrol parameters execute 36000 balanced21 runs andfocus only on the one-way pooled effects of our proceduraland behavioral (all agents are focused or naive) parametersWe generate comprehensive graphs and tables to account forthe obtained results

Please notice the slight change in the size of the parameterspace for the procedural parameters and in the values ofthe nonprocedural ones for the first and second type ofresults (refer to Tables 1 and 2 respectively) Since we wantto have a finer idea of the marginal effects of each of theprocedural parameters on how well a group decides onproposals we make the value jumps for each parameter (1)

Complexity 15

() (U T) = (02 18) (b) ( pa m nD) = (005 05 00 2 02) (＝) ０ＬＧＮＬＭ ＣＨ (＜) (＝) Ｈ＞ tD = 3

(d) (５ ４) = (02 14) (e) ( pa m nD) = (01 03 05 3 02) () ０ＬＧＮＬＭ ＣＨ (＞) () Ｈ＞ tD = 6

Figure 7 From left to right common opinion trajectories and distributions for dyadic deliberative and mixed opinion dynamics within thestructure of different decision-making processes

small enough to detect significant differences in the estimatesand (2) meaningful for interpretation For the fixed one-valued parameters in Table 2 we use Table 1 to set them totheir minimum mean or median values

41 Comments on Dyadic and Deliberative Opinions Dynam-ics In this subsection we present our first results regardingthe deliberative opinion dynamics described in Equation (6)and its differences with the dyadic dynamics (Equation (5))in terms of our observations

411 Qualitative Analysis of Pair-Wise Opinion DynamicsPair-wise opinion dynamics alone produces multiclusterstable convergencewith variable cluster sizes (see Figures 7(a)and 7(d)) Insight on the complexity of these dynamics wasdiscussed in Section 321

In Figures 7(a) and 7(d) agents discuss randomly withone another in pairs Clusters in the extremes form quicklysince either agents that are close to the extremes attractone another or are convinced to stay close to the extremesby interacting with other agents they disagree with Otheragents with near-extreme positions may be simply attractedto one of the several moderate foci of agents in the opiniondistribution On the other hand moderate agents are eitherattracted to the closest opinion focus or pushed towards theextremes of the distribution Over time agents in the centralfocus either attract other agents into it or ignore the opinions

of the extreme agents that interact with them The result ofthese dynamics yields amulticlustered convergencewhere theopinion foci are at least at distance 119880 from one another andonly the extreme foci (clusters of agents with |119900119894| ge 075) areat a distance greater than or equal to 119879 from one anotherThefoci at the bounds of the opinion distribution are very stableand in analogy to the (12119889)-rule in assimilation bounded-confidence models [33] the number of clusters that form isroughly equal to 119880minus1412 Qualitative Analysis of Deliberative Opinion DynamicsQualitatively deliberative opinion dynamics yield a looseunipolar convergence of opinions near the center (centralconvergence see Figure 7(b)) near the center of either the leftor right portions of the distribution (group polarization seeFigure 7(e)) or to a sparse bipolar opinion distribution withclusters at the center and at the extremes (see Figure 7(c)) ifmixed with pair-wise interactions This is probably a reasonto expect the variance of opinions and the proportion ofextremists (almost perfectly correlated 120588 = 089 119901 lt 0001)to be low for these scenarios The side towards which theopinion cluster skews depends essentially on the valence ofthe first argument that is collectively accepted

In Figures 7(e) and 7(b) agents that update their opin-ions do so at the same time and only when deliberationis successful In these scenarios every time a decision iscollectively accepted agents update their opinions towards

16 Complexity

Table 3 Ordinary least squares estimates for pair-wise interactions scenarios for most of the observed metrics

Dependent variable119881119886119903(119900) 119878ℎ 119890119888 119894119903(1) (2) (3) (4)Constant 0610lowastlowastlowast 4429lowastlowastlowast 0472lowastlowastlowast 0000

(0002) (0206) (0003) (0000)

119880 = 06 (119880 = 02) minus0012lowastlowastlowast 2981lowastlowastlowast 0061lowastlowastlowast 0000(0002) (0260) (0004) (0000)

119879 = 18 (ref 119879 = 14) minus0264lowastlowastlowast minus3261lowastlowastlowast 0004 0000(0002) (0260) (0004) (0000)

120572 = 066 (ref 120572 = 05) 0002 minus0059 minus0100lowastlowastlowast 0000(0002) (0184) (0003) (0000)

(119880 119879) = (06 18) (ref (119880 119879) = (02 14)) minus0324lowastlowastlowast minus3728lowastlowastlowast minus0033lowastlowastlowast 0000(0003) (0368) (0006) (0000)

Observations 1440 1440 1440 1440R2 0983 0389 0529Adjusted R2 0983 0388 0527Residual Std Error (df = 1435) 0032 3492 0053 0000F Statistic (df = 4 1435) 21 272740lowastlowastlowast 228686lowastlowastlowast 402326lowastlowastlowastNote lowastplt01 lowastlowastplt005 lowastlowastlowastplt001

the argumentrsquos defended position Hence if there is votingand if agents happen to obtain 119894119899 proposal arguments theyinexorably cluster towards one or the other side of the opiniondistribution resulting in opinion convergence remeniscent ofgroup polarization (see Figure 7(e)) Furthermore as agentscluster in the same side of the opinion spectrum the argu-ments that may be advanced in deliberation are fewer andmore skewed towards that one side of the opinion spectrumAgents become more easily persuaded in deliberation andonly by arguments supporting one side of the spectrumConvergence of opinions towards one loose opinion focusbecomes faster and more certain If there is no vote (seeFigure 7(b)) the dynamics are similar but for the fact thatit is the uniform randomness of the proposal argument thatguarantees a central convergence of opinions

413 Quantitative Analysis of Pair-Wise Opinion DynamicsThe three parameters that allow for quantitative analysis are119880 and119879 the agentsrsquo attitude structure and themajority quotavoting rule 120572 Table 3 supports the claim that going from 120572 =05 to 120572 = 066 does not affect the variance nor the shifts ofopinions However judgment accuracy is significantly higherwhen accepting proposals becomes more difficult This israther surprising but also intuitive since decisions that aretaken using more restrictive methods of scrutiny should bemore accurate

Concerning the attitude structures the interaction termbetween 119880 and 119879 shows that 119880 has a small effect on thevariance of opinions given that 119879 is low This shows thateven when the assimilation threshold is high a sufficientlysmall rejection threshold can make its one-way effect vanishHowever high levels of both 119880 and 119879 reduce the variance ofopinions dramatically In contrast when it comes to overallchanges in opinion the magnitude of 119880rsquos one-way marginal

effect increases the shifts of opinions whereas 119879 lowers itThis can be interpreted as having the rejection dynamics pushmoderate-to-extreme agents to positions that are close tothe extremes and very moderate agents to the center of thedistribution Not too many quantitative changes (on average)can therefore be observedThe fact that higher119880 is associatedwith faster convergence of opinions seems to imply lowerjudgment accuracy At some point voting for the same type ofproposals for too longmay result inmany collectivemistakesThis is an interesting conclusion to consider in the analysis ofthe mixed model Finally the interaction term shows that 119879has no strong impact on the metric

414 Quantitative Analysis of Deliberative OpinionDynamicsSeven parameters can be considered for quantitative anal-ysis in this dynamics the parameters of interest recalledin Section 312 the size of the argument sacks (119896) agentbehavior during deliberation the strength of the deliberationdynamics (120574) and the measure sensitivity to deliberatedresults (119901119886) The most ldquounexpectedrdquo result obtained fromthe deliberation dynamics is that it shows how deliberationcontributes to the proportion of extremists and the varianceof opinions It happens that having more agents discussproposals and having more of these discussions increasesthe variance of opinions and the proportion of extremistswhile lowering judgment inaccuracy and coherence For theformer the fact that far too many arguments are advancedduring deliberation plummets the chances that the centralargument is accepted hence deliberation does not moveopinions too often For the latter we observe that only ldquobigrdquodifferences (going from the lowest value of a parameter to thehighest) in protocol actually prove to have a preponderantimpact on the metrics These observations hint at a possibletrade-off between judgment accuracy and the variance of

Complexity 17

Table 4 Ordinary least squares estimates for the deliberative interactions scenarios for most of the observed metrics

Dependent variable119881119886119903(119900) 119878ℎ 119890119888 119894119903(1) (2) (3) (4)Constant 0032lowastlowastlowast 13092lowastlowastlowast 0370lowastlowastlowast 1007lowastlowastlowast

(0001) (0406) (0001) (0001)

119898 = 3 (ref119898 = 1) 0012lowastlowastlowast minus2062lowastlowastlowast minus0012lowastlowastlowast minus0001(0001) (0276) (0001) (0001)

119898 = 6 (ref119898 = 1) 0023lowastlowastlowast minus3285lowastlowastlowast minus0018lowastlowastlowast minus0004lowastlowastlowast(0001) (0276) (0001) (0001)

119899119863 = 002 (ref 119899119863 = 001) 0007lowastlowastlowast minus0965lowastlowastlowast minus0006lowastlowastlowast minus0002lowastlowast(0001) (0276) (0001) (0001)

119899119863 = 005 (ref 119899119863 = 001) 0020lowastlowastlowast minus3316lowastlowastlowast minus0015lowastlowastlowast minus0004lowastlowastlowast(0001) (0276) (0001) (0001)

120572 = 05 (ref 120572 = 0) 0146lowastlowastlowast 68932lowastlowastlowast 0068lowastlowastlowast minus0507lowastlowastlowast(0001) (0276) (0001) (0001)

120572 = 066 (ref 120572 = 0) 0291lowastlowastlowast minus13842lowastlowastlowast 00004 minus0000(0001) (0276) (0001) (0001)

119896 = 8 (ref 119896 = 4) 0004lowastlowastlowast minus0815lowastlowastlowast minus0002lowastlowastlowast minus0001(0001) (0276) (0001) (0001)

119896 = 16 (ref 119896 = 4) 0008lowastlowastlowast minus1118lowastlowastlowast minus0004lowastlowastlowast minus0003lowastlowastlowast(0001) (0276) (0001) (0001)

119861119890ℎ119886V119894119900119903 = ldquoFocusedrdquo (ref ldquoNaiverdquo) 0015lowastlowastlowast minus1980lowastlowastlowast minus0012lowastlowastlowast minus0004lowastlowastlowast(00005) (0225) (00004) (0001)

120574 = 01 (ref 120574 = 005) minus0020lowastlowastlowast 5762lowastlowastlowast minus00003 minus0001(0001) (0276) (0001) (0001)

120574 = 02 (ref 120574 = 005) minus0025lowastlowastlowast 8648lowastlowastlowast minus00003 minus0001(0001) (0276) (0001) (0001)

119901119886 = 05 (ref 119901119886 = 03) minus0013lowastlowastlowast 1581lowastlowastlowast 00004 00003(00005) (0225) (00004) (0001)

Observations 19440 19440 19440 19440R2 0931 0844 0562 0958Adjusted R2 0931 0844 0562 0958Residual Std Error (df = 19427) 0033 15701 0030 0050F Statistic (df = 12 19427) 21 844790lowastlowastlowast 8 733563lowastlowastlowast 2 079585lowastlowastlowast 36 761250lowastlowastlowastNote lowastplt01 lowastlowastplt005 lowastlowastlowastplt001

opinions that may be of interest in the mixed discussionmodel

Group coherence shifts of opinions and the varianceof opinions are highly affected by the voting thresholdthe coherence metric is reduced by half when allowing forclassic majority voting shifts explode and the variance ofopinions increases dramatically This is not surprising sincethe majority quota rule works as a buffer to the success ofdecision-making processes and in consequence it hampersthe deliberative opinion updates For the shifts of opinions itis clear that the two-third majority voting rule results in littleopinion change and that the simple majority rule inducesgroup polarization As expected there are less extremists inscenarios where there is no voting and the shifts of opinionare more likely to happen when agents are more sensitiveto deliberation When we include voting in the system

the grouprsquos judgment accuracy decreases due to the correctdeliberated decisions that are vetoed by a salient opinion-agent majority Strangely the effect only holds for the simplemajority decision rule Lastly the proportion of individualsthat participate in the debate and the minimal amount ofdebates have similar marginal effects on all the metrics (seeTable 4)

415 Comparing Deliberative and Pair-Wise Opinion Dynam-ics in a Nuthshell Qualitatively both models yield similaroutcomes but for the fact that deliberation never producesopinion polarization with both types of extremists (seeFigure 7) Furthermore clusters in the deliberative interac-tions model are less homogeneous in size and multiclusterconvergence occurs only when deliberation is rarely suc-cessful which happens sporadically given the calibration of

18 Complexity

Table 5 Mean difference 095 confidence intervals for metrics by model comparison

Model Metric119881119886119903(119900) 119901119903119900119901119890119909 119878ℎ 119890119888 119894119903Dyadic vs Deliberative [0191 0216] [0170 0197] [-2948 -2828] [0074 0082] [0165 0172]Mixed vs Deliberative [0142 0147] [0150 0155] [-2085 -1976] [0003 0002] [-0038 -0030]Mixed vs Dyadic [-0072 -0046] [-0045 -0018] [8332 8859] [-0079 -0071] [-0205 -0201]the model On the other hand pair-wise opinion dynamicsproduces multicluster convergence most of the time andcentral convergence only in the infrequent cases where thelatitude of acceptance is large

The comparison of these two scenarios in terms ofthe metrics is straightforward Whilst dyadic interactionsallow for more variability in opinions through pair-wiseinteractions and more mistakes in judging proposals delib-erative updates simply act as an antagonist force On aver-age the deliberative interaction model produces opiniondistributions with smaller variance and yields steady stateswith higher judgment accuracy a higher shift statistic andnaturally lower group coherence since only proposals that aredeliberated 119894119899 are eligible for voting (see Table 5)

42 Articulating Dyadic and Deliberative Social Interactionsin Opinion Dynamics So far we have considered scenarioswhere argumentation had no place and where individualsmade decisions according to opinions that were purelyconstructed from pair-wise discussions We also observedcases in which agents formed their opinions only by inte-grating deliberated proposals into their opinion updates Thescenarios we present in this section combine the two pre-ceding scenarios to account for both the effects of pair-wisediscussion and the effects of deliberation on opinions judg-ment accuracy and group coherence From the precedingobservations we conclude the following deliberative opiniondynamics reduces the variance and the proportion of extrem-ists because it shifts opinions greatly and polarizes groupsby eliminating at least one type of extremist the one whoseopinions are initially opposed to the deliberated resultsor it brings opinions close to neutrality (see Figure 7(b))This may not be desired in a group since the diversity andstability of opinions are necessary for deliberation to makesense and lead to the collective acceptance of controversialproposals Dyadic opinion dynamics on the other handtend to increase the variance of opinions and the numberof extremists while having a lesser aggregated effect on theshifts of opinion This is an immediate result of the pair-wiseinteraction dynamics inasmuch as it polarizes individualsvery quickly

To sum up in the mixed interactions model deliberationhinders the effect of voting on opinions by making someproposals that would normally be submitted for voting noteligible for voting It may also undo opinion changes dueto pair-wise interactions The mixed model can be seen asthe deliberation opinion dynamics model in which pair-wiseinteractions occur in between collective discussion and mayaffect which arguments are advanced during deliberationEquivalently it can be seen as the dyadic opinion dynamics

model in which deliberation accounts for endogenousldquoshocksrdquo on the opinion distribution

421 Qualitative Analysis of the Mixed Dynamics In Figures7(c) and 7(f) agents update their opinions through delibera-tion and pair-wise interactions Opinion trajectories visualsshow a combination of both previously described opiniontrajectories Over time as agents discuss with one anotheropinion clusters form as in the pair-wise or dyadic opiniondynamics Deliberation may disrupt the formation of theseclusters and three different situations may occur

(i) Deliberation disrupts the formation of the opinionclusters but not sufficiently to jeopardize pair-wiseopinion cluster formation (eg for high values 119905119863)Two phenomena may be responsible for it (1) thevariability of the proposal argumentrsquos support for theprinciple P and (2) too few (successive) deliberationsteps that result in opinion updates Once the opinionclusters are formed successful deliberation foreshad-ows the merging of the previously formed clustersmay bring extremists closer to moderate agents andthanks to the assimilation component of the pair-wisedynamics allow bigger and more moderate opinionclusters to form (refer to Figure 7(c))

(ii) Deliberation significantly disrupts the pair-wisedynamic formation of clusters This may happenwhen the requirements for accepting proposals areweak in the decision-making process (eg low 120572) Inthis case one has either central convergence sinceall agents follow the deliberated results in the samemanner or 3-cluster convergence with two extremeopinion foci In the latter situation the extremegroups form quickly (like in the pair-wise model) anddeliberation cannot pull extreme agents away fromthe extreme foci they are in The main reason for thisis that immediately after an extreme agent attempts toleave her foci through deliberation updates she getspulled or pushed back into it by means of a dyadicinteraction with another extreme agent

(iii) Deliberation does not disrupt the formation of theopinions cluster This is the rare case scenario forvery demanding decision-making processes in termsof inclusion (eg 119899119863) and argument acceptability (eg120572) Opinions converge as in the pair-wise opiniondynamics model (refer to Figures 7(a) and 7(d))

Last when there is voting (120572 = 0) the position ofthe moderate opinion cluster is determined by the first

Complexity 19

minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

Var(o)

sh

ec

ir

Var(o)

propex

sh

ec

ir

1

098

minus02

minus004

001

1

minus014

minus001

minus003

1

026

minus028

1

minus057 1

ＪＬＩＪ

Ｒ

Figure 8 Significant correlation estimates for themetrics of interestin the mixed opinion dyamics scenarios

accepted proposal argument andor the opinionmajority thatis formed from previous dyadic interactions (see Figure 7(f))

422 Relating the Metrics of Interest in Mixed OpinionDynamics The first thing to notice is that for all of thesescenarios all the metrics are significantly correlated (seeFigure 8) The variance of opinions and the proportion ofextremists are almost perfectly correlated To some extentthis means that a high variance of opinions known thatthe mean of opinions is statistically null equates to havingmany extremists in the population be towards one or theother side of the opinion distribution (eg extreme bipolar-ization) Similarly we find a negative and strong correlationbetween the judgment inaccuracymeasure and the coherencecoefficient (see Figure 8)mdashwhen judgment inaccuracy is high(low) coherence to deliberation is low (high) Cases likethese are telling since they may be related to sequences ofcollective decisions in which agents do not make too manymistakes when judging proposal arguments and still makedeliberated decisions that are on average in line with theirprinciples

The shift statistic interestingly is negatively correlatedwith the variance of opinions with judgment accuracy andwith the coherence coefficient In a way this means thathaving more moderate agents conduces on average to moremistakes in judging arguments and to less coherent collectivedecisions An explanation for this phenomenon could be thatas agents reach consensus in opinion important (possiblyextreme) arguments for correctly judging argument propos-als are not advanced in the deliberation arenawhich results inmore mistakes and in possibly accepting proposal argumentsthat after voting are not meant to be accepted Indeed the

negative correlation between the variance of opinions andjudgment inaccuracy points towards this direction since onaverage the more shifts there are the smaller the variance ofopinions is (see Figure 8)

423 Comparing Mixed Dynamics with Other MonolithicDynamics of Interactions Qualitatively opinion trajectoriesin the mixed model are more chaotic than in the pair-wise and deliberative interaction models (see Figure 7) Asanalyzed previously deliberation dynamics disrupt the pair-wise dynamics and create situations in which central andmulticlustered convergence (with two clusters at the extremesof the distribution) are possible at the expense of more(pair-wise dynamics) or less (deliberative dynamics) clustersIt tolerates the survival of very small isolated group ofagents that the previous models rarely tolerate in a hundredcollective decision-making steps (see Figure 7(c))

One of the most interesting observations one can makeof this family of models regards how groups change (in termsof our metrics) when dyadic social interactions are addedto deliberative interactions and vice-versa When comparingthe mixed interactions with only deliberative ones shiftsagentsrsquo aggregated change in opinions are on average lesspronounced The variance of opinions and the proportionof extremists are significantly higher for the former than forthe latter group coherence in contrast is lower with respectto the scenarios where opinions are formed only throughdeliberation Judgment accuracy follows that same trendA reason for this might be that mixed dynamics are fasterto yield semistable clusters andor opinion convergence Inconsequence the potential argument pool for deliberativeexchange shrinks faster and deliberation becomes less effec-tive over time which results in more collective mistakes Theoutcomeon coherence is predictable inasmuch as deliberativeinteractions produce group polarization If a group agrees inopinion debates are sterile because themembers of the groupwill most likely refrain from attacking the proposal argu-ments they all support and will work together to disparagethe proposal arguments they do not support (see Table 5)

In the same manner we report that the proportion ofextremists and the variance of opinions are barely lower inthe pair-wise interaction model when including deliberationin the decision-making process Shifts of opinions are alsobarely lower when there is no deliberation This can indicatethat deliberation does not undo the changes in opiniondue to pair-wise interactions nor does it amplifies them Incontrast when we consider the labeling-based metrics themixed interaction model ensures lower judgment inaccuracybecause of deliberation and lower group coherence (seeTable 5)

In sum the mixed discussion model is a compromisebetween the pair-wise interaction and deliberative opiniondynamic models in that it offers some guarantees in termsof judgment accuracy and coherence while allowing forldquoreasonablerdquo variance in the opinion pool It establishes somekind of trade-off between labeling-based metrics and thevariance of opinions that may be of interest for decision-making process design

20 Complexity

Table 6 Summary of sign or relative marginal effects for all varying parameters on dyadic deliberative and mixed interactions models inand for all experiments

Metrics Parameters119905119863 119898 119899119863 120572 119879 119880 119901119886 120574 119896 119861119890ℎ119886V119894119900119903119881119886119903(119900) XX X X O X X X X X X119901119903119900119901119890119909 XX X X O X X X X X X119878ℎ XX X X O X X X X X119890119888 XX X X XO X XOO XOO X X119894119903 XX X X O XO O X OOX XOO X XNota gray symbols correspond to dyadic dynamics light gray to deliberative dynamics and bold black to mixed dynamics Given a scenario the symbol 119883stands for constant parameter and 119874 for insignificant marginal effect and as parameters grow for positive (negative) marginal effect for the parameterFrom left to right at the parameter level the leftmost arrow corresponds to the scenario with the strongest (in absolute value) marginal effect of the parameter(with respect to to its factors) Example Let 119901 be a parameter and 119874 stands for ldquo119901 has a significant positive effect in dyadic discussion scenario negativein the deliberative one and insignificant in mixed discussion The positive effect in dyadic discussion is greater than the absolute value of the negative one inthe purely deliberative scenariordquo

424 Comments on the Differences between the Three Modelsfor OLS Regression Estimates The significance of most of theregression parameters estimated for the mixed interactionsscenarios are similar to those estimated for the pure dyadicanddeliberative scenariosNevertheless themagnitude of theeffects is different insomuch that the dynamics themselvesare different So instead of commenting the linear regressionestimates of the mixed model we compare when possiblethe standardized significant coefficients of the regressors ofthe mixed model with those of the pair-wise and deliberativeinteractionmodels (see Table 6 for a summary of the results)

In mixed interactions for instance the influence of119898 onthe opinion distribution is much more preponderant than inthe deliberative dynamics scenarios Going from 119898 = 1 to119898 = 3 increases the standard variation units of the variance ofopinions by twice as much as the one in the deliberative sce-nariosThis in particular is due to the design of the decision-making process If 119898 is higher then the number of socialinfluence steps (119905119863) in the decision process is also higher(compared to 119905119863 = 0 in the deliberative case) If we decideto consider these steps as contributing to the variance ofopinions then the effect of119898 on the variance of opinions hasto be stronger in the mixed interactions model In contrastthe number of agents participating in the debate has a weakereffect on the variance of opinions in the mixed interactionsmodel than in the deliberative dynamics specially for 119899119863 =005 (more than twice as much) The difference may comefrom the fact that in mixed discussions the part of theargument pool used in deliberation is more restrictive thanin the deliberative case We can argue that since pair-wisedynamics may get individuals closer to one another aftersome deliberation the same happens to the arguments thatindividuals will most likely use in deliberation Taking thisinto account means that deliberation will at many instancesbe unfruitful and have no effect on opinions thus failing tomoderate extreme opinions andor polarize the group In thesame direction a high number of participants in deliberationcan make the deliberation dynamics stiff In this case thedistribution of opinion is close to the uniform distributionwhich has a relatively high-to-moderate variancewith respectto the reference results

The strength of the deliberative dynamics has a positivestronger effect on the variance of opinions in themixedmodelthan in the deliberative model when 120574 = 02 and a negativeweaker one when 120574 = 01 A sound interpretation of thisresult is that when deliberation is successful and its effect onopinions does not get agents to move sufficiently away fromtheir current positions agents are pulled or pushed back totheir previously held opinions the variance of opinions doesnot changemuchOn the other hand if the effect of successfuldeliberation is strong opinions can change sufficiently tomake agents much more prone to assimilation into moderateopinion foci the variance of opinions falls

The deliberative opinion dynamics model is much moresensitive to 120572 than the mixed discussion model Differ-ences in marginal effects can double The probability ofattraction to deliberated results (119901119886) for instance has astrongmoderating effect (negative) on agentsrsquo opinions in thedeliberative opinion dynamics but a meager one (two-fold)in the mixed and dyadic dynamic models This is becauseopinion updates in the deliberative model depend only onthat parameter whereas in the mixed model more numeroustypes of interactions make opinion updates possible

In terms of judgment accuracy all parameters except thesize of the argument sacks (119896) have a significant slightlystronger effect on the deliberative dynamics than in themixedmodel In effect the difference may be attributed to theupshot of pair-wise interactions on the distribution of opin-ions and on the arguments that would be advanced duringdeliberation The result respecting parameter 119896 however iscounterintuitive Since argument sacks are static scenarioswhere shifts are higher should make 119896 more important fordeliberation to succeed and it is precisely in the scenarios ofpure deliberative dynamics that we find the biggest opinionshifts This may point to the hypothesis that in deliberativeinteractions agents change opinions substantially but do notoften change their adherence to the principle (go from anegative opinion to a positive one or vice versa) when thereis no group polarization

43 Sensitivity Analysis for the Procedural Parameters ofInterest Weperform a sensitivity analysis of the observationson the parameters of interest in themixed interactionsmodel

Complexity 21

We extend the parameter domains of the pure dyadic anddeliberative scenarios as we make it explicit in Table 2

431 Minimum Number of Debates (119898) We observe that theminimum number of debates has a significant well-observedeffect on all of our metrics excluding group coherence Forthe variance of opinions we can observe that the moredebates there are the bigger the value of the metric isalthough the higher 119899119863 and 119905119863 are the weaker the overalleffect of119898 becomes (Figures 9(d) and 9(a)) Furthermore themarginal effect of increasing minimal debates is decreasingan additional deliberation step requirement increases thevariance of opinions but less and less as it grows 119898 coupledwith 119899119863 has a shy S-shaped effect (see Figure 9(d)) on thevariance of opinions meaning that it has a stronger effectwhen 119899119863 is low which suggests a possible trade-off betweenthe number of arguments accepted in the debate and thenumber of debates required before accepting a proposal

In contrast the shifts in opinion are less and less likelyas 119898 grows and this is independent of the variations of theother parameters Again the effect is marginally decreasingand is only truly significant when 120572 = 05 An explanationto these effects may be linked to the design of the systemFirst the variance of opinions is higher when deliberation isasked for because the more deliberation steps there are thehigher the chances are that the central proposal argumentis deemed unacceptable specially when agents are focusedFurthermoremechanically speaking increasing theminimalamount of debates implies that whenever a decision is to betaken at least 119898 times 119905119863 time steps have to take place and if atany moment a proposal argument is considered undecided119905119863 time steps are added to the process So unless the debatedoes not yield 119906119899119889 labels for proposal arguments (highlyunlikely considering that 120590 = grounded semantics) the morenondeliberation steps there are in the decision process thehigher the variance of opinions is Concerning the shiftswhen 120572 = 05 either the system is too stiff to accept anyproposal argument and opinions do not changemuch or theeffects of pair-wise discussion and deliberation cancel out insuch a way that individual shifts are minimal (Figure 9(b))

On the side of labeling-based metrics the more debatesare asked for the more accurate a group is in its judgmentthe effect being smaller as 119898 grows When agents are naivethe effect is quasilinear while when they are focused thestrongest effects of acquiring more deliberation are foundwhen levels of deliberation are already low (Figure 9(f))Thiscan be explained by the fact that the more debates thereare in the decision process the closer one gets to the idealargumentation framework

432 Proportion of the Population in Deliberation Steps (119899119863)Like for 119898 119899119863 has a significant effect on the proportionof extremists and on the shifts and variance of opinions(Figure 9(e)) This may result from the fact that being ableto put more arguments in play at the same debate step mayincrease the odds of revealing cycles around the proposalargument in119860119865120576 Given that we use grounded semantics thearguments in the cycles are labeled 119906119899119889 and in consequence

debates are more often postponed when 119899119863 is higher Post-poning debates in turn increases the number of nondebatessteps in the decision process which increases the varianceof opinions and limits over time the moderating effect ofdeliberation Moreover the effect of this parameter is verydependent on the value of 120572 (Figure 9(e)) For shifts forinstance 120572 = 05 makes the effect of 119899119863 negative while120572 = 0 makes it positive but to a lesser extent Furthermorefor higher requirements of deliberation (119898) adding moreindividuals to the deliberation process has weaker effectson the variance of opinions and on the other metrics It isalso quite interesting to notice that it has the same effect onagents independently of agent behavior during deliberationThis is a surprising result as one would have expected morefocused agents in a deliberation arena to cause a raise inthe proportion of extremists as they play to knock outopposing proposal arguments and thereof reducing the effectof deliberation (lowers variance) on the whole population

Finally similar to119898 addingmore people into the deliber-ation process increases judgment accuracy (see Figure 9(c))and has no clear effect if any on group coherence (seeFigure 9(i))

433 Steps between Deliberation Steps (119905119863) In all configu-rations 119905119863 increases the proportion of extremists (variance)and decreases the shifts of opinions among agents Theshifts and the effect on the variance of opinions are presentfor 120572 = 05 insignificant otherwise (Figure 9(b)) 119905119863 ishighly linked to 119898 by construction When 119898 = 1 thecurve linking the variance of opinions and 119905119863 is convex As119898 increases the curve becomes more and more concavewhich means that 119905119863 has a more important effect on theopinion distribution as collective decisions take longer to beachieved This seems counterintuitive but in reality it reflectsthemultiplicative relationship between deliberation and pair-wise interactions and the semantics in the model If 119898 is lowand 119905119863 is high the effective number of pair-wise interactionsis on average smaller in the deliberation process Hencethe structure of the decision-making process hinders anyincrease in the variance of opinions If119898 is high the oppositeeffect is observed Additionally since the grounded semanticsyields few 119894119899 arguments with respect to other admissibility-based semantics getting closer to the ideal argumentationframeworkmaymake it difficult to obtain the expected effectsof deliberative discussion on agents Lower 119898 constrains thevariance of opinions by making deliberation more influentialon opinions

In terms of judgment accuracy and coherence we observethat 119905119863 does not explain coherence yet it tends to decreasejudgment inaccuracy This may be due to a more variedargument pool in the debate which in turn may be aconsequence of a higher variance of opinions or vice versa

434 Acceptability Voting Quota (120572) By far it is the mostinfluential parameter in our study It changes the directionand the intensity of the effect of all the other proceduralparameters and by construction heavily constrains the roadto accepting a proposal (see Figure 9)

22 Complexity

015

020

025

030

035

040

1 2 3 4 5 6

12345

Values of Ｇ

(a) 119905119863 times 119901119903119900119901119890119909 by119898

0

10

20

30

40

50

1 2 3 4 5 6

005066

Values of

(b) 119905119863 times 119878ℎ by 120572

040

045

050

055

060

1 2 3 4 5

001002003004005

Values of Ｈ＄

(c) 119898 times 119890119888 by 119899119863

015

020

025

030

035

040

1 2 3 4 5

001002003004005

Values of Ｈ＄

(d) 119898 times 119901119903119900119901119890119909 by 119899119863

0

10

20

30

40

50

001 002 003 004 005

00005066

Values of

(e) 119899119863 times 119878ℎ by 120572

040

045

050

055

060

1 2 3 4 5

BehaviorNaiveFocused

(f) 119898 times 119890119888 by 119887119890ℎ119886V119894119900119903

015

020

025

030

035

040

0 05 066Behavior

NaiveFocused

(g) 120572 times 119901119903119900119901119890119909 by 119887119890ℎ119886V119894119900119903

05

06

07

08

09

10

1 2 3 4 5 6

00005066

Values of

(h) 119905119863 times 119894119903 by 120572

08700

08725

08750

08775

08800

001 002 003 004 005

BehaviorNaiveFocused

(i) 119899119863 times 119894119903 by 119887119890ℎ119886V119894119900119903

Figure 9 Curves of mean observations for different parameters and metrics for 36000 runs Bars are 095 confidence intervals Plots withthe same observables (ordinate) except for 119894119903 are on the same scale

Complexity 23

The higher the requirement for accepting the proposal isthe higher the proportion of extremists and the variance ofopinions are and the lower the shift statistic is In few words120572 constrains the world to dyadic discussion or throws it intoa process in which deliberation is much more importantthan dyadic interactions This is why for any 120572 = 05 oneeither gives too much weight to deliberated results whichshadows pair-wise interactions or too little weight in thesense that no deliberated result is ever accepted and thusnever integrated into the opinions of the agents that havevoted for it The reason for the latter is that the latitudeacceptance is too low and thereafter pair-wise interactionsare not enough to unevenly polarize the population in such away that deliberated proposal arguments are voted favorablyby a non-negligeable majority Shifts for 120572 = 0 are stable fordifferent levels of 119905119863 which implies that deliberation keeps incheck all the shifts of opinions related to an increase in thenumber of pair-wise interactions For 120572 = 05 deliberationis less likely to be successful and shifts tend towards the shiftlevels for 120572 = 0 (see Figure 9(b))

Concerning labeling-basedmetrics120572 entirely determinesthe coherence statistic For 120572 = 05 there is no difference incoherence because of how coherence is defined it is maximalwhen 120572 = 0 and strangely maximal when 120572 = 066(see Figure 9(h)) Surprisingly 120572 has no effect on judgmentaccuracy One would have expected a higher 120572 to increasejudgment accuracy since agents would hardly collectivelyaccept a proposal argument discussed during deliberationProposal arguments have given how the argument lattice isgenerated a higher probability of being rejected than of beingaccepted in the ideal argumentation framework

435 A Word on Focused and Naive Agents Focused andnaive specifications for agent behavior are an importantparameter in the decision process since they model howagents choose arguments in the deliberation arena In the sce-narios where agents are focused agents show equal or highervariance of opinions and are therefore more extreme (seeFigure 9(g)) independently of the voting requirements Shiftsof opinion are less likely as well We can see this happeningbecause focused agents knock out proposal arguments moreoften than naive agents provided that 119899119863 is low

One could have expected naive agents to yield morecoherent decisions since they argue sincerely However thisseems not to be the case (see Figure 9(i)) Agents are unable todeliberate in away thatmakes deliberation reflect their votingintentions either because they do not have the necessaryarguments to do so or because such arguments do not exist

In terms of judgment accuracy focused agents do betterthan naive agents in finding the ideal labels for the proposalarguments This is likely because when reconstructing theframework focused agents take into account the deliberatedproposal argument and choose themost pertinent argumentsin their sacks that relate to the state of affairs at the deliber-ation arena In fine debates result in a better approximationofL119886119887120576(119868) even though one could have believed the oppositethat is focused agents are ready to sacrifice a correct collectivelabel of a proposal argument in the goal of defending theirpositions

44 Points of Discussion In this subsection we discuss theresults obtained from the simulations We attempt to makesense of them from a social sciences perspective

441 Opinion Consensus and Dissensus The deliberativeopinion dynamics corresponds to the simulated scenarios inwhich the variance of opinions is at its lowest and judgmentaccuracy at its highest Dyadic opinion dynamics are associ-ated with the scenarios in which the variance of opinions is atits highest judgment accuracy at its lowest and coherence atitsmaximumpossible levelThemixed interactions dynamicsis a combination of these aspects for all the metrics exceptfor coherence at which it is the worsemdashdeliberation leadsto some kind of loose opinion consensus that is reinforcedby pair-wise interactions Likewise the mixed discussionmodel confirms some results exposed in [16] expressingthat opinions and voting intentions in a deliberative contextldquooften changerdquo That being said and more particularly indeliberative opinion dynamics outputs align well with twointeresting results on consensus formation given by aninterpretation of Moscovici and Doisersquos [15] work Sunsteinrsquos[14] account on deliberation and group polarization andsocial impact theory [4] The first states that mild consensusis reached in deliberation as procedure disengages agentsthe second is that within discussion groups majorities tendto become larger and opinions more extrememdashnot on thebasis of competing arguments but of the initial balance ofopinions After several decision processes we find opiniondistributions that reflect these observations even though weare not able to precisely coin the set of parameters that yieldssuch distributions (see Figure 7)

Although differences between mixed and dyadic opiniondynamics in terms of the variance of opinions and shiftsare slim the former does much better in terms of judg-ment accuracy In this sense deliberation does moderateopinions but not as much when non-deliberation steps arerelatively frequent throughout the decision-making processIn terms of Moscovici and Doise theory of consensus [15]this makes sense since pair-wise discussion engages or ego-involve agents in the discussion and pushes to convergenceof opinion at the extremes of the opinion distributionHowever what remains unclear is the reason why proceduralparameters that constrain deliberation do not disengageagents from discussion and thus result in a mild consensusas also expressed by Moscovici and Doise [15] A possibleexplanation of this phenomenon may stem from the factthat deliberation in the model is always accompanied by aseries of ego-involving discussions that shadowor counter themoderating effect of deliberation Not only this deliberationcould also be responsible for crystallizing the opinions ofextremists for it favors one or the other part of the distributionof opinions as explained in [14] and to some extent in [13]Furthermore the mixed discussion model gives an exampleof polarization that results from a minority of dissenterson one side of an issue acceding to views of the majorityrsquosside [4] as opposed to the strong polarization observed indyadic interactions Situations in which the status quo orthe undecidability of the proposal argument lingers cannotpossibly yield situations of moderation In cognitive science

24 Complexity

one can be inclined to believe that the longer a deliberationprocess is the more likely individuals will take an immutablestance on the situation In view of the assumption that longdecision-making processes can be more costly for a group orsociety as a whole agents may feel forced to take a stancejust for the sake of ending the process The stance that isadopted would therefore depend on pair-wise interactionsamong individuals and not on deliberation since deliberationtakes too long to be conclusive

442 For Hypothetical Recommendations in Decision-MakingProcess Elaboration On another tune results from themodelsuggest that procedural parameters have marginally decreas-ing positive or negative effects on the metrics of interest Ina hypothetical policy analysis perspective this may indicatethat after choosing a correct number of individuals andimposing the right amount of debates one can attain rea-sonable levels of judgment accuracy coherence and varianceof opinions As accounted in [18] deliberation increasesknowledge (collective knowledge as well) and therefore itheightens the chances of making correct decisions If onebelieves that social welfare is linked to deliberation andeventually to the metrics studied previously then proceduralparameters can both be used as an instrument to attaindesired levels of social welfare and reinforce the notionof legitimacy of collective decisions It can also provide ajustification for choosing a deliberation regime rather thananother on the basis of how important (or urgent) a topic ofdiscussion is Indeed if we adopt the claim that deliberationis a forerunner of welfare then any state in which deliberativecues are at their maximum (highest number of participantsmany deliberation steps before making a decision and asfrequent as possible) has to be mapped to the highest attain-able social welfare In this case is a situation where judgmentaccuracy is at its highest yet the variance of opinions or theproportion of extremists at its lowest an ideal situation Wecannot say for sure since strong opinion consensus and longdecision-making processes in many situations may result ina loss of welfare rather than in a gain of it

Another interesting result in procedure for deliberationis the very similar estimates we find for the effects of theminimum number (119898) of debates necessary to make adecision on a proposal and the proportion (119899119863) of individualsparticipating in deliberation on judgment accuracy In thescope of social welfare this raises the question of whetheradvocating for longer decision processes by demandingmoredeliberation steps (higher 119898) rather than bigger debates(higher 119899119863) may increase welfare Depending on the objec-tives of the decider of the deliberation procedure he or shemay choose to concentrate on one or the other A recom-mendation we can assert from the mixed social interactionsmodel is that if one prefers low to high variance of opinionsandor high to low coherence the decider is better off ifshe focuses on increasing the size of the debate rather thanorganizing debates very frequently One of the reasons forthis might be that when making bigger debates all agents aretruly considered as equals in the decision-making processIn organizing many debates individuals that are chosen toparticipate in many of these will be more influential and will

somehow bias the set of arguments in the deliberation arenaThemajority rulemodeled through120572 also determines the sizeof the shifts and the distribution of opinions Because a bigger120572 is associated with higher levels of extremism and opinionvariance deliberated cues are harder to accept and thereforeit becomesmore difficult for agents to reach concensus andormoderate their opinions

On the other hand and further into the idea of legitimacythe goal of deliberative processes is the agreement amongall participants This agreement stands for the right or bestdecision from a formal or procedural perspective and it is alsothe best from a substantive point of view In other words thelegitimacy of decisions not only derives fromwell-establishedprocedural requirements (respect of the protocol) but alsobegs for the fulfillment of two essential conditions (1) satisfy-ing the procedural requirements for a correct procedure (for-mal legitimacy) and (2) the rational acceptability of the resultsof this procedure (substantive acceptability) Legitimacy canbe then obtained from the coherence statistic scrutinizedin the model insofar as it measures how well agents acceptthe results of a correct procedure (eg deliberation) So fora proposal to be legitimate one has to choose parametersthat would maximize group coherence in decision-makingprocesses

As deliberation may give rise to consensus in the delib-eration arena it may also sow the seeds of dissensus inthe group Our model clearly illustrates this phenomenonthrough the procedural parameters of deliberation and themetrics observed From another perspective in deliberativedemocracy dissensus in collective decision-making may bedesirable According to Landermore and Pagersquos accounts [36]lack of consensus can be perceived as having agents withalternative ideas that combined with other ideas can providea better approximation of the ideal status of an argumentTheauthors in [36] call this dissensus ldquopositive dissensusrdquo andit makes normative sense when solving complex collectivedecision-making problems Our model captures the ideaof having normative requirements to define a good andlegitimate collective decision requirements of high consen-sual deliberative accuracy (judgment accuracy) of legitimacy(coherence) and of diversity (variance of opinions) are alltaken into account and for such we can conclude that ourmodel is successful in describing a sound process of collectivedecision-making with interpretable outcomes

Of course the present simulation results and analysis donot prove that all opinion dynamics can be accounted forby such simple processes like the one presented here Theydo however shed light on the expressiveness of combiningdifferent paradigms to account formore interpretablemodelsand pave the way to creating original models of the sortThe simulations suggest the desirability of discovering theconsequences of relatively simple laws of communication atdifferent levels (micro meso and macro) to determine whatstill needs to be explained

5 Related Work

We see our model as a contribution to the influence andopinion dynamics field in ABM and a pragmatic application

Complexity 25

of abstract argumentation theory To our knowledge we areunaware of existing literature on agent-based modeling thatexplicitly relates collective choice and the notions of deliber-ation and opinion diffusion through abstract argumentationas we have done it

Most models in the literature of opinion diffusion areinterested in opinions because these have an influence incollective decision and in questions of social order Forinstance in [3] the authors are interested in consensus andin how a group collectively chooses among two alternativesIn other models authors are interested in the emergenceof extremism [8] and on the distribution of opinions whenextremists are introduced in the population [5 33 37] whileother authors coin the notion of opinion polarization as anemergent property of the system [8 12] They show usingmodels of ldquobounded confidencerdquo and opinion diffusion withtrust that three different kinds of steady states (unipolarbipolar and central) are possible depending on whetheragents are sufficiently uncertain about their opinions andsufficiently connected andor a certain proportion of indi-viduals are already extreme Recent articles on informationand opinion dynamics stress the importance of governanceand government intervention in the spread of emotions(opinions) in the frangibility of social consensus systemIn [10] the authors show that government intervention inthe spread of negative emotions can lead to an even fasterspread of negative emotions (single-peeked convergence) andin a faster collapse of the social consensus system Opinionas a function of trust is studied in [12] and relates to howindividuals form their opinions on the basis of how muchthey trust agents in their networks

Another stream of discrete opinion dynamics modelsthat explain the emergence of opinion can be found in theliterature Computer multiagent models of attitude changebased on Latanersquos social impact theory are presented in [4] Inthe theory an agent changes her opinion on the basis of theinformational impact she is subject to which depends on thepersuasiveness (strength) supportiveness and immediacy(group structure) of the environment and information shereceives The simulations predict two emergent groups ofphenomena the shifting of attitudes towards incompletelypolarized equilibria and the formation of coherent clusteringof subgroups with deviant attitudes Similarly but in acontinuous more argumentative fashion Mas and Flache[11] present a model of opinion diffusion using argumentsArguments are considered to be for or against a proposal andare given an agent-dependant relevance in dyadic persuasiveinteractions Arguments also determine the agentsrsquo opinionsand their dissimilarities which posits the rules of pair-wiseencounters in theirmodel as agents are assumed to form theiropinions according to the arguments they own They showthat in an argumentative opinion diffusion model the onlyobservable steady states are bipolarization and consensusand that bipolarization can emerge from interactions amongsimilar agents In other words bipolarization of opinionsis possible with homophily and without negative influenceif there is argumentation They subsequently compare theirresults to experimental data In [38] a similar approach toargumentation and opinion formation is taken but for the fact

that they introduce different types of arguments explicitlyapply social judgment theory [22] and use survey data toestablish model-to-real-world comparisons

Closer to opinion formation and abstract argumentationGabbriellini and Torroni [39] are the first to have origi-nally and soundly merged opinion diffusion and abstractargumentation They define an agentrsquos opinion as a functionof the arguments she holds and the attack relation amongthem They devised a focused peer-to-peer dialogue systemof persuasion (NetArg) inspired from Mercier and Sperberrsquosargumentative theory of reasoning [40] which used onlyabstract argumentation to study opinion polarization andopinion dynamics Moreover they considered networks andthe notions of trust and epistemic vigilance to define adynamics for knowledge revision and trust itself whichwe clearly lack in our model They showed that if a con-servative belief operator in argumentation was applied byagents when they reasoned then their dialogue protocoldid not increase polarization among agents That said theyused the model to study the effect of Groenvetterrsquos weaklink theory in the spreading of arguments and were notparticularly interested in notions like judgment accuracy orcoherence nor in tackling questions related to collectivedecision-making procedures Their system is very expressiveand helps position our work in the litterature our work is atthe frontier of pure argumentative opinion diffusion modelsopinion diffusion with arguments and continuous bounded-confidence models of opinion diffusion a la Deffuant

On another note work on collective cognitive conver-gence [41] and opinion sharing [19] shows that consensustowards a certain ldquocorrectrdquo opinion or cognitive state is alwayspossible yet dependent on noise variability and awarenessof agents In [20] the authors show that learning about anexogenous correct state of the world (represented by bits)under confidence was possible but only if the agents werenot too confident In a homogeneous population they showthat the higher the confidence the worse the learningmdashthevery confident agents do not learn the properties of the truestate of the world and disrupt the learning process of the lessconfident ones Collective cognitive convergence can be seenas a result of deliberation in truth-seeking models

When it comes to abstract argumentation theory we takean approach that wires two types of dialogues that are well-studied in the literature persuasion dialogues [42] and delib-eration dialogues [43] Another interesting line of work is theone on mechanism design [44] or the problem of devisingan argumentation protocol where strategic argumentation isnot a liability for success in debates We tackle mechanismdesign in a different way though Instead of consideringstrategy-proofness we are interested in how differences inprotocol can result in epistemologically ldquobetterrdquo collectivechoices and guarantee that opinion distributions are favorablefor deliberation For a survey on persuasion dialogue see[42]

Work on agent-based argumentation usually assumes thatthe semantic relationship between arguments is fixed Inother words if two individuals were to put two argumentsin the public arena such that one logically attacked the otherthen everyone would agree that such attack exists [45 46]

26 Complexity

Other models which do not make this restrictive assumptioncan also be found in the literature [27 47] and derive fromthe class or family of opponent models [27 48] in which twoopposing sides attempt to win a dialogue Ourmodel is in theintersection of these two but the frame combining opiniondiffusion of the kind and argumentation remains new

The idea of mixing interpersonal influence and verticalcommunication however is not original It is described andimplemented in innovation diffusion models such as [5 49]In both vertical communication is modeled as exogenoustransparent informationmdashagents are aware of the existenceof an innovation thus triggering several processes of choiceand stabilization of opinions Also in [50] an Eulerian modelis implemented to show the effect of media and exogenousinformation on opinion distributions With respect to thispoint the originality of our work is in that the informationemitted as vertical communication is endogenous It is issuedfrom a deliberation model that agents shape on the basisof their opinions arguments and behavior In the spirit of[51] where the authors control for the design of verticalcommunication (to whom the vertical communication isaddressed its timing how it influences agents opinions) wecontrol for the process generating the information for ourown set of variables length of debates majority quotas andthe frequency at which discussions take place

6 Conclusion and Perspectives

The main objective of this article was to build a bridgewhere decision-making argumentation and opinion dif-fusion could come together We proposed a model thatcombined abstract argumentation theory and a boundedconfidence opinion diffusion model and showed to whatextent it could explain variance of opinions extremismcoherence in collective decisions and judgment accuracyon arguments given an ideal state of full information Thesecond objective was to show in what way governance pro-viding agents with arenas of discussion and deliberation wasimportant in addressing the success of collective decision-making processes and the quality of their outcomes

The model revealed that allowing for more deliberationtime though in lower frequency and allowing for widerparticipation in deliberation increased the variance of opin-ions and the proportion of extremists in a group Theseobservations are consistent with the combination of resultsfound in [9] and in [13 14] which stress that deliberationmay polarize groups and may have a meager effect on theshifts of opinions and inconsistent with [16] where it isargued that opinions tend to moderate after deliberationIt happens that deliberation alone did moderate opinionyet when integrated into a more complex system in whichindividuals were allowed to interact with one another andnot all that was deliberated was accepted its influence wasshadowed by other more individualistic dynamics

Undeniably the grounded semantics played an importantrole in the weak effect of deliberation since it models askeptical way of reasoning over arguments For agentsaccepting arguments thus updating their opinions accord-ingly happened not too often Nevertheless asking for more

deliberation did increase judgment accuracy as observed in[18] yet in a marginally decreasing fashion We showed thatvoting within the deliberation protocol not only increasedthe proportion of extremists and the variance of opinions ina group but also determined how coherent deliberation andvoting procedures were with each other Lastly we showedthat agents that are focused judged arguments better hadmore stable opinions and constituted a group with higherproportion of extremists than their naive counterparts

In terms of governance the model says that there is per-haps no trade-off between extremismand judgment accuracyInstead it asserts that the higher the variance of opinions thecloser the group gets to the correct decisions This situationmay be interpreted as Landermore and Pagersquos notion of ldquoposi-tive dissensusrdquo (in complex collective decision-making tasks)[36] as opposed to consensus as a normative requirementfor correct collective decision-making If a decider has toorganize deliberation to legitimize a public policy dependingon what kind of world he or she wants different set ofparameters may be chosen to account for different levels oflegitimacy and correctness For instance one may ask for adecision to be legitimate to respect a certain level of coherenceor of judgment accuracy and to be discussed by a sundryof different-opinionated peoples If so deliberation has tobe made more often more participants have to be includedin debates or in deliberation instances and not too manytime steps should be introduced in between two deliberationinstances So the model may capture normative ideas todefine correct and legitimate collective decisions and tomake recommendations on those grounds Requirements ofhigh consensual deliberative accuracy (judgment accuracy)legitimacy (coherence) and diversity (variance of opinions)are all taken into account and for such a reason the model israther successful in describing a sound process of collectivedecision-making with interpretable outcomes

61 Extensions of the Model and Other Ideas Our model canbe extended in many ways Value-based abstract argumen-tation frameworks (VAFs) for example as an extension ofDungrsquos argumentation framework provide a formal descrip-tion of the process of decision-making in which argumentsare given values and audiences (set of agents) ignore attacksbetween arguments on the basis of their preferences over val-ues Social abstract argumentation frameworks (SAFs) alsoprovide an interesting and seemingly convenient formalismfor our model In SAFs agents vote on the acceptability ofarguments and the resulting labeling is a combination of thearguments that are voted for and labeling-based semantics Inour model we apply the same rule yet only for one argumentand given precise agent-based votingmechanisms and votingrules which are eventually parametrized

To consider our model as a specification of a mixed VAFand SAF model may be of great interest for future workThis formalism can provide solid foundations to describeexplain interpret and eventually extend the results foundhere from an argument-based perspective On those groundsfuture work will focus on creating a hybrid argumentationframeworkmdashwith its semanticsmdashthat takes heed of bothvalues or principles and voting

Complexity 27

Concerning other possible extensions of the model anatural reflex would be to argue on the validity and therobustness of the mix between the bounded confidenceopinion diffusion models and deliberation Consequentiallyadding deliberation to different bounded confidence modelsof opinion diffusion such as the Hegelsmann-Krause model[37] or to models of social impact theory like in [4] canbe of great interest for future work We acknowledge that afiner understanding of argumentation frameworks randomlattices and their respective applications for this modelare necessary Testing for different labeling-based semanticsor even trading labeling-based semantics for scoring-basedsemantics (semantics that defines acceptability on scoresgiven to arguments) such as debate semantics and ranking-based semantics (semantics that yield a preorder) may beinteresting to consider in future work

More into modeling deliberation it is not inadequateto believe that agents may also advance proposals duringdeliberation and replace to some extent the proposal forwhich the deliberation is taking place Endogenizing theprocess that produces the type of argument to be deliberatedon may be interesting to explore either by having agentsstrategically replace proposals at certain moments duringthe debate or having the central authority attempt to chooseproposals such that the probability of having them acceptedis high Agents may also be thought to be autoorganizing(independent of a central authority) and concede on the sizeof the instances of deliberation To consider differences insemantics across agents or even endogenizing the semanticson how urgent or important the proposal that is beingdiscussed is can also be an interesting direction to take

Last but not least many nontrivial modifications of ourmodel are possible and most should include better-thoughtdeliberation protocols It may be interesting to design andobserve protocols in which deliberation only affects agentsthat are actually debating for example Trust network effectsmultidimensionality of opinions new processes of argumentexchange or learning among agents are notions to furtherdevelop in order to make the model more realistic and relaxsome unreliable assumptions In sum the model is to berefined and extended with the objective of either studyingconcrete cases of deliberative polling and opinion dynamicsor implementing more intuitive thought experiments

Appendix

We present below some of the pseudocode we used toimplement the model

Algorithms

(a) Algorithm 1 implements the dyadic influence in themodel It describes how two agents interact andupdate their opinions

(b) Algorithm 2 implements the deliberative influence inthe model which describes how agents update theiropinions in sight of the accepted deliberated proposalarguments

(c) Algorithm 3 is used to obtain the grounded labelingof an argumentation framework

(d) Algorithm 4 provides an overview of a run in thesimulations119872 is the number of proposal argumentsto discuss which we set to 100 in the model

Each object in the system is indexed by a119908ℎ119900 number Hencea set of agents is annotated by a set of natural numbers Weuse this fact in the implementation of the algorithm 119874119887119895119890119888119905119894denotes the object of 119908ℎ119900 number 119894 Agentsrsquo opinions (119900) arecalibrated so that 119900 isin [minus1 1]Data Availability

The program and data used to support the findings of thisstudy are available from the corresponding author uponrequest

Disclosure

A preliminary version of this work was presented anddiscussed at the conference Bridging the Gap between FormalArgumentation and Actual Human Reasoning on October 4-5 2018 at BochumGermanyThe authors thank the audiencefor their valuable comments and suggestions

Conflicts of Interest

The authors declare that they have no conflicts of interestregarding the publication of this article

Acknowledgments

The contribution of Gabriella Pigozzi was supported by theDeutsche Forschungsgemeinschaft (DFG) and the CzechScience Foundation (GACR) as part of the joint project FromShared Evidence to Group Attitudes (RO 45486-1)

Endnotes

1 In other words the superiority of the ldquoright answerrdquo orlabeling in our case will appear as such to all

2 All agents hold the same opinion3 By sensitivity we mean how much an agent is ready to

change her opinion on the sole fact that an informationalcue has been debated on

4 The assumption is strong but this trust among agentsmay be rooted from a genuine individual motivation toreach a democratic consensus

5 We adopted a rule of thumbs that says that if an agentis neutral with respect to the principle then she willvote for any proposal that is presented to her duringdeliberation

6 If the argument is not unique she chooses one atrandom

7 If there are more than one candidate the agent will favorthe argument at minimal distance from 119868

28 Complexity

Require Vector of values for parameters (119881 = (120583 119880 119879))Ensure Successful interactions among agents119896 larr997888 0119873119890 larr997888 0

while 119896 le |119873|2 do ⊳Fill119873119890 with |119873|2 agents119896 larr997888 119896 + 1119873119890 larr997888 119877119886119899119889119900119898(|119873| + 1) ⊳Pseudo-random generator Generates integer between 0 and |119873|119873119903 larr997888 0 120575119894119895 larr997888 0 ⊳Variable to store differences in opinionfor all 119894 isin 119873119890 doRandomly pick a number 119895 isin 119873(119873119890 cup 119873119903)119873119903 larr997888 119873119903 cup 119895 120575119894119895 larr997888 |119900119894 minus 119900119895|procedure dyadic-update(119894 119895) ⊳Two agents 119894 and 119895 discuss and update opinions1199001015840 larr997888 119900119894 (120583119895 119880119895 119879119895) larr997888 (120583119894 119880119894 119879119894) larr997888 (120583 119880 119879) ⊳Save agent irsquos opinion before updating and homogenize agents

if 120575119894119895 lt 119880119894 then119900119894 larr997888 120583119894119900119895 + (1 minus 120583119894)119900119894if 120575119894119895 gt 119879119894 then119900119894 larr997888 (1 + 120583119894)119900119894 minus 120583119894119900119895if 120575119894119895 lt 119880119895 then119900119895 larr997888 1205831198951199001015840 + (1 minus 120583119895)119900119895if 120575119894119895 gt 119879119895 then119900119895 larr997888 (1 + 120583119895)119900119895 minus 1205831198951199001015840

Algorithm 1 Dyadic social influence

Require Vector of values for parameters (119881 = (120574 119901119886 119901119903))Ensure Successful interaction between proposal arguments and agents

for all 119894 isin 119873 doprocedure Probability-Update(V119886 119901119886 119901119903) ⊳Update individual probabilities of change due to deliberated cues119909119894 larr997888 119901|V119886minus119900119894 |2119886 119910119894 larr997888 1199012|V119886minus119900119894 |119903119902 larr997888 0if L119886119887(119868) = 119894119899 then

procedure deliberative-update(119894 119886) ⊳Agent i argument 119886120574119894 larr997888 120574 119902 larr997888 119906 sim U(0 1) ⊳ 119906 is a realization of a uniform random variableif 119902 lt 119909119894 then119900119894 larr997888 120574119894119900119894 + (1 minus 120574119894)V119886break

if 119902 lt 119910119894(1 minus 119909119894) then119900119894 larr997888 (1 + 120574119894)119900119894 minus 120574119894V119886Algorithm 2 Deliberation influence

8 In terms of computational complexity naive agentsmaketheirmoves in119874(|A119894| log(|A119894|)) while focused agents doat least as bad

9 Tables have memory Therefore debates are not neces-sarily trees

10 If there are less than (119899119863 times |119873|)2 agents for any of bothgroups the CA summons all of them to the deliberationtable

11 Or careful consideration of the viewpoints of otherswhich implies that citizens keep an open mind and donot reject arguments outright

12 A ldquohealthyrdquo consensus refers to the case in which manyconsensual decisions are taken but opinion-wise agentsdo not agree with each other Quotes are used on theword healthy because some political science theorists

(eg [36]) believe that in certain situations agreement inopinions for deliberation is epistemologically damaging

13 Simulations on groups of |119873| = 800 and |119873| = 400agents shows that scaling the population has an effect onall themetrics but it is confounding depending onwhichcouples of attitudes structures are considered (119880 119879) Westick to |119873| = 400 as in [9] for comparability andsimulation running time

14 By balanced wemean with as many arguments with V119886 lt0 as with V119886 gt 0

15 Experiments on the system suggest that a bigger(smaller) argument pool 119872 = 1000 (119872 = 200) isassociated with a higher (lower) variance of opinions(and extremism) and higher (lower) judgment accuracyFor the rest of the observations 119872 seem to have noeffect We choose a middle value for 119872 Similarly theargument pool chosen to be balanced or uniformly

Complexity 29

Require Argumentation framework 119860119865 = (AR)Ensure Assignment of labels to arguments from the grounded labeling of 119860119865L(A)

L0(A) larr997888 (0 0 0)119896 larr997888 0do119894119899 L119896+1(A) larr997888 119894119899 L119896(A) cup 119886 | 119886 is not labeled 119894119899 L119896(A) 119886119899119889 forall119887 isin A 119904119905 119887 R 119886 then 119887 isin 119900119906119905 L119896(A)119900119906119905 L119896+1(A) larr997888 119900119906119905 L119896(A) cup 119886 | 119886 is not labeled 119894119899 L119896(A) 119886119899119889 exist119887 isin A 119904119905 119887 R 119886 and 119887 isin 119894119899 L119896+1(A)

while L119896+1(A) = L119896(A) = 0Algorithm 3 Solve argumentation frameworks (with grounded semantics)

Require Vector of values for parameters (119881)Ensure Vector of statistics (119880)

procedure Set-parameters(119881) ⊳Initialize parameters for scenarioprocedure Init-objs(119899119898 119896 ) ⊳Create119898 arguments 119899 agents and 119860119865120576procedure Solve-Argumentation-Framework(119860119865) ⊳Assigns the epistemic labelingL120576 toA119894 larr997888 1119905 larr997888 1while 119894 le 119872 do ⊳ 119872 the number of arguments to discuss119868119894 larr997888 119866119890119899 119886119903119892(1) ⊳Generates proposal argument 119868119860 119894119905 larr997888 (119868119894 0)L119886119887(119868119894) larr997888 119906119899119889 ⊳Initialize label for 119868119895 larr997888 1 ⊳Initialize debate counter 119895while ((L119886119887(119868) = 119906119899119889 or 119898 le 119895) and 119895 le 119898) doif 119905 equiv 0(119898119900119889 119905119863) then

procedure build-argumentation-framework(119899119863 119887119890ℎ119886V119894119900119903) ⊳Build the argument graph agents deliberate(119861119894119905 119878119894119905) larr997888 (119883 119884) sub (⋃119897isin119873119899119863 A119897119894119905 ⋃119897isin119873119899119863 R119897119894119905) ⊳ 119873119899119863 sub 119873 are agents sampled by the CA119860 119894119905 larr997888 119889(119860 119894119905 (119861119894119905 119878119894119905))procedure Solve-Argumentation-Framework(A119889) ⊳FindL(A119889)L119886119887(119868119894) larr997888 L119886119887119863(119868119894)119895 larr997888 119895 + 1

else119896 larr997888 1while 119896 le 119905119863 doprocedure dyadic-social-influence(120583 119879 119880) ⊳Agents discuss one-to-one119896 larr997888 119896 + 1119905 larr997888 119905 + 1

if 120572 gt 0 andL119886119887(119868119894) = 119900119906119905 then ⊳Agents vote for the proposalif (|119897 isin 119873|119900119897 times V119868119894 ge 0| ge 120572 times 119899) then

L119886119887(119868119894) larr997888 119894119899else

L119886119887(119868119894) larr997888 119900119906119905if L119886119887(119868119894) = 119894119899 thenprocedure Deliberation-influence(120574 119901119886 119901119903) ⊳Agents are influenced by the result of the decision process119894 larr997888 119894 + 1119905 larr997888 119905 + 1119880 larr997888 119862119900119898119901119906119905119890 119878119905119886119905119894119904119905119894119888119904(119881)

return 119880 ⊳Get the statisticsAlgorithm 4 Run for one scenario

generated does not seem to play a significant explanatoryrole on any of the metrics

16 Experiments show that themaximumnumber of debatesis not an important parameter even if it is an intuitiveone when considering decision-making processes Inpractice debates end after at most 119898 + 1 deliberation

steps and even then it happens rarely However to avoidunpleasant surprises like infinite debates on a completeargument graph containing the proposal argument weadded the constraint 119898 Its value is precisely 7 becausethe biggest minimum number of debates we studyis 6

30 Complexity

17 To ensure comparability with the deliberative andmixedopinion dynamics models on judgment accuracy andcoherence agents vote on proposal arguments every 119905119863times(119898+1) 119905119863 and119898does not explain variability in judgmentaccuracy nor in coherence

18 Student t-tests for independent samples are robust tothe violation of two hypothesis on the distribution ofthe samples normality and homogeneity of variancesas long as the sample size is big enough (central limittheorem) and if the difference in size of the comparedsamples is smallrespectively

19 Studentrsquos test version forwhen variances are not assumedequal and the difference in size of the compared samplesis big

20 We do not correct for homoskedasticity nor normality oferrors in the estimates

21 Balanced as the same number of runs per scenario orsuch that no significant correlations are found betweenparameters

References

[1] R Axelrod ldquoThe dissemination of culture a model withlocal convergence and global polarizationrdquo Journal of ConflictResolution vol 41 no 2 pp 203ndash226 1997

[2] D M Rousseau Reinforcing the MicroMacro Bridge Organiza-tionalThinking and Pluralistic Vehicles SAGEPublications SageLos Angeles Calif USA 2011

[3] S Galam and SMoscovici ldquoTowards a theory of collective phe-nomena consensus and attitude changes in groupsrdquo EuropeanJournal of Social Psychology vol 21 no 1 pp 49ndash74 1991

[4] A Nowak J Szamrej and B Latane ldquoFrom private attitude topublic opinion a dynamic theory of social impactrdquo Psychologi-cal Review vol 97 no 3 pp 362ndash376 1990

[5] G Deffuant S Huet and F Amblard ldquoAn individual-basedmodel of innovation diffusion mixing social value and individ-ual benefitrdquo American Journal of Sociology vol 110 no 4 pp1041ndash1069 2005

[6] G Deffuant D Neau F Amblard and G Weisbuch ldquoMixingbeliefs among interacting agentsrdquoAdvances in Complex Systems(ACS) vol 3 no 01n04 pp 87ndash98 2000

[7] J Rouchier P Tubaro and C Emery ldquoOpinion transmission inorganizations an agent-based modeling approachrdquo Computa-tional and Mathematical OrganizationTheory vol 20 no 3 pp252ndash277 2014

[8] M Meadows and D Cliff ldquoThe relative disagreement model ofopinion dynamics where do extremists come fromrdquo in Pro-ceedings of the International Workshop on Self-Organizing Sys-tems Lecture Notes in Computer Science pp 66ndash77 Springer2013

[9] W Jager and F Amblard ldquoUniformity bipolarization and plu-riformity captured as generic stylized behavior with an agent-based simulation model of attitude changerdquo Computational ampMathematical Organization Theory vol 10 no 4 pp 295ndash3032005

[10] Y-F Zhang H-Y Duan and Z-L Geng ldquoEvolutionary mech-anism of frangibility in social consensus system based onnegative emotions spreadrdquo Complexity vol 2017 Article ID4037049 8 pages 2017

[11] A Flache andMWMacy ldquoSmall worlds and cultural polariza-tionrdquoThe Journal of Mathematical Sociology vol 35 no 1-3 pp146ndash176 2011

[12] A Tsang and K Larson ldquoOpinion dynamics of skepticalagentsrdquo in Proceedings of the 13th International Conference onAutonomous Agents and Multiagent Systems AAMAS 2014 pp277ndash284 International Foundation forAutonomousAgents andMultiagent Systems France May 2014

[13] K M Hansen Deliberative democracy and opinion formationUniversity Press of Southern Denmark Odense 2004

[14] C R Sunstein ldquoThe law of group polarizationrdquo Journal ofPolitical Philosophy vol 10 no 2 pp 175ndash195 2002 WileyOnline Library

[15] S Moscovici and W Doise Dissensions et Consensus unetheorie generale des decisions collectives Presses Universitairesde France-PUF 1992

[16] J S Fishkin and R C Luskin ldquoExperimenting with a demo-cratic ideal deliberative polling and public opinionrdquo ActaPolitica vol 40 no 3 pp 284ndash298 2005

[17] R C Luskin J S Fishkin and S Iyengar Considered Opinionson US Foreign Policy Face-To-Face versus Online Delibera-tive Polling International Communication Association NewOrleans La USA 2004

[18] J Barabas ldquoHowdeliberation affects policy opinionsrdquoAmericanPolitical Science Review vol 98 no 4 pp 687ndash701 2004

[19] O Pryymak A Rogers and N R Jennings ldquoEfficient opinionsharing in large decentralised teamsrdquo in Proceedings of the 11thInternational Conference on Autonomous Agents andMultiagentSystems vol 1 pp 543ndash550 Spain June 2012

[20] J Rouchier and E Tanimura ldquoWhen overconfident agents slowdown collective learningrdquo Simulation vol 88 no 1 pp 33ndash492012

[21] D Miller ldquoDeliberative democracy and social choicerdquo PoliticalStudies vol 40 S1 pp 54ndash67 1992

[22] M Sherif and C I Hovland Social Judgment Assimilation andContrast Effects in Communication and Attitude Change YaleUniversity Press 1961

[23] P M Dung ldquoOn the acceptability of arguments and its funda-mental role in nonmonotonic reasoning logic programmingand n-person gamesrdquo Artificial Intelligence vol 77 no 2 pp321ndash357 1995

[24] M Caminada and G Pigozzi ldquoOn judgment aggregation inabstract argumentationrdquo Autonomous Agents and Multi-AgentSystems vol 22 no 1 pp 64ndash102 2011

[25] P Baroni M Caminada andM Giacomin ldquoAn introduction toargumentation semanticsrdquoThe Knowledge Engineering Reviewvol 26 no 4 pp 365ndash410 2011

[26] S Chambers ldquoDeliberative democratic theoryrdquo Annual Reviewof Political Science vol 6 no 1 pp 307ndash326 2003

[27] E Bonzon andNMaudet ldquoOn the outcomes of multiparty per-suasionrdquo in Proceedings of the 10th International Conference onAutonomous Agents and Multiagent Systems 2011 AAMAS 2011vol 1 pp 47ndash54 International Foundation for AutonomousAgents and Multiagent Systems Taiwan May 2011

[28] J Ranciere Disagreement Politics and Philosophy University ofMinnesota Press 1999

[29] T Bench-Capon and K Atkinson ldquoAbstract argumentation andvaluesrdquo in Argumentation in Artificial Intelligence pp 45ndash64Springer 2009

[30] M Caminada ldquoOn the issue of reinstatement in argumenta-tionrdquo in Proceedings of the European Workshop on Logics in

Complexity 31

Artificial Intelligence vol 4160 of Lecture Notes in ComputerScience pp 111ndash123 Springer Berlin Heidelberg

[31] GAVreeswijk ldquoAn algorithm to computeminimally groundedand admissible defence sets in argument systemsrdquo in Pro-ceedings of the 2006 conference on Computational Models ofArgument Proceedings of COMMA 2006 vol 144 pp 109ndash120IOS Press Amsterdam The Netherlands 2006

[32] C W Sherif M Kelly H L Rodgers et al ldquoPersonal involve-ment social judgment and actionrdquo Journal of Personality andSocial Psychology vol 27 no 3 pp 311ndash328 1973

[33] G Deffuant F Amblard G Weisbuch and T Faure ldquoHow canextremism prevail A study based on the relative agreementinteraction modelrdquo Journal of Artificial Societies and SocialSimulation vol 5 no 4 p 1 2002

[34] E Awad M W Caminada G Pigozzi M Podlaszewski andI Rahwan ldquoPareto optimality and strategy-proofness in groupargument evaluationrdquo Journal of Logic and Computation vol 27no 8 pp 2581ndash2609 2017

[35] D Grossi and G Pigozzi ldquoJudgment aggregation a primerrdquoSynthesis Lectures on Artificial Intelligence and Machine Learn-ing vol 8 no 2 pp 1ndash151 2014

[36] H Landemore and S E Page ldquoDeliberation and disagreementproblem solving prediction and positive dissensusrdquo PoliticsPhilosophy and Economics vol 14 no 3 pp 229ndash254 2015

[37] R Hegselmann and U Krause ldquoOpinion dynamics andbounded confidence models analysis and simulationrdquo Journalof Artificial Societies and Social Simulation vol 5 no 3 2002

[38] A Stefanelli and R Seidl ldquoModerate and polarized opinionsUsing empirical data for an agent-based simulationrdquo inProceed-ings of the Social Simulation Conference 2014

[39] S Gabbriellini and P Torroni ldquoA new framework for ABMsbased on argumentative reasoningrdquo in Advances in SocialSimulation vol 229 pp 25ndash36 Springer Berlin Germany 2014

[40] H Mercier and D Sperber ldquoWhy do humans reason Argu-ments for an argumentative theoryrdquo Behavioral and BrainSciences vol 34 no 2 pp 57ndash74 2011

[41] H V Parunak T C Belding R Hilscher and S BruecknerldquoModeling and managing collective cognitive convergencerdquoin In Proceedings of the 7th iNternational Joint Conference onAutonomous Agents and Multiagent Systems vol 3 pp 1505ndash1508 International Foundation for Autonomous Agents andMultiagent Systems 2008

[42] H Prakken ldquoFormal systems for persuasion dialoguerdquo TheKnowledge Engineering Review vol 21 no 2 pp 163ndash188 2006

[43] K Atkinson T Bench-Capon and P Mcburney ldquoA dialoguegame protocol for multi-agent argument over proposals foractionrdquoAutonomousAgents andMulti-Agent Systems vol 11 no2 pp 153ndash171 2005

[44] M Thimm and A J Garcıa ldquoClassification and strategicalissues of argumentation games on structured argumentationframeworksrdquo in Proceedings of the 9th International Conferenceon Autonomous Agents and Multiagent Systems vol 1 pp 1247ndash1254 International Foundation for Autonomous Agents andMultiagent Systems Canada 2010

[45] I Rahwan and K Larson ldquoMechanism design for abstractargumentationrdquo in Proceedings of the 7th International JointConference on Autonomous Agents and Multiagent SystemsAAMAS 2008 pp 1031ndash1038 International Foundation forAutonomous Agents and Multiagent Systems Portugal May2008

[46] H Prakken R Riveret A Rotolo and G Sartor ldquoHeuristics inArgumentation A Game-Theoretical Investigationrdquo EuropeanUniversity Institute 2008

[47] N Hadidi Y Dimopoulos and P Moraitis ldquoTactics and con-cessions for argumentation-based negotiationrdquo in Proceedingsof the COMMA vol 245 pp 285ndash296 2012

[48] N Oren and T J Norman ldquoArguing using opponent modelsrdquoin Proceedings of the International Workshop on Argumentationin Multi-Agent Systems pp 160ndash174 Springer 2009

[49] S Delre W Jager T Bijmolt and M Janssen ldquoTargeting andtiming promotional activities An agent-based model for thetakeoff of new productsrdquo Journal of Business Research vol 60no 8 pp 826ndash835 2007

[50] A Mirtabatabaei P Jia and F Bullo ldquoEulerian opinion dynam-ics with bounded confidence and exogenous inputsrdquo SIAMJournal on Applied Dynamical Systems vol 13 no 1 pp 425ndash446 2014

[51] V Barbet N Guiraud V Laperriere and J Rouchier Hagglingon Values Towards Consensus or Trouble 2019

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Complexity 5

(i) conflict-free if ∄119886 119887 isin A st (119886 119887) isin R and 119886 119887 isin119894119899L(A)

(ii) admissible if conflict-free and forall119886 isin AL119886119887(119886) =119906119899119889 997904rArr 119886 is legallyL119886119887(119886)

(iii) complete if admissible and forall119886 isin AL119886119887(119886) =119906119899119889 997904rArr 119886 is legally labeled 119906119899119889

(iv) preferred if complete and if it maximizes 119894119899L(A)(v) grounded if complete and if it maximizes 119906119899119889L(A)

We choose admissibility-based semantics because for onethey supply a comprehensive model of collective reasoningthey allow for a meaningful parameterization of credulousand skeptical collective reasoning and last they are relativelyeasy to interpret since the differences between them arewell documented in the literature (eg [18]) Additionallyand in contrast to other rank-based or graded semantics inabstract argumentation admissibility semantics assume thatall arguments have the same weight

223 Abstract Argumentation and Discursive Situations inCollective Decision-Making Abstract argumentation is aconvenient formalism for argument-based decision-makingsince it ignores difficulties relative to the nature generationand number of arguments and posits the possibility of usinggraph theoretic tools to model (collective) reasoning in aclear coherent and easy way [31] Example 2 below providesa simple overview of a debate between two agents over theacceptability of the proposal ldquoTax the richrdquo

Example 2 Let P = ldquoTax the richrdquo and P = ldquoLiberalismrdquoFigure 1 represents the abstract argumentation frameworkobtained from the following arguments

(i) 119868 = 119888 = ldquoOnly the rich should be taxed because theypossess most of the capital in the countryrdquo

(ii) 119887 = ldquoIf you only tax the rich the rich will leave andthen yoursquoll have no one to taxrdquo

(iii) 119886 = ldquoThe rich will not leave because they have theirlivelihoods here and it would cost themmore to leavethan to pay the taxesrdquo

Agent 1 advances the proposal argument 119888 agent 2 arguesthat given the justification 119888 for P P is not tenable Agent1 defends his proposal by advancing 119886 and defeating theargument 119887The conclusion is that holding 119888 and 119886 is a tenableposition so the rich should be taxed Notice that by simplyaccepting 119888 and taking no position on 119887 nor 119886 is conflict-freeand still corresponds to the position advocating that the richshould be taxed

Abstract argumentation comes as an immediate appli-cation to our model in the construction of an ideal argu-mentation framework that loosely models Habermasrsquo idealspeech situation The ideal speech situation is important inour work because it corresponds to a normative state that isin practice difficult to reach and that allows us to observehow different deliberation protocols affect deliberative out-comes

In the model a label given to an argument provideda semantics 120590 is said to be ideal if it is obtained from astate of affairs in which all arguments are presented duringdeliberation We call ideal or consensual argumentationframework the argumentation framework (AR) = 119860119865120576containing all arguments in the system and all consensualattacks among them In a multiagent approach a consensualattack between two arguments 119886 119887 isin A is a couple (119886 119887)such that (119886 119887) isin R if and only if a certain majority ofagents recognizes that such attack exists In our model allagents agree on the attack relation over 119860119865120576 It follows thatif two agents advance two distinct arguments 119886 119887 isin A suchthat (119886 119887) isin R then all agents will recognize that suchconflict exists An immediate consequence of this assumptionis that all deliberated results are consensual even if an opinionconsensus2 on the principle is not necessarily reached

23 Deliberative Social Agents In this section we presentthe agents in our model We define them on the basisof their opinions on the principle P the arguments theypossess their knowledge of the relationship among themtheir behavior during deliberation and their sensitivity todeliberated proposals

231 Agents in Dyadic Social Interactions Every agent 119894 hasan opinion a relative position or degree of adherence 119900119894 isin[minus1 1] to a principle P and a couple (119880119894 119879119894) isin [0 2] times [0 2](119880119894 le 119879119894) of latitudes of acceptance and rejection respectivelyof informational cues They live in [0 2] since 0 and 2 arerespectively the minimum and maximum distances of anytwo informational cues in the system The idea behind thecouple (119880119894 119879119894) is that there exist levels of relative tolerancefrom which informational cues have either an attractive ora repulsive effect on the individual [22 32] An 119900119894 close to 1implies that agent 119894 fully supports the principle P and closeto -1 that she rejects principleP or equivalently fully supportsnotP Moreover if an agent 119894rsquos opinion is such that |119900119894| ge 075119894 is considered to have an extreme opinion a moderate oneotherwise 119880119894 may be considered as 119894rsquos uncertainty about herown opinion [33] or as the limit below which the objectshe judges may attract her 119879119894 may be seen as 119894rsquos bound oftolerance from which informational cues disgust her andconfirmher initial positionDifferent combinations of (119880119894119879119894)can be associated with agent 119894rsquos ego or personal involvementin discussion processes as it is vividly explained in [32]Finally agents are assumed to be sincere and precise whencommunicating their positions to one another (no noise inthe interactions)

232 Agents in Deliberative Social Interactions Agents voteand participate in deliberation because they are aware ofthe potential changes an accepted proposal may induce inthe opinion of the group After all proposals promote aprinciple that potentially leads to a shift in other agentsrsquoopinions Agentsrsquo incentive to participate in deliberation isbased on the idea that every single one of them wants tomake her point across and at the same time reach a correctcollective decision Agents are endowed with a probability

6 Complexity

-1 0 1

Arguments

Adherence to principle

Figure 2 Representation of agents and their arguments on theopinion or adherence to the principles spectrum The rectangleson the agents are their argument sacks Arguments are depicted bydots and arrows from the arguments to the sacks depict both thearguments they hold in the opinion spectrum and the number ofarguments they hold 119896119894 = 4 and 120601119890119894 = 075 and 120601119898119894 = 05 for allagent 119894119909119894 = 119891(119889(119900119894 V119868) 119901119886) of being attracted to a deliberated cueand a probability of being repulsed by it 119910119894 = 119892(119889(119900119894 V119868) 119901119903)The two probabilities are assumed to be independent Theyare a function of the distance between the opinion andthe proposal argument and of the grouprsquos sensitivity3 todeliberation (119901119886 119901119903) The former is decreasing on 119889(119900119894 V119868)and increasing on 119901119886 while the latter is increasing on both119889(119900119894 V119868) and 119901119903 Agents are also assumed

(i) to be capable of assessing the degree of support for Pof all arguments

(ii) to trust4 one another when they utter informationalcues

233 Agents and Voting At the end of the decision-makingprocess agents vote on whether they agree or not on the pro-posal argument that has been discussed during deliberationVoting for an agent is the expression of her opinion in thefinal phase of the decision-making process An agent 119894 is saidto vote favorably for a proposal P of justification argument119868 if and only if V119868 times 119900119894 ge 0 or equivalently if the proposalargument does not adhere to the opposite of the principle 119894supports5

234 Agent Knowledge of Arguments Let A be a finite setof arguments Each agent 119894 has a sack of arguments A119894 subA of size |A119894| = 119896119894 whose content reflects her relativeposition 119900119894 on P (see Figure 2) For extreme-opinion agentsa proportion 120601119890 ge 05 of the 119896 arguments in their sacks areof the same adherence to the principle than their opinions

whereas for moderate agents a proportion 120601119898 le 120601119890 oftheir 119896 arguments are of the valence of their opinions Thehypothesis feeds from results found in [22] stating that ldquoanindividual places a verbal statement on an issue both in termsof the itemrsquos relative proximity to his own position and thelatitude which is acceptable to him around that focal point ofacceptancerdquo

The arguments in an agent 119894rsquos sack are those that sheknows how to use and advance in a deliberative interac-tion The size of the sack represents agent 119894rsquos ability tocommunicate in a deliberative context It follows that eachagent 119894 possesses partial knowledge (119870119894 sub A times A) of theattack relation between the arguments in A and of 119860119865120576which she derives from the arguments in her sack andwhat she observes in deliberation Knowledge of argumentsis assumed to be ldquoattack-orientedrdquomdashan agent 119894 knows anattack (119886 119887) isin A119894 times A if she knows upon observationthat neither she nor the group can rationally accept 119886alongside 119887 She may be aware of the existing defense relationamong arguments when concatenating the information sheaccumulates on their attack relation during deliberationShe may use this information strategically in deliberativecontexts

Let 119889119894119904119905 A times A 997888rarr N be the length of the shortestpath between two elements in the argumentation frameworkinduced by an agent 119894rsquos knowledge119870119894 119894 sees an argument 119886 asan attacker (defender) of an argument 119887 if119889119894119904119905(119886 119887) equiv 1mod 2(119889119894119904119905(119886 119887) equiv 0mod 2) An argument 119886 that is at an odddistance from another argument 119887 attacks it since either itdirectly attacks the argument or it attacks a defender of theargument Likewise if 119886 is at a pair distance of 119887 it attacksan attacker of 119887 and thus defends it If such path does notexist then the distance is undefined and the agent does notsee either argument as an attacker or defender of the otherFurther let AF119889 = (A119889R119889) denote the current state ofaffairs in deliberation An agent 119894rsquos knowledge ofR is the setof attacks and defenses she can infer from the arguments sheknows (119860119905119905119894119863119890119891119894) and the attacks and defenses she can inferfrom deliberation (119860119905119905119889119894 119863119890119891119889119894 )

119870119894 = 119860119905119905119894 cup 119860119905119905119889119894 cup 119863119890119891119894 cup 119863119890119891119889119894 (2)

where

(i) 119860119905119905119894 = (119886 119887) isin A119894 timesA119894 | 119889119894119904119905(119886 119887) equiv 1mod 2)(ii) 119863119890119891119894 = (119886 119887) isin A119894 timesA119894 | 119889119894119904119905(119886 119887) equiv 0mod 2(iii) 119860119905119905119889119894 = (119886 119887) isin (A119894 cup A119889) times A119889 | 119889119894119904119905(119886 119887) equiv1mod 2)(iv) 119863119890119891119889119894 = (119886 119887) isin (A119894 cup A119889) times A119889 | 119889119894119904119905(119886 119887) equiv0mod 2

Notice that agents have no restriction on the amount ofinformation they can carry and use during deliberation Themodel implies that agents have the ability to use and processall information on the attack relations if they could observethe attacks and that they all have equally high cognitivecapacity Agent knowledge resets at the end of each decision-making process but argument sacks stay untouched In otherwords argument sacks are static in the model

Complexity 7

235 Agent Behavior during Deliberation Agents maybehave in twodifferentways in deliberationTheymay behavenaively or focusedly Naive agents use deliberation to voicetheir opinions on the principle through arguments Focusedagents strategically argue in favor of proposal arguments thatsupport the principle they favor In both cases agents advancearguments in terms of the argumentsrsquo relative proximity totheir own positions [22]

Let 119886119894 isin A119894 denote the argument an agent advancesin a debate 119889 (of a deliberation process 119863119868) over a centralargument 119868 and V119886 the valence of an argument 119886 isin A Let

(i) 119891119894(119886) equiv (119900119894 times V119886 gt 0 or 119900119894 = V119886 = 0) the formula thatis satisfied if the argument 119886 is of the same valence asthe opinion of agent 119894

(ii) 119892119894(119886) equiv (119886 119887 isin A119894A119889 and forall119887(|119900119894 minus V119887| gt |119900119894 minus V119886|)) theformula that is satisfied only if the argument 119886 is in 119894rsquossack (A119894) and not in the debate and is the closest interms of adherence to the principle P to 119894rsquos opinion

(iii) ℎ119894(119886) equiv (119891119894(119886) and 119892119894(119886)) the conjunction of thepreceding formuli indicating that an argument 119886 is inagent 119894rsquos sack and not in the debate of the same signas 119900119894 and closest to 119900119894 in absolute value

Naive agents choose 119886119894 = 119886 such that ℎ119894(119886) is true given anyproposal argument 119868 Agents that behave strategically or tosay focusedly choose their moves in the debate as follows(119873119868119871 indicates no move or null argument)

(a) if agent 119894rsquos position with respect to P is of the samesign as the proposal argumentrsquos adherence to P (119891119894(119868)) sheadvances an argument 119886 such that

(i) ℎ119894(119886) and ((119886 119868) isin 119863119890119891119889119894 )) is true If 119886119894 = 119873119868119871(i1) notℎ119894(119886) and 119892119894(119886) and ((119886 119868) isin 119863119890119891119889119894 ) (chooses anargument not of the same sign and closest to heropinion and defending 119868) is true If 119886119894 = 119873119868119871(ii) ℎ119894(119886) and ((exist119888 isin A119894)((119888 119868) isin 119860119905119905119889119894 and (119886 119888) isin 119860119905119905119894))(anticipates an attack from an argument 119888 on 119868 andadvances an argument that attacks the argument 119888)6is true If 119886119894 = 119873119868119871(ii1) notℎ119894(119886)and119892119894(119886)and ((exist119888 isin A119894)((119888 119868) isin 119860119905119905119889119894 and(119886 119888) isin119860119905119905119894)) is true If 119886119894 = 119873119868119871(iii) ℎ119894(119886) and ((119886 119868) notin 119860119905119905119889119894 ) (simply does not attack 119868)is true If 119886119894 = 119873119868119871(iii1)notℎ119894(119886)and119892119894(119886)and((119886 119868) notin 119860119905119905119889119894 ) is true If 119886119894 = 119873119868119871(iv) 119886119894 = 119873119868119871 (the agent will advance no argument)

(b) if agent 119894rsquos position with respect toP is of the oppositesign of the proposal argumentrsquos adherence to P (not119891119894(119868)) sheplays the argument 119886 such that

(i) ℎ119894(119886) and ((119886 119868) isin 119860119905119905119889119894 ) 7 If 119886119894 = 119873119868119871(i1) notℎ119894(119886)and119892119894(119886)and ((119886 119868) isin 119860119905119905119889119894 ) is true If 119886119894 = 119873119868119871(ii) ℎ119894(119886) and ((exist119887 isin A119894)((119887 119868) isin 119863119890119891119889119894 and (119886 119887) isin 119860119905119905119894))(anticipates a defense from an argument 119888 to 119868 andadvances an argument that attacks 119888) is true If 119886119894 =119873119868119871

(ii1)notℎ119894(119886)and119892119894(119886)and((exist119887 isin A119894)((119887 119868) isin 119863119890119891119889119894 and(119886 119887) isin119860119905119905119894)) is true If 119886119894 = 119873119868119871(iii) ℎ119894(119886) and ((forall119887 isin A119889)((119887 119868) isin 119860119905119905119889119894 and (119886 119887) notin 119860119905119905119889119894 ))(avoids attacking any attacker of 119868) If 119886119894 = 119873119868119871(iii1)notℎ119894(119886)and119892119894(119886)and((forall119887 isin A119889)((119887 119868) isin 119860119905119905119889119894 and(119886 119887) notin119860119905119905119889119894 )) If 119886119894 = 119873119868119871(iv) 119886119894 = 119873119868119871

In English a focused agent that is opposed to the proposalattempts to either attack it directly or attack an argument thatis defending it whereas an agent that agrees with the proposalattempts to either defend it or avoid attacking it altogetherThis behavioral approach is similar to debate protocols inabstract argumentation but for the fact that agents mayadvance arguments that do not result in an ldquoadvancementrdquoof the debate In our case the debates stop for other reasons(see Section 24)

Two important points deserve to be highlighted Firstlyfocused agents with small argument sacks will regularly findthemselves applying rules (119894119894119894) or (119894V) In effect if they donot have enough arguments to infer attacks and defensestheir behavior in deliberation is likely to be similar to that ofthe naive agents Secondly the naive and focused behavioralassumptions presented here are very different in terms ofcomputational complexity The first assumes that the agentobserves the debate but what she sees has no effect on hercourse of action ie the argument she voices is uniquelydetermined by how she feels about the principle behindthe proposal argument In the second an agent debatesessentially to knock out any proposal argument she disagreeswith8

24 Tables for Deliberation Tables are the physical or virtualplaces where the exchange of arguments occurs Agentsdeliberate on these tables to determine whether the proposalargument is acceptable or not and are thus subdued bythe deliberation procedure imposed by the tablersquos centralauthority (CA) [27] The CA decides on the structure andlength of the collective decision-making process and thedeliberation procedure It controls the percentage of agentsfrom the population that actively participates in the debate(119899119863) and the labeling-based semantics used to extract accept-able arguments from the framework (120590) The percentage 119899119863denotes either the proportion of agents in the population thatgets to advance arguments in a population-scale deliberationor an independent sample of agents summoned to activelyparticipate in the debate 120590 stands for the procedure used toconclude on the epistemic status of the proposal argumentand the arguments advanced during deliberation

The CA has the ability to stop debates at will using a stoprule that inherently depends on the number of debates thatought to take place before a decision is deemed sufficientlydiscussed (119898) the maximum number of debates that cantake place before abandoning deliberation (119898) and the labelgiven to the proposal argument 119868 (119904119903(119898119898L119886119887(119868))) 119904119903is a Boolean function whose value ldquotruerdquo signals the callfor a vote andor the end of the decision-making process119898 is associated with a minimal dialectical or epistemic

8 Complexity

requirement to consider a proposal for voting and to a lowerbound of the length of the deliberation process 119898 providesan upper bound Moreover the CA controls the size of thetime interval between debates (119905119863) the collective decisionrule (eg if there is voting on the deliberated proposals) andthe majority quota rule for accepting proposal arguments(120572) in the decision-making process 119905119863 may represent thefrequency or density of pair-wise interactions in a decision-making process

241 The Construction of a Decision-Making Process Adeliberation process or debate in ourmodel is a tool to obtainlabels for proposal arguments that are as close as possible tothe ideal or consensual ones To define a deliberation processformally we introduce the notion of debate step as constituentof a deliberation process Informally a debate step is a timestep in which a debate occurs Formally and more event-oriented a debate step (1198891198731015840 1198611015840 1198781015840) of a debate 119863119868 on 119868 isa quadruplet composed of a set of agents 1198731015840 sub 119873 a set ofarguments 1198611015840 sub ⋃119894isin1198731015840 A119894 a set of attack relations 1198781015840 sub Rand a mapping

119889 (AF 2A times 2R) 997888rarr AF

(119860119865 (1198611015840 1198781015840)) 997891997888rarr 1198601198651015840 = (119861 cup 1198611015840 119878 ⊔ 1198781015840) (3)

that adds the arguments in 1198611015840 and some attacks in 1198781015840 to aframework 119860119865 = (119861 119878) and where 119878 ⊔ 1198781015840 = (119886 119887) isin 119878 cup 1198781015840 |119886 119887 isin 119861 cup 1198611015840 In the same spirit we define a deliberationprocess119863119868 on a proposal argument 119868 as a sequence of debatesteps (119889119873119904 119861119904 119878119904)119904gt0 such that1198730 = 0 and

119889((119904minus1⋃119895=0

119861119895 119904minus1⨆119895=0

119878119895) (119861119904 119878119904)) sub (AR) (4)

with

(i) 1198610 = 119868 (the initial argument is the proposal argu-ment)

(ii) 1198780 = 0 (no reflexive attack from 119868 to 119868 is allowed)(iii) (forall119904 gt 0)(⋃119895lt119904 119861119895cap119861119904 = 0) (any newly added argument

to the framework has yet to be added to it)(iv) (forall119904 gt 0)(⨆119895lt119904 119878119895 cap 119878119904 = 0) (any newly considered

attack among arguments cannot be declared amongarguments that are not in the framework)

(v) lim119904997888rarrinfin(⋃119904gt0 119861119904 ⋃119904gt0 119878119904) sube (AR) (the system isstable no arguments are created during deliberation)

Finally let L119886119887119889(119868) denote the label given to argument 119868 ata debate step 119889 during a deliberation process 119863119868 A decision-making process notedD119868 = (119899119863 119905119863 119904119903(119898119898L119886119887119889(119868)) 120572 120590)over a proposalwhose justification argument is 119868 is a sequenceof debate and nondebate steps (119911119905)119905isinN 119911119905 isin 119889 119889 such that

(i) 119889 are debate steps and 119889 nondebate steps that corre-spond to time steps at which there are no debates

(ii) If 119905(mod 119905119863) equiv 0 then 119911119905 = 119889 and 119889 otherwise

For Against

Table

CA

in out

out out in

out in

proparg

Figure 3 A representation of a table of deliberation betweentwo groups of 3 agents with 4 arguments each Each agent hasadvanced an argument Agents tacitly agree on the attacks betweenthe arguments The CA determines the labeling using the groundedlabeling-based semantics

d d d d d d d d d d d d dTime 0 1 2 3 4 5 6 7 8 9 10 11 12I

Figure 4 A decision-making processD119868 or collective decision stepstarting at 119905 = 0 composed of debate (119889) and nondebate (119889) substepsDebate steps are shaded The number of nondebate steps betweendebates is 119905119863 = 3 the number of debates in the decision-makingprocess is119898 = 4

(iii) The subsequence (119911119897119905)119897119905isinN of 119911119905 with 119897119905 = 119905 such that119905(mod 119905119863) equiv 0 is a deliberation process with |1198731015840| =119899119863 times |119873|(iv) 119904119903(sdot sdotL119886119887119889(119868)) pursues the deliberation process as

long asL119886119887119889(119868) = 119906119899119889 under the semantics 120590(v) The length of the sequence |119911119905| or equivalently the

duration of the process is somewhere in the interval⟦119898119905119863 119898119905119863⟧(vi) The final labeling for the proposal argument 119868

L119886119887(119868) is determined by a combination of its delib-erated labelL119886119887119889(119868) and a majority vote of majorityquota 120572

A decision-making process ends when a final decision hasbeen taken concerning the acceptability of the proposal Peg 119868 has been deemed unacceptable (L119886119887119889(119868) = 119900119906119905) or amajority of agents have voted againstP For a representationof a deliberative interaction on a table see Figure 3 for one ofa decision-making process see Figure 4

Please note that every collective decision can be seen asa ldquotime steprdquo in as much as it describes how agents updateopinions and make collective decisions In the model thedecision-making process and the parameters define the sub-steps or events that occur within the time step Henceforth

Complexity 9

Fix proceduralsm m α nD tD σ Init count m larr 0

Given proposal anargument I is randomly generated

nD times |N | agents arerandomly picked

m larr m + 1

Begin decision processon proposal at time t

t larr t + 1Deliberation step m

larr 0

t larr t + 1dyadic opinionupdate Eq (5)tm larr tm + 1

m ge m

tm gt tD

ℒd (I) = in

m ge m

yields ℒ(I)

ℒ(I) = in Deliberationupdate Eq (6)

End decision

yes

no

no

no

yes

yes

no

yes

yes

yes

no

no

I

Vote on ℒd(I)

ℒd (I) = und

ＪＬＩ＝ＭＭ I

Init count tmyields ℒd(I)

Figure 5 Process chart for a decision processD119868 on a proposalP 119905 corresponds to time and 119898 (119905119898) corresponds to the number of debate(non-debate) steps that have taken place in the deliberation process (in between two debate steps) over time for the mixed discussion modelWe useL119886119887(119868) andL(119868) interchangeably in the figure for presentation

comparing simulations on the basis of the different decision-making processes translates into comparing these collectivedecision steps

242 Deliberation Protocol To define the deliberation proto-col held in the table either as a consequence of the definitionprocess or as a statement we assume the following

(i) Agents may decide not to contribute to the debate(ii) All agents have the same probability conditional to

their opinions of being picked to participate in adebate or deliberation step

(iii) There is no restriction on the number of times eachagent can participate in a deliberation process

(iv) Each agent may only place one argument per debatestep

(v) Arguments that have already been advanced in thedebate may attack newly placed arguments9

The deliberation or debate protocol goes as follows

(1) The CA randomly generates and makes public acentral argument or proposal argument 119868 and informsall agents about the rules of the decision-makingprocess

(2) The CA randomly draws two sets of (119899119863 times |119873|)2agents with divergent views10 onP and P It mergesthem to create the set of debaters1198731015840

(3) Each agent 119894 advances an argument from her sackA119894 The CA makes sure that there are no repeatingarguments (agents already take argument repetitioninto account)

(4) The CA establishes the debate step 119889rsquos argumentationframework and computes its labelingL120590

119889(A119889) using120590(5) If the computed label for 119868 is L119886119887119889(119868) = 119906119899119889

or the number of debates steps is inferior or equalto 119898 at time 119905 then the CA stops the debate andresumes it at the (119905 + 119905119863 + 1)rsquoth time step by repeating(1) (2) (3) (4) and (5)

(6) Let L119886119887(119868) be the final label given to the proposalargument 119868 If voting is allowed then if more than120572 times |119873| agents agree with 119868 119868 is accepted (L119886119887(119868) =119894119899) it is rejected if strictly less than 120572times |119873| agree withit (L119886119887(119868) = 119900119906119905) When there is a tie ifL119886119887119889(119868) =119906119899119889 then L119886119887(119868) = 119900119906119905 if L119886119887119889(119868) = 119894119899 thenL119886119887(119868) = 119894119899 When voting is not part of the decision-making process (120572 = 0)L119886119887119889(119868) = L119886119887(119868)

One important point to notice about this protocol is that itinduces a stop rule 119904119903 (cf steps (5) and (6) of protocol) forthe decision process and thus it always ends Either agentsdebate and agree on the proposalrsquos acceptability throughargumentation or after 119898 debate steps they directly voteon it so that a decision concerning the proposal is alwaysreached Refer to Figure 5 for a sweeping description of thedeliberation protocol and the decision-making process

10 Complexity

-1

-1 1

1

Latitude of acceptance

Latitude of rejection Agent i

Agent j

Figure 6 A dyadic interaction between agents 119894 and 119895 as expressedin (5) Agent 119894rsquos opinion falls within agent 119895rsquos latitude of noncommit-ment and conversely Neither agent affects the opinion of the other

25 Opinion Dynamics for Social Interactions In this subsec-tion we describe the opinion diffusion model based on pair-wise interactions among agents and deliberation Sherif rsquosand Hovlandrsquos social judgment theory [22] motivates part ofour approach It describes how individualsrsquo opinions changeon the basis of their attitude structures Attitude structuresrefer to the relative scope width or latitude of categoriesused by individuals when evaluating information namelythe latitudes of acceptance rejection and noncommitment[32] The idea behind this theory is that individuals changetheir positions only in accordance to how far or close thecommunicative cues they receive are from (to) their anchorpositions It holds that if communicative cues are far (close)from (to) an agentrsquos position say over her latitude of rejection(acceptance) then the agent shifts her position away from(towards) the position defended by the cues In the casewherethe cues fall within the agentrsquos latitude of noncommitmenther position does not change (see Figure 6 and Equation (5))

251 Pair-Wise or Dyadic Opinion Dynamics As agentsmay communicate and deliberate collectively they may alsoengage in one-to-one conversations with other agents toponder on their positions We loosely associate this type ofcommunication with dyadic nonargumentative exchange ordiscussion based on fallacious arguments and persuasionIn the light of social judgment theory and the descriptionof agents in the system we model pair-wise symmetricinteractions following the opinion dynamicsmodel in [9] Anagent 119895rsquos influence on an agent 119894rsquos opinion at time 119905 is governedby the ensuing difference equation

119900119905+1119894 =

119900119905119894 + 120583119894 (119900119905119895 minus 119900119905119894) 119894119891 10038161003816100381610038161003816119900119905119894 minus 11990011990511989510038161003816100381610038161003816 lt 119880119894119900119905119894 minus 120583119894 (119900119905119895 minus 119900119905119894) 119894119891 1003816100381610038161003816119900119905119894 minus 119900119905119894 1003816100381610038161003816 gt 119879119894119900119905119894 119900119905ℎ119890119903119908119894119904119890(5)

where the parameter 120583119894 isin [0 05] controls for the strengthof the attraction and repulsion in social influence and 119879119894and 119880119894 are the latitudes of rejection and acceptance for agent119894 respectively The parameter 120583119894 may be thought of as therelative importance agent 119894 gives to the opinions of her peersor analogously the weight she gives to her own opinionwhen updating 120583 lt 05 means that agents will never givemore weight to the opinions of other agents than to their

own (egocentric bias) When two agents 119894 and 119895 discuss iffor instance 119895 advances an informational cue (argumentpersuasion tactic) and it happens to be close enough to119894rsquos opinion anchor then 119894 shifts her opinion towards thedirection of 119895rsquos informational cue The symmetric influencefrom agent 119894 to agent 119895 takes effect in the same way (seeFigure 6)

We add to the dyadic dynamics a rule of encounters ateach step of pair-wise interactions each agent meets exactlyone other agent at random This rule may be associated withrandom day-to-day encounters among agents Steps of socialinfluence correspond to nondebate steps (119889) in decision-making processes

252 Deliberative Opinion Dynamics We define an opinionupdate equation that links the proposal arguments that areadvanced during deliberation and opinions We combine theuncertain and probabilistic nature of the effect of delibera-tion have it a moderating [15 16] or polarizing [14] one witha mechanism similar to the one in social judgment theorybased on the distance between different informational cuesand opinion anchors [32] The probabilistic modeling of theopinion update can be related to deliberation encouragingopen-mindedness11 during collective discussion [18] Fromthis choice it follows that deliberated cues can affect even themost extreme of agents as opposed to some classic opiniondiffusion models (eg [5 8]) where once agents becomeextremists they may no longer become moderate

LetP be a proposal V119905119868 the proposal argument 119868rsquos level ofsupport for a principle P and 119900119905119894 an agent 119894rsquos opinion at time119905 Then given the difference 120575119905119894 = (V119905119868 minus 119900119905119894)2 we define 119894rsquosprobability of being attracted to a decisionrsquos informational cue119868 by 119909119905119894 = 119901|120575119905119894 |119886 where 119901119886 denotes a general probability param-eter that characterizes how important deliberated results arefor the group The parameter may also be interpreted asthe grouprsquos tendency to be swayed by a decisional majoritySimilarly we define 119894rsquos probability of being repulsed from adecision informational cue 119868 by 119910119905119894 = 1199011|120575119905119894 |119903 where 119901119903 denotesa general probability parameter that characterizes the grouprsquosdislike of deliberated results 119901119903 can also be thought of as thegrouprsquos distrust of the system symbolized by deliberation anddemocracy

Let L119886119887(119868) denote the epistemic status of the proposalargument 119868 at the end of a decision process over P everyagent 119894 updates her opinion as follows

(i) IfL119886119887(119868) = 119894119899

119900119905+1119894 =

119900119905119894 + 2120574119894120575119905119894 119908119894119905ℎ 119901119903119900119887119886119887119894119897119894119905119910 119909119905119894119900119905119894 minus 2120574119894120575t119894 119908119894119905ℎ 119901119903119900119887119886119887119894119897119894119905119910 119910119905119894119900119905119894 119908119894119905ℎ 119901119903119900119887119886119887119894119897119894119905119910 1 minus 119909119905119894 minus 119910119905119894

(6)

(ii) IfL119886119887(119868) = 119900119906119905 119900119905+1119894 = 119900119905119894where 120574119894 isin [0 05] is the strength of repulsion and attractionin the dynamicThemeaning of 120574119894 is analogous to the one of120583119894in the dyadic interactionsmodel but for the observation that 119894

Complexity 11

weights her opinion to an argument rather than to an opinionThe interpretation of this dynamics is straightforward Ifthe deliberated proposal argument is close to an agentrsquosopinion then it is very likely that the agent shifts its opiniontowards it Please note that the probability 119909119905119894 (119910119905119894 ) that anagent is attracted (repelled) to (from) an accepted deliberatedproposal argument is bounded below (above) by 119901119886 (119901119903)(equivalently 119909119905119894 isin [119901119886 1] 119910119905119894 isin [0 119901119903]) Steps of deliberativeopinion dynamics are associated with debate steps in which aconcluding collective decision is made

253 Mixed Opinion Dynamics Opinion dynamics in thecontext of decision-making processes can be best described asa combination of the two preceding dynamics In this opiniondynamics model agents are engaged in a democratic systemin which they deliberate vote for proposals and occasionallydiscuss with one another on the principle supported by theproposalsThe opinion dynamics for an agent 119894 can be writtenas a sequence of dyadic and deliberative opinion updates

119900119905+1119894 =

119864119902119906119886119905119894119900119899 (5) 119894119891 119905 equiv 0 (119898119900119889119905119863) and 119904119903 (sdot) = 119891119886119897119904119890119864119902119906119886119905119894119900119899 (6) 119894119891 119905 equiv 0 (119898119900119889119905119863) and 119904119903 (sdot) = 119905119903119906119890119900119905119894 119900119905ℎ119890119903119908119894119904119890

(7)

where 119904119903(sdot) is the decision processrsquos stop rule 119905119863 the numberof time steps between each deliberation step Equation (5)describes the dynamics for the dyadic interactions and Equa-tion (6) posits the changes in opinions due to the deliberatedcues Equation (5) applies when agents are not deliberatingand when each agent encounters another agent to exchangeinformation that may lead to local opinion updates Whenthere is deliberation a handful of agents deliberate and ifthey reach the stop condition imposed by the table all agentsvote for the proposal They update (see Equation (6)) theiropinions on the basis of the proposal argumentrsquos support forthe principle and the result of the scrutiny Otherwise thereis no change in the opinions of the agents A vote indicatesthe end of the decision-making process

3 Experiments and Calibration

In this section we introduce the metrics that enable us toobserve the simulations and characterize the calibration ofthe model

31 Observations and Initialization In this section wedescribe the simulations and the protocol used to test andobserve the results of our model We introduce the metricsof interest explain the calibration of the model describesome results obtained from the simulation data used forcalibration and conclude with a brief discussion on theexpected outcomes of the mixed opinion dynamics model

311 Metrics or Statistics of Interest Let 119878 denote the end ofa simulation We are interested in the effect of deliberationand of the modelrsquos procedural parameters on the followingmetrics or statistics

(i) Variance of opinions (119881119886119903(119900)) the variance ofopinions at time 119878 Since opinions live in the unionof the positive and negative unit intervals 119881119886119903(119900) isin[0 1] The higher the variance of the distribution isthe more ldquodiverserdquo the opinions are(ii) Proportion of extremists in the population(119901119903119900119901119890119909) the proportion of agents in the populationwith opinions such that |119900119894| ge 075 A high proportionof extremists makes ldquohealthyrdquo consensus12 difficult toreach and deliberation more or less informative(iii) Shifts of opinions (119878ℎ) [33] statistic that measuresthe aggregated change in individual opinion at time 119878with respect to aggregated individual opinion at thebeginning of the simulation

119878ℎ = 2sum119894isin119873 10038161003816100381610038161199000119894 minus 1199001198781198941003816100381610038161003816max119894isin1198731199000119894 minusmin119894isin1198731199000119894 (8)

119878ℎ is positive and bounded above by 2|119873| since 119900119894 isin[minus1 1] A low shift statistic implies that the processhas a small impact on opinions(iv) Judgment or consensual inaccuracy (119890119888) consen-sual accuracy of a group consists of an ad hoc statisticmeasuring a grouprsquos ability to infer correct labels forproposal arguments from a decision-making processgiven the ideal consensual labeling based on full-informationL120590

120576 (A) We use a Hamming-based dis-tance on labelings [34] to define the statistic

119890119888 = 1|I| sum119868isinI1198861198681003816100381610038161003816L119886119887120576 (119868) = L119886119887 (119868)1003816100381610038161003816 (9)

whereI is the set of all discussed proposal argumentsand 119886119868 = 12 ifL119886119887120576(119868) = 119906119899119889 or ifL119886119887(119868) = 119906119899119889119886119868 = 1 otherwise119890119888 lives in the interval [0 1] An inaccuracy statisticclose to 1 indicates that agents subdued to a particulardecision-making process make many mistakes injudging the labels of the proposal arguments Notethat when there is voting all proposal arguments arelabeled either 119894119899 or 119900119906119905 after deliberation(v) Coherence (119894119903) letL119886119887119863(119868) be the label obtainedfor 119868 from the deliberation process without votingThe coherence statistic measures how well votingresults adjust to results obtained during deliberationonly We use the proportion of arguments that havebeen labeled 119894119899 in the debate and that agents havevoted favorably for

119894119903 = 1003816100381610038161003816119868 isin I | L119886119887119863 (119868) = 119894119899 andL119886119887 (119868) = 11989411989910038161003816100381610038161003816100381610038161003816119868 isin I | L119886119887119863 (119868) = 1198941198991003816100381610038161003816 (10)

The coherence statisticrsquos domain is [0 1] If after 119872debates no deliberated central argument has beenlabeled 119894119899 or voting is not part of the procedure ofcollective decision-making then the statistic equals

12 Complexity

1 or said differently agents are perfectly coherentThis comes for the fact that if the central argumentis labeled 119900119906119905 then it is not even eligible for voting iflabeled 119906119899119889 then the debate will always be coherentwith the preferences of the agents since the result willbe a simple aggregation of their votes If the statisticequals 0 then none of the proposal arguments labeled119894119899 are voted in favorably by the agents It follows that ahigh coherence statistic implies that when agents votefor acceptable proposal arguments the results of thescrutiny reflect the consensual rationality expressedin the deliberation process

312 Parameters of Interest We recall the parameters ofinterest in our study that are linked to the structure of thedecision-making processes

(i) 119898 the minimum number of time steps in which adebate or a scrutiny occurs in a decision-making pro-cess before a final deliberated decision is submitted toa vote

(ii) 119899119863 the number of agents that deliberate as a propor-tion of the population

(iii) 119905119863 the number of time steps between debates inwhich pair-wise interactions among agents mayoccur

(iv) 120572 the proportional majority requirement for theacceptance of a proposal When 120572 = 0 it stands forno voting ldquoany proposal argument that is labeled 119894119899during deliberation is acceptedrdquo

Please recall that in terms of the definition of a decision-making process each parameter controls either the length orthe content of the sequence D119868 (119905119863 119898) or the rules that areapplied during the debate steps (120572 119899119863) (refer to Figure 4)313 Initialization All |119873| = 400 agents start off with anopinion 119900119894 drawn from a uniform distribution U(minus1 1) 13Given 119900119894 every agent 119894 randomly draws a setA119894 (|A119894| = 119896) ofarguments from a balanced14 argument pool A of119872 = 600nonneutral (V119886 = 0) arguments on the basis of 119900119894 if |119900119894| lt 025(the opinion is moderate) then agent 119894 randomly fills halfof her argument sack with arguments such that V119886 lt 0 andthe other half with arguments such that V119886 gt 0 (120601119898119894 = 05)Otherwise she randomly fills 120601119890119894 ge 05 of her sack witharguments such that V119886 and 119900119894 are of the same sign and theremaining (1 minus 120601119890119894 ) with arguments of the opposite sign15

Like for the opinion of agents every argument 119886rsquosadherence to the principle P V119886 is drawn from a uniformdistribution U(minus1 1) The attack relation that gives birth tothe ideal argumentation framework (119860119865120576) is established onthe basis of the V119886rsquos and the argumentsrsquo epistemic reach 119890119903

Let 120579(119886 119887 119903) = (|V119886 + V119887|2)119903 be an auxiliary positivereal-valued function that takes two arguments (119886 119887) and apositive real number (119903 gt 0) as inputThe probability that anyargument 119886 isin A creates a link to any other argument 119887 isin A

is given by the equation 119901 = 119901120579(1198861198872)119890 and thus argumentssupporting opposing views of the principle always attack each

other We fix 119901119890 the epistemic correlation parameter to 015If 119901119890 was greater than 015 then arguments of the same signwould attack each other too often (and focused agents wouldbe rarely incited to advance favorable arguments duringdeliberation) If 119901119890 was lower then the arguments of thesame type would induce an almost empty graph which isunrealistic The number of arguments that any argument 119886can attack is bounded by 119886rsquos epistemic reach that we fix to15 A higher value of epistemic reach makes the argumentlattice too conflicting and nearly bipartite A lower epistemicreachmakes the lattice not sufficiently conflicting in the sensethat too many arguments can attack the proposal argumentrelatively to the few that can defend it If any argument attacksthe proposal argument the chances that another argumentattacks the attacker are low Hence the proposal argumentis almost never accepted and deliberative opinion updatesbecome rare

The arguments in the resulting argumentation frame-work 119860119865120576 are given a permanent labelingL120590

120576 (A) (in shortL120576) using120590 = grounded semanticsWe choose the groundedlabeling-based semantics because it provides a unique admis-sible labeling respects minimal rationality constraints whilesimplifying the model and models a skeptic approach toaccepting arguments (see [23 25 31]) as it maximizes thecardinality of the set 119906119899119889 L120576 To some extent if we considerproposals as committing because they may guide collectiveaction choosing a skeptic semantics seems reasonable insofaras it labels an argument 119894119899 or 119900119906119905 only if it has no reasonto label it 119906119899119889 For instance if a proposal is a public policythat requires large amounts of resources and engages tofuture course of action then it may also determine policycycles and heavily burden a group and its future decisionsIn important situations like these it seems reasonable tolengthen deliberative cues and ask for more demanding andldquogroundedrdquo criteria for policy argument acceptability

On the proposalrsquos side we create an argument 119868 whoseadherence to P is also drawn from aU(minus1 1) and we label itL119886119887(119868) = 119906119899119889 119868 is interpreted as being the main argumentjustifying the discussed proposal P and V119905119868 as its supportfor P at time 119905 119868 cannot attack other arguments but otherarguments can attack it For each argument 119886 isin A a directedarc or attack from 119886 to 119868 is activated with probability 119901119901 =003120579(11988611986802) When 119901119901rsquos lower bound is too high the proposalargument is almost always defeated and thus deliberativeopinion updates do not happen When 119901119901rsquos lower boundis lower than 003 then the opposite occursmdashthe proposalargument is always accepted due to an absence of attackstowards it Finally we set the maximum number of debatesto119898 = 7 for the decision-making processes stop rules1632 Calibration and Simulation Protocol In this subsectionwe discuss the calibration of the models the terminationconditions for runs in the deliberative and mixed model andthe expected outcomes in terms of the observations

321 Calibrating Dyadic Opinion Dynamics Dyadic opiniondynamics correspond to a space of parameters in whichargumentation and deliberation spaces are not taken into

Complexity 13

Table 1 Parameters and parameter types and their domains used to compare dyadic collective and mixed opinion dynamics

Length ofD119868 Decisional in119863119868 Social influence Deliberation influence Cognitive119905119863 119898 119898 119899119863 120572 120583 119879 119880 119901119886 119901119903 120574 119896 119861119890ℎ119886V1198941199001199031 3 6 1 3 6 7 001 002 005 0 05 066 01 14 18 02 06 03 05 01 005 01 02 4 8 16 119873119886119894V119890 119865119900119888119906119904119890119889account Deliberated arguments and informational cues haveno effect on agent opinion but agents still vote for theproposal argument17 We use and calibrate this model forcomparability in terms of all our metrics since they arenot explicitly observed in [9] a reference opinion dynamicsmodel Furthermore for simplicity and comparability againwe suppose that agents are homogeneous in terms of theirattitude structures and opinion weightings We fix 120583119894 =120583 119879119894 = 119879 and 119880119894 = 119880 for all 119894 isin 119873 and for some triplet(120583 119880 119879) isin [0 05] times [0 2] times [119880 2]

Experiments suggest that the strength of the dyadic inter-actions is an explanatory factor of the time of convergence(120583 lt 05) and of the dyamicsrsquo steady states Bigger values of 120583speed up convergence towards a stable set of opinion clustersand affect the size and the relative position of these clustersin the opinion distribution 120583 however does not play a majorrole in the number of opinion clusters observed at the steadystate of the dynamics

We set 120583 to the value given to it in Jagger et alrsquos opiniondynamics model (120583 = 01) [9] For one it is the smallestreal number for which most results found in [9] hold and thedynamics are smooth secondly it seems to be a reasonableldquostrengthrdquo of influence considering that we would not wantpair-wise social interactions to completely shadow or maskan eventual effect of deliberation in the opinion dynamics

For different values of attitude structures 119879 and 119880 weget exactly the same results as in [9] in terms of the numberand density of the opinion clusters The different couplesof values of 119880 and 119879 model the size of the latitude ofnoncommitment (119879 minus 119880) and thus the agentsrsquo propensityto update opinions Whenever 119879 minus 119880 le 02 regularitiesin convergence and opinion clustering appear If 119880 gt 1there is always central convergence if 119879 lt 1 there is alwaysbipolar convergence with clusters of similar sizes For 119879 minus119880 gt 02 relatively low values of 119879 (119879 lt 1) and relativelyhigh values of 119880 always yield extreme bipolar convergencewith a relatively small cluster of moderate agents (bipolar-central convergence) For 119879 sufficiently high (119879 ge 12) and119880 sufficiently low (119880 le 08) in other words not very ego-involved agents there is a bigger spectrum of steady statesFor this reason we studied the metrics of interest for thesevalue differences and kept the couples of (119880 119879) depicted inTable 1 For these couples central bipolar bipolar-centralconvergence (3 clusters) and multicluster convergence (4 or5 clusters) are possible and result frommeaningfully differentattitude structures Judgment accuracy varies significantlyacross the four different attitude structures we retain and soit happens with the variance of opinions These scenariosprovides us with reference results from which to build upon

322 Calibrating Deliberative Social Interactions Delibera-tive social interactions correspond to social situations in

which individuals are not influenced by pair-wise discussionwith peers but are sensitive to collectively deliberated infor-mational cues On these terms parameters such as the sizeof agentsrsquo argument sacks (119896119894) the distribution of argumentsin them (120601119894) the minimum number of debates (119898) andthe proportion of debaters from the population (119899119863) are nolonger mute We assume for simplicity that the formulaexist120601119890exist119896forall119894(119896119894 = 119896 and 120601119890119894 = 120601119890 and 120601119898119894 = 05) is true with 119894 isin 119873for some value 120601119890 isin [05 1] and for 119896 isin N

The chosen domains for these parameters (see Table 1)are justified by the size of the population and the size of theargument pool (119872 = 600) which we set from the start If 119896is too big (119896 ge 32) andor too many agents are allowed inthe debate (119899119863 ge 02) then individuals always disprove thecentral argument and deliberation is never or rarely takeninto account in the update of opinions Similar effects occurwhen 119898 is high yet we only calibrate it with respect to themean running time of a simulation and the values of 119905119863Different distributions of 120601119890119894 seem to significantly or directlyexplain none of the statistics we analyze so we convenientlyset 120601119890119894 = 075 We chose the parameter 119896 to be always divisibleby 4 for it is convenient given the initial conditions regardingthe argument distribution in the agentsrsquo argument sacks

We fix the value space for the parameters 120574 119901119886 119901119903 as inTable 1 to account for different intensities of the effect ofdeliberative voting retroaction on the system For examplethe scenarios where the three parameters are at their lowestvalues correspond to the scenarios with the smallest fixedprior effect of deliberation in the model Increasing any ofthese parameters should mechanically be associated witha world in which agents are more sensitive to collectivedecisions The values for 119901119886 are taken from [13] wherethe author states that by means of deliberation 7 to28 of individuals changed their opinion from agreeingto disagreeing or vice versa on a referendum question forDenmarkrsquos participation in the EuroWe choose two differentvalues for 119901119886 that account for high (03) and very high (05)sensitivity to deliberation in a group 119901119903 on the other handis taken to be small (equal to 01) since we believe that thereexist agents that will always go against the reached consensusTheir numbers are meager

For simplicity we assume that for all 119894 isin 119873 120574119894 = 120574 forsome 120574 isin [0 05] In effect we posit that the heterogeneityof agents in the deliberative context only derives from theirarguments and their opinions We decide to calibrate 120574 inallusion to the strength of the social dynamics in pair-wiseinteractions In doing so we limit ourselves to consider threedifferent scenarios in which deliberation has respectivelyhalf equal and twice as much opinion-shifting power thanone-to-one social influence Finally we include the accept-ability voting quota rule 120572 whose domain is inspired fromclassical majority rules [35] observed empirically (Table 1)

14 Complexity

Table 2 Parameters and their domains used to study the effect of the procedural parameters on the metrics

Procedural parameters Other parameters119905119863 119898 119899119863 120572 119898 120583 119879 119880 119901119886 119901119903 120574 119896 119861119890ℎ119886V1198941199001199031 2 3 4 5 6 1 2 3 4 5 001 002 003 004 005 0 05 066 7 01 16 02 03 005 02 12 119873119886119894V119890 119865119900119888119906119904119890119889

323 Calibrating the Mixed Social Interactions Model Themixed interaction model ascribes to a parameter space inwhich the effect of collective choices on our metrics isnontrivial and where deliberation and voting on proposalsdetermine their acceptability Since the mixed model isequivalent to periodic iterations of pair-wise and collectivesocial interactions the calibration of the parameters in bothpreceding models holds (see Table 1 for the calibration)The first reason for this is comparability the second is thatthe previous calibration takes into account the fact thatboth dynamics are going to be combined We include thefrequency of debates by adding the parameter 119905119863 whichcontrols for the amount of pair-wise interaction amongagents between two distinct deliberation steps

324 Termination Conditions for Runs Simulations stoponce 100 proposals are deliberated on andor voted for inother words after 100 collective decision steps have occuredThe number of proposals discussed seems arbitrary yet it ishigh enough to observe the effects of deliberation on opiniondistributions and on other metrics related to coherence andjudgment accuracy We choose the number of runs as afunction of our research questions which give relevance tothe procedural parameters in the mixed model rather than tothe parameters set to describe the population and its behavior

325 Expected Outcomes of the Simulations We expect moredeliberation in terms of higher 119898 and 119899119863 to increasejudgment accuracy and at the same time to reduce thevariance of opinions In turn a smaller variance of opinionsimplies that the argument pool for deliberation is smaller andtherefore judgment accuracy should be lower Moreovervariance increasing dyadic interactions (or rejection) shouldalso increase judgment accuracy and coherence since bipo-larization and dissensus foster argument diversity in delibera-tionThe coherence statistic should be stronger in simulationsin which central convergence appears quickly and agentsare naive Attitude structures sensitivity to informationalcues and weights given to deliberated cues that makes pathsto bipolar convergence smaller or to central convergencelonger should be associated with higher judgment accuracyShifts of opinion should be more visible in scenarios inwhich extreme agents are pulled away from the extremesHence the sensitivity to deliberated cues (119901119886 119901119903) and 120574should explain the shifts When crossed with high values of119880 and 119879 the shifting power of deliberation should be at itshighest

In the end we do not know how these mechanismswill play out The results of the subsequent experimentsgive us insight on the interplay between the aforementionedeffects

4 Results

Before pointing at any result obtained from the mixedinteractions model we describe and compare the dyadicand deliberative interactions models in the parameter spaceobtained from the calibration in Section 3 (see Table 1)Primarily we require our results to describe two orthogonaltypes of opinion dynamics the pair-wise dyadic and thedeliberative The latter comprises scenarios in which indi-viduals do not influence each other by means of pair-wisediscussion and only update their opinions from delibera-tion the former scenarios where only pair-wise interactionsamong agents determine the dynamics We describe both onthe parameter space given in Table 1 by performing at least20 simulations per scenario Quantitative results from thepair-wise interaction model alone are described during thecalibration and are not treated heremdashdeliberative parametershave no effect on it Moreover we show how these differentmodel specifications yield qualitatively different opiniondistributions and compare them on the basis of the metricsof interest To this end we perform independent two-sampleStudentrsquos t-tests18 or Welch t-tests19 and compute confidenceintervals at a 095 level of confidence Last but not least wecomment ordinary least squares (OLS) regression estimatesto account for the direction and magnitude20 of the effectsof the parameters on the metrics Estimates are declaredsignificant at a 5 level of risk

We perform the same analysis on the mixed interactionsmodel as we compare it to the pair-wise dyadic and deliber-ative interactions models and discuss the marginal effects ofthe parameters on the metrics More precisely we obtain twodifferent types of results for the mixed interactions modelthe first compares the mixed scenarios to their monolithiccounterparts with respect to each metric and with respect tothe regression estimates the second gives a clear idea of themarginal effect of each governance parameter (or parameterof interest) on our observations In the first we allow controland procedural parameters to vary on a parameter spacesimilar to those of the dyadic and deliberative opiniondynamics (see Table 1) In the second we fix all of ourcontrol parameters execute 36000 balanced21 runs andfocus only on the one-way pooled effects of our proceduraland behavioral (all agents are focused or naive) parametersWe generate comprehensive graphs and tables to account forthe obtained results

Please notice the slight change in the size of the parameterspace for the procedural parameters and in the values ofthe nonprocedural ones for the first and second type ofresults (refer to Tables 1 and 2 respectively) Since we wantto have a finer idea of the marginal effects of each of theprocedural parameters on how well a group decides onproposals we make the value jumps for each parameter (1)

Complexity 15

() (U T) = (02 18) (b) ( pa m nD) = (005 05 00 2 02) (＝) ０ＬＧＮＬＭ ＣＨ (＜) (＝) Ｈ＞ tD = 3

(d) (５ ４) = (02 14) (e) ( pa m nD) = (01 03 05 3 02) () ０ＬＧＮＬＭ ＣＨ (＞) () Ｈ＞ tD = 6

Figure 7 From left to right common opinion trajectories and distributions for dyadic deliberative and mixed opinion dynamics within thestructure of different decision-making processes

small enough to detect significant differences in the estimatesand (2) meaningful for interpretation For the fixed one-valued parameters in Table 2 we use Table 1 to set them totheir minimum mean or median values

41 Comments on Dyadic and Deliberative Opinions Dynam-ics In this subsection we present our first results regardingthe deliberative opinion dynamics described in Equation (6)and its differences with the dyadic dynamics (Equation (5))in terms of our observations

411 Qualitative Analysis of Pair-Wise Opinion DynamicsPair-wise opinion dynamics alone produces multiclusterstable convergencewith variable cluster sizes (see Figures 7(a)and 7(d)) Insight on the complexity of these dynamics wasdiscussed in Section 321

In Figures 7(a) and 7(d) agents discuss randomly withone another in pairs Clusters in the extremes form quicklysince either agents that are close to the extremes attractone another or are convinced to stay close to the extremesby interacting with other agents they disagree with Otheragents with near-extreme positions may be simply attractedto one of the several moderate foci of agents in the opiniondistribution On the other hand moderate agents are eitherattracted to the closest opinion focus or pushed towards theextremes of the distribution Over time agents in the centralfocus either attract other agents into it or ignore the opinions

of the extreme agents that interact with them The result ofthese dynamics yields amulticlustered convergencewhere theopinion foci are at least at distance 119880 from one another andonly the extreme foci (clusters of agents with |119900119894| ge 075) areat a distance greater than or equal to 119879 from one anotherThefoci at the bounds of the opinion distribution are very stableand in analogy to the (12119889)-rule in assimilation bounded-confidence models [33] the number of clusters that form isroughly equal to 119880minus1412 Qualitative Analysis of Deliberative Opinion DynamicsQualitatively deliberative opinion dynamics yield a looseunipolar convergence of opinions near the center (centralconvergence see Figure 7(b)) near the center of either the leftor right portions of the distribution (group polarization seeFigure 7(e)) or to a sparse bipolar opinion distribution withclusters at the center and at the extremes (see Figure 7(c)) ifmixed with pair-wise interactions This is probably a reasonto expect the variance of opinions and the proportion ofextremists (almost perfectly correlated 120588 = 089 119901 lt 0001)to be low for these scenarios The side towards which theopinion cluster skews depends essentially on the valence ofthe first argument that is collectively accepted

In Figures 7(e) and 7(b) agents that update their opin-ions do so at the same time and only when deliberationis successful In these scenarios every time a decision iscollectively accepted agents update their opinions towards

16 Complexity

Table 3 Ordinary least squares estimates for pair-wise interactions scenarios for most of the observed metrics

Dependent variable119881119886119903(119900) 119878ℎ 119890119888 119894119903(1) (2) (3) (4)Constant 0610lowastlowastlowast 4429lowastlowastlowast 0472lowastlowastlowast 0000

(0002) (0206) (0003) (0000)

119880 = 06 (119880 = 02) minus0012lowastlowastlowast 2981lowastlowastlowast 0061lowastlowastlowast 0000(0002) (0260) (0004) (0000)

119879 = 18 (ref 119879 = 14) minus0264lowastlowastlowast minus3261lowastlowastlowast 0004 0000(0002) (0260) (0004) (0000)

120572 = 066 (ref 120572 = 05) 0002 minus0059 minus0100lowastlowastlowast 0000(0002) (0184) (0003) (0000)

(119880 119879) = (06 18) (ref (119880 119879) = (02 14)) minus0324lowastlowastlowast minus3728lowastlowastlowast minus0033lowastlowastlowast 0000(0003) (0368) (0006) (0000)

Observations 1440 1440 1440 1440R2 0983 0389 0529Adjusted R2 0983 0388 0527Residual Std Error (df = 1435) 0032 3492 0053 0000F Statistic (df = 4 1435) 21 272740lowastlowastlowast 228686lowastlowastlowast 402326lowastlowastlowastNote lowastplt01 lowastlowastplt005 lowastlowastlowastplt001

the argumentrsquos defended position Hence if there is votingand if agents happen to obtain 119894119899 proposal arguments theyinexorably cluster towards one or the other side of the opiniondistribution resulting in opinion convergence remeniscent ofgroup polarization (see Figure 7(e)) Furthermore as agentscluster in the same side of the opinion spectrum the argu-ments that may be advanced in deliberation are fewer andmore skewed towards that one side of the opinion spectrumAgents become more easily persuaded in deliberation andonly by arguments supporting one side of the spectrumConvergence of opinions towards one loose opinion focusbecomes faster and more certain If there is no vote (seeFigure 7(b)) the dynamics are similar but for the fact thatit is the uniform randomness of the proposal argument thatguarantees a central convergence of opinions

413 Quantitative Analysis of Pair-Wise Opinion DynamicsThe three parameters that allow for quantitative analysis are119880 and119879 the agentsrsquo attitude structure and themajority quotavoting rule 120572 Table 3 supports the claim that going from 120572 =05 to 120572 = 066 does not affect the variance nor the shifts ofopinions However judgment accuracy is significantly higherwhen accepting proposals becomes more difficult This israther surprising but also intuitive since decisions that aretaken using more restrictive methods of scrutiny should bemore accurate

Concerning the attitude structures the interaction termbetween 119880 and 119879 shows that 119880 has a small effect on thevariance of opinions given that 119879 is low This shows thateven when the assimilation threshold is high a sufficientlysmall rejection threshold can make its one-way effect vanishHowever high levels of both 119880 and 119879 reduce the variance ofopinions dramatically In contrast when it comes to overallchanges in opinion the magnitude of 119880rsquos one-way marginal

effect increases the shifts of opinions whereas 119879 lowers itThis can be interpreted as having the rejection dynamics pushmoderate-to-extreme agents to positions that are close tothe extremes and very moderate agents to the center of thedistribution Not too many quantitative changes (on average)can therefore be observedThe fact that higher119880 is associatedwith faster convergence of opinions seems to imply lowerjudgment accuracy At some point voting for the same type ofproposals for too longmay result inmany collectivemistakesThis is an interesting conclusion to consider in the analysis ofthe mixed model Finally the interaction term shows that 119879has no strong impact on the metric

414 Quantitative Analysis of Deliberative OpinionDynamicsSeven parameters can be considered for quantitative anal-ysis in this dynamics the parameters of interest recalledin Section 312 the size of the argument sacks (119896) agentbehavior during deliberation the strength of the deliberationdynamics (120574) and the measure sensitivity to deliberatedresults (119901119886) The most ldquounexpectedrdquo result obtained fromthe deliberation dynamics is that it shows how deliberationcontributes to the proportion of extremists and the varianceof opinions It happens that having more agents discussproposals and having more of these discussions increasesthe variance of opinions and the proportion of extremistswhile lowering judgment inaccuracy and coherence For theformer the fact that far too many arguments are advancedduring deliberation plummets the chances that the centralargument is accepted hence deliberation does not moveopinions too often For the latter we observe that only ldquobigrdquodifferences (going from the lowest value of a parameter to thehighest) in protocol actually prove to have a preponderantimpact on the metrics These observations hint at a possibletrade-off between judgment accuracy and the variance of

Complexity 17

Table 4 Ordinary least squares estimates for the deliberative interactions scenarios for most of the observed metrics

Dependent variable119881119886119903(119900) 119878ℎ 119890119888 119894119903(1) (2) (3) (4)Constant 0032lowastlowastlowast 13092lowastlowastlowast 0370lowastlowastlowast 1007lowastlowastlowast

(0001) (0406) (0001) (0001)

119898 = 3 (ref119898 = 1) 0012lowastlowastlowast minus2062lowastlowastlowast minus0012lowastlowastlowast minus0001(0001) (0276) (0001) (0001)

119898 = 6 (ref119898 = 1) 0023lowastlowastlowast minus3285lowastlowastlowast minus0018lowastlowastlowast minus0004lowastlowastlowast(0001) (0276) (0001) (0001)

119899119863 = 002 (ref 119899119863 = 001) 0007lowastlowastlowast minus0965lowastlowastlowast minus0006lowastlowastlowast minus0002lowastlowast(0001) (0276) (0001) (0001)

119899119863 = 005 (ref 119899119863 = 001) 0020lowastlowastlowast minus3316lowastlowastlowast minus0015lowastlowastlowast minus0004lowastlowastlowast(0001) (0276) (0001) (0001)

120572 = 05 (ref 120572 = 0) 0146lowastlowastlowast 68932lowastlowastlowast 0068lowastlowastlowast minus0507lowastlowastlowast(0001) (0276) (0001) (0001)

120572 = 066 (ref 120572 = 0) 0291lowastlowastlowast minus13842lowastlowastlowast 00004 minus0000(0001) (0276) (0001) (0001)

119896 = 8 (ref 119896 = 4) 0004lowastlowastlowast minus0815lowastlowastlowast minus0002lowastlowastlowast minus0001(0001) (0276) (0001) (0001)

119896 = 16 (ref 119896 = 4) 0008lowastlowastlowast minus1118lowastlowastlowast minus0004lowastlowastlowast minus0003lowastlowastlowast(0001) (0276) (0001) (0001)

119861119890ℎ119886V119894119900119903 = ldquoFocusedrdquo (ref ldquoNaiverdquo) 0015lowastlowastlowast minus1980lowastlowastlowast minus0012lowastlowastlowast minus0004lowastlowastlowast(00005) (0225) (00004) (0001)

120574 = 01 (ref 120574 = 005) minus0020lowastlowastlowast 5762lowastlowastlowast minus00003 minus0001(0001) (0276) (0001) (0001)

120574 = 02 (ref 120574 = 005) minus0025lowastlowastlowast 8648lowastlowastlowast minus00003 minus0001(0001) (0276) (0001) (0001)

119901119886 = 05 (ref 119901119886 = 03) minus0013lowastlowastlowast 1581lowastlowastlowast 00004 00003(00005) (0225) (00004) (0001)

Observations 19440 19440 19440 19440R2 0931 0844 0562 0958Adjusted R2 0931 0844 0562 0958Residual Std Error (df = 19427) 0033 15701 0030 0050F Statistic (df = 12 19427) 21 844790lowastlowastlowast 8 733563lowastlowastlowast 2 079585lowastlowastlowast 36 761250lowastlowastlowastNote lowastplt01 lowastlowastplt005 lowastlowastlowastplt001

opinions that may be of interest in the mixed discussionmodel

Group coherence shifts of opinions and the varianceof opinions are highly affected by the voting thresholdthe coherence metric is reduced by half when allowing forclassic majority voting shifts explode and the variance ofopinions increases dramatically This is not surprising sincethe majority quota rule works as a buffer to the success ofdecision-making processes and in consequence it hampersthe deliberative opinion updates For the shifts of opinions itis clear that the two-third majority voting rule results in littleopinion change and that the simple majority rule inducesgroup polarization As expected there are less extremists inscenarios where there is no voting and the shifts of opinionare more likely to happen when agents are more sensitiveto deliberation When we include voting in the system

the grouprsquos judgment accuracy decreases due to the correctdeliberated decisions that are vetoed by a salient opinion-agent majority Strangely the effect only holds for the simplemajority decision rule Lastly the proportion of individualsthat participate in the debate and the minimal amount ofdebates have similar marginal effects on all the metrics (seeTable 4)

415 Comparing Deliberative and Pair-Wise Opinion Dynam-ics in a Nuthshell Qualitatively both models yield similaroutcomes but for the fact that deliberation never producesopinion polarization with both types of extremists (seeFigure 7) Furthermore clusters in the deliberative interac-tions model are less homogeneous in size and multiclusterconvergence occurs only when deliberation is rarely suc-cessful which happens sporadically given the calibration of

18 Complexity

Table 5 Mean difference 095 confidence intervals for metrics by model comparison

Model Metric119881119886119903(119900) 119901119903119900119901119890119909 119878ℎ 119890119888 119894119903Dyadic vs Deliberative [0191 0216] [0170 0197] [-2948 -2828] [0074 0082] [0165 0172]Mixed vs Deliberative [0142 0147] [0150 0155] [-2085 -1976] [0003 0002] [-0038 -0030]Mixed vs Dyadic [-0072 -0046] [-0045 -0018] [8332 8859] [-0079 -0071] [-0205 -0201]the model On the other hand pair-wise opinion dynamicsproduces multicluster convergence most of the time andcentral convergence only in the infrequent cases where thelatitude of acceptance is large

The comparison of these two scenarios in terms ofthe metrics is straightforward Whilst dyadic interactionsallow for more variability in opinions through pair-wiseinteractions and more mistakes in judging proposals delib-erative updates simply act as an antagonist force On aver-age the deliberative interaction model produces opiniondistributions with smaller variance and yields steady stateswith higher judgment accuracy a higher shift statistic andnaturally lower group coherence since only proposals that aredeliberated 119894119899 are eligible for voting (see Table 5)

42 Articulating Dyadic and Deliberative Social Interactionsin Opinion Dynamics So far we have considered scenarioswhere argumentation had no place and where individualsmade decisions according to opinions that were purelyconstructed from pair-wise discussions We also observedcases in which agents formed their opinions only by inte-grating deliberated proposals into their opinion updates Thescenarios we present in this section combine the two pre-ceding scenarios to account for both the effects of pair-wisediscussion and the effects of deliberation on opinions judg-ment accuracy and group coherence From the precedingobservations we conclude the following deliberative opiniondynamics reduces the variance and the proportion of extrem-ists because it shifts opinions greatly and polarizes groupsby eliminating at least one type of extremist the one whoseopinions are initially opposed to the deliberated resultsor it brings opinions close to neutrality (see Figure 7(b))This may not be desired in a group since the diversity andstability of opinions are necessary for deliberation to makesense and lead to the collective acceptance of controversialproposals Dyadic opinion dynamics on the other handtend to increase the variance of opinions and the numberof extremists while having a lesser aggregated effect on theshifts of opinion This is an immediate result of the pair-wiseinteraction dynamics inasmuch as it polarizes individualsvery quickly

To sum up in the mixed interactions model deliberationhinders the effect of voting on opinions by making someproposals that would normally be submitted for voting noteligible for voting It may also undo opinion changes dueto pair-wise interactions The mixed model can be seen asthe deliberation opinion dynamics model in which pair-wiseinteractions occur in between collective discussion and mayaffect which arguments are advanced during deliberationEquivalently it can be seen as the dyadic opinion dynamics

model in which deliberation accounts for endogenousldquoshocksrdquo on the opinion distribution

421 Qualitative Analysis of the Mixed Dynamics In Figures7(c) and 7(f) agents update their opinions through delibera-tion and pair-wise interactions Opinion trajectories visualsshow a combination of both previously described opiniontrajectories Over time as agents discuss with one anotheropinion clusters form as in the pair-wise or dyadic opiniondynamics Deliberation may disrupt the formation of theseclusters and three different situations may occur

(i) Deliberation disrupts the formation of the opinionclusters but not sufficiently to jeopardize pair-wiseopinion cluster formation (eg for high values 119905119863)Two phenomena may be responsible for it (1) thevariability of the proposal argumentrsquos support for theprinciple P and (2) too few (successive) deliberationsteps that result in opinion updates Once the opinionclusters are formed successful deliberation foreshad-ows the merging of the previously formed clustersmay bring extremists closer to moderate agents andthanks to the assimilation component of the pair-wisedynamics allow bigger and more moderate opinionclusters to form (refer to Figure 7(c))

(ii) Deliberation significantly disrupts the pair-wisedynamic formation of clusters This may happenwhen the requirements for accepting proposals areweak in the decision-making process (eg low 120572) Inthis case one has either central convergence sinceall agents follow the deliberated results in the samemanner or 3-cluster convergence with two extremeopinion foci In the latter situation the extremegroups form quickly (like in the pair-wise model) anddeliberation cannot pull extreme agents away fromthe extreme foci they are in The main reason for thisis that immediately after an extreme agent attempts toleave her foci through deliberation updates she getspulled or pushed back into it by means of a dyadicinteraction with another extreme agent

(iii) Deliberation does not disrupt the formation of theopinions cluster This is the rare case scenario forvery demanding decision-making processes in termsof inclusion (eg 119899119863) and argument acceptability (eg120572) Opinions converge as in the pair-wise opiniondynamics model (refer to Figures 7(a) and 7(d))

Last when there is voting (120572 = 0) the position ofthe moderate opinion cluster is determined by the first

Complexity 19

minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

Var(o)

sh

ec

ir

Var(o)

propex

sh

ec

ir

1

098

minus02

minus004

001

1

minus014

minus001

minus003

1

026

minus028

1

minus057 1

ＪＬＩＪ

Ｒ

Figure 8 Significant correlation estimates for themetrics of interestin the mixed opinion dyamics scenarios

accepted proposal argument andor the opinionmajority thatis formed from previous dyadic interactions (see Figure 7(f))

422 Relating the Metrics of Interest in Mixed OpinionDynamics The first thing to notice is that for all of thesescenarios all the metrics are significantly correlated (seeFigure 8) The variance of opinions and the proportion ofextremists are almost perfectly correlated To some extentthis means that a high variance of opinions known thatthe mean of opinions is statistically null equates to havingmany extremists in the population be towards one or theother side of the opinion distribution (eg extreme bipolar-ization) Similarly we find a negative and strong correlationbetween the judgment inaccuracymeasure and the coherencecoefficient (see Figure 8)mdashwhen judgment inaccuracy is high(low) coherence to deliberation is low (high) Cases likethese are telling since they may be related to sequences ofcollective decisions in which agents do not make too manymistakes when judging proposal arguments and still makedeliberated decisions that are on average in line with theirprinciples

The shift statistic interestingly is negatively correlatedwith the variance of opinions with judgment accuracy andwith the coherence coefficient In a way this means thathaving more moderate agents conduces on average to moremistakes in judging arguments and to less coherent collectivedecisions An explanation for this phenomenon could be thatas agents reach consensus in opinion important (possiblyextreme) arguments for correctly judging argument propos-als are not advanced in the deliberation arenawhich results inmore mistakes and in possibly accepting proposal argumentsthat after voting are not meant to be accepted Indeed the

negative correlation between the variance of opinions andjudgment inaccuracy points towards this direction since onaverage the more shifts there are the smaller the variance ofopinions is (see Figure 8)

423 Comparing Mixed Dynamics with Other MonolithicDynamics of Interactions Qualitatively opinion trajectoriesin the mixed model are more chaotic than in the pair-wise and deliberative interaction models (see Figure 7) Asanalyzed previously deliberation dynamics disrupt the pair-wise dynamics and create situations in which central andmulticlustered convergence (with two clusters at the extremesof the distribution) are possible at the expense of more(pair-wise dynamics) or less (deliberative dynamics) clustersIt tolerates the survival of very small isolated group ofagents that the previous models rarely tolerate in a hundredcollective decision-making steps (see Figure 7(c))

One of the most interesting observations one can makeof this family of models regards how groups change (in termsof our metrics) when dyadic social interactions are addedto deliberative interactions and vice-versa When comparingthe mixed interactions with only deliberative ones shiftsagentsrsquo aggregated change in opinions are on average lesspronounced The variance of opinions and the proportionof extremists are significantly higher for the former than forthe latter group coherence in contrast is lower with respectto the scenarios where opinions are formed only throughdeliberation Judgment accuracy follows that same trendA reason for this might be that mixed dynamics are fasterto yield semistable clusters andor opinion convergence Inconsequence the potential argument pool for deliberativeexchange shrinks faster and deliberation becomes less effec-tive over time which results in more collective mistakes Theoutcomeon coherence is predictable inasmuch as deliberativeinteractions produce group polarization If a group agrees inopinion debates are sterile because themembers of the groupwill most likely refrain from attacking the proposal argu-ments they all support and will work together to disparagethe proposal arguments they do not support (see Table 5)

In the same manner we report that the proportion ofextremists and the variance of opinions are barely lower inthe pair-wise interaction model when including deliberationin the decision-making process Shifts of opinions are alsobarely lower when there is no deliberation This can indicatethat deliberation does not undo the changes in opiniondue to pair-wise interactions nor does it amplifies them Incontrast when we consider the labeling-based metrics themixed interaction model ensures lower judgment inaccuracybecause of deliberation and lower group coherence (seeTable 5)

In sum the mixed discussion model is a compromisebetween the pair-wise interaction and deliberative opiniondynamic models in that it offers some guarantees in termsof judgment accuracy and coherence while allowing forldquoreasonablerdquo variance in the opinion pool It establishes somekind of trade-off between labeling-based metrics and thevariance of opinions that may be of interest for decision-making process design

20 Complexity

Table 6 Summary of sign or relative marginal effects for all varying parameters on dyadic deliberative and mixed interactions models inand for all experiments

Metrics Parameters119905119863 119898 119899119863 120572 119879 119880 119901119886 120574 119896 119861119890ℎ119886V119894119900119903119881119886119903(119900) XX X X O X X X X X X119901119903119900119901119890119909 XX X X O X X X X X X119878ℎ XX X X O X X X X X119890119888 XX X X XO X XOO XOO X X119894119903 XX X X O XO O X OOX XOO X XNota gray symbols correspond to dyadic dynamics light gray to deliberative dynamics and bold black to mixed dynamics Given a scenario the symbol 119883stands for constant parameter and 119874 for insignificant marginal effect and as parameters grow for positive (negative) marginal effect for the parameterFrom left to right at the parameter level the leftmost arrow corresponds to the scenario with the strongest (in absolute value) marginal effect of the parameter(with respect to to its factors) Example Let 119901 be a parameter and 119874 stands for ldquo119901 has a significant positive effect in dyadic discussion scenario negativein the deliberative one and insignificant in mixed discussion The positive effect in dyadic discussion is greater than the absolute value of the negative one inthe purely deliberative scenariordquo

424 Comments on the Differences between the Three Modelsfor OLS Regression Estimates The significance of most of theregression parameters estimated for the mixed interactionsscenarios are similar to those estimated for the pure dyadicanddeliberative scenariosNevertheless themagnitude of theeffects is different insomuch that the dynamics themselvesare different So instead of commenting the linear regressionestimates of the mixed model we compare when possiblethe standardized significant coefficients of the regressors ofthe mixed model with those of the pair-wise and deliberativeinteractionmodels (see Table 6 for a summary of the results)

In mixed interactions for instance the influence of119898 onthe opinion distribution is much more preponderant than inthe deliberative dynamics scenarios Going from 119898 = 1 to119898 = 3 increases the standard variation units of the variance ofopinions by twice as much as the one in the deliberative sce-nariosThis in particular is due to the design of the decision-making process If 119898 is higher then the number of socialinfluence steps (119905119863) in the decision process is also higher(compared to 119905119863 = 0 in the deliberative case) If we decideto consider these steps as contributing to the variance ofopinions then the effect of119898 on the variance of opinions hasto be stronger in the mixed interactions model In contrastthe number of agents participating in the debate has a weakereffect on the variance of opinions in the mixed interactionsmodel than in the deliberative dynamics specially for 119899119863 =005 (more than twice as much) The difference may comefrom the fact that in mixed discussions the part of theargument pool used in deliberation is more restrictive thanin the deliberative case We can argue that since pair-wisedynamics may get individuals closer to one another aftersome deliberation the same happens to the arguments thatindividuals will most likely use in deliberation Taking thisinto account means that deliberation will at many instancesbe unfruitful and have no effect on opinions thus failing tomoderate extreme opinions andor polarize the group In thesame direction a high number of participants in deliberationcan make the deliberation dynamics stiff In this case thedistribution of opinion is close to the uniform distributionwhich has a relatively high-to-moderate variancewith respectto the reference results

The strength of the deliberative dynamics has a positivestronger effect on the variance of opinions in themixedmodelthan in the deliberative model when 120574 = 02 and a negativeweaker one when 120574 = 01 A sound interpretation of thisresult is that when deliberation is successful and its effect onopinions does not get agents to move sufficiently away fromtheir current positions agents are pulled or pushed back totheir previously held opinions the variance of opinions doesnot changemuchOn the other hand if the effect of successfuldeliberation is strong opinions can change sufficiently tomake agents much more prone to assimilation into moderateopinion foci the variance of opinions falls

The deliberative opinion dynamics model is much moresensitive to 120572 than the mixed discussion model Differ-ences in marginal effects can double The probability ofattraction to deliberated results (119901119886) for instance has astrongmoderating effect (negative) on agentsrsquo opinions in thedeliberative opinion dynamics but a meager one (two-fold)in the mixed and dyadic dynamic models This is becauseopinion updates in the deliberative model depend only onthat parameter whereas in the mixed model more numeroustypes of interactions make opinion updates possible

In terms of judgment accuracy all parameters except thesize of the argument sacks (119896) have a significant slightlystronger effect on the deliberative dynamics than in themixedmodel In effect the difference may be attributed to theupshot of pair-wise interactions on the distribution of opin-ions and on the arguments that would be advanced duringdeliberation The result respecting parameter 119896 however iscounterintuitive Since argument sacks are static scenarioswhere shifts are higher should make 119896 more important fordeliberation to succeed and it is precisely in the scenarios ofpure deliberative dynamics that we find the biggest opinionshifts This may point to the hypothesis that in deliberativeinteractions agents change opinions substantially but do notoften change their adherence to the principle (go from anegative opinion to a positive one or vice versa) when thereis no group polarization

43 Sensitivity Analysis for the Procedural Parameters ofInterest Weperform a sensitivity analysis of the observationson the parameters of interest in themixed interactionsmodel

Complexity 21

We extend the parameter domains of the pure dyadic anddeliberative scenarios as we make it explicit in Table 2

431 Minimum Number of Debates (119898) We observe that theminimum number of debates has a significant well-observedeffect on all of our metrics excluding group coherence Forthe variance of opinions we can observe that the moredebates there are the bigger the value of the metric isalthough the higher 119899119863 and 119905119863 are the weaker the overalleffect of119898 becomes (Figures 9(d) and 9(a)) Furthermore themarginal effect of increasing minimal debates is decreasingan additional deliberation step requirement increases thevariance of opinions but less and less as it grows 119898 coupledwith 119899119863 has a shy S-shaped effect (see Figure 9(d)) on thevariance of opinions meaning that it has a stronger effectwhen 119899119863 is low which suggests a possible trade-off betweenthe number of arguments accepted in the debate and thenumber of debates required before accepting a proposal

In contrast the shifts in opinion are less and less likelyas 119898 grows and this is independent of the variations of theother parameters Again the effect is marginally decreasingand is only truly significant when 120572 = 05 An explanationto these effects may be linked to the design of the systemFirst the variance of opinions is higher when deliberation isasked for because the more deliberation steps there are thehigher the chances are that the central proposal argumentis deemed unacceptable specially when agents are focusedFurthermoremechanically speaking increasing theminimalamount of debates implies that whenever a decision is to betaken at least 119898 times 119905119863 time steps have to take place and if atany moment a proposal argument is considered undecided119905119863 time steps are added to the process So unless the debatedoes not yield 119906119899119889 labels for proposal arguments (highlyunlikely considering that 120590 = grounded semantics) the morenondeliberation steps there are in the decision process thehigher the variance of opinions is Concerning the shiftswhen 120572 = 05 either the system is too stiff to accept anyproposal argument and opinions do not changemuch or theeffects of pair-wise discussion and deliberation cancel out insuch a way that individual shifts are minimal (Figure 9(b))

On the side of labeling-based metrics the more debatesare asked for the more accurate a group is in its judgmentthe effect being smaller as 119898 grows When agents are naivethe effect is quasilinear while when they are focused thestrongest effects of acquiring more deliberation are foundwhen levels of deliberation are already low (Figure 9(f))Thiscan be explained by the fact that the more debates thereare in the decision process the closer one gets to the idealargumentation framework

432 Proportion of the Population in Deliberation Steps (119899119863)Like for 119898 119899119863 has a significant effect on the proportionof extremists and on the shifts and variance of opinions(Figure 9(e)) This may result from the fact that being ableto put more arguments in play at the same debate step mayincrease the odds of revealing cycles around the proposalargument in119860119865120576 Given that we use grounded semantics thearguments in the cycles are labeled 119906119899119889 and in consequence

debates are more often postponed when 119899119863 is higher Post-poning debates in turn increases the number of nondebatessteps in the decision process which increases the varianceof opinions and limits over time the moderating effect ofdeliberation Moreover the effect of this parameter is verydependent on the value of 120572 (Figure 9(e)) For shifts forinstance 120572 = 05 makes the effect of 119899119863 negative while120572 = 0 makes it positive but to a lesser extent Furthermorefor higher requirements of deliberation (119898) adding moreindividuals to the deliberation process has weaker effectson the variance of opinions and on the other metrics It isalso quite interesting to notice that it has the same effect onagents independently of agent behavior during deliberationThis is a surprising result as one would have expected morefocused agents in a deliberation arena to cause a raise inthe proportion of extremists as they play to knock outopposing proposal arguments and thereof reducing the effectof deliberation (lowers variance) on the whole population

Finally similar to119898 addingmore people into the deliber-ation process increases judgment accuracy (see Figure 9(c))and has no clear effect if any on group coherence (seeFigure 9(i))

433 Steps between Deliberation Steps (119905119863) In all configu-rations 119905119863 increases the proportion of extremists (variance)and decreases the shifts of opinions among agents Theshifts and the effect on the variance of opinions are presentfor 120572 = 05 insignificant otherwise (Figure 9(b)) 119905119863 ishighly linked to 119898 by construction When 119898 = 1 thecurve linking the variance of opinions and 119905119863 is convex As119898 increases the curve becomes more and more concavewhich means that 119905119863 has a more important effect on theopinion distribution as collective decisions take longer to beachieved This seems counterintuitive but in reality it reflectsthemultiplicative relationship between deliberation and pair-wise interactions and the semantics in the model If 119898 is lowand 119905119863 is high the effective number of pair-wise interactionsis on average smaller in the deliberation process Hencethe structure of the decision-making process hinders anyincrease in the variance of opinions If119898 is high the oppositeeffect is observed Additionally since the grounded semanticsyields few 119894119899 arguments with respect to other admissibility-based semantics getting closer to the ideal argumentationframeworkmaymake it difficult to obtain the expected effectsof deliberative discussion on agents Lower 119898 constrains thevariance of opinions by making deliberation more influentialon opinions

In terms of judgment accuracy and coherence we observethat 119905119863 does not explain coherence yet it tends to decreasejudgment inaccuracy This may be due to a more variedargument pool in the debate which in turn may be aconsequence of a higher variance of opinions or vice versa

434 Acceptability Voting Quota (120572) By far it is the mostinfluential parameter in our study It changes the directionand the intensity of the effect of all the other proceduralparameters and by construction heavily constrains the roadto accepting a proposal (see Figure 9)

22 Complexity

015

020

025

030

035

040

1 2 3 4 5 6

12345

Values of Ｇ

(a) 119905119863 times 119901119903119900119901119890119909 by119898

0

10

20

30

40

50

1 2 3 4 5 6

005066

Values of

(b) 119905119863 times 119878ℎ by 120572

040

045

050

055

060

1 2 3 4 5

001002003004005

Values of Ｈ＄

(c) 119898 times 119890119888 by 119899119863

015

020

025

030

035

040

1 2 3 4 5

001002003004005

Values of Ｈ＄

(d) 119898 times 119901119903119900119901119890119909 by 119899119863

0

10

20

30

40

50

001 002 003 004 005

00005066

Values of

(e) 119899119863 times 119878ℎ by 120572

040

045

050

055

060

1 2 3 4 5

BehaviorNaiveFocused

(f) 119898 times 119890119888 by 119887119890ℎ119886V119894119900119903

015

020

025

030

035

040

0 05 066Behavior

NaiveFocused

(g) 120572 times 119901119903119900119901119890119909 by 119887119890ℎ119886V119894119900119903

05

06

07

08

09

10

1 2 3 4 5 6

00005066

Values of

(h) 119905119863 times 119894119903 by 120572

08700

08725

08750

08775

08800

001 002 003 004 005

BehaviorNaiveFocused

(i) 119899119863 times 119894119903 by 119887119890ℎ119886V119894119900119903

Figure 9 Curves of mean observations for different parameters and metrics for 36000 runs Bars are 095 confidence intervals Plots withthe same observables (ordinate) except for 119894119903 are on the same scale

Complexity 23

The higher the requirement for accepting the proposal isthe higher the proportion of extremists and the variance ofopinions are and the lower the shift statistic is In few words120572 constrains the world to dyadic discussion or throws it intoa process in which deliberation is much more importantthan dyadic interactions This is why for any 120572 = 05 oneeither gives too much weight to deliberated results whichshadows pair-wise interactions or too little weight in thesense that no deliberated result is ever accepted and thusnever integrated into the opinions of the agents that havevoted for it The reason for the latter is that the latitudeacceptance is too low and thereafter pair-wise interactionsare not enough to unevenly polarize the population in such away that deliberated proposal arguments are voted favorablyby a non-negligeable majority Shifts for 120572 = 0 are stable fordifferent levels of 119905119863 which implies that deliberation keeps incheck all the shifts of opinions related to an increase in thenumber of pair-wise interactions For 120572 = 05 deliberationis less likely to be successful and shifts tend towards the shiftlevels for 120572 = 0 (see Figure 9(b))

Concerning labeling-basedmetrics120572 entirely determinesthe coherence statistic For 120572 = 05 there is no difference incoherence because of how coherence is defined it is maximalwhen 120572 = 0 and strangely maximal when 120572 = 066(see Figure 9(h)) Surprisingly 120572 has no effect on judgmentaccuracy One would have expected a higher 120572 to increasejudgment accuracy since agents would hardly collectivelyaccept a proposal argument discussed during deliberationProposal arguments have given how the argument lattice isgenerated a higher probability of being rejected than of beingaccepted in the ideal argumentation framework

435 A Word on Focused and Naive Agents Focused andnaive specifications for agent behavior are an importantparameter in the decision process since they model howagents choose arguments in the deliberation arena In the sce-narios where agents are focused agents show equal or highervariance of opinions and are therefore more extreme (seeFigure 9(g)) independently of the voting requirements Shiftsof opinion are less likely as well We can see this happeningbecause focused agents knock out proposal arguments moreoften than naive agents provided that 119899119863 is low

One could have expected naive agents to yield morecoherent decisions since they argue sincerely However thisseems not to be the case (see Figure 9(i)) Agents are unable todeliberate in away thatmakes deliberation reflect their votingintentions either because they do not have the necessaryarguments to do so or because such arguments do not exist

In terms of judgment accuracy focused agents do betterthan naive agents in finding the ideal labels for the proposalarguments This is likely because when reconstructing theframework focused agents take into account the deliberatedproposal argument and choose themost pertinent argumentsin their sacks that relate to the state of affairs at the deliber-ation arena In fine debates result in a better approximationofL119886119887120576(119868) even though one could have believed the oppositethat is focused agents are ready to sacrifice a correct collectivelabel of a proposal argument in the goal of defending theirpositions

44 Points of Discussion In this subsection we discuss theresults obtained from the simulations We attempt to makesense of them from a social sciences perspective

441 Opinion Consensus and Dissensus The deliberativeopinion dynamics corresponds to the simulated scenarios inwhich the variance of opinions is at its lowest and judgmentaccuracy at its highest Dyadic opinion dynamics are associ-ated with the scenarios in which the variance of opinions is atits highest judgment accuracy at its lowest and coherence atitsmaximumpossible levelThemixed interactions dynamicsis a combination of these aspects for all the metrics exceptfor coherence at which it is the worsemdashdeliberation leadsto some kind of loose opinion consensus that is reinforcedby pair-wise interactions Likewise the mixed discussionmodel confirms some results exposed in [16] expressingthat opinions and voting intentions in a deliberative contextldquooften changerdquo That being said and more particularly indeliberative opinion dynamics outputs align well with twointeresting results on consensus formation given by aninterpretation of Moscovici and Doisersquos [15] work Sunsteinrsquos[14] account on deliberation and group polarization andsocial impact theory [4] The first states that mild consensusis reached in deliberation as procedure disengages agentsthe second is that within discussion groups majorities tendto become larger and opinions more extrememdashnot on thebasis of competing arguments but of the initial balance ofopinions After several decision processes we find opiniondistributions that reflect these observations even though weare not able to precisely coin the set of parameters that yieldssuch distributions (see Figure 7)

Although differences between mixed and dyadic opiniondynamics in terms of the variance of opinions and shiftsare slim the former does much better in terms of judg-ment accuracy In this sense deliberation does moderateopinions but not as much when non-deliberation steps arerelatively frequent throughout the decision-making processIn terms of Moscovici and Doise theory of consensus [15]this makes sense since pair-wise discussion engages or ego-involve agents in the discussion and pushes to convergenceof opinion at the extremes of the opinion distributionHowever what remains unclear is the reason why proceduralparameters that constrain deliberation do not disengageagents from discussion and thus result in a mild consensusas also expressed by Moscovici and Doise [15] A possibleexplanation of this phenomenon may stem from the factthat deliberation in the model is always accompanied by aseries of ego-involving discussions that shadowor counter themoderating effect of deliberation Not only this deliberationcould also be responsible for crystallizing the opinions ofextremists for it favors one or the other part of the distributionof opinions as explained in [14] and to some extent in [13]Furthermore the mixed discussion model gives an exampleof polarization that results from a minority of dissenterson one side of an issue acceding to views of the majorityrsquosside [4] as opposed to the strong polarization observed indyadic interactions Situations in which the status quo orthe undecidability of the proposal argument lingers cannotpossibly yield situations of moderation In cognitive science

24 Complexity

one can be inclined to believe that the longer a deliberationprocess is the more likely individuals will take an immutablestance on the situation In view of the assumption that longdecision-making processes can be more costly for a group orsociety as a whole agents may feel forced to take a stancejust for the sake of ending the process The stance that isadopted would therefore depend on pair-wise interactionsamong individuals and not on deliberation since deliberationtakes too long to be conclusive

442 For Hypothetical Recommendations in Decision-MakingProcess Elaboration On another tune results from themodelsuggest that procedural parameters have marginally decreas-ing positive or negative effects on the metrics of interest Ina hypothetical policy analysis perspective this may indicatethat after choosing a correct number of individuals andimposing the right amount of debates one can attain rea-sonable levels of judgment accuracy coherence and varianceof opinions As accounted in [18] deliberation increasesknowledge (collective knowledge as well) and therefore itheightens the chances of making correct decisions If onebelieves that social welfare is linked to deliberation andeventually to the metrics studied previously then proceduralparameters can both be used as an instrument to attaindesired levels of social welfare and reinforce the notionof legitimacy of collective decisions It can also provide ajustification for choosing a deliberation regime rather thananother on the basis of how important (or urgent) a topic ofdiscussion is Indeed if we adopt the claim that deliberationis a forerunner of welfare then any state in which deliberativecues are at their maximum (highest number of participantsmany deliberation steps before making a decision and asfrequent as possible) has to be mapped to the highest attain-able social welfare In this case is a situation where judgmentaccuracy is at its highest yet the variance of opinions or theproportion of extremists at its lowest an ideal situation Wecannot say for sure since strong opinion consensus and longdecision-making processes in many situations may result ina loss of welfare rather than in a gain of it

Another interesting result in procedure for deliberationis the very similar estimates we find for the effects of theminimum number (119898) of debates necessary to make adecision on a proposal and the proportion (119899119863) of individualsparticipating in deliberation on judgment accuracy In thescope of social welfare this raises the question of whetheradvocating for longer decision processes by demandingmoredeliberation steps (higher 119898) rather than bigger debates(higher 119899119863) may increase welfare Depending on the objec-tives of the decider of the deliberation procedure he or shemay choose to concentrate on one or the other A recom-mendation we can assert from the mixed social interactionsmodel is that if one prefers low to high variance of opinionsandor high to low coherence the decider is better off ifshe focuses on increasing the size of the debate rather thanorganizing debates very frequently One of the reasons forthis might be that when making bigger debates all agents aretruly considered as equals in the decision-making processIn organizing many debates individuals that are chosen toparticipate in many of these will be more influential and will

somehow bias the set of arguments in the deliberation arenaThemajority rulemodeled through120572 also determines the sizeof the shifts and the distribution of opinions Because a bigger120572 is associated with higher levels of extremism and opinionvariance deliberated cues are harder to accept and thereforeit becomesmore difficult for agents to reach concensus andormoderate their opinions

On the other hand and further into the idea of legitimacythe goal of deliberative processes is the agreement amongall participants This agreement stands for the right or bestdecision from a formal or procedural perspective and it is alsothe best from a substantive point of view In other words thelegitimacy of decisions not only derives fromwell-establishedprocedural requirements (respect of the protocol) but alsobegs for the fulfillment of two essential conditions (1) satisfy-ing the procedural requirements for a correct procedure (for-mal legitimacy) and (2) the rational acceptability of the resultsof this procedure (substantive acceptability) Legitimacy canbe then obtained from the coherence statistic scrutinizedin the model insofar as it measures how well agents acceptthe results of a correct procedure (eg deliberation) So fora proposal to be legitimate one has to choose parametersthat would maximize group coherence in decision-makingprocesses

As deliberation may give rise to consensus in the delib-eration arena it may also sow the seeds of dissensus inthe group Our model clearly illustrates this phenomenonthrough the procedural parameters of deliberation and themetrics observed From another perspective in deliberativedemocracy dissensus in collective decision-making may bedesirable According to Landermore and Pagersquos accounts [36]lack of consensus can be perceived as having agents withalternative ideas that combined with other ideas can providea better approximation of the ideal status of an argumentTheauthors in [36] call this dissensus ldquopositive dissensusrdquo andit makes normative sense when solving complex collectivedecision-making problems Our model captures the ideaof having normative requirements to define a good andlegitimate collective decision requirements of high consen-sual deliberative accuracy (judgment accuracy) of legitimacy(coherence) and of diversity (variance of opinions) are alltaken into account and for such we can conclude that ourmodel is successful in describing a sound process of collectivedecision-making with interpretable outcomes

Of course the present simulation results and analysis donot prove that all opinion dynamics can be accounted forby such simple processes like the one presented here Theydo however shed light on the expressiveness of combiningdifferent paradigms to account formore interpretablemodelsand pave the way to creating original models of the sortThe simulations suggest the desirability of discovering theconsequences of relatively simple laws of communication atdifferent levels (micro meso and macro) to determine whatstill needs to be explained

5 Related Work

We see our model as a contribution to the influence andopinion dynamics field in ABM and a pragmatic application

Complexity 25

of abstract argumentation theory To our knowledge we areunaware of existing literature on agent-based modeling thatexplicitly relates collective choice and the notions of deliber-ation and opinion diffusion through abstract argumentationas we have done it

Most models in the literature of opinion diffusion areinterested in opinions because these have an influence incollective decision and in questions of social order Forinstance in [3] the authors are interested in consensus andin how a group collectively chooses among two alternativesIn other models authors are interested in the emergenceof extremism [8] and on the distribution of opinions whenextremists are introduced in the population [5 33 37] whileother authors coin the notion of opinion polarization as anemergent property of the system [8 12] They show usingmodels of ldquobounded confidencerdquo and opinion diffusion withtrust that three different kinds of steady states (unipolarbipolar and central) are possible depending on whetheragents are sufficiently uncertain about their opinions andsufficiently connected andor a certain proportion of indi-viduals are already extreme Recent articles on informationand opinion dynamics stress the importance of governanceand government intervention in the spread of emotions(opinions) in the frangibility of social consensus systemIn [10] the authors show that government intervention inthe spread of negative emotions can lead to an even fasterspread of negative emotions (single-peeked convergence) andin a faster collapse of the social consensus system Opinionas a function of trust is studied in [12] and relates to howindividuals form their opinions on the basis of how muchthey trust agents in their networks

Another stream of discrete opinion dynamics modelsthat explain the emergence of opinion can be found in theliterature Computer multiagent models of attitude changebased on Latanersquos social impact theory are presented in [4] Inthe theory an agent changes her opinion on the basis of theinformational impact she is subject to which depends on thepersuasiveness (strength) supportiveness and immediacy(group structure) of the environment and information shereceives The simulations predict two emergent groups ofphenomena the shifting of attitudes towards incompletelypolarized equilibria and the formation of coherent clusteringof subgroups with deviant attitudes Similarly but in acontinuous more argumentative fashion Mas and Flache[11] present a model of opinion diffusion using argumentsArguments are considered to be for or against a proposal andare given an agent-dependant relevance in dyadic persuasiveinteractions Arguments also determine the agentsrsquo opinionsand their dissimilarities which posits the rules of pair-wiseencounters in theirmodel as agents are assumed to form theiropinions according to the arguments they own They showthat in an argumentative opinion diffusion model the onlyobservable steady states are bipolarization and consensusand that bipolarization can emerge from interactions amongsimilar agents In other words bipolarization of opinionsis possible with homophily and without negative influenceif there is argumentation They subsequently compare theirresults to experimental data In [38] a similar approach toargumentation and opinion formation is taken but for the fact

that they introduce different types of arguments explicitlyapply social judgment theory [22] and use survey data toestablish model-to-real-world comparisons

Closer to opinion formation and abstract argumentationGabbriellini and Torroni [39] are the first to have origi-nally and soundly merged opinion diffusion and abstractargumentation They define an agentrsquos opinion as a functionof the arguments she holds and the attack relation amongthem They devised a focused peer-to-peer dialogue systemof persuasion (NetArg) inspired from Mercier and Sperberrsquosargumentative theory of reasoning [40] which used onlyabstract argumentation to study opinion polarization andopinion dynamics Moreover they considered networks andthe notions of trust and epistemic vigilance to define adynamics for knowledge revision and trust itself whichwe clearly lack in our model They showed that if a con-servative belief operator in argumentation was applied byagents when they reasoned then their dialogue protocoldid not increase polarization among agents That said theyused the model to study the effect of Groenvetterrsquos weaklink theory in the spreading of arguments and were notparticularly interested in notions like judgment accuracy orcoherence nor in tackling questions related to collectivedecision-making procedures Their system is very expressiveand helps position our work in the litterature our work is atthe frontier of pure argumentative opinion diffusion modelsopinion diffusion with arguments and continuous bounded-confidence models of opinion diffusion a la Deffuant

On another note work on collective cognitive conver-gence [41] and opinion sharing [19] shows that consensustowards a certain ldquocorrectrdquo opinion or cognitive state is alwayspossible yet dependent on noise variability and awarenessof agents In [20] the authors show that learning about anexogenous correct state of the world (represented by bits)under confidence was possible but only if the agents werenot too confident In a homogeneous population they showthat the higher the confidence the worse the learningmdashthevery confident agents do not learn the properties of the truestate of the world and disrupt the learning process of the lessconfident ones Collective cognitive convergence can be seenas a result of deliberation in truth-seeking models

When it comes to abstract argumentation theory we takean approach that wires two types of dialogues that are well-studied in the literature persuasion dialogues [42] and delib-eration dialogues [43] Another interesting line of work is theone on mechanism design [44] or the problem of devisingan argumentation protocol where strategic argumentation isnot a liability for success in debates We tackle mechanismdesign in a different way though Instead of consideringstrategy-proofness we are interested in how differences inprotocol can result in epistemologically ldquobetterrdquo collectivechoices and guarantee that opinion distributions are favorablefor deliberation For a survey on persuasion dialogue see[42]

Work on agent-based argumentation usually assumes thatthe semantic relationship between arguments is fixed Inother words if two individuals were to put two argumentsin the public arena such that one logically attacked the otherthen everyone would agree that such attack exists [45 46]

26 Complexity

Other models which do not make this restrictive assumptioncan also be found in the literature [27 47] and derive fromthe class or family of opponent models [27 48] in which twoopposing sides attempt to win a dialogue Ourmodel is in theintersection of these two but the frame combining opiniondiffusion of the kind and argumentation remains new

The idea of mixing interpersonal influence and verticalcommunication however is not original It is described andimplemented in innovation diffusion models such as [5 49]In both vertical communication is modeled as exogenoustransparent informationmdashagents are aware of the existenceof an innovation thus triggering several processes of choiceand stabilization of opinions Also in [50] an Eulerian modelis implemented to show the effect of media and exogenousinformation on opinion distributions With respect to thispoint the originality of our work is in that the informationemitted as vertical communication is endogenous It is issuedfrom a deliberation model that agents shape on the basisof their opinions arguments and behavior In the spirit of[51] where the authors control for the design of verticalcommunication (to whom the vertical communication isaddressed its timing how it influences agents opinions) wecontrol for the process generating the information for ourown set of variables length of debates majority quotas andthe frequency at which discussions take place

6 Conclusion and Perspectives

The main objective of this article was to build a bridgewhere decision-making argumentation and opinion dif-fusion could come together We proposed a model thatcombined abstract argumentation theory and a boundedconfidence opinion diffusion model and showed to whatextent it could explain variance of opinions extremismcoherence in collective decisions and judgment accuracyon arguments given an ideal state of full information Thesecond objective was to show in what way governance pro-viding agents with arenas of discussion and deliberation wasimportant in addressing the success of collective decision-making processes and the quality of their outcomes

The model revealed that allowing for more deliberationtime though in lower frequency and allowing for widerparticipation in deliberation increased the variance of opin-ions and the proportion of extremists in a group Theseobservations are consistent with the combination of resultsfound in [9] and in [13 14] which stress that deliberationmay polarize groups and may have a meager effect on theshifts of opinions and inconsistent with [16] where it isargued that opinions tend to moderate after deliberationIt happens that deliberation alone did moderate opinionyet when integrated into a more complex system in whichindividuals were allowed to interact with one another andnot all that was deliberated was accepted its influence wasshadowed by other more individualistic dynamics

Undeniably the grounded semantics played an importantrole in the weak effect of deliberation since it models askeptical way of reasoning over arguments For agentsaccepting arguments thus updating their opinions accord-ingly happened not too often Nevertheless asking for more

deliberation did increase judgment accuracy as observed in[18] yet in a marginally decreasing fashion We showed thatvoting within the deliberation protocol not only increasedthe proportion of extremists and the variance of opinions ina group but also determined how coherent deliberation andvoting procedures were with each other Lastly we showedthat agents that are focused judged arguments better hadmore stable opinions and constituted a group with higherproportion of extremists than their naive counterparts

In terms of governance the model says that there is per-haps no trade-off between extremismand judgment accuracyInstead it asserts that the higher the variance of opinions thecloser the group gets to the correct decisions This situationmay be interpreted as Landermore and Pagersquos notion of ldquoposi-tive dissensusrdquo (in complex collective decision-making tasks)[36] as opposed to consensus as a normative requirementfor correct collective decision-making If a decider has toorganize deliberation to legitimize a public policy dependingon what kind of world he or she wants different set ofparameters may be chosen to account for different levels oflegitimacy and correctness For instance one may ask for adecision to be legitimate to respect a certain level of coherenceor of judgment accuracy and to be discussed by a sundryof different-opinionated peoples If so deliberation has tobe made more often more participants have to be includedin debates or in deliberation instances and not too manytime steps should be introduced in between two deliberationinstances So the model may capture normative ideas todefine correct and legitimate collective decisions and tomake recommendations on those grounds Requirements ofhigh consensual deliberative accuracy (judgment accuracy)legitimacy (coherence) and diversity (variance of opinions)are all taken into account and for such a reason the model israther successful in describing a sound process of collectivedecision-making with interpretable outcomes

61 Extensions of the Model and Other Ideas Our model canbe extended in many ways Value-based abstract argumen-tation frameworks (VAFs) for example as an extension ofDungrsquos argumentation framework provide a formal descrip-tion of the process of decision-making in which argumentsare given values and audiences (set of agents) ignore attacksbetween arguments on the basis of their preferences over val-ues Social abstract argumentation frameworks (SAFs) alsoprovide an interesting and seemingly convenient formalismfor our model In SAFs agents vote on the acceptability ofarguments and the resulting labeling is a combination of thearguments that are voted for and labeling-based semantics Inour model we apply the same rule yet only for one argumentand given precise agent-based votingmechanisms and votingrules which are eventually parametrized

To consider our model as a specification of a mixed VAFand SAF model may be of great interest for future workThis formalism can provide solid foundations to describeexplain interpret and eventually extend the results foundhere from an argument-based perspective On those groundsfuture work will focus on creating a hybrid argumentationframeworkmdashwith its semanticsmdashthat takes heed of bothvalues or principles and voting

Complexity 27

Concerning other possible extensions of the model anatural reflex would be to argue on the validity and therobustness of the mix between the bounded confidenceopinion diffusion models and deliberation Consequentiallyadding deliberation to different bounded confidence modelsof opinion diffusion such as the Hegelsmann-Krause model[37] or to models of social impact theory like in [4] canbe of great interest for future work We acknowledge that afiner understanding of argumentation frameworks randomlattices and their respective applications for this modelare necessary Testing for different labeling-based semanticsor even trading labeling-based semantics for scoring-basedsemantics (semantics that defines acceptability on scoresgiven to arguments) such as debate semantics and ranking-based semantics (semantics that yield a preorder) may beinteresting to consider in future work

More into modeling deliberation it is not inadequateto believe that agents may also advance proposals duringdeliberation and replace to some extent the proposal forwhich the deliberation is taking place Endogenizing theprocess that produces the type of argument to be deliberatedon may be interesting to explore either by having agentsstrategically replace proposals at certain moments duringthe debate or having the central authority attempt to chooseproposals such that the probability of having them acceptedis high Agents may also be thought to be autoorganizing(independent of a central authority) and concede on the sizeof the instances of deliberation To consider differences insemantics across agents or even endogenizing the semanticson how urgent or important the proposal that is beingdiscussed is can also be an interesting direction to take

Last but not least many nontrivial modifications of ourmodel are possible and most should include better-thoughtdeliberation protocols It may be interesting to design andobserve protocols in which deliberation only affects agentsthat are actually debating for example Trust network effectsmultidimensionality of opinions new processes of argumentexchange or learning among agents are notions to furtherdevelop in order to make the model more realistic and relaxsome unreliable assumptions In sum the model is to berefined and extended with the objective of either studyingconcrete cases of deliberative polling and opinion dynamicsor implementing more intuitive thought experiments

Appendix

We present below some of the pseudocode we used toimplement the model

Algorithms

(a) Algorithm 1 implements the dyadic influence in themodel It describes how two agents interact andupdate their opinions

(b) Algorithm 2 implements the deliberative influence inthe model which describes how agents update theiropinions in sight of the accepted deliberated proposalarguments

(c) Algorithm 3 is used to obtain the grounded labelingof an argumentation framework

(d) Algorithm 4 provides an overview of a run in thesimulations119872 is the number of proposal argumentsto discuss which we set to 100 in the model

Each object in the system is indexed by a119908ℎ119900 number Hencea set of agents is annotated by a set of natural numbers Weuse this fact in the implementation of the algorithm 119874119887119895119890119888119905119894denotes the object of 119908ℎ119900 number 119894 Agentsrsquo opinions (119900) arecalibrated so that 119900 isin [minus1 1]Data Availability

The program and data used to support the findings of thisstudy are available from the corresponding author uponrequest

Disclosure

A preliminary version of this work was presented anddiscussed at the conference Bridging the Gap between FormalArgumentation and Actual Human Reasoning on October 4-5 2018 at BochumGermanyThe authors thank the audiencefor their valuable comments and suggestions

Conflicts of Interest

The authors declare that they have no conflicts of interestregarding the publication of this article

Acknowledgments

The contribution of Gabriella Pigozzi was supported by theDeutsche Forschungsgemeinschaft (DFG) and the CzechScience Foundation (GACR) as part of the joint project FromShared Evidence to Group Attitudes (RO 45486-1)

Endnotes

1 In other words the superiority of the ldquoright answerrdquo orlabeling in our case will appear as such to all

2 All agents hold the same opinion3 By sensitivity we mean how much an agent is ready to

change her opinion on the sole fact that an informationalcue has been debated on

4 The assumption is strong but this trust among agentsmay be rooted from a genuine individual motivation toreach a democratic consensus

5 We adopted a rule of thumbs that says that if an agentis neutral with respect to the principle then she willvote for any proposal that is presented to her duringdeliberation

6 If the argument is not unique she chooses one atrandom

7 If there are more than one candidate the agent will favorthe argument at minimal distance from 119868

28 Complexity

Require Vector of values for parameters (119881 = (120583 119880 119879))Ensure Successful interactions among agents119896 larr997888 0119873119890 larr997888 0

while 119896 le |119873|2 do ⊳Fill119873119890 with |119873|2 agents119896 larr997888 119896 + 1119873119890 larr997888 119877119886119899119889119900119898(|119873| + 1) ⊳Pseudo-random generator Generates integer between 0 and |119873|119873119903 larr997888 0 120575119894119895 larr997888 0 ⊳Variable to store differences in opinionfor all 119894 isin 119873119890 doRandomly pick a number 119895 isin 119873(119873119890 cup 119873119903)119873119903 larr997888 119873119903 cup 119895 120575119894119895 larr997888 |119900119894 minus 119900119895|procedure dyadic-update(119894 119895) ⊳Two agents 119894 and 119895 discuss and update opinions1199001015840 larr997888 119900119894 (120583119895 119880119895 119879119895) larr997888 (120583119894 119880119894 119879119894) larr997888 (120583 119880 119879) ⊳Save agent irsquos opinion before updating and homogenize agents

if 120575119894119895 lt 119880119894 then119900119894 larr997888 120583119894119900119895 + (1 minus 120583119894)119900119894if 120575119894119895 gt 119879119894 then119900119894 larr997888 (1 + 120583119894)119900119894 minus 120583119894119900119895if 120575119894119895 lt 119880119895 then119900119895 larr997888 1205831198951199001015840 + (1 minus 120583119895)119900119895if 120575119894119895 gt 119879119895 then119900119895 larr997888 (1 + 120583119895)119900119895 minus 1205831198951199001015840

Algorithm 1 Dyadic social influence

Require Vector of values for parameters (119881 = (120574 119901119886 119901119903))Ensure Successful interaction between proposal arguments and agents

for all 119894 isin 119873 doprocedure Probability-Update(V119886 119901119886 119901119903) ⊳Update individual probabilities of change due to deliberated cues119909119894 larr997888 119901|V119886minus119900119894 |2119886 119910119894 larr997888 1199012|V119886minus119900119894 |119903119902 larr997888 0if L119886119887(119868) = 119894119899 then

procedure deliberative-update(119894 119886) ⊳Agent i argument 119886120574119894 larr997888 120574 119902 larr997888 119906 sim U(0 1) ⊳ 119906 is a realization of a uniform random variableif 119902 lt 119909119894 then119900119894 larr997888 120574119894119900119894 + (1 minus 120574119894)V119886break

if 119902 lt 119910119894(1 minus 119909119894) then119900119894 larr997888 (1 + 120574119894)119900119894 minus 120574119894V119886Algorithm 2 Deliberation influence

8 In terms of computational complexity naive agentsmaketheirmoves in119874(|A119894| log(|A119894|)) while focused agents doat least as bad

9 Tables have memory Therefore debates are not neces-sarily trees

10 If there are less than (119899119863 times |119873|)2 agents for any of bothgroups the CA summons all of them to the deliberationtable

11 Or careful consideration of the viewpoints of otherswhich implies that citizens keep an open mind and donot reject arguments outright

12 A ldquohealthyrdquo consensus refers to the case in which manyconsensual decisions are taken but opinion-wise agentsdo not agree with each other Quotes are used on theword healthy because some political science theorists

(eg [36]) believe that in certain situations agreement inopinions for deliberation is epistemologically damaging

13 Simulations on groups of |119873| = 800 and |119873| = 400agents shows that scaling the population has an effect onall themetrics but it is confounding depending onwhichcouples of attitudes structures are considered (119880 119879) Westick to |119873| = 400 as in [9] for comparability andsimulation running time

14 By balanced wemean with as many arguments with V119886 lt0 as with V119886 gt 0

15 Experiments on the system suggest that a bigger(smaller) argument pool 119872 = 1000 (119872 = 200) isassociated with a higher (lower) variance of opinions(and extremism) and higher (lower) judgment accuracyFor the rest of the observations 119872 seem to have noeffect We choose a middle value for 119872 Similarly theargument pool chosen to be balanced or uniformly

Complexity 29

Require Argumentation framework 119860119865 = (AR)Ensure Assignment of labels to arguments from the grounded labeling of 119860119865L(A)

L0(A) larr997888 (0 0 0)119896 larr997888 0do119894119899 L119896+1(A) larr997888 119894119899 L119896(A) cup 119886 | 119886 is not labeled 119894119899 L119896(A) 119886119899119889 forall119887 isin A 119904119905 119887 R 119886 then 119887 isin 119900119906119905 L119896(A)119900119906119905 L119896+1(A) larr997888 119900119906119905 L119896(A) cup 119886 | 119886 is not labeled 119894119899 L119896(A) 119886119899119889 exist119887 isin A 119904119905 119887 R 119886 and 119887 isin 119894119899 L119896+1(A)

while L119896+1(A) = L119896(A) = 0Algorithm 3 Solve argumentation frameworks (with grounded semantics)

Require Vector of values for parameters (119881)Ensure Vector of statistics (119880)

procedure Set-parameters(119881) ⊳Initialize parameters for scenarioprocedure Init-objs(119899119898 119896 ) ⊳Create119898 arguments 119899 agents and 119860119865120576procedure Solve-Argumentation-Framework(119860119865) ⊳Assigns the epistemic labelingL120576 toA119894 larr997888 1119905 larr997888 1while 119894 le 119872 do ⊳ 119872 the number of arguments to discuss119868119894 larr997888 119866119890119899 119886119903119892(1) ⊳Generates proposal argument 119868119860 119894119905 larr997888 (119868119894 0)L119886119887(119868119894) larr997888 119906119899119889 ⊳Initialize label for 119868119895 larr997888 1 ⊳Initialize debate counter 119895while ((L119886119887(119868) = 119906119899119889 or 119898 le 119895) and 119895 le 119898) doif 119905 equiv 0(119898119900119889 119905119863) then

procedure build-argumentation-framework(119899119863 119887119890ℎ119886V119894119900119903) ⊳Build the argument graph agents deliberate(119861119894119905 119878119894119905) larr997888 (119883 119884) sub (⋃119897isin119873119899119863 A119897119894119905 ⋃119897isin119873119899119863 R119897119894119905) ⊳ 119873119899119863 sub 119873 are agents sampled by the CA119860 119894119905 larr997888 119889(119860 119894119905 (119861119894119905 119878119894119905))procedure Solve-Argumentation-Framework(A119889) ⊳FindL(A119889)L119886119887(119868119894) larr997888 L119886119887119863(119868119894)119895 larr997888 119895 + 1

else119896 larr997888 1while 119896 le 119905119863 doprocedure dyadic-social-influence(120583 119879 119880) ⊳Agents discuss one-to-one119896 larr997888 119896 + 1119905 larr997888 119905 + 1

if 120572 gt 0 andL119886119887(119868119894) = 119900119906119905 then ⊳Agents vote for the proposalif (|119897 isin 119873|119900119897 times V119868119894 ge 0| ge 120572 times 119899) then

L119886119887(119868119894) larr997888 119894119899else

L119886119887(119868119894) larr997888 119900119906119905if L119886119887(119868119894) = 119894119899 thenprocedure Deliberation-influence(120574 119901119886 119901119903) ⊳Agents are influenced by the result of the decision process119894 larr997888 119894 + 1119905 larr997888 119905 + 1119880 larr997888 119862119900119898119901119906119905119890 119878119905119886119905119894119904119905119894119888119904(119881)

return 119880 ⊳Get the statisticsAlgorithm 4 Run for one scenario

generated does not seem to play a significant explanatoryrole on any of the metrics

16 Experiments show that themaximumnumber of debatesis not an important parameter even if it is an intuitiveone when considering decision-making processes Inpractice debates end after at most 119898 + 1 deliberation

steps and even then it happens rarely However to avoidunpleasant surprises like infinite debates on a completeargument graph containing the proposal argument weadded the constraint 119898 Its value is precisely 7 becausethe biggest minimum number of debates we studyis 6

30 Complexity

17 To ensure comparability with the deliberative andmixedopinion dynamics models on judgment accuracy andcoherence agents vote on proposal arguments every 119905119863times(119898+1) 119905119863 and119898does not explain variability in judgmentaccuracy nor in coherence

18 Student t-tests for independent samples are robust tothe violation of two hypothesis on the distribution ofthe samples normality and homogeneity of variancesas long as the sample size is big enough (central limittheorem) and if the difference in size of the comparedsamples is smallrespectively

19 Studentrsquos test version forwhen variances are not assumedequal and the difference in size of the compared samplesis big

20 We do not correct for homoskedasticity nor normality oferrors in the estimates

21 Balanced as the same number of runs per scenario orsuch that no significant correlations are found betweenparameters

References

[1] R Axelrod ldquoThe dissemination of culture a model withlocal convergence and global polarizationrdquo Journal of ConflictResolution vol 41 no 2 pp 203ndash226 1997

[2] D M Rousseau Reinforcing the MicroMacro Bridge Organiza-tionalThinking and Pluralistic Vehicles SAGEPublications SageLos Angeles Calif USA 2011

[3] S Galam and SMoscovici ldquoTowards a theory of collective phe-nomena consensus and attitude changes in groupsrdquo EuropeanJournal of Social Psychology vol 21 no 1 pp 49ndash74 1991

[4] A Nowak J Szamrej and B Latane ldquoFrom private attitude topublic opinion a dynamic theory of social impactrdquo Psychologi-cal Review vol 97 no 3 pp 362ndash376 1990

[5] G Deffuant S Huet and F Amblard ldquoAn individual-basedmodel of innovation diffusion mixing social value and individ-ual benefitrdquo American Journal of Sociology vol 110 no 4 pp1041ndash1069 2005

[6] G Deffuant D Neau F Amblard and G Weisbuch ldquoMixingbeliefs among interacting agentsrdquoAdvances in Complex Systems(ACS) vol 3 no 01n04 pp 87ndash98 2000

[7] J Rouchier P Tubaro and C Emery ldquoOpinion transmission inorganizations an agent-based modeling approachrdquo Computa-tional and Mathematical OrganizationTheory vol 20 no 3 pp252ndash277 2014

[8] M Meadows and D Cliff ldquoThe relative disagreement model ofopinion dynamics where do extremists come fromrdquo in Pro-ceedings of the International Workshop on Self-Organizing Sys-tems Lecture Notes in Computer Science pp 66ndash77 Springer2013

[9] W Jager and F Amblard ldquoUniformity bipolarization and plu-riformity captured as generic stylized behavior with an agent-based simulation model of attitude changerdquo Computational ampMathematical Organization Theory vol 10 no 4 pp 295ndash3032005

[10] Y-F Zhang H-Y Duan and Z-L Geng ldquoEvolutionary mech-anism of frangibility in social consensus system based onnegative emotions spreadrdquo Complexity vol 2017 Article ID4037049 8 pages 2017

[11] A Flache andMWMacy ldquoSmall worlds and cultural polariza-tionrdquoThe Journal of Mathematical Sociology vol 35 no 1-3 pp146ndash176 2011

[12] A Tsang and K Larson ldquoOpinion dynamics of skepticalagentsrdquo in Proceedings of the 13th International Conference onAutonomous Agents and Multiagent Systems AAMAS 2014 pp277ndash284 International Foundation forAutonomousAgents andMultiagent Systems France May 2014

[13] K M Hansen Deliberative democracy and opinion formationUniversity Press of Southern Denmark Odense 2004

[14] C R Sunstein ldquoThe law of group polarizationrdquo Journal ofPolitical Philosophy vol 10 no 2 pp 175ndash195 2002 WileyOnline Library

[15] S Moscovici and W Doise Dissensions et Consensus unetheorie generale des decisions collectives Presses Universitairesde France-PUF 1992

[16] J S Fishkin and R C Luskin ldquoExperimenting with a demo-cratic ideal deliberative polling and public opinionrdquo ActaPolitica vol 40 no 3 pp 284ndash298 2005

[17] R C Luskin J S Fishkin and S Iyengar Considered Opinionson US Foreign Policy Face-To-Face versus Online Delibera-tive Polling International Communication Association NewOrleans La USA 2004

[18] J Barabas ldquoHowdeliberation affects policy opinionsrdquoAmericanPolitical Science Review vol 98 no 4 pp 687ndash701 2004

[19] O Pryymak A Rogers and N R Jennings ldquoEfficient opinionsharing in large decentralised teamsrdquo in Proceedings of the 11thInternational Conference on Autonomous Agents andMultiagentSystems vol 1 pp 543ndash550 Spain June 2012

[20] J Rouchier and E Tanimura ldquoWhen overconfident agents slowdown collective learningrdquo Simulation vol 88 no 1 pp 33ndash492012

[21] D Miller ldquoDeliberative democracy and social choicerdquo PoliticalStudies vol 40 S1 pp 54ndash67 1992

[22] M Sherif and C I Hovland Social Judgment Assimilation andContrast Effects in Communication and Attitude Change YaleUniversity Press 1961

[23] P M Dung ldquoOn the acceptability of arguments and its funda-mental role in nonmonotonic reasoning logic programmingand n-person gamesrdquo Artificial Intelligence vol 77 no 2 pp321ndash357 1995

[24] M Caminada and G Pigozzi ldquoOn judgment aggregation inabstract argumentationrdquo Autonomous Agents and Multi-AgentSystems vol 22 no 1 pp 64ndash102 2011

[25] P Baroni M Caminada andM Giacomin ldquoAn introduction toargumentation semanticsrdquoThe Knowledge Engineering Reviewvol 26 no 4 pp 365ndash410 2011

[26] S Chambers ldquoDeliberative democratic theoryrdquo Annual Reviewof Political Science vol 6 no 1 pp 307ndash326 2003

[27] E Bonzon andNMaudet ldquoOn the outcomes of multiparty per-suasionrdquo in Proceedings of the 10th International Conference onAutonomous Agents and Multiagent Systems 2011 AAMAS 2011vol 1 pp 47ndash54 International Foundation for AutonomousAgents and Multiagent Systems Taiwan May 2011

[28] J Ranciere Disagreement Politics and Philosophy University ofMinnesota Press 1999

[29] T Bench-Capon and K Atkinson ldquoAbstract argumentation andvaluesrdquo in Argumentation in Artificial Intelligence pp 45ndash64Springer 2009

[30] M Caminada ldquoOn the issue of reinstatement in argumenta-tionrdquo in Proceedings of the European Workshop on Logics in

Complexity 31

Artificial Intelligence vol 4160 of Lecture Notes in ComputerScience pp 111ndash123 Springer Berlin Heidelberg

[31] GAVreeswijk ldquoAn algorithm to computeminimally groundedand admissible defence sets in argument systemsrdquo in Pro-ceedings of the 2006 conference on Computational Models ofArgument Proceedings of COMMA 2006 vol 144 pp 109ndash120IOS Press Amsterdam The Netherlands 2006

[32] C W Sherif M Kelly H L Rodgers et al ldquoPersonal involve-ment social judgment and actionrdquo Journal of Personality andSocial Psychology vol 27 no 3 pp 311ndash328 1973

[33] G Deffuant F Amblard G Weisbuch and T Faure ldquoHow canextremism prevail A study based on the relative agreementinteraction modelrdquo Journal of Artificial Societies and SocialSimulation vol 5 no 4 p 1 2002

[34] E Awad M W Caminada G Pigozzi M Podlaszewski andI Rahwan ldquoPareto optimality and strategy-proofness in groupargument evaluationrdquo Journal of Logic and Computation vol 27no 8 pp 2581ndash2609 2017

[35] D Grossi and G Pigozzi ldquoJudgment aggregation a primerrdquoSynthesis Lectures on Artificial Intelligence and Machine Learn-ing vol 8 no 2 pp 1ndash151 2014

[36] H Landemore and S E Page ldquoDeliberation and disagreementproblem solving prediction and positive dissensusrdquo PoliticsPhilosophy and Economics vol 14 no 3 pp 229ndash254 2015

[37] R Hegselmann and U Krause ldquoOpinion dynamics andbounded confidence models analysis and simulationrdquo Journalof Artificial Societies and Social Simulation vol 5 no 3 2002

[38] A Stefanelli and R Seidl ldquoModerate and polarized opinionsUsing empirical data for an agent-based simulationrdquo inProceed-ings of the Social Simulation Conference 2014

[39] S Gabbriellini and P Torroni ldquoA new framework for ABMsbased on argumentative reasoningrdquo in Advances in SocialSimulation vol 229 pp 25ndash36 Springer Berlin Germany 2014

[40] H Mercier and D Sperber ldquoWhy do humans reason Argu-ments for an argumentative theoryrdquo Behavioral and BrainSciences vol 34 no 2 pp 57ndash74 2011

[41] H V Parunak T C Belding R Hilscher and S BruecknerldquoModeling and managing collective cognitive convergencerdquoin In Proceedings of the 7th iNternational Joint Conference onAutonomous Agents and Multiagent Systems vol 3 pp 1505ndash1508 International Foundation for Autonomous Agents andMultiagent Systems 2008

[42] H Prakken ldquoFormal systems for persuasion dialoguerdquo TheKnowledge Engineering Review vol 21 no 2 pp 163ndash188 2006

[43] K Atkinson T Bench-Capon and P Mcburney ldquoA dialoguegame protocol for multi-agent argument over proposals foractionrdquoAutonomousAgents andMulti-Agent Systems vol 11 no2 pp 153ndash171 2005

[44] M Thimm and A J Garcıa ldquoClassification and strategicalissues of argumentation games on structured argumentationframeworksrdquo in Proceedings of the 9th International Conferenceon Autonomous Agents and Multiagent Systems vol 1 pp 1247ndash1254 International Foundation for Autonomous Agents andMultiagent Systems Canada 2010

[45] I Rahwan and K Larson ldquoMechanism design for abstractargumentationrdquo in Proceedings of the 7th International JointConference on Autonomous Agents and Multiagent SystemsAAMAS 2008 pp 1031ndash1038 International Foundation forAutonomous Agents and Multiagent Systems Portugal May2008

[46] H Prakken R Riveret A Rotolo and G Sartor ldquoHeuristics inArgumentation A Game-Theoretical Investigationrdquo EuropeanUniversity Institute 2008

[47] N Hadidi Y Dimopoulos and P Moraitis ldquoTactics and con-cessions for argumentation-based negotiationrdquo in Proceedingsof the COMMA vol 245 pp 285ndash296 2012

[48] N Oren and T J Norman ldquoArguing using opponent modelsrdquoin Proceedings of the International Workshop on Argumentationin Multi-Agent Systems pp 160ndash174 Springer 2009

[49] S Delre W Jager T Bijmolt and M Janssen ldquoTargeting andtiming promotional activities An agent-based model for thetakeoff of new productsrdquo Journal of Business Research vol 60no 8 pp 826ndash835 2007

[50] A Mirtabatabaei P Jia and F Bullo ldquoEulerian opinion dynam-ics with bounded confidence and exogenous inputsrdquo SIAMJournal on Applied Dynamical Systems vol 13 no 1 pp 425ndash446 2014

[51] V Barbet N Guiraud V Laperriere and J Rouchier Hagglingon Values Towards Consensus or Trouble 2019

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Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

6 Complexity

-1 0 1

Arguments

Adherence to principle

Figure 2 Representation of agents and their arguments on theopinion or adherence to the principles spectrum The rectangleson the agents are their argument sacks Arguments are depicted bydots and arrows from the arguments to the sacks depict both thearguments they hold in the opinion spectrum and the number ofarguments they hold 119896119894 = 4 and 120601119890119894 = 075 and 120601119898119894 = 05 for allagent 119894119909119894 = 119891(119889(119900119894 V119868) 119901119886) of being attracted to a deliberated cueand a probability of being repulsed by it 119910119894 = 119892(119889(119900119894 V119868) 119901119903)The two probabilities are assumed to be independent Theyare a function of the distance between the opinion andthe proposal argument and of the grouprsquos sensitivity3 todeliberation (119901119886 119901119903) The former is decreasing on 119889(119900119894 V119868)and increasing on 119901119886 while the latter is increasing on both119889(119900119894 V119868) and 119901119903 Agents are also assumed

(i) to be capable of assessing the degree of support for Pof all arguments

(ii) to trust4 one another when they utter informationalcues

233 Agents and Voting At the end of the decision-makingprocess agents vote on whether they agree or not on the pro-posal argument that has been discussed during deliberationVoting for an agent is the expression of her opinion in thefinal phase of the decision-making process An agent 119894 is saidto vote favorably for a proposal P of justification argument119868 if and only if V119868 times 119900119894 ge 0 or equivalently if the proposalargument does not adhere to the opposite of the principle 119894supports5

234 Agent Knowledge of Arguments Let A be a finite setof arguments Each agent 119894 has a sack of arguments A119894 subA of size |A119894| = 119896119894 whose content reflects her relativeposition 119900119894 on P (see Figure 2) For extreme-opinion agentsa proportion 120601119890 ge 05 of the 119896 arguments in their sacks areof the same adherence to the principle than their opinions

whereas for moderate agents a proportion 120601119898 le 120601119890 oftheir 119896 arguments are of the valence of their opinions Thehypothesis feeds from results found in [22] stating that ldquoanindividual places a verbal statement on an issue both in termsof the itemrsquos relative proximity to his own position and thelatitude which is acceptable to him around that focal point ofacceptancerdquo

The arguments in an agent 119894rsquos sack are those that sheknows how to use and advance in a deliberative interac-tion The size of the sack represents agent 119894rsquos ability tocommunicate in a deliberative context It follows that eachagent 119894 possesses partial knowledge (119870119894 sub A times A) of theattack relation between the arguments in A and of 119860119865120576which she derives from the arguments in her sack andwhat she observes in deliberation Knowledge of argumentsis assumed to be ldquoattack-orientedrdquomdashan agent 119894 knows anattack (119886 119887) isin A119894 times A if she knows upon observationthat neither she nor the group can rationally accept 119886alongside 119887 She may be aware of the existing defense relationamong arguments when concatenating the information sheaccumulates on their attack relation during deliberationShe may use this information strategically in deliberativecontexts

Let 119889119894119904119905 A times A 997888rarr N be the length of the shortestpath between two elements in the argumentation frameworkinduced by an agent 119894rsquos knowledge119870119894 119894 sees an argument 119886 asan attacker (defender) of an argument 119887 if119889119894119904119905(119886 119887) equiv 1mod 2(119889119894119904119905(119886 119887) equiv 0mod 2) An argument 119886 that is at an odddistance from another argument 119887 attacks it since either itdirectly attacks the argument or it attacks a defender of theargument Likewise if 119886 is at a pair distance of 119887 it attacksan attacker of 119887 and thus defends it If such path does notexist then the distance is undefined and the agent does notsee either argument as an attacker or defender of the otherFurther let AF119889 = (A119889R119889) denote the current state ofaffairs in deliberation An agent 119894rsquos knowledge ofR is the setof attacks and defenses she can infer from the arguments sheknows (119860119905119905119894119863119890119891119894) and the attacks and defenses she can inferfrom deliberation (119860119905119905119889119894 119863119890119891119889119894 )

119870119894 = 119860119905119905119894 cup 119860119905119905119889119894 cup 119863119890119891119894 cup 119863119890119891119889119894 (2)

where

(i) 119860119905119905119894 = (119886 119887) isin A119894 timesA119894 | 119889119894119904119905(119886 119887) equiv 1mod 2)(ii) 119863119890119891119894 = (119886 119887) isin A119894 timesA119894 | 119889119894119904119905(119886 119887) equiv 0mod 2(iii) 119860119905119905119889119894 = (119886 119887) isin (A119894 cup A119889) times A119889 | 119889119894119904119905(119886 119887) equiv1mod 2)(iv) 119863119890119891119889119894 = (119886 119887) isin (A119894 cup A119889) times A119889 | 119889119894119904119905(119886 119887) equiv0mod 2

Notice that agents have no restriction on the amount ofinformation they can carry and use during deliberation Themodel implies that agents have the ability to use and processall information on the attack relations if they could observethe attacks and that they all have equally high cognitivecapacity Agent knowledge resets at the end of each decision-making process but argument sacks stay untouched In otherwords argument sacks are static in the model

Complexity 7

235 Agent Behavior during Deliberation Agents maybehave in twodifferentways in deliberationTheymay behavenaively or focusedly Naive agents use deliberation to voicetheir opinions on the principle through arguments Focusedagents strategically argue in favor of proposal arguments thatsupport the principle they favor In both cases agents advancearguments in terms of the argumentsrsquo relative proximity totheir own positions [22]

Let 119886119894 isin A119894 denote the argument an agent advancesin a debate 119889 (of a deliberation process 119863119868) over a centralargument 119868 and V119886 the valence of an argument 119886 isin A Let

(i) 119891119894(119886) equiv (119900119894 times V119886 gt 0 or 119900119894 = V119886 = 0) the formula thatis satisfied if the argument 119886 is of the same valence asthe opinion of agent 119894

(ii) 119892119894(119886) equiv (119886 119887 isin A119894A119889 and forall119887(|119900119894 minus V119887| gt |119900119894 minus V119886|)) theformula that is satisfied only if the argument 119886 is in 119894rsquossack (A119894) and not in the debate and is the closest interms of adherence to the principle P to 119894rsquos opinion

(iii) ℎ119894(119886) equiv (119891119894(119886) and 119892119894(119886)) the conjunction of thepreceding formuli indicating that an argument 119886 is inagent 119894rsquos sack and not in the debate of the same signas 119900119894 and closest to 119900119894 in absolute value

Naive agents choose 119886119894 = 119886 such that ℎ119894(119886) is true given anyproposal argument 119868 Agents that behave strategically or tosay focusedly choose their moves in the debate as follows(119873119868119871 indicates no move or null argument)

(a) if agent 119894rsquos position with respect to P is of the samesign as the proposal argumentrsquos adherence to P (119891119894(119868)) sheadvances an argument 119886 such that

(i) ℎ119894(119886) and ((119886 119868) isin 119863119890119891119889119894 )) is true If 119886119894 = 119873119868119871(i1) notℎ119894(119886) and 119892119894(119886) and ((119886 119868) isin 119863119890119891119889119894 ) (chooses anargument not of the same sign and closest to heropinion and defending 119868) is true If 119886119894 = 119873119868119871(ii) ℎ119894(119886) and ((exist119888 isin A119894)((119888 119868) isin 119860119905119905119889119894 and (119886 119888) isin 119860119905119905119894))(anticipates an attack from an argument 119888 on 119868 andadvances an argument that attacks the argument 119888)6is true If 119886119894 = 119873119868119871(ii1) notℎ119894(119886)and119892119894(119886)and ((exist119888 isin A119894)((119888 119868) isin 119860119905119905119889119894 and(119886 119888) isin119860119905119905119894)) is true If 119886119894 = 119873119868119871(iii) ℎ119894(119886) and ((119886 119868) notin 119860119905119905119889119894 ) (simply does not attack 119868)is true If 119886119894 = 119873119868119871(iii1)notℎ119894(119886)and119892119894(119886)and((119886 119868) notin 119860119905119905119889119894 ) is true If 119886119894 = 119873119868119871(iv) 119886119894 = 119873119868119871 (the agent will advance no argument)

(b) if agent 119894rsquos position with respect toP is of the oppositesign of the proposal argumentrsquos adherence to P (not119891119894(119868)) sheplays the argument 119886 such that

(i) ℎ119894(119886) and ((119886 119868) isin 119860119905119905119889119894 ) 7 If 119886119894 = 119873119868119871(i1) notℎ119894(119886)and119892119894(119886)and ((119886 119868) isin 119860119905119905119889119894 ) is true If 119886119894 = 119873119868119871(ii) ℎ119894(119886) and ((exist119887 isin A119894)((119887 119868) isin 119863119890119891119889119894 and (119886 119887) isin 119860119905119905119894))(anticipates a defense from an argument 119888 to 119868 andadvances an argument that attacks 119888) is true If 119886119894 =119873119868119871

(ii1)notℎ119894(119886)and119892119894(119886)and((exist119887 isin A119894)((119887 119868) isin 119863119890119891119889119894 and(119886 119887) isin119860119905119905119894)) is true If 119886119894 = 119873119868119871(iii) ℎ119894(119886) and ((forall119887 isin A119889)((119887 119868) isin 119860119905119905119889119894 and (119886 119887) notin 119860119905119905119889119894 ))(avoids attacking any attacker of 119868) If 119886119894 = 119873119868119871(iii1)notℎ119894(119886)and119892119894(119886)and((forall119887 isin A119889)((119887 119868) isin 119860119905119905119889119894 and(119886 119887) notin119860119905119905119889119894 )) If 119886119894 = 119873119868119871(iv) 119886119894 = 119873119868119871

In English a focused agent that is opposed to the proposalattempts to either attack it directly or attack an argument thatis defending it whereas an agent that agrees with the proposalattempts to either defend it or avoid attacking it altogetherThis behavioral approach is similar to debate protocols inabstract argumentation but for the fact that agents mayadvance arguments that do not result in an ldquoadvancementrdquoof the debate In our case the debates stop for other reasons(see Section 24)

Two important points deserve to be highlighted Firstlyfocused agents with small argument sacks will regularly findthemselves applying rules (119894119894119894) or (119894V) In effect if they donot have enough arguments to infer attacks and defensestheir behavior in deliberation is likely to be similar to that ofthe naive agents Secondly the naive and focused behavioralassumptions presented here are very different in terms ofcomputational complexity The first assumes that the agentobserves the debate but what she sees has no effect on hercourse of action ie the argument she voices is uniquelydetermined by how she feels about the principle behindthe proposal argument In the second an agent debatesessentially to knock out any proposal argument she disagreeswith8

24 Tables for Deliberation Tables are the physical or virtualplaces where the exchange of arguments occurs Agentsdeliberate on these tables to determine whether the proposalargument is acceptable or not and are thus subdued bythe deliberation procedure imposed by the tablersquos centralauthority (CA) [27] The CA decides on the structure andlength of the collective decision-making process and thedeliberation procedure It controls the percentage of agentsfrom the population that actively participates in the debate(119899119863) and the labeling-based semantics used to extract accept-able arguments from the framework (120590) The percentage 119899119863denotes either the proportion of agents in the population thatgets to advance arguments in a population-scale deliberationor an independent sample of agents summoned to activelyparticipate in the debate 120590 stands for the procedure used toconclude on the epistemic status of the proposal argumentand the arguments advanced during deliberation

The CA has the ability to stop debates at will using a stoprule that inherently depends on the number of debates thatought to take place before a decision is deemed sufficientlydiscussed (119898) the maximum number of debates that cantake place before abandoning deliberation (119898) and the labelgiven to the proposal argument 119868 (119904119903(119898119898L119886119887(119868))) 119904119903is a Boolean function whose value ldquotruerdquo signals the callfor a vote andor the end of the decision-making process119898 is associated with a minimal dialectical or epistemic

8 Complexity

requirement to consider a proposal for voting and to a lowerbound of the length of the deliberation process 119898 providesan upper bound Moreover the CA controls the size of thetime interval between debates (119905119863) the collective decisionrule (eg if there is voting on the deliberated proposals) andthe majority quota rule for accepting proposal arguments(120572) in the decision-making process 119905119863 may represent thefrequency or density of pair-wise interactions in a decision-making process

241 The Construction of a Decision-Making Process Adeliberation process or debate in ourmodel is a tool to obtainlabels for proposal arguments that are as close as possible tothe ideal or consensual ones To define a deliberation processformally we introduce the notion of debate step as constituentof a deliberation process Informally a debate step is a timestep in which a debate occurs Formally and more event-oriented a debate step (1198891198731015840 1198611015840 1198781015840) of a debate 119863119868 on 119868 isa quadruplet composed of a set of agents 1198731015840 sub 119873 a set ofarguments 1198611015840 sub ⋃119894isin1198731015840 A119894 a set of attack relations 1198781015840 sub Rand a mapping

119889 (AF 2A times 2R) 997888rarr AF

(119860119865 (1198611015840 1198781015840)) 997891997888rarr 1198601198651015840 = (119861 cup 1198611015840 119878 ⊔ 1198781015840) (3)

that adds the arguments in 1198611015840 and some attacks in 1198781015840 to aframework 119860119865 = (119861 119878) and where 119878 ⊔ 1198781015840 = (119886 119887) isin 119878 cup 1198781015840 |119886 119887 isin 119861 cup 1198611015840 In the same spirit we define a deliberationprocess119863119868 on a proposal argument 119868 as a sequence of debatesteps (119889119873119904 119861119904 119878119904)119904gt0 such that1198730 = 0 and

119889((119904minus1⋃119895=0

119861119895 119904minus1⨆119895=0

119878119895) (119861119904 119878119904)) sub (AR) (4)

with

(i) 1198610 = 119868 (the initial argument is the proposal argu-ment)

(ii) 1198780 = 0 (no reflexive attack from 119868 to 119868 is allowed)(iii) (forall119904 gt 0)(⋃119895lt119904 119861119895cap119861119904 = 0) (any newly added argument

to the framework has yet to be added to it)(iv) (forall119904 gt 0)(⨆119895lt119904 119878119895 cap 119878119904 = 0) (any newly considered

attack among arguments cannot be declared amongarguments that are not in the framework)

(v) lim119904997888rarrinfin(⋃119904gt0 119861119904 ⋃119904gt0 119878119904) sube (AR) (the system isstable no arguments are created during deliberation)

Finally let L119886119887119889(119868) denote the label given to argument 119868 ata debate step 119889 during a deliberation process 119863119868 A decision-making process notedD119868 = (119899119863 119905119863 119904119903(119898119898L119886119887119889(119868)) 120572 120590)over a proposalwhose justification argument is 119868 is a sequenceof debate and nondebate steps (119911119905)119905isinN 119911119905 isin 119889 119889 such that

(i) 119889 are debate steps and 119889 nondebate steps that corre-spond to time steps at which there are no debates

(ii) If 119905(mod 119905119863) equiv 0 then 119911119905 = 119889 and 119889 otherwise

For Against

Table

CA

in out

out out in

out in

proparg

Figure 3 A representation of a table of deliberation betweentwo groups of 3 agents with 4 arguments each Each agent hasadvanced an argument Agents tacitly agree on the attacks betweenthe arguments The CA determines the labeling using the groundedlabeling-based semantics

d d d d d d d d d d d d dTime 0 1 2 3 4 5 6 7 8 9 10 11 12I

Figure 4 A decision-making processD119868 or collective decision stepstarting at 119905 = 0 composed of debate (119889) and nondebate (119889) substepsDebate steps are shaded The number of nondebate steps betweendebates is 119905119863 = 3 the number of debates in the decision-makingprocess is119898 = 4

(iii) The subsequence (119911119897119905)119897119905isinN of 119911119905 with 119897119905 = 119905 such that119905(mod 119905119863) equiv 0 is a deliberation process with |1198731015840| =119899119863 times |119873|(iv) 119904119903(sdot sdotL119886119887119889(119868)) pursues the deliberation process as

long asL119886119887119889(119868) = 119906119899119889 under the semantics 120590(v) The length of the sequence |119911119905| or equivalently the

duration of the process is somewhere in the interval⟦119898119905119863 119898119905119863⟧(vi) The final labeling for the proposal argument 119868

L119886119887(119868) is determined by a combination of its delib-erated labelL119886119887119889(119868) and a majority vote of majorityquota 120572

A decision-making process ends when a final decision hasbeen taken concerning the acceptability of the proposal Peg 119868 has been deemed unacceptable (L119886119887119889(119868) = 119900119906119905) or amajority of agents have voted againstP For a representationof a deliberative interaction on a table see Figure 3 for one ofa decision-making process see Figure 4

Please note that every collective decision can be seen asa ldquotime steprdquo in as much as it describes how agents updateopinions and make collective decisions In the model thedecision-making process and the parameters define the sub-steps or events that occur within the time step Henceforth

Complexity 9

Fix proceduralsm m α nD tD σ Init count m larr 0

Given proposal anargument I is randomly generated

nD times |N | agents arerandomly picked

m larr m + 1

Begin decision processon proposal at time t

t larr t + 1Deliberation step m

larr 0

t larr t + 1dyadic opinionupdate Eq (5)tm larr tm + 1

m ge m

tm gt tD

ℒd (I) = in

m ge m

yields ℒ(I)

ℒ(I) = in Deliberationupdate Eq (6)

End decision

yes

no

no

no

yes

yes

no

yes

yes

yes

no

no

I

Vote on ℒd(I)

ℒd (I) = und

ＪＬＩ＝ＭＭ I

Init count tmyields ℒd(I)

Figure 5 Process chart for a decision processD119868 on a proposalP 119905 corresponds to time and 119898 (119905119898) corresponds to the number of debate(non-debate) steps that have taken place in the deliberation process (in between two debate steps) over time for the mixed discussion modelWe useL119886119887(119868) andL(119868) interchangeably in the figure for presentation

comparing simulations on the basis of the different decision-making processes translates into comparing these collectivedecision steps

242 Deliberation Protocol To define the deliberation proto-col held in the table either as a consequence of the definitionprocess or as a statement we assume the following

(i) Agents may decide not to contribute to the debate(ii) All agents have the same probability conditional to

their opinions of being picked to participate in adebate or deliberation step

(iii) There is no restriction on the number of times eachagent can participate in a deliberation process

(iv) Each agent may only place one argument per debatestep

(v) Arguments that have already been advanced in thedebate may attack newly placed arguments9

The deliberation or debate protocol goes as follows

(1) The CA randomly generates and makes public acentral argument or proposal argument 119868 and informsall agents about the rules of the decision-makingprocess

(2) The CA randomly draws two sets of (119899119863 times |119873|)2agents with divergent views10 onP and P It mergesthem to create the set of debaters1198731015840

(3) Each agent 119894 advances an argument from her sackA119894 The CA makes sure that there are no repeatingarguments (agents already take argument repetitioninto account)

(4) The CA establishes the debate step 119889rsquos argumentationframework and computes its labelingL120590

119889(A119889) using120590(5) If the computed label for 119868 is L119886119887119889(119868) = 119906119899119889

or the number of debates steps is inferior or equalto 119898 at time 119905 then the CA stops the debate andresumes it at the (119905 + 119905119863 + 1)rsquoth time step by repeating(1) (2) (3) (4) and (5)

(6) Let L119886119887(119868) be the final label given to the proposalargument 119868 If voting is allowed then if more than120572 times |119873| agents agree with 119868 119868 is accepted (L119886119887(119868) =119894119899) it is rejected if strictly less than 120572times |119873| agree withit (L119886119887(119868) = 119900119906119905) When there is a tie ifL119886119887119889(119868) =119906119899119889 then L119886119887(119868) = 119900119906119905 if L119886119887119889(119868) = 119894119899 thenL119886119887(119868) = 119894119899 When voting is not part of the decision-making process (120572 = 0)L119886119887119889(119868) = L119886119887(119868)

One important point to notice about this protocol is that itinduces a stop rule 119904119903 (cf steps (5) and (6) of protocol) forthe decision process and thus it always ends Either agentsdebate and agree on the proposalrsquos acceptability throughargumentation or after 119898 debate steps they directly voteon it so that a decision concerning the proposal is alwaysreached Refer to Figure 5 for a sweeping description of thedeliberation protocol and the decision-making process

10 Complexity

-1

-1 1

1

Latitude of acceptance

Latitude of rejection Agent i

Agent j

Figure 6 A dyadic interaction between agents 119894 and 119895 as expressedin (5) Agent 119894rsquos opinion falls within agent 119895rsquos latitude of noncommit-ment and conversely Neither agent affects the opinion of the other

25 Opinion Dynamics for Social Interactions In this subsec-tion we describe the opinion diffusion model based on pair-wise interactions among agents and deliberation Sherif rsquosand Hovlandrsquos social judgment theory [22] motivates part ofour approach It describes how individualsrsquo opinions changeon the basis of their attitude structures Attitude structuresrefer to the relative scope width or latitude of categoriesused by individuals when evaluating information namelythe latitudes of acceptance rejection and noncommitment[32] The idea behind this theory is that individuals changetheir positions only in accordance to how far or close thecommunicative cues they receive are from (to) their anchorpositions It holds that if communicative cues are far (close)from (to) an agentrsquos position say over her latitude of rejection(acceptance) then the agent shifts her position away from(towards) the position defended by the cues In the casewherethe cues fall within the agentrsquos latitude of noncommitmenther position does not change (see Figure 6 and Equation (5))

251 Pair-Wise or Dyadic Opinion Dynamics As agentsmay communicate and deliberate collectively they may alsoengage in one-to-one conversations with other agents toponder on their positions We loosely associate this type ofcommunication with dyadic nonargumentative exchange ordiscussion based on fallacious arguments and persuasionIn the light of social judgment theory and the descriptionof agents in the system we model pair-wise symmetricinteractions following the opinion dynamicsmodel in [9] Anagent 119895rsquos influence on an agent 119894rsquos opinion at time 119905 is governedby the ensuing difference equation

119900119905+1119894 =

119900119905119894 + 120583119894 (119900119905119895 minus 119900119905119894) 119894119891 10038161003816100381610038161003816119900119905119894 minus 11990011990511989510038161003816100381610038161003816 lt 119880119894119900119905119894 minus 120583119894 (119900119905119895 minus 119900119905119894) 119894119891 1003816100381610038161003816119900119905119894 minus 119900119905119894 1003816100381610038161003816 gt 119879119894119900119905119894 119900119905ℎ119890119903119908119894119904119890(5)

where the parameter 120583119894 isin [0 05] controls for the strengthof the attraction and repulsion in social influence and 119879119894and 119880119894 are the latitudes of rejection and acceptance for agent119894 respectively The parameter 120583119894 may be thought of as therelative importance agent 119894 gives to the opinions of her peersor analogously the weight she gives to her own opinionwhen updating 120583 lt 05 means that agents will never givemore weight to the opinions of other agents than to their

own (egocentric bias) When two agents 119894 and 119895 discuss iffor instance 119895 advances an informational cue (argumentpersuasion tactic) and it happens to be close enough to119894rsquos opinion anchor then 119894 shifts her opinion towards thedirection of 119895rsquos informational cue The symmetric influencefrom agent 119894 to agent 119895 takes effect in the same way (seeFigure 6)

We add to the dyadic dynamics a rule of encounters ateach step of pair-wise interactions each agent meets exactlyone other agent at random This rule may be associated withrandom day-to-day encounters among agents Steps of socialinfluence correspond to nondebate steps (119889) in decision-making processes

252 Deliberative Opinion Dynamics We define an opinionupdate equation that links the proposal arguments that areadvanced during deliberation and opinions We combine theuncertain and probabilistic nature of the effect of delibera-tion have it a moderating [15 16] or polarizing [14] one witha mechanism similar to the one in social judgment theorybased on the distance between different informational cuesand opinion anchors [32] The probabilistic modeling of theopinion update can be related to deliberation encouragingopen-mindedness11 during collective discussion [18] Fromthis choice it follows that deliberated cues can affect even themost extreme of agents as opposed to some classic opiniondiffusion models (eg [5 8]) where once agents becomeextremists they may no longer become moderate

LetP be a proposal V119905119868 the proposal argument 119868rsquos level ofsupport for a principle P and 119900119905119894 an agent 119894rsquos opinion at time119905 Then given the difference 120575119905119894 = (V119905119868 minus 119900119905119894)2 we define 119894rsquosprobability of being attracted to a decisionrsquos informational cue119868 by 119909119905119894 = 119901|120575119905119894 |119886 where 119901119886 denotes a general probability param-eter that characterizes how important deliberated results arefor the group The parameter may also be interpreted asthe grouprsquos tendency to be swayed by a decisional majoritySimilarly we define 119894rsquos probability of being repulsed from adecision informational cue 119868 by 119910119905119894 = 1199011|120575119905119894 |119903 where 119901119903 denotesa general probability parameter that characterizes the grouprsquosdislike of deliberated results 119901119903 can also be thought of as thegrouprsquos distrust of the system symbolized by deliberation anddemocracy

Let L119886119887(119868) denote the epistemic status of the proposalargument 119868 at the end of a decision process over P everyagent 119894 updates her opinion as follows

(i) IfL119886119887(119868) = 119894119899

119900119905+1119894 =

119900119905119894 + 2120574119894120575119905119894 119908119894119905ℎ 119901119903119900119887119886119887119894119897119894119905119910 119909119905119894119900119905119894 minus 2120574119894120575t119894 119908119894119905ℎ 119901119903119900119887119886119887119894119897119894119905119910 119910119905119894119900119905119894 119908119894119905ℎ 119901119903119900119887119886119887119894119897119894119905119910 1 minus 119909119905119894 minus 119910119905119894

(6)

(ii) IfL119886119887(119868) = 119900119906119905 119900119905+1119894 = 119900119905119894where 120574119894 isin [0 05] is the strength of repulsion and attractionin the dynamicThemeaning of 120574119894 is analogous to the one of120583119894in the dyadic interactionsmodel but for the observation that 119894

Complexity 11

weights her opinion to an argument rather than to an opinionThe interpretation of this dynamics is straightforward Ifthe deliberated proposal argument is close to an agentrsquosopinion then it is very likely that the agent shifts its opiniontowards it Please note that the probability 119909119905119894 (119910119905119894 ) that anagent is attracted (repelled) to (from) an accepted deliberatedproposal argument is bounded below (above) by 119901119886 (119901119903)(equivalently 119909119905119894 isin [119901119886 1] 119910119905119894 isin [0 119901119903]) Steps of deliberativeopinion dynamics are associated with debate steps in which aconcluding collective decision is made

253 Mixed Opinion Dynamics Opinion dynamics in thecontext of decision-making processes can be best described asa combination of the two preceding dynamics In this opiniondynamics model agents are engaged in a democratic systemin which they deliberate vote for proposals and occasionallydiscuss with one another on the principle supported by theproposalsThe opinion dynamics for an agent 119894 can be writtenas a sequence of dyadic and deliberative opinion updates

119900119905+1119894 =

119864119902119906119886119905119894119900119899 (5) 119894119891 119905 equiv 0 (119898119900119889119905119863) and 119904119903 (sdot) = 119891119886119897119904119890119864119902119906119886119905119894119900119899 (6) 119894119891 119905 equiv 0 (119898119900119889119905119863) and 119904119903 (sdot) = 119905119903119906119890119900119905119894 119900119905ℎ119890119903119908119894119904119890

(7)

where 119904119903(sdot) is the decision processrsquos stop rule 119905119863 the numberof time steps between each deliberation step Equation (5)describes the dynamics for the dyadic interactions and Equa-tion (6) posits the changes in opinions due to the deliberatedcues Equation (5) applies when agents are not deliberatingand when each agent encounters another agent to exchangeinformation that may lead to local opinion updates Whenthere is deliberation a handful of agents deliberate and ifthey reach the stop condition imposed by the table all agentsvote for the proposal They update (see Equation (6)) theiropinions on the basis of the proposal argumentrsquos support forthe principle and the result of the scrutiny Otherwise thereis no change in the opinions of the agents A vote indicatesthe end of the decision-making process

3 Experiments and Calibration

In this section we introduce the metrics that enable us toobserve the simulations and characterize the calibration ofthe model

31 Observations and Initialization In this section wedescribe the simulations and the protocol used to test andobserve the results of our model We introduce the metricsof interest explain the calibration of the model describesome results obtained from the simulation data used forcalibration and conclude with a brief discussion on theexpected outcomes of the mixed opinion dynamics model

311 Metrics or Statistics of Interest Let 119878 denote the end ofa simulation We are interested in the effect of deliberationand of the modelrsquos procedural parameters on the followingmetrics or statistics

(i) Variance of opinions (119881119886119903(119900)) the variance ofopinions at time 119878 Since opinions live in the unionof the positive and negative unit intervals 119881119886119903(119900) isin[0 1] The higher the variance of the distribution isthe more ldquodiverserdquo the opinions are(ii) Proportion of extremists in the population(119901119903119900119901119890119909) the proportion of agents in the populationwith opinions such that |119900119894| ge 075 A high proportionof extremists makes ldquohealthyrdquo consensus12 difficult toreach and deliberation more or less informative(iii) Shifts of opinions (119878ℎ) [33] statistic that measuresthe aggregated change in individual opinion at time 119878with respect to aggregated individual opinion at thebeginning of the simulation

119878ℎ = 2sum119894isin119873 10038161003816100381610038161199000119894 minus 1199001198781198941003816100381610038161003816max119894isin1198731199000119894 minusmin119894isin1198731199000119894 (8)

119878ℎ is positive and bounded above by 2|119873| since 119900119894 isin[minus1 1] A low shift statistic implies that the processhas a small impact on opinions(iv) Judgment or consensual inaccuracy (119890119888) consen-sual accuracy of a group consists of an ad hoc statisticmeasuring a grouprsquos ability to infer correct labels forproposal arguments from a decision-making processgiven the ideal consensual labeling based on full-informationL120590

120576 (A) We use a Hamming-based dis-tance on labelings [34] to define the statistic

119890119888 = 1|I| sum119868isinI1198861198681003816100381610038161003816L119886119887120576 (119868) = L119886119887 (119868)1003816100381610038161003816 (9)

whereI is the set of all discussed proposal argumentsand 119886119868 = 12 ifL119886119887120576(119868) = 119906119899119889 or ifL119886119887(119868) = 119906119899119889119886119868 = 1 otherwise119890119888 lives in the interval [0 1] An inaccuracy statisticclose to 1 indicates that agents subdued to a particulardecision-making process make many mistakes injudging the labels of the proposal arguments Notethat when there is voting all proposal arguments arelabeled either 119894119899 or 119900119906119905 after deliberation(v) Coherence (119894119903) letL119886119887119863(119868) be the label obtainedfor 119868 from the deliberation process without votingThe coherence statistic measures how well votingresults adjust to results obtained during deliberationonly We use the proportion of arguments that havebeen labeled 119894119899 in the debate and that agents havevoted favorably for

119894119903 = 1003816100381610038161003816119868 isin I | L119886119887119863 (119868) = 119894119899 andL119886119887 (119868) = 11989411989910038161003816100381610038161003816100381610038161003816119868 isin I | L119886119887119863 (119868) = 1198941198991003816100381610038161003816 (10)

The coherence statisticrsquos domain is [0 1] If after 119872debates no deliberated central argument has beenlabeled 119894119899 or voting is not part of the procedure ofcollective decision-making then the statistic equals

12 Complexity

1 or said differently agents are perfectly coherentThis comes for the fact that if the central argumentis labeled 119900119906119905 then it is not even eligible for voting iflabeled 119906119899119889 then the debate will always be coherentwith the preferences of the agents since the result willbe a simple aggregation of their votes If the statisticequals 0 then none of the proposal arguments labeled119894119899 are voted in favorably by the agents It follows that ahigh coherence statistic implies that when agents votefor acceptable proposal arguments the results of thescrutiny reflect the consensual rationality expressedin the deliberation process

312 Parameters of Interest We recall the parameters ofinterest in our study that are linked to the structure of thedecision-making processes

(i) 119898 the minimum number of time steps in which adebate or a scrutiny occurs in a decision-making pro-cess before a final deliberated decision is submitted toa vote

(ii) 119899119863 the number of agents that deliberate as a propor-tion of the population

(iii) 119905119863 the number of time steps between debates inwhich pair-wise interactions among agents mayoccur

(iv) 120572 the proportional majority requirement for theacceptance of a proposal When 120572 = 0 it stands forno voting ldquoany proposal argument that is labeled 119894119899during deliberation is acceptedrdquo

Please recall that in terms of the definition of a decision-making process each parameter controls either the length orthe content of the sequence D119868 (119905119863 119898) or the rules that areapplied during the debate steps (120572 119899119863) (refer to Figure 4)313 Initialization All |119873| = 400 agents start off with anopinion 119900119894 drawn from a uniform distribution U(minus1 1) 13Given 119900119894 every agent 119894 randomly draws a setA119894 (|A119894| = 119896) ofarguments from a balanced14 argument pool A of119872 = 600nonneutral (V119886 = 0) arguments on the basis of 119900119894 if |119900119894| lt 025(the opinion is moderate) then agent 119894 randomly fills halfof her argument sack with arguments such that V119886 lt 0 andthe other half with arguments such that V119886 gt 0 (120601119898119894 = 05)Otherwise she randomly fills 120601119890119894 ge 05 of her sack witharguments such that V119886 and 119900119894 are of the same sign and theremaining (1 minus 120601119890119894 ) with arguments of the opposite sign15

Like for the opinion of agents every argument 119886rsquosadherence to the principle P V119886 is drawn from a uniformdistribution U(minus1 1) The attack relation that gives birth tothe ideal argumentation framework (119860119865120576) is established onthe basis of the V119886rsquos and the argumentsrsquo epistemic reach 119890119903

Let 120579(119886 119887 119903) = (|V119886 + V119887|2)119903 be an auxiliary positivereal-valued function that takes two arguments (119886 119887) and apositive real number (119903 gt 0) as inputThe probability that anyargument 119886 isin A creates a link to any other argument 119887 isin A

is given by the equation 119901 = 119901120579(1198861198872)119890 and thus argumentssupporting opposing views of the principle always attack each

other We fix 119901119890 the epistemic correlation parameter to 015If 119901119890 was greater than 015 then arguments of the same signwould attack each other too often (and focused agents wouldbe rarely incited to advance favorable arguments duringdeliberation) If 119901119890 was lower then the arguments of thesame type would induce an almost empty graph which isunrealistic The number of arguments that any argument 119886can attack is bounded by 119886rsquos epistemic reach that we fix to15 A higher value of epistemic reach makes the argumentlattice too conflicting and nearly bipartite A lower epistemicreachmakes the lattice not sufficiently conflicting in the sensethat too many arguments can attack the proposal argumentrelatively to the few that can defend it If any argument attacksthe proposal argument the chances that another argumentattacks the attacker are low Hence the proposal argumentis almost never accepted and deliberative opinion updatesbecome rare

The arguments in the resulting argumentation frame-work 119860119865120576 are given a permanent labelingL120590

120576 (A) (in shortL120576) using120590 = grounded semanticsWe choose the groundedlabeling-based semantics because it provides a unique admis-sible labeling respects minimal rationality constraints whilesimplifying the model and models a skeptic approach toaccepting arguments (see [23 25 31]) as it maximizes thecardinality of the set 119906119899119889 L120576 To some extent if we considerproposals as committing because they may guide collectiveaction choosing a skeptic semantics seems reasonable insofaras it labels an argument 119894119899 or 119900119906119905 only if it has no reasonto label it 119906119899119889 For instance if a proposal is a public policythat requires large amounts of resources and engages tofuture course of action then it may also determine policycycles and heavily burden a group and its future decisionsIn important situations like these it seems reasonable tolengthen deliberative cues and ask for more demanding andldquogroundedrdquo criteria for policy argument acceptability

On the proposalrsquos side we create an argument 119868 whoseadherence to P is also drawn from aU(minus1 1) and we label itL119886119887(119868) = 119906119899119889 119868 is interpreted as being the main argumentjustifying the discussed proposal P and V119905119868 as its supportfor P at time 119905 119868 cannot attack other arguments but otherarguments can attack it For each argument 119886 isin A a directedarc or attack from 119886 to 119868 is activated with probability 119901119901 =003120579(11988611986802) When 119901119901rsquos lower bound is too high the proposalargument is almost always defeated and thus deliberativeopinion updates do not happen When 119901119901rsquos lower boundis lower than 003 then the opposite occursmdashthe proposalargument is always accepted due to an absence of attackstowards it Finally we set the maximum number of debatesto119898 = 7 for the decision-making processes stop rules1632 Calibration and Simulation Protocol In this subsectionwe discuss the calibration of the models the terminationconditions for runs in the deliberative and mixed model andthe expected outcomes in terms of the observations

321 Calibrating Dyadic Opinion Dynamics Dyadic opiniondynamics correspond to a space of parameters in whichargumentation and deliberation spaces are not taken into

Complexity 13

Table 1 Parameters and parameter types and their domains used to compare dyadic collective and mixed opinion dynamics

Length ofD119868 Decisional in119863119868 Social influence Deliberation influence Cognitive119905119863 119898 119898 119899119863 120572 120583 119879 119880 119901119886 119901119903 120574 119896 119861119890ℎ119886V1198941199001199031 3 6 1 3 6 7 001 002 005 0 05 066 01 14 18 02 06 03 05 01 005 01 02 4 8 16 119873119886119894V119890 119865119900119888119906119904119890119889account Deliberated arguments and informational cues haveno effect on agent opinion but agents still vote for theproposal argument17 We use and calibrate this model forcomparability in terms of all our metrics since they arenot explicitly observed in [9] a reference opinion dynamicsmodel Furthermore for simplicity and comparability againwe suppose that agents are homogeneous in terms of theirattitude structures and opinion weightings We fix 120583119894 =120583 119879119894 = 119879 and 119880119894 = 119880 for all 119894 isin 119873 and for some triplet(120583 119880 119879) isin [0 05] times [0 2] times [119880 2]

Experiments suggest that the strength of the dyadic inter-actions is an explanatory factor of the time of convergence(120583 lt 05) and of the dyamicsrsquo steady states Bigger values of 120583speed up convergence towards a stable set of opinion clustersand affect the size and the relative position of these clustersin the opinion distribution 120583 however does not play a majorrole in the number of opinion clusters observed at the steadystate of the dynamics

We set 120583 to the value given to it in Jagger et alrsquos opiniondynamics model (120583 = 01) [9] For one it is the smallestreal number for which most results found in [9] hold and thedynamics are smooth secondly it seems to be a reasonableldquostrengthrdquo of influence considering that we would not wantpair-wise social interactions to completely shadow or maskan eventual effect of deliberation in the opinion dynamics

For different values of attitude structures 119879 and 119880 weget exactly the same results as in [9] in terms of the numberand density of the opinion clusters The different couplesof values of 119880 and 119879 model the size of the latitude ofnoncommitment (119879 minus 119880) and thus the agentsrsquo propensityto update opinions Whenever 119879 minus 119880 le 02 regularitiesin convergence and opinion clustering appear If 119880 gt 1there is always central convergence if 119879 lt 1 there is alwaysbipolar convergence with clusters of similar sizes For 119879 minus119880 gt 02 relatively low values of 119879 (119879 lt 1) and relativelyhigh values of 119880 always yield extreme bipolar convergencewith a relatively small cluster of moderate agents (bipolar-central convergence) For 119879 sufficiently high (119879 ge 12) and119880 sufficiently low (119880 le 08) in other words not very ego-involved agents there is a bigger spectrum of steady statesFor this reason we studied the metrics of interest for thesevalue differences and kept the couples of (119880 119879) depicted inTable 1 For these couples central bipolar bipolar-centralconvergence (3 clusters) and multicluster convergence (4 or5 clusters) are possible and result frommeaningfully differentattitude structures Judgment accuracy varies significantlyacross the four different attitude structures we retain and soit happens with the variance of opinions These scenariosprovides us with reference results from which to build upon

322 Calibrating Deliberative Social Interactions Delibera-tive social interactions correspond to social situations in

which individuals are not influenced by pair-wise discussionwith peers but are sensitive to collectively deliberated infor-mational cues On these terms parameters such as the sizeof agentsrsquo argument sacks (119896119894) the distribution of argumentsin them (120601119894) the minimum number of debates (119898) andthe proportion of debaters from the population (119899119863) are nolonger mute We assume for simplicity that the formulaexist120601119890exist119896forall119894(119896119894 = 119896 and 120601119890119894 = 120601119890 and 120601119898119894 = 05) is true with 119894 isin 119873for some value 120601119890 isin [05 1] and for 119896 isin N

The chosen domains for these parameters (see Table 1)are justified by the size of the population and the size of theargument pool (119872 = 600) which we set from the start If 119896is too big (119896 ge 32) andor too many agents are allowed inthe debate (119899119863 ge 02) then individuals always disprove thecentral argument and deliberation is never or rarely takeninto account in the update of opinions Similar effects occurwhen 119898 is high yet we only calibrate it with respect to themean running time of a simulation and the values of 119905119863Different distributions of 120601119890119894 seem to significantly or directlyexplain none of the statistics we analyze so we convenientlyset 120601119890119894 = 075 We chose the parameter 119896 to be always divisibleby 4 for it is convenient given the initial conditions regardingthe argument distribution in the agentsrsquo argument sacks

We fix the value space for the parameters 120574 119901119886 119901119903 as inTable 1 to account for different intensities of the effect ofdeliberative voting retroaction on the system For examplethe scenarios where the three parameters are at their lowestvalues correspond to the scenarios with the smallest fixedprior effect of deliberation in the model Increasing any ofthese parameters should mechanically be associated witha world in which agents are more sensitive to collectivedecisions The values for 119901119886 are taken from [13] wherethe author states that by means of deliberation 7 to28 of individuals changed their opinion from agreeingto disagreeing or vice versa on a referendum question forDenmarkrsquos participation in the EuroWe choose two differentvalues for 119901119886 that account for high (03) and very high (05)sensitivity to deliberation in a group 119901119903 on the other handis taken to be small (equal to 01) since we believe that thereexist agents that will always go against the reached consensusTheir numbers are meager

For simplicity we assume that for all 119894 isin 119873 120574119894 = 120574 forsome 120574 isin [0 05] In effect we posit that the heterogeneityof agents in the deliberative context only derives from theirarguments and their opinions We decide to calibrate 120574 inallusion to the strength of the social dynamics in pair-wiseinteractions In doing so we limit ourselves to consider threedifferent scenarios in which deliberation has respectivelyhalf equal and twice as much opinion-shifting power thanone-to-one social influence Finally we include the accept-ability voting quota rule 120572 whose domain is inspired fromclassical majority rules [35] observed empirically (Table 1)

14 Complexity

Table 2 Parameters and their domains used to study the effect of the procedural parameters on the metrics

Procedural parameters Other parameters119905119863 119898 119899119863 120572 119898 120583 119879 119880 119901119886 119901119903 120574 119896 119861119890ℎ119886V1198941199001199031 2 3 4 5 6 1 2 3 4 5 001 002 003 004 005 0 05 066 7 01 16 02 03 005 02 12 119873119886119894V119890 119865119900119888119906119904119890119889

323 Calibrating the Mixed Social Interactions Model Themixed interaction model ascribes to a parameter space inwhich the effect of collective choices on our metrics isnontrivial and where deliberation and voting on proposalsdetermine their acceptability Since the mixed model isequivalent to periodic iterations of pair-wise and collectivesocial interactions the calibration of the parameters in bothpreceding models holds (see Table 1 for the calibration)The first reason for this is comparability the second is thatthe previous calibration takes into account the fact thatboth dynamics are going to be combined We include thefrequency of debates by adding the parameter 119905119863 whichcontrols for the amount of pair-wise interaction amongagents between two distinct deliberation steps

324 Termination Conditions for Runs Simulations stoponce 100 proposals are deliberated on andor voted for inother words after 100 collective decision steps have occuredThe number of proposals discussed seems arbitrary yet it ishigh enough to observe the effects of deliberation on opiniondistributions and on other metrics related to coherence andjudgment accuracy We choose the number of runs as afunction of our research questions which give relevance tothe procedural parameters in the mixed model rather than tothe parameters set to describe the population and its behavior

325 Expected Outcomes of the Simulations We expect moredeliberation in terms of higher 119898 and 119899119863 to increasejudgment accuracy and at the same time to reduce thevariance of opinions In turn a smaller variance of opinionsimplies that the argument pool for deliberation is smaller andtherefore judgment accuracy should be lower Moreovervariance increasing dyadic interactions (or rejection) shouldalso increase judgment accuracy and coherence since bipo-larization and dissensus foster argument diversity in delibera-tionThe coherence statistic should be stronger in simulationsin which central convergence appears quickly and agentsare naive Attitude structures sensitivity to informationalcues and weights given to deliberated cues that makes pathsto bipolar convergence smaller or to central convergencelonger should be associated with higher judgment accuracyShifts of opinion should be more visible in scenarios inwhich extreme agents are pulled away from the extremesHence the sensitivity to deliberated cues (119901119886 119901119903) and 120574should explain the shifts When crossed with high values of119880 and 119879 the shifting power of deliberation should be at itshighest

In the end we do not know how these mechanismswill play out The results of the subsequent experimentsgive us insight on the interplay between the aforementionedeffects

4 Results

Before pointing at any result obtained from the mixedinteractions model we describe and compare the dyadicand deliberative interactions models in the parameter spaceobtained from the calibration in Section 3 (see Table 1)Primarily we require our results to describe two orthogonaltypes of opinion dynamics the pair-wise dyadic and thedeliberative The latter comprises scenarios in which indi-viduals do not influence each other by means of pair-wisediscussion and only update their opinions from delibera-tion the former scenarios where only pair-wise interactionsamong agents determine the dynamics We describe both onthe parameter space given in Table 1 by performing at least20 simulations per scenario Quantitative results from thepair-wise interaction model alone are described during thecalibration and are not treated heremdashdeliberative parametershave no effect on it Moreover we show how these differentmodel specifications yield qualitatively different opiniondistributions and compare them on the basis of the metricsof interest To this end we perform independent two-sampleStudentrsquos t-tests18 or Welch t-tests19 and compute confidenceintervals at a 095 level of confidence Last but not least wecomment ordinary least squares (OLS) regression estimatesto account for the direction and magnitude20 of the effectsof the parameters on the metrics Estimates are declaredsignificant at a 5 level of risk

We perform the same analysis on the mixed interactionsmodel as we compare it to the pair-wise dyadic and deliber-ative interactions models and discuss the marginal effects ofthe parameters on the metrics More precisely we obtain twodifferent types of results for the mixed interactions modelthe first compares the mixed scenarios to their monolithiccounterparts with respect to each metric and with respect tothe regression estimates the second gives a clear idea of themarginal effect of each governance parameter (or parameterof interest) on our observations In the first we allow controland procedural parameters to vary on a parameter spacesimilar to those of the dyadic and deliberative opiniondynamics (see Table 1) In the second we fix all of ourcontrol parameters execute 36000 balanced21 runs andfocus only on the one-way pooled effects of our proceduraland behavioral (all agents are focused or naive) parametersWe generate comprehensive graphs and tables to account forthe obtained results

Please notice the slight change in the size of the parameterspace for the procedural parameters and in the values ofthe nonprocedural ones for the first and second type ofresults (refer to Tables 1 and 2 respectively) Since we wantto have a finer idea of the marginal effects of each of theprocedural parameters on how well a group decides onproposals we make the value jumps for each parameter (1)

Complexity 15

() (U T) = (02 18) (b) ( pa m nD) = (005 05 00 2 02) (＝) ０ＬＧＮＬＭ ＣＨ (＜) (＝) Ｈ＞ tD = 3

(d) (５ ４) = (02 14) (e) ( pa m nD) = (01 03 05 3 02) () ０ＬＧＮＬＭ ＣＨ (＞) () Ｈ＞ tD = 6

Figure 7 From left to right common opinion trajectories and distributions for dyadic deliberative and mixed opinion dynamics within thestructure of different decision-making processes

small enough to detect significant differences in the estimatesand (2) meaningful for interpretation For the fixed one-valued parameters in Table 2 we use Table 1 to set them totheir minimum mean or median values

41 Comments on Dyadic and Deliberative Opinions Dynam-ics In this subsection we present our first results regardingthe deliberative opinion dynamics described in Equation (6)and its differences with the dyadic dynamics (Equation (5))in terms of our observations

411 Qualitative Analysis of Pair-Wise Opinion DynamicsPair-wise opinion dynamics alone produces multiclusterstable convergencewith variable cluster sizes (see Figures 7(a)and 7(d)) Insight on the complexity of these dynamics wasdiscussed in Section 321

In Figures 7(a) and 7(d) agents discuss randomly withone another in pairs Clusters in the extremes form quicklysince either agents that are close to the extremes attractone another or are convinced to stay close to the extremesby interacting with other agents they disagree with Otheragents with near-extreme positions may be simply attractedto one of the several moderate foci of agents in the opiniondistribution On the other hand moderate agents are eitherattracted to the closest opinion focus or pushed towards theextremes of the distribution Over time agents in the centralfocus either attract other agents into it or ignore the opinions

of the extreme agents that interact with them The result ofthese dynamics yields amulticlustered convergencewhere theopinion foci are at least at distance 119880 from one another andonly the extreme foci (clusters of agents with |119900119894| ge 075) areat a distance greater than or equal to 119879 from one anotherThefoci at the bounds of the opinion distribution are very stableand in analogy to the (12119889)-rule in assimilation bounded-confidence models [33] the number of clusters that form isroughly equal to 119880minus1412 Qualitative Analysis of Deliberative Opinion DynamicsQualitatively deliberative opinion dynamics yield a looseunipolar convergence of opinions near the center (centralconvergence see Figure 7(b)) near the center of either the leftor right portions of the distribution (group polarization seeFigure 7(e)) or to a sparse bipolar opinion distribution withclusters at the center and at the extremes (see Figure 7(c)) ifmixed with pair-wise interactions This is probably a reasonto expect the variance of opinions and the proportion ofextremists (almost perfectly correlated 120588 = 089 119901 lt 0001)to be low for these scenarios The side towards which theopinion cluster skews depends essentially on the valence ofthe first argument that is collectively accepted

In Figures 7(e) and 7(b) agents that update their opin-ions do so at the same time and only when deliberationis successful In these scenarios every time a decision iscollectively accepted agents update their opinions towards

16 Complexity

Table 3 Ordinary least squares estimates for pair-wise interactions scenarios for most of the observed metrics

Dependent variable119881119886119903(119900) 119878ℎ 119890119888 119894119903(1) (2) (3) (4)Constant 0610lowastlowastlowast 4429lowastlowastlowast 0472lowastlowastlowast 0000

(0002) (0206) (0003) (0000)

119880 = 06 (119880 = 02) minus0012lowastlowastlowast 2981lowastlowastlowast 0061lowastlowastlowast 0000(0002) (0260) (0004) (0000)

119879 = 18 (ref 119879 = 14) minus0264lowastlowastlowast minus3261lowastlowastlowast 0004 0000(0002) (0260) (0004) (0000)

120572 = 066 (ref 120572 = 05) 0002 minus0059 minus0100lowastlowastlowast 0000(0002) (0184) (0003) (0000)

(119880 119879) = (06 18) (ref (119880 119879) = (02 14)) minus0324lowastlowastlowast minus3728lowastlowastlowast minus0033lowastlowastlowast 0000(0003) (0368) (0006) (0000)

Observations 1440 1440 1440 1440R2 0983 0389 0529Adjusted R2 0983 0388 0527Residual Std Error (df = 1435) 0032 3492 0053 0000F Statistic (df = 4 1435) 21 272740lowastlowastlowast 228686lowastlowastlowast 402326lowastlowastlowastNote lowastplt01 lowastlowastplt005 lowastlowastlowastplt001

the argumentrsquos defended position Hence if there is votingand if agents happen to obtain 119894119899 proposal arguments theyinexorably cluster towards one or the other side of the opiniondistribution resulting in opinion convergence remeniscent ofgroup polarization (see Figure 7(e)) Furthermore as agentscluster in the same side of the opinion spectrum the argu-ments that may be advanced in deliberation are fewer andmore skewed towards that one side of the opinion spectrumAgents become more easily persuaded in deliberation andonly by arguments supporting one side of the spectrumConvergence of opinions towards one loose opinion focusbecomes faster and more certain If there is no vote (seeFigure 7(b)) the dynamics are similar but for the fact thatit is the uniform randomness of the proposal argument thatguarantees a central convergence of opinions

413 Quantitative Analysis of Pair-Wise Opinion DynamicsThe three parameters that allow for quantitative analysis are119880 and119879 the agentsrsquo attitude structure and themajority quotavoting rule 120572 Table 3 supports the claim that going from 120572 =05 to 120572 = 066 does not affect the variance nor the shifts ofopinions However judgment accuracy is significantly higherwhen accepting proposals becomes more difficult This israther surprising but also intuitive since decisions that aretaken using more restrictive methods of scrutiny should bemore accurate

Concerning the attitude structures the interaction termbetween 119880 and 119879 shows that 119880 has a small effect on thevariance of opinions given that 119879 is low This shows thateven when the assimilation threshold is high a sufficientlysmall rejection threshold can make its one-way effect vanishHowever high levels of both 119880 and 119879 reduce the variance ofopinions dramatically In contrast when it comes to overallchanges in opinion the magnitude of 119880rsquos one-way marginal

effect increases the shifts of opinions whereas 119879 lowers itThis can be interpreted as having the rejection dynamics pushmoderate-to-extreme agents to positions that are close tothe extremes and very moderate agents to the center of thedistribution Not too many quantitative changes (on average)can therefore be observedThe fact that higher119880 is associatedwith faster convergence of opinions seems to imply lowerjudgment accuracy At some point voting for the same type ofproposals for too longmay result inmany collectivemistakesThis is an interesting conclusion to consider in the analysis ofthe mixed model Finally the interaction term shows that 119879has no strong impact on the metric

414 Quantitative Analysis of Deliberative OpinionDynamicsSeven parameters can be considered for quantitative anal-ysis in this dynamics the parameters of interest recalledin Section 312 the size of the argument sacks (119896) agentbehavior during deliberation the strength of the deliberationdynamics (120574) and the measure sensitivity to deliberatedresults (119901119886) The most ldquounexpectedrdquo result obtained fromthe deliberation dynamics is that it shows how deliberationcontributes to the proportion of extremists and the varianceof opinions It happens that having more agents discussproposals and having more of these discussions increasesthe variance of opinions and the proportion of extremistswhile lowering judgment inaccuracy and coherence For theformer the fact that far too many arguments are advancedduring deliberation plummets the chances that the centralargument is accepted hence deliberation does not moveopinions too often For the latter we observe that only ldquobigrdquodifferences (going from the lowest value of a parameter to thehighest) in protocol actually prove to have a preponderantimpact on the metrics These observations hint at a possibletrade-off between judgment accuracy and the variance of

Complexity 17

Table 4 Ordinary least squares estimates for the deliberative interactions scenarios for most of the observed metrics

Dependent variable119881119886119903(119900) 119878ℎ 119890119888 119894119903(1) (2) (3) (4)Constant 0032lowastlowastlowast 13092lowastlowastlowast 0370lowastlowastlowast 1007lowastlowastlowast

(0001) (0406) (0001) (0001)

119898 = 3 (ref119898 = 1) 0012lowastlowastlowast minus2062lowastlowastlowast minus0012lowastlowastlowast minus0001(0001) (0276) (0001) (0001)

119898 = 6 (ref119898 = 1) 0023lowastlowastlowast minus3285lowastlowastlowast minus0018lowastlowastlowast minus0004lowastlowastlowast(0001) (0276) (0001) (0001)

119899119863 = 002 (ref 119899119863 = 001) 0007lowastlowastlowast minus0965lowastlowastlowast minus0006lowastlowastlowast minus0002lowastlowast(0001) (0276) (0001) (0001)

119899119863 = 005 (ref 119899119863 = 001) 0020lowastlowastlowast minus3316lowastlowastlowast minus0015lowastlowastlowast minus0004lowastlowastlowast(0001) (0276) (0001) (0001)

120572 = 05 (ref 120572 = 0) 0146lowastlowastlowast 68932lowastlowastlowast 0068lowastlowastlowast minus0507lowastlowastlowast(0001) (0276) (0001) (0001)

120572 = 066 (ref 120572 = 0) 0291lowastlowastlowast minus13842lowastlowastlowast 00004 minus0000(0001) (0276) (0001) (0001)

119896 = 8 (ref 119896 = 4) 0004lowastlowastlowast minus0815lowastlowastlowast minus0002lowastlowastlowast minus0001(0001) (0276) (0001) (0001)

119896 = 16 (ref 119896 = 4) 0008lowastlowastlowast minus1118lowastlowastlowast minus0004lowastlowastlowast minus0003lowastlowastlowast(0001) (0276) (0001) (0001)

119861119890ℎ119886V119894119900119903 = ldquoFocusedrdquo (ref ldquoNaiverdquo) 0015lowastlowastlowast minus1980lowastlowastlowast minus0012lowastlowastlowast minus0004lowastlowastlowast(00005) (0225) (00004) (0001)

120574 = 01 (ref 120574 = 005) minus0020lowastlowastlowast 5762lowastlowastlowast minus00003 minus0001(0001) (0276) (0001) (0001)

120574 = 02 (ref 120574 = 005) minus0025lowastlowastlowast 8648lowastlowastlowast minus00003 minus0001(0001) (0276) (0001) (0001)

119901119886 = 05 (ref 119901119886 = 03) minus0013lowastlowastlowast 1581lowastlowastlowast 00004 00003(00005) (0225) (00004) (0001)

Observations 19440 19440 19440 19440R2 0931 0844 0562 0958Adjusted R2 0931 0844 0562 0958Residual Std Error (df = 19427) 0033 15701 0030 0050F Statistic (df = 12 19427) 21 844790lowastlowastlowast 8 733563lowastlowastlowast 2 079585lowastlowastlowast 36 761250lowastlowastlowastNote lowastplt01 lowastlowastplt005 lowastlowastlowastplt001

opinions that may be of interest in the mixed discussionmodel

Group coherence shifts of opinions and the varianceof opinions are highly affected by the voting thresholdthe coherence metric is reduced by half when allowing forclassic majority voting shifts explode and the variance ofopinions increases dramatically This is not surprising sincethe majority quota rule works as a buffer to the success ofdecision-making processes and in consequence it hampersthe deliberative opinion updates For the shifts of opinions itis clear that the two-third majority voting rule results in littleopinion change and that the simple majority rule inducesgroup polarization As expected there are less extremists inscenarios where there is no voting and the shifts of opinionare more likely to happen when agents are more sensitiveto deliberation When we include voting in the system

the grouprsquos judgment accuracy decreases due to the correctdeliberated decisions that are vetoed by a salient opinion-agent majority Strangely the effect only holds for the simplemajority decision rule Lastly the proportion of individualsthat participate in the debate and the minimal amount ofdebates have similar marginal effects on all the metrics (seeTable 4)

415 Comparing Deliberative and Pair-Wise Opinion Dynam-ics in a Nuthshell Qualitatively both models yield similaroutcomes but for the fact that deliberation never producesopinion polarization with both types of extremists (seeFigure 7) Furthermore clusters in the deliberative interac-tions model are less homogeneous in size and multiclusterconvergence occurs only when deliberation is rarely suc-cessful which happens sporadically given the calibration of

18 Complexity

Table 5 Mean difference 095 confidence intervals for metrics by model comparison

Model Metric119881119886119903(119900) 119901119903119900119901119890119909 119878ℎ 119890119888 119894119903Dyadic vs Deliberative [0191 0216] [0170 0197] [-2948 -2828] [0074 0082] [0165 0172]Mixed vs Deliberative [0142 0147] [0150 0155] [-2085 -1976] [0003 0002] [-0038 -0030]Mixed vs Dyadic [-0072 -0046] [-0045 -0018] [8332 8859] [-0079 -0071] [-0205 -0201]the model On the other hand pair-wise opinion dynamicsproduces multicluster convergence most of the time andcentral convergence only in the infrequent cases where thelatitude of acceptance is large

The comparison of these two scenarios in terms ofthe metrics is straightforward Whilst dyadic interactionsallow for more variability in opinions through pair-wiseinteractions and more mistakes in judging proposals delib-erative updates simply act as an antagonist force On aver-age the deliberative interaction model produces opiniondistributions with smaller variance and yields steady stateswith higher judgment accuracy a higher shift statistic andnaturally lower group coherence since only proposals that aredeliberated 119894119899 are eligible for voting (see Table 5)

42 Articulating Dyadic and Deliberative Social Interactionsin Opinion Dynamics So far we have considered scenarioswhere argumentation had no place and where individualsmade decisions according to opinions that were purelyconstructed from pair-wise discussions We also observedcases in which agents formed their opinions only by inte-grating deliberated proposals into their opinion updates Thescenarios we present in this section combine the two pre-ceding scenarios to account for both the effects of pair-wisediscussion and the effects of deliberation on opinions judg-ment accuracy and group coherence From the precedingobservations we conclude the following deliberative opiniondynamics reduces the variance and the proportion of extrem-ists because it shifts opinions greatly and polarizes groupsby eliminating at least one type of extremist the one whoseopinions are initially opposed to the deliberated resultsor it brings opinions close to neutrality (see Figure 7(b))This may not be desired in a group since the diversity andstability of opinions are necessary for deliberation to makesense and lead to the collective acceptance of controversialproposals Dyadic opinion dynamics on the other handtend to increase the variance of opinions and the numberof extremists while having a lesser aggregated effect on theshifts of opinion This is an immediate result of the pair-wiseinteraction dynamics inasmuch as it polarizes individualsvery quickly

To sum up in the mixed interactions model deliberationhinders the effect of voting on opinions by making someproposals that would normally be submitted for voting noteligible for voting It may also undo opinion changes dueto pair-wise interactions The mixed model can be seen asthe deliberation opinion dynamics model in which pair-wiseinteractions occur in between collective discussion and mayaffect which arguments are advanced during deliberationEquivalently it can be seen as the dyadic opinion dynamics

model in which deliberation accounts for endogenousldquoshocksrdquo on the opinion distribution

421 Qualitative Analysis of the Mixed Dynamics In Figures7(c) and 7(f) agents update their opinions through delibera-tion and pair-wise interactions Opinion trajectories visualsshow a combination of both previously described opiniontrajectories Over time as agents discuss with one anotheropinion clusters form as in the pair-wise or dyadic opiniondynamics Deliberation may disrupt the formation of theseclusters and three different situations may occur

(i) Deliberation disrupts the formation of the opinionclusters but not sufficiently to jeopardize pair-wiseopinion cluster formation (eg for high values 119905119863)Two phenomena may be responsible for it (1) thevariability of the proposal argumentrsquos support for theprinciple P and (2) too few (successive) deliberationsteps that result in opinion updates Once the opinionclusters are formed successful deliberation foreshad-ows the merging of the previously formed clustersmay bring extremists closer to moderate agents andthanks to the assimilation component of the pair-wisedynamics allow bigger and more moderate opinionclusters to form (refer to Figure 7(c))

(ii) Deliberation significantly disrupts the pair-wisedynamic formation of clusters This may happenwhen the requirements for accepting proposals areweak in the decision-making process (eg low 120572) Inthis case one has either central convergence sinceall agents follow the deliberated results in the samemanner or 3-cluster convergence with two extremeopinion foci In the latter situation the extremegroups form quickly (like in the pair-wise model) anddeliberation cannot pull extreme agents away fromthe extreme foci they are in The main reason for thisis that immediately after an extreme agent attempts toleave her foci through deliberation updates she getspulled or pushed back into it by means of a dyadicinteraction with another extreme agent

(iii) Deliberation does not disrupt the formation of theopinions cluster This is the rare case scenario forvery demanding decision-making processes in termsof inclusion (eg 119899119863) and argument acceptability (eg120572) Opinions converge as in the pair-wise opiniondynamics model (refer to Figures 7(a) and 7(d))

Last when there is voting (120572 = 0) the position ofthe moderate opinion cluster is determined by the first

Complexity 19

minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

Var(o)

sh

ec

ir

Var(o)

propex

sh

ec

ir

1

098

minus02

minus004

001

1

minus014

minus001

minus003

1

026

minus028

1

minus057 1

ＪＬＩＪ

Ｒ

Figure 8 Significant correlation estimates for themetrics of interestin the mixed opinion dyamics scenarios

accepted proposal argument andor the opinionmajority thatis formed from previous dyadic interactions (see Figure 7(f))

422 Relating the Metrics of Interest in Mixed OpinionDynamics The first thing to notice is that for all of thesescenarios all the metrics are significantly correlated (seeFigure 8) The variance of opinions and the proportion ofextremists are almost perfectly correlated To some extentthis means that a high variance of opinions known thatthe mean of opinions is statistically null equates to havingmany extremists in the population be towards one or theother side of the opinion distribution (eg extreme bipolar-ization) Similarly we find a negative and strong correlationbetween the judgment inaccuracymeasure and the coherencecoefficient (see Figure 8)mdashwhen judgment inaccuracy is high(low) coherence to deliberation is low (high) Cases likethese are telling since they may be related to sequences ofcollective decisions in which agents do not make too manymistakes when judging proposal arguments and still makedeliberated decisions that are on average in line with theirprinciples

The shift statistic interestingly is negatively correlatedwith the variance of opinions with judgment accuracy andwith the coherence coefficient In a way this means thathaving more moderate agents conduces on average to moremistakes in judging arguments and to less coherent collectivedecisions An explanation for this phenomenon could be thatas agents reach consensus in opinion important (possiblyextreme) arguments for correctly judging argument propos-als are not advanced in the deliberation arenawhich results inmore mistakes and in possibly accepting proposal argumentsthat after voting are not meant to be accepted Indeed the

negative correlation between the variance of opinions andjudgment inaccuracy points towards this direction since onaverage the more shifts there are the smaller the variance ofopinions is (see Figure 8)

423 Comparing Mixed Dynamics with Other MonolithicDynamics of Interactions Qualitatively opinion trajectoriesin the mixed model are more chaotic than in the pair-wise and deliberative interaction models (see Figure 7) Asanalyzed previously deliberation dynamics disrupt the pair-wise dynamics and create situations in which central andmulticlustered convergence (with two clusters at the extremesof the distribution) are possible at the expense of more(pair-wise dynamics) or less (deliberative dynamics) clustersIt tolerates the survival of very small isolated group ofagents that the previous models rarely tolerate in a hundredcollective decision-making steps (see Figure 7(c))

One of the most interesting observations one can makeof this family of models regards how groups change (in termsof our metrics) when dyadic social interactions are addedto deliberative interactions and vice-versa When comparingthe mixed interactions with only deliberative ones shiftsagentsrsquo aggregated change in opinions are on average lesspronounced The variance of opinions and the proportionof extremists are significantly higher for the former than forthe latter group coherence in contrast is lower with respectto the scenarios where opinions are formed only throughdeliberation Judgment accuracy follows that same trendA reason for this might be that mixed dynamics are fasterto yield semistable clusters andor opinion convergence Inconsequence the potential argument pool for deliberativeexchange shrinks faster and deliberation becomes less effec-tive over time which results in more collective mistakes Theoutcomeon coherence is predictable inasmuch as deliberativeinteractions produce group polarization If a group agrees inopinion debates are sterile because themembers of the groupwill most likely refrain from attacking the proposal argu-ments they all support and will work together to disparagethe proposal arguments they do not support (see Table 5)

In the same manner we report that the proportion ofextremists and the variance of opinions are barely lower inthe pair-wise interaction model when including deliberationin the decision-making process Shifts of opinions are alsobarely lower when there is no deliberation This can indicatethat deliberation does not undo the changes in opiniondue to pair-wise interactions nor does it amplifies them Incontrast when we consider the labeling-based metrics themixed interaction model ensures lower judgment inaccuracybecause of deliberation and lower group coherence (seeTable 5)

In sum the mixed discussion model is a compromisebetween the pair-wise interaction and deliberative opiniondynamic models in that it offers some guarantees in termsof judgment accuracy and coherence while allowing forldquoreasonablerdquo variance in the opinion pool It establishes somekind of trade-off between labeling-based metrics and thevariance of opinions that may be of interest for decision-making process design

20 Complexity

Table 6 Summary of sign or relative marginal effects for all varying parameters on dyadic deliberative and mixed interactions models inand for all experiments

Metrics Parameters119905119863 119898 119899119863 120572 119879 119880 119901119886 120574 119896 119861119890ℎ119886V119894119900119903119881119886119903(119900) XX X X O X X X X X X119901119903119900119901119890119909 XX X X O X X X X X X119878ℎ XX X X O X X X X X119890119888 XX X X XO X XOO XOO X X119894119903 XX X X O XO O X OOX XOO X XNota gray symbols correspond to dyadic dynamics light gray to deliberative dynamics and bold black to mixed dynamics Given a scenario the symbol 119883stands for constant parameter and 119874 for insignificant marginal effect and as parameters grow for positive (negative) marginal effect for the parameterFrom left to right at the parameter level the leftmost arrow corresponds to the scenario with the strongest (in absolute value) marginal effect of the parameter(with respect to to its factors) Example Let 119901 be a parameter and 119874 stands for ldquo119901 has a significant positive effect in dyadic discussion scenario negativein the deliberative one and insignificant in mixed discussion The positive effect in dyadic discussion is greater than the absolute value of the negative one inthe purely deliberative scenariordquo

424 Comments on the Differences between the Three Modelsfor OLS Regression Estimates The significance of most of theregression parameters estimated for the mixed interactionsscenarios are similar to those estimated for the pure dyadicanddeliberative scenariosNevertheless themagnitude of theeffects is different insomuch that the dynamics themselvesare different So instead of commenting the linear regressionestimates of the mixed model we compare when possiblethe standardized significant coefficients of the regressors ofthe mixed model with those of the pair-wise and deliberativeinteractionmodels (see Table 6 for a summary of the results)

In mixed interactions for instance the influence of119898 onthe opinion distribution is much more preponderant than inthe deliberative dynamics scenarios Going from 119898 = 1 to119898 = 3 increases the standard variation units of the variance ofopinions by twice as much as the one in the deliberative sce-nariosThis in particular is due to the design of the decision-making process If 119898 is higher then the number of socialinfluence steps (119905119863) in the decision process is also higher(compared to 119905119863 = 0 in the deliberative case) If we decideto consider these steps as contributing to the variance ofopinions then the effect of119898 on the variance of opinions hasto be stronger in the mixed interactions model In contrastthe number of agents participating in the debate has a weakereffect on the variance of opinions in the mixed interactionsmodel than in the deliberative dynamics specially for 119899119863 =005 (more than twice as much) The difference may comefrom the fact that in mixed discussions the part of theargument pool used in deliberation is more restrictive thanin the deliberative case We can argue that since pair-wisedynamics may get individuals closer to one another aftersome deliberation the same happens to the arguments thatindividuals will most likely use in deliberation Taking thisinto account means that deliberation will at many instancesbe unfruitful and have no effect on opinions thus failing tomoderate extreme opinions andor polarize the group In thesame direction a high number of participants in deliberationcan make the deliberation dynamics stiff In this case thedistribution of opinion is close to the uniform distributionwhich has a relatively high-to-moderate variancewith respectto the reference results

The strength of the deliberative dynamics has a positivestronger effect on the variance of opinions in themixedmodelthan in the deliberative model when 120574 = 02 and a negativeweaker one when 120574 = 01 A sound interpretation of thisresult is that when deliberation is successful and its effect onopinions does not get agents to move sufficiently away fromtheir current positions agents are pulled or pushed back totheir previously held opinions the variance of opinions doesnot changemuchOn the other hand if the effect of successfuldeliberation is strong opinions can change sufficiently tomake agents much more prone to assimilation into moderateopinion foci the variance of opinions falls

The deliberative opinion dynamics model is much moresensitive to 120572 than the mixed discussion model Differ-ences in marginal effects can double The probability ofattraction to deliberated results (119901119886) for instance has astrongmoderating effect (negative) on agentsrsquo opinions in thedeliberative opinion dynamics but a meager one (two-fold)in the mixed and dyadic dynamic models This is becauseopinion updates in the deliberative model depend only onthat parameter whereas in the mixed model more numeroustypes of interactions make opinion updates possible

In terms of judgment accuracy all parameters except thesize of the argument sacks (119896) have a significant slightlystronger effect on the deliberative dynamics than in themixedmodel In effect the difference may be attributed to theupshot of pair-wise interactions on the distribution of opin-ions and on the arguments that would be advanced duringdeliberation The result respecting parameter 119896 however iscounterintuitive Since argument sacks are static scenarioswhere shifts are higher should make 119896 more important fordeliberation to succeed and it is precisely in the scenarios ofpure deliberative dynamics that we find the biggest opinionshifts This may point to the hypothesis that in deliberativeinteractions agents change opinions substantially but do notoften change their adherence to the principle (go from anegative opinion to a positive one or vice versa) when thereis no group polarization

43 Sensitivity Analysis for the Procedural Parameters ofInterest Weperform a sensitivity analysis of the observationson the parameters of interest in themixed interactionsmodel

Complexity 21

We extend the parameter domains of the pure dyadic anddeliberative scenarios as we make it explicit in Table 2

431 Minimum Number of Debates (119898) We observe that theminimum number of debates has a significant well-observedeffect on all of our metrics excluding group coherence Forthe variance of opinions we can observe that the moredebates there are the bigger the value of the metric isalthough the higher 119899119863 and 119905119863 are the weaker the overalleffect of119898 becomes (Figures 9(d) and 9(a)) Furthermore themarginal effect of increasing minimal debates is decreasingan additional deliberation step requirement increases thevariance of opinions but less and less as it grows 119898 coupledwith 119899119863 has a shy S-shaped effect (see Figure 9(d)) on thevariance of opinions meaning that it has a stronger effectwhen 119899119863 is low which suggests a possible trade-off betweenthe number of arguments accepted in the debate and thenumber of debates required before accepting a proposal

In contrast the shifts in opinion are less and less likelyas 119898 grows and this is independent of the variations of theother parameters Again the effect is marginally decreasingand is only truly significant when 120572 = 05 An explanationto these effects may be linked to the design of the systemFirst the variance of opinions is higher when deliberation isasked for because the more deliberation steps there are thehigher the chances are that the central proposal argumentis deemed unacceptable specially when agents are focusedFurthermoremechanically speaking increasing theminimalamount of debates implies that whenever a decision is to betaken at least 119898 times 119905119863 time steps have to take place and if atany moment a proposal argument is considered undecided119905119863 time steps are added to the process So unless the debatedoes not yield 119906119899119889 labels for proposal arguments (highlyunlikely considering that 120590 = grounded semantics) the morenondeliberation steps there are in the decision process thehigher the variance of opinions is Concerning the shiftswhen 120572 = 05 either the system is too stiff to accept anyproposal argument and opinions do not changemuch or theeffects of pair-wise discussion and deliberation cancel out insuch a way that individual shifts are minimal (Figure 9(b))

On the side of labeling-based metrics the more debatesare asked for the more accurate a group is in its judgmentthe effect being smaller as 119898 grows When agents are naivethe effect is quasilinear while when they are focused thestrongest effects of acquiring more deliberation are foundwhen levels of deliberation are already low (Figure 9(f))Thiscan be explained by the fact that the more debates thereare in the decision process the closer one gets to the idealargumentation framework

432 Proportion of the Population in Deliberation Steps (119899119863)Like for 119898 119899119863 has a significant effect on the proportionof extremists and on the shifts and variance of opinions(Figure 9(e)) This may result from the fact that being ableto put more arguments in play at the same debate step mayincrease the odds of revealing cycles around the proposalargument in119860119865120576 Given that we use grounded semantics thearguments in the cycles are labeled 119906119899119889 and in consequence

debates are more often postponed when 119899119863 is higher Post-poning debates in turn increases the number of nondebatessteps in the decision process which increases the varianceof opinions and limits over time the moderating effect ofdeliberation Moreover the effect of this parameter is verydependent on the value of 120572 (Figure 9(e)) For shifts forinstance 120572 = 05 makes the effect of 119899119863 negative while120572 = 0 makes it positive but to a lesser extent Furthermorefor higher requirements of deliberation (119898) adding moreindividuals to the deliberation process has weaker effectson the variance of opinions and on the other metrics It isalso quite interesting to notice that it has the same effect onagents independently of agent behavior during deliberationThis is a surprising result as one would have expected morefocused agents in a deliberation arena to cause a raise inthe proportion of extremists as they play to knock outopposing proposal arguments and thereof reducing the effectof deliberation (lowers variance) on the whole population

Finally similar to119898 addingmore people into the deliber-ation process increases judgment accuracy (see Figure 9(c))and has no clear effect if any on group coherence (seeFigure 9(i))

433 Steps between Deliberation Steps (119905119863) In all configu-rations 119905119863 increases the proportion of extremists (variance)and decreases the shifts of opinions among agents Theshifts and the effect on the variance of opinions are presentfor 120572 = 05 insignificant otherwise (Figure 9(b)) 119905119863 ishighly linked to 119898 by construction When 119898 = 1 thecurve linking the variance of opinions and 119905119863 is convex As119898 increases the curve becomes more and more concavewhich means that 119905119863 has a more important effect on theopinion distribution as collective decisions take longer to beachieved This seems counterintuitive but in reality it reflectsthemultiplicative relationship between deliberation and pair-wise interactions and the semantics in the model If 119898 is lowand 119905119863 is high the effective number of pair-wise interactionsis on average smaller in the deliberation process Hencethe structure of the decision-making process hinders anyincrease in the variance of opinions If119898 is high the oppositeeffect is observed Additionally since the grounded semanticsyields few 119894119899 arguments with respect to other admissibility-based semantics getting closer to the ideal argumentationframeworkmaymake it difficult to obtain the expected effectsof deliberative discussion on agents Lower 119898 constrains thevariance of opinions by making deliberation more influentialon opinions

In terms of judgment accuracy and coherence we observethat 119905119863 does not explain coherence yet it tends to decreasejudgment inaccuracy This may be due to a more variedargument pool in the debate which in turn may be aconsequence of a higher variance of opinions or vice versa

434 Acceptability Voting Quota (120572) By far it is the mostinfluential parameter in our study It changes the directionand the intensity of the effect of all the other proceduralparameters and by construction heavily constrains the roadto accepting a proposal (see Figure 9)

22 Complexity

015

020

025

030

035

040

1 2 3 4 5 6

12345

Values of Ｇ

(a) 119905119863 times 119901119903119900119901119890119909 by119898

0

10

20

30

40

50

1 2 3 4 5 6

005066

Values of

(b) 119905119863 times 119878ℎ by 120572

040

045

050

055

060

1 2 3 4 5

001002003004005

Values of Ｈ＄

(c) 119898 times 119890119888 by 119899119863

015

020

025

030

035

040

1 2 3 4 5

001002003004005

Values of Ｈ＄

(d) 119898 times 119901119903119900119901119890119909 by 119899119863

0

10

20

30

40

50

001 002 003 004 005

00005066

Values of

(e) 119899119863 times 119878ℎ by 120572

040

045

050

055

060

1 2 3 4 5

BehaviorNaiveFocused

(f) 119898 times 119890119888 by 119887119890ℎ119886V119894119900119903

015

020

025

030

035

040

0 05 066Behavior

NaiveFocused

(g) 120572 times 119901119903119900119901119890119909 by 119887119890ℎ119886V119894119900119903

05

06

07

08

09

10

1 2 3 4 5 6

00005066

Values of

(h) 119905119863 times 119894119903 by 120572

08700

08725

08750

08775

08800

001 002 003 004 005

BehaviorNaiveFocused

(i) 119899119863 times 119894119903 by 119887119890ℎ119886V119894119900119903

Figure 9 Curves of mean observations for different parameters and metrics for 36000 runs Bars are 095 confidence intervals Plots withthe same observables (ordinate) except for 119894119903 are on the same scale

Complexity 23

The higher the requirement for accepting the proposal isthe higher the proportion of extremists and the variance ofopinions are and the lower the shift statistic is In few words120572 constrains the world to dyadic discussion or throws it intoa process in which deliberation is much more importantthan dyadic interactions This is why for any 120572 = 05 oneeither gives too much weight to deliberated results whichshadows pair-wise interactions or too little weight in thesense that no deliberated result is ever accepted and thusnever integrated into the opinions of the agents that havevoted for it The reason for the latter is that the latitudeacceptance is too low and thereafter pair-wise interactionsare not enough to unevenly polarize the population in such away that deliberated proposal arguments are voted favorablyby a non-negligeable majority Shifts for 120572 = 0 are stable fordifferent levels of 119905119863 which implies that deliberation keeps incheck all the shifts of opinions related to an increase in thenumber of pair-wise interactions For 120572 = 05 deliberationis less likely to be successful and shifts tend towards the shiftlevels for 120572 = 0 (see Figure 9(b))

Concerning labeling-basedmetrics120572 entirely determinesthe coherence statistic For 120572 = 05 there is no difference incoherence because of how coherence is defined it is maximalwhen 120572 = 0 and strangely maximal when 120572 = 066(see Figure 9(h)) Surprisingly 120572 has no effect on judgmentaccuracy One would have expected a higher 120572 to increasejudgment accuracy since agents would hardly collectivelyaccept a proposal argument discussed during deliberationProposal arguments have given how the argument lattice isgenerated a higher probability of being rejected than of beingaccepted in the ideal argumentation framework

435 A Word on Focused and Naive Agents Focused andnaive specifications for agent behavior are an importantparameter in the decision process since they model howagents choose arguments in the deliberation arena In the sce-narios where agents are focused agents show equal or highervariance of opinions and are therefore more extreme (seeFigure 9(g)) independently of the voting requirements Shiftsof opinion are less likely as well We can see this happeningbecause focused agents knock out proposal arguments moreoften than naive agents provided that 119899119863 is low

One could have expected naive agents to yield morecoherent decisions since they argue sincerely However thisseems not to be the case (see Figure 9(i)) Agents are unable todeliberate in away thatmakes deliberation reflect their votingintentions either because they do not have the necessaryarguments to do so or because such arguments do not exist

In terms of judgment accuracy focused agents do betterthan naive agents in finding the ideal labels for the proposalarguments This is likely because when reconstructing theframework focused agents take into account the deliberatedproposal argument and choose themost pertinent argumentsin their sacks that relate to the state of affairs at the deliber-ation arena In fine debates result in a better approximationofL119886119887120576(119868) even though one could have believed the oppositethat is focused agents are ready to sacrifice a correct collectivelabel of a proposal argument in the goal of defending theirpositions

44 Points of Discussion In this subsection we discuss theresults obtained from the simulations We attempt to makesense of them from a social sciences perspective

441 Opinion Consensus and Dissensus The deliberativeopinion dynamics corresponds to the simulated scenarios inwhich the variance of opinions is at its lowest and judgmentaccuracy at its highest Dyadic opinion dynamics are associ-ated with the scenarios in which the variance of opinions is atits highest judgment accuracy at its lowest and coherence atitsmaximumpossible levelThemixed interactions dynamicsis a combination of these aspects for all the metrics exceptfor coherence at which it is the worsemdashdeliberation leadsto some kind of loose opinion consensus that is reinforcedby pair-wise interactions Likewise the mixed discussionmodel confirms some results exposed in [16] expressingthat opinions and voting intentions in a deliberative contextldquooften changerdquo That being said and more particularly indeliberative opinion dynamics outputs align well with twointeresting results on consensus formation given by aninterpretation of Moscovici and Doisersquos [15] work Sunsteinrsquos[14] account on deliberation and group polarization andsocial impact theory [4] The first states that mild consensusis reached in deliberation as procedure disengages agentsthe second is that within discussion groups majorities tendto become larger and opinions more extrememdashnot on thebasis of competing arguments but of the initial balance ofopinions After several decision processes we find opiniondistributions that reflect these observations even though weare not able to precisely coin the set of parameters that yieldssuch distributions (see Figure 7)

Although differences between mixed and dyadic opiniondynamics in terms of the variance of opinions and shiftsare slim the former does much better in terms of judg-ment accuracy In this sense deliberation does moderateopinions but not as much when non-deliberation steps arerelatively frequent throughout the decision-making processIn terms of Moscovici and Doise theory of consensus [15]this makes sense since pair-wise discussion engages or ego-involve agents in the discussion and pushes to convergenceof opinion at the extremes of the opinion distributionHowever what remains unclear is the reason why proceduralparameters that constrain deliberation do not disengageagents from discussion and thus result in a mild consensusas also expressed by Moscovici and Doise [15] A possibleexplanation of this phenomenon may stem from the factthat deliberation in the model is always accompanied by aseries of ego-involving discussions that shadowor counter themoderating effect of deliberation Not only this deliberationcould also be responsible for crystallizing the opinions ofextremists for it favors one or the other part of the distributionof opinions as explained in [14] and to some extent in [13]Furthermore the mixed discussion model gives an exampleof polarization that results from a minority of dissenterson one side of an issue acceding to views of the majorityrsquosside [4] as opposed to the strong polarization observed indyadic interactions Situations in which the status quo orthe undecidability of the proposal argument lingers cannotpossibly yield situations of moderation In cognitive science

24 Complexity

one can be inclined to believe that the longer a deliberationprocess is the more likely individuals will take an immutablestance on the situation In view of the assumption that longdecision-making processes can be more costly for a group orsociety as a whole agents may feel forced to take a stancejust for the sake of ending the process The stance that isadopted would therefore depend on pair-wise interactionsamong individuals and not on deliberation since deliberationtakes too long to be conclusive

442 For Hypothetical Recommendations in Decision-MakingProcess Elaboration On another tune results from themodelsuggest that procedural parameters have marginally decreas-ing positive or negative effects on the metrics of interest Ina hypothetical policy analysis perspective this may indicatethat after choosing a correct number of individuals andimposing the right amount of debates one can attain rea-sonable levels of judgment accuracy coherence and varianceof opinions As accounted in [18] deliberation increasesknowledge (collective knowledge as well) and therefore itheightens the chances of making correct decisions If onebelieves that social welfare is linked to deliberation andeventually to the metrics studied previously then proceduralparameters can both be used as an instrument to attaindesired levels of social welfare and reinforce the notionof legitimacy of collective decisions It can also provide ajustification for choosing a deliberation regime rather thananother on the basis of how important (or urgent) a topic ofdiscussion is Indeed if we adopt the claim that deliberationis a forerunner of welfare then any state in which deliberativecues are at their maximum (highest number of participantsmany deliberation steps before making a decision and asfrequent as possible) has to be mapped to the highest attain-able social welfare In this case is a situation where judgmentaccuracy is at its highest yet the variance of opinions or theproportion of extremists at its lowest an ideal situation Wecannot say for sure since strong opinion consensus and longdecision-making processes in many situations may result ina loss of welfare rather than in a gain of it

Another interesting result in procedure for deliberationis the very similar estimates we find for the effects of theminimum number (119898) of debates necessary to make adecision on a proposal and the proportion (119899119863) of individualsparticipating in deliberation on judgment accuracy In thescope of social welfare this raises the question of whetheradvocating for longer decision processes by demandingmoredeliberation steps (higher 119898) rather than bigger debates(higher 119899119863) may increase welfare Depending on the objec-tives of the decider of the deliberation procedure he or shemay choose to concentrate on one or the other A recom-mendation we can assert from the mixed social interactionsmodel is that if one prefers low to high variance of opinionsandor high to low coherence the decider is better off ifshe focuses on increasing the size of the debate rather thanorganizing debates very frequently One of the reasons forthis might be that when making bigger debates all agents aretruly considered as equals in the decision-making processIn organizing many debates individuals that are chosen toparticipate in many of these will be more influential and will

somehow bias the set of arguments in the deliberation arenaThemajority rulemodeled through120572 also determines the sizeof the shifts and the distribution of opinions Because a bigger120572 is associated with higher levels of extremism and opinionvariance deliberated cues are harder to accept and thereforeit becomesmore difficult for agents to reach concensus andormoderate their opinions

On the other hand and further into the idea of legitimacythe goal of deliberative processes is the agreement amongall participants This agreement stands for the right or bestdecision from a formal or procedural perspective and it is alsothe best from a substantive point of view In other words thelegitimacy of decisions not only derives fromwell-establishedprocedural requirements (respect of the protocol) but alsobegs for the fulfillment of two essential conditions (1) satisfy-ing the procedural requirements for a correct procedure (for-mal legitimacy) and (2) the rational acceptability of the resultsof this procedure (substantive acceptability) Legitimacy canbe then obtained from the coherence statistic scrutinizedin the model insofar as it measures how well agents acceptthe results of a correct procedure (eg deliberation) So fora proposal to be legitimate one has to choose parametersthat would maximize group coherence in decision-makingprocesses

As deliberation may give rise to consensus in the delib-eration arena it may also sow the seeds of dissensus inthe group Our model clearly illustrates this phenomenonthrough the procedural parameters of deliberation and themetrics observed From another perspective in deliberativedemocracy dissensus in collective decision-making may bedesirable According to Landermore and Pagersquos accounts [36]lack of consensus can be perceived as having agents withalternative ideas that combined with other ideas can providea better approximation of the ideal status of an argumentTheauthors in [36] call this dissensus ldquopositive dissensusrdquo andit makes normative sense when solving complex collectivedecision-making problems Our model captures the ideaof having normative requirements to define a good andlegitimate collective decision requirements of high consen-sual deliberative accuracy (judgment accuracy) of legitimacy(coherence) and of diversity (variance of opinions) are alltaken into account and for such we can conclude that ourmodel is successful in describing a sound process of collectivedecision-making with interpretable outcomes

Of course the present simulation results and analysis donot prove that all opinion dynamics can be accounted forby such simple processes like the one presented here Theydo however shed light on the expressiveness of combiningdifferent paradigms to account formore interpretablemodelsand pave the way to creating original models of the sortThe simulations suggest the desirability of discovering theconsequences of relatively simple laws of communication atdifferent levels (micro meso and macro) to determine whatstill needs to be explained

5 Related Work

We see our model as a contribution to the influence andopinion dynamics field in ABM and a pragmatic application

Complexity 25

of abstract argumentation theory To our knowledge we areunaware of existing literature on agent-based modeling thatexplicitly relates collective choice and the notions of deliber-ation and opinion diffusion through abstract argumentationas we have done it

Most models in the literature of opinion diffusion areinterested in opinions because these have an influence incollective decision and in questions of social order Forinstance in [3] the authors are interested in consensus andin how a group collectively chooses among two alternativesIn other models authors are interested in the emergenceof extremism [8] and on the distribution of opinions whenextremists are introduced in the population [5 33 37] whileother authors coin the notion of opinion polarization as anemergent property of the system [8 12] They show usingmodels of ldquobounded confidencerdquo and opinion diffusion withtrust that three different kinds of steady states (unipolarbipolar and central) are possible depending on whetheragents are sufficiently uncertain about their opinions andsufficiently connected andor a certain proportion of indi-viduals are already extreme Recent articles on informationand opinion dynamics stress the importance of governanceand government intervention in the spread of emotions(opinions) in the frangibility of social consensus systemIn [10] the authors show that government intervention inthe spread of negative emotions can lead to an even fasterspread of negative emotions (single-peeked convergence) andin a faster collapse of the social consensus system Opinionas a function of trust is studied in [12] and relates to howindividuals form their opinions on the basis of how muchthey trust agents in their networks

Another stream of discrete opinion dynamics modelsthat explain the emergence of opinion can be found in theliterature Computer multiagent models of attitude changebased on Latanersquos social impact theory are presented in [4] Inthe theory an agent changes her opinion on the basis of theinformational impact she is subject to which depends on thepersuasiveness (strength) supportiveness and immediacy(group structure) of the environment and information shereceives The simulations predict two emergent groups ofphenomena the shifting of attitudes towards incompletelypolarized equilibria and the formation of coherent clusteringof subgroups with deviant attitudes Similarly but in acontinuous more argumentative fashion Mas and Flache[11] present a model of opinion diffusion using argumentsArguments are considered to be for or against a proposal andare given an agent-dependant relevance in dyadic persuasiveinteractions Arguments also determine the agentsrsquo opinionsand their dissimilarities which posits the rules of pair-wiseencounters in theirmodel as agents are assumed to form theiropinions according to the arguments they own They showthat in an argumentative opinion diffusion model the onlyobservable steady states are bipolarization and consensusand that bipolarization can emerge from interactions amongsimilar agents In other words bipolarization of opinionsis possible with homophily and without negative influenceif there is argumentation They subsequently compare theirresults to experimental data In [38] a similar approach toargumentation and opinion formation is taken but for the fact

that they introduce different types of arguments explicitlyapply social judgment theory [22] and use survey data toestablish model-to-real-world comparisons

Closer to opinion formation and abstract argumentationGabbriellini and Torroni [39] are the first to have origi-nally and soundly merged opinion diffusion and abstractargumentation They define an agentrsquos opinion as a functionof the arguments she holds and the attack relation amongthem They devised a focused peer-to-peer dialogue systemof persuasion (NetArg) inspired from Mercier and Sperberrsquosargumentative theory of reasoning [40] which used onlyabstract argumentation to study opinion polarization andopinion dynamics Moreover they considered networks andthe notions of trust and epistemic vigilance to define adynamics for knowledge revision and trust itself whichwe clearly lack in our model They showed that if a con-servative belief operator in argumentation was applied byagents when they reasoned then their dialogue protocoldid not increase polarization among agents That said theyused the model to study the effect of Groenvetterrsquos weaklink theory in the spreading of arguments and were notparticularly interested in notions like judgment accuracy orcoherence nor in tackling questions related to collectivedecision-making procedures Their system is very expressiveand helps position our work in the litterature our work is atthe frontier of pure argumentative opinion diffusion modelsopinion diffusion with arguments and continuous bounded-confidence models of opinion diffusion a la Deffuant

On another note work on collective cognitive conver-gence [41] and opinion sharing [19] shows that consensustowards a certain ldquocorrectrdquo opinion or cognitive state is alwayspossible yet dependent on noise variability and awarenessof agents In [20] the authors show that learning about anexogenous correct state of the world (represented by bits)under confidence was possible but only if the agents werenot too confident In a homogeneous population they showthat the higher the confidence the worse the learningmdashthevery confident agents do not learn the properties of the truestate of the world and disrupt the learning process of the lessconfident ones Collective cognitive convergence can be seenas a result of deliberation in truth-seeking models

When it comes to abstract argumentation theory we takean approach that wires two types of dialogues that are well-studied in the literature persuasion dialogues [42] and delib-eration dialogues [43] Another interesting line of work is theone on mechanism design [44] or the problem of devisingan argumentation protocol where strategic argumentation isnot a liability for success in debates We tackle mechanismdesign in a different way though Instead of consideringstrategy-proofness we are interested in how differences inprotocol can result in epistemologically ldquobetterrdquo collectivechoices and guarantee that opinion distributions are favorablefor deliberation For a survey on persuasion dialogue see[42]

Work on agent-based argumentation usually assumes thatthe semantic relationship between arguments is fixed Inother words if two individuals were to put two argumentsin the public arena such that one logically attacked the otherthen everyone would agree that such attack exists [45 46]

26 Complexity

Other models which do not make this restrictive assumptioncan also be found in the literature [27 47] and derive fromthe class or family of opponent models [27 48] in which twoopposing sides attempt to win a dialogue Ourmodel is in theintersection of these two but the frame combining opiniondiffusion of the kind and argumentation remains new

The idea of mixing interpersonal influence and verticalcommunication however is not original It is described andimplemented in innovation diffusion models such as [5 49]In both vertical communication is modeled as exogenoustransparent informationmdashagents are aware of the existenceof an innovation thus triggering several processes of choiceand stabilization of opinions Also in [50] an Eulerian modelis implemented to show the effect of media and exogenousinformation on opinion distributions With respect to thispoint the originality of our work is in that the informationemitted as vertical communication is endogenous It is issuedfrom a deliberation model that agents shape on the basisof their opinions arguments and behavior In the spirit of[51] where the authors control for the design of verticalcommunication (to whom the vertical communication isaddressed its timing how it influences agents opinions) wecontrol for the process generating the information for ourown set of variables length of debates majority quotas andthe frequency at which discussions take place

6 Conclusion and Perspectives

The main objective of this article was to build a bridgewhere decision-making argumentation and opinion dif-fusion could come together We proposed a model thatcombined abstract argumentation theory and a boundedconfidence opinion diffusion model and showed to whatextent it could explain variance of opinions extremismcoherence in collective decisions and judgment accuracyon arguments given an ideal state of full information Thesecond objective was to show in what way governance pro-viding agents with arenas of discussion and deliberation wasimportant in addressing the success of collective decision-making processes and the quality of their outcomes

The model revealed that allowing for more deliberationtime though in lower frequency and allowing for widerparticipation in deliberation increased the variance of opin-ions and the proportion of extremists in a group Theseobservations are consistent with the combination of resultsfound in [9] and in [13 14] which stress that deliberationmay polarize groups and may have a meager effect on theshifts of opinions and inconsistent with [16] where it isargued that opinions tend to moderate after deliberationIt happens that deliberation alone did moderate opinionyet when integrated into a more complex system in whichindividuals were allowed to interact with one another andnot all that was deliberated was accepted its influence wasshadowed by other more individualistic dynamics

Undeniably the grounded semantics played an importantrole in the weak effect of deliberation since it models askeptical way of reasoning over arguments For agentsaccepting arguments thus updating their opinions accord-ingly happened not too often Nevertheless asking for more

deliberation did increase judgment accuracy as observed in[18] yet in a marginally decreasing fashion We showed thatvoting within the deliberation protocol not only increasedthe proportion of extremists and the variance of opinions ina group but also determined how coherent deliberation andvoting procedures were with each other Lastly we showedthat agents that are focused judged arguments better hadmore stable opinions and constituted a group with higherproportion of extremists than their naive counterparts

In terms of governance the model says that there is per-haps no trade-off between extremismand judgment accuracyInstead it asserts that the higher the variance of opinions thecloser the group gets to the correct decisions This situationmay be interpreted as Landermore and Pagersquos notion of ldquoposi-tive dissensusrdquo (in complex collective decision-making tasks)[36] as opposed to consensus as a normative requirementfor correct collective decision-making If a decider has toorganize deliberation to legitimize a public policy dependingon what kind of world he or she wants different set ofparameters may be chosen to account for different levels oflegitimacy and correctness For instance one may ask for adecision to be legitimate to respect a certain level of coherenceor of judgment accuracy and to be discussed by a sundryof different-opinionated peoples If so deliberation has tobe made more often more participants have to be includedin debates or in deliberation instances and not too manytime steps should be introduced in between two deliberationinstances So the model may capture normative ideas todefine correct and legitimate collective decisions and tomake recommendations on those grounds Requirements ofhigh consensual deliberative accuracy (judgment accuracy)legitimacy (coherence) and diversity (variance of opinions)are all taken into account and for such a reason the model israther successful in describing a sound process of collectivedecision-making with interpretable outcomes

61 Extensions of the Model and Other Ideas Our model canbe extended in many ways Value-based abstract argumen-tation frameworks (VAFs) for example as an extension ofDungrsquos argumentation framework provide a formal descrip-tion of the process of decision-making in which argumentsare given values and audiences (set of agents) ignore attacksbetween arguments on the basis of their preferences over val-ues Social abstract argumentation frameworks (SAFs) alsoprovide an interesting and seemingly convenient formalismfor our model In SAFs agents vote on the acceptability ofarguments and the resulting labeling is a combination of thearguments that are voted for and labeling-based semantics Inour model we apply the same rule yet only for one argumentand given precise agent-based votingmechanisms and votingrules which are eventually parametrized

To consider our model as a specification of a mixed VAFand SAF model may be of great interest for future workThis formalism can provide solid foundations to describeexplain interpret and eventually extend the results foundhere from an argument-based perspective On those groundsfuture work will focus on creating a hybrid argumentationframeworkmdashwith its semanticsmdashthat takes heed of bothvalues or principles and voting

Complexity 27

Concerning other possible extensions of the model anatural reflex would be to argue on the validity and therobustness of the mix between the bounded confidenceopinion diffusion models and deliberation Consequentiallyadding deliberation to different bounded confidence modelsof opinion diffusion such as the Hegelsmann-Krause model[37] or to models of social impact theory like in [4] canbe of great interest for future work We acknowledge that afiner understanding of argumentation frameworks randomlattices and their respective applications for this modelare necessary Testing for different labeling-based semanticsor even trading labeling-based semantics for scoring-basedsemantics (semantics that defines acceptability on scoresgiven to arguments) such as debate semantics and ranking-based semantics (semantics that yield a preorder) may beinteresting to consider in future work

More into modeling deliberation it is not inadequateto believe that agents may also advance proposals duringdeliberation and replace to some extent the proposal forwhich the deliberation is taking place Endogenizing theprocess that produces the type of argument to be deliberatedon may be interesting to explore either by having agentsstrategically replace proposals at certain moments duringthe debate or having the central authority attempt to chooseproposals such that the probability of having them acceptedis high Agents may also be thought to be autoorganizing(independent of a central authority) and concede on the sizeof the instances of deliberation To consider differences insemantics across agents or even endogenizing the semanticson how urgent or important the proposal that is beingdiscussed is can also be an interesting direction to take

Last but not least many nontrivial modifications of ourmodel are possible and most should include better-thoughtdeliberation protocols It may be interesting to design andobserve protocols in which deliberation only affects agentsthat are actually debating for example Trust network effectsmultidimensionality of opinions new processes of argumentexchange or learning among agents are notions to furtherdevelop in order to make the model more realistic and relaxsome unreliable assumptions In sum the model is to berefined and extended with the objective of either studyingconcrete cases of deliberative polling and opinion dynamicsor implementing more intuitive thought experiments

Appendix

We present below some of the pseudocode we used toimplement the model

Algorithms

(a) Algorithm 1 implements the dyadic influence in themodel It describes how two agents interact andupdate their opinions

(b) Algorithm 2 implements the deliberative influence inthe model which describes how agents update theiropinions in sight of the accepted deliberated proposalarguments

(c) Algorithm 3 is used to obtain the grounded labelingof an argumentation framework

(d) Algorithm 4 provides an overview of a run in thesimulations119872 is the number of proposal argumentsto discuss which we set to 100 in the model

Each object in the system is indexed by a119908ℎ119900 number Hencea set of agents is annotated by a set of natural numbers Weuse this fact in the implementation of the algorithm 119874119887119895119890119888119905119894denotes the object of 119908ℎ119900 number 119894 Agentsrsquo opinions (119900) arecalibrated so that 119900 isin [minus1 1]Data Availability

The program and data used to support the findings of thisstudy are available from the corresponding author uponrequest

Disclosure

A preliminary version of this work was presented anddiscussed at the conference Bridging the Gap between FormalArgumentation and Actual Human Reasoning on October 4-5 2018 at BochumGermanyThe authors thank the audiencefor their valuable comments and suggestions

Conflicts of Interest

The authors declare that they have no conflicts of interestregarding the publication of this article

Acknowledgments

The contribution of Gabriella Pigozzi was supported by theDeutsche Forschungsgemeinschaft (DFG) and the CzechScience Foundation (GACR) as part of the joint project FromShared Evidence to Group Attitudes (RO 45486-1)

Endnotes

1 In other words the superiority of the ldquoright answerrdquo orlabeling in our case will appear as such to all

2 All agents hold the same opinion3 By sensitivity we mean how much an agent is ready to

change her opinion on the sole fact that an informationalcue has been debated on

4 The assumption is strong but this trust among agentsmay be rooted from a genuine individual motivation toreach a democratic consensus

5 We adopted a rule of thumbs that says that if an agentis neutral with respect to the principle then she willvote for any proposal that is presented to her duringdeliberation

6 If the argument is not unique she chooses one atrandom

7 If there are more than one candidate the agent will favorthe argument at minimal distance from 119868

28 Complexity

Require Vector of values for parameters (119881 = (120583 119880 119879))Ensure Successful interactions among agents119896 larr997888 0119873119890 larr997888 0

while 119896 le |119873|2 do ⊳Fill119873119890 with |119873|2 agents119896 larr997888 119896 + 1119873119890 larr997888 119877119886119899119889119900119898(|119873| + 1) ⊳Pseudo-random generator Generates integer between 0 and |119873|119873119903 larr997888 0 120575119894119895 larr997888 0 ⊳Variable to store differences in opinionfor all 119894 isin 119873119890 doRandomly pick a number 119895 isin 119873(119873119890 cup 119873119903)119873119903 larr997888 119873119903 cup 119895 120575119894119895 larr997888 |119900119894 minus 119900119895|procedure dyadic-update(119894 119895) ⊳Two agents 119894 and 119895 discuss and update opinions1199001015840 larr997888 119900119894 (120583119895 119880119895 119879119895) larr997888 (120583119894 119880119894 119879119894) larr997888 (120583 119880 119879) ⊳Save agent irsquos opinion before updating and homogenize agents

if 120575119894119895 lt 119880119894 then119900119894 larr997888 120583119894119900119895 + (1 minus 120583119894)119900119894if 120575119894119895 gt 119879119894 then119900119894 larr997888 (1 + 120583119894)119900119894 minus 120583119894119900119895if 120575119894119895 lt 119880119895 then119900119895 larr997888 1205831198951199001015840 + (1 minus 120583119895)119900119895if 120575119894119895 gt 119879119895 then119900119895 larr997888 (1 + 120583119895)119900119895 minus 1205831198951199001015840

Algorithm 1 Dyadic social influence

Require Vector of values for parameters (119881 = (120574 119901119886 119901119903))Ensure Successful interaction between proposal arguments and agents

for all 119894 isin 119873 doprocedure Probability-Update(V119886 119901119886 119901119903) ⊳Update individual probabilities of change due to deliberated cues119909119894 larr997888 119901|V119886minus119900119894 |2119886 119910119894 larr997888 1199012|V119886minus119900119894 |119903119902 larr997888 0if L119886119887(119868) = 119894119899 then

procedure deliberative-update(119894 119886) ⊳Agent i argument 119886120574119894 larr997888 120574 119902 larr997888 119906 sim U(0 1) ⊳ 119906 is a realization of a uniform random variableif 119902 lt 119909119894 then119900119894 larr997888 120574119894119900119894 + (1 minus 120574119894)V119886break

if 119902 lt 119910119894(1 minus 119909119894) then119900119894 larr997888 (1 + 120574119894)119900119894 minus 120574119894V119886Algorithm 2 Deliberation influence

8 In terms of computational complexity naive agentsmaketheirmoves in119874(|A119894| log(|A119894|)) while focused agents doat least as bad

9 Tables have memory Therefore debates are not neces-sarily trees

10 If there are less than (119899119863 times |119873|)2 agents for any of bothgroups the CA summons all of them to the deliberationtable

11 Or careful consideration of the viewpoints of otherswhich implies that citizens keep an open mind and donot reject arguments outright

12 A ldquohealthyrdquo consensus refers to the case in which manyconsensual decisions are taken but opinion-wise agentsdo not agree with each other Quotes are used on theword healthy because some political science theorists

(eg [36]) believe that in certain situations agreement inopinions for deliberation is epistemologically damaging

13 Simulations on groups of |119873| = 800 and |119873| = 400agents shows that scaling the population has an effect onall themetrics but it is confounding depending onwhichcouples of attitudes structures are considered (119880 119879) Westick to |119873| = 400 as in [9] for comparability andsimulation running time

14 By balanced wemean with as many arguments with V119886 lt0 as with V119886 gt 0

15 Experiments on the system suggest that a bigger(smaller) argument pool 119872 = 1000 (119872 = 200) isassociated with a higher (lower) variance of opinions(and extremism) and higher (lower) judgment accuracyFor the rest of the observations 119872 seem to have noeffect We choose a middle value for 119872 Similarly theargument pool chosen to be balanced or uniformly

Complexity 29

Require Argumentation framework 119860119865 = (AR)Ensure Assignment of labels to arguments from the grounded labeling of 119860119865L(A)

L0(A) larr997888 (0 0 0)119896 larr997888 0do119894119899 L119896+1(A) larr997888 119894119899 L119896(A) cup 119886 | 119886 is not labeled 119894119899 L119896(A) 119886119899119889 forall119887 isin A 119904119905 119887 R 119886 then 119887 isin 119900119906119905 L119896(A)119900119906119905 L119896+1(A) larr997888 119900119906119905 L119896(A) cup 119886 | 119886 is not labeled 119894119899 L119896(A) 119886119899119889 exist119887 isin A 119904119905 119887 R 119886 and 119887 isin 119894119899 L119896+1(A)

while L119896+1(A) = L119896(A) = 0Algorithm 3 Solve argumentation frameworks (with grounded semantics)

Require Vector of values for parameters (119881)Ensure Vector of statistics (119880)

procedure Set-parameters(119881) ⊳Initialize parameters for scenarioprocedure Init-objs(119899119898 119896 ) ⊳Create119898 arguments 119899 agents and 119860119865120576procedure Solve-Argumentation-Framework(119860119865) ⊳Assigns the epistemic labelingL120576 toA119894 larr997888 1119905 larr997888 1while 119894 le 119872 do ⊳ 119872 the number of arguments to discuss119868119894 larr997888 119866119890119899 119886119903119892(1) ⊳Generates proposal argument 119868119860 119894119905 larr997888 (119868119894 0)L119886119887(119868119894) larr997888 119906119899119889 ⊳Initialize label for 119868119895 larr997888 1 ⊳Initialize debate counter 119895while ((L119886119887(119868) = 119906119899119889 or 119898 le 119895) and 119895 le 119898) doif 119905 equiv 0(119898119900119889 119905119863) then

procedure build-argumentation-framework(119899119863 119887119890ℎ119886V119894119900119903) ⊳Build the argument graph agents deliberate(119861119894119905 119878119894119905) larr997888 (119883 119884) sub (⋃119897isin119873119899119863 A119897119894119905 ⋃119897isin119873119899119863 R119897119894119905) ⊳ 119873119899119863 sub 119873 are agents sampled by the CA119860 119894119905 larr997888 119889(119860 119894119905 (119861119894119905 119878119894119905))procedure Solve-Argumentation-Framework(A119889) ⊳FindL(A119889)L119886119887(119868119894) larr997888 L119886119887119863(119868119894)119895 larr997888 119895 + 1

else119896 larr997888 1while 119896 le 119905119863 doprocedure dyadic-social-influence(120583 119879 119880) ⊳Agents discuss one-to-one119896 larr997888 119896 + 1119905 larr997888 119905 + 1

if 120572 gt 0 andL119886119887(119868119894) = 119900119906119905 then ⊳Agents vote for the proposalif (|119897 isin 119873|119900119897 times V119868119894 ge 0| ge 120572 times 119899) then

L119886119887(119868119894) larr997888 119894119899else

L119886119887(119868119894) larr997888 119900119906119905if L119886119887(119868119894) = 119894119899 thenprocedure Deliberation-influence(120574 119901119886 119901119903) ⊳Agents are influenced by the result of the decision process119894 larr997888 119894 + 1119905 larr997888 119905 + 1119880 larr997888 119862119900119898119901119906119905119890 119878119905119886119905119894119904119905119894119888119904(119881)

return 119880 ⊳Get the statisticsAlgorithm 4 Run for one scenario

generated does not seem to play a significant explanatoryrole on any of the metrics

16 Experiments show that themaximumnumber of debatesis not an important parameter even if it is an intuitiveone when considering decision-making processes Inpractice debates end after at most 119898 + 1 deliberation

steps and even then it happens rarely However to avoidunpleasant surprises like infinite debates on a completeargument graph containing the proposal argument weadded the constraint 119898 Its value is precisely 7 becausethe biggest minimum number of debates we studyis 6

30 Complexity

17 To ensure comparability with the deliberative andmixedopinion dynamics models on judgment accuracy andcoherence agents vote on proposal arguments every 119905119863times(119898+1) 119905119863 and119898does not explain variability in judgmentaccuracy nor in coherence

18 Student t-tests for independent samples are robust tothe violation of two hypothesis on the distribution ofthe samples normality and homogeneity of variancesas long as the sample size is big enough (central limittheorem) and if the difference in size of the comparedsamples is smallrespectively

19 Studentrsquos test version forwhen variances are not assumedequal and the difference in size of the compared samplesis big

20 We do not correct for homoskedasticity nor normality oferrors in the estimates

21 Balanced as the same number of runs per scenario orsuch that no significant correlations are found betweenparameters

References

[1] R Axelrod ldquoThe dissemination of culture a model withlocal convergence and global polarizationrdquo Journal of ConflictResolution vol 41 no 2 pp 203ndash226 1997

[2] D M Rousseau Reinforcing the MicroMacro Bridge Organiza-tionalThinking and Pluralistic Vehicles SAGEPublications SageLos Angeles Calif USA 2011

[3] S Galam and SMoscovici ldquoTowards a theory of collective phe-nomena consensus and attitude changes in groupsrdquo EuropeanJournal of Social Psychology vol 21 no 1 pp 49ndash74 1991

[4] A Nowak J Szamrej and B Latane ldquoFrom private attitude topublic opinion a dynamic theory of social impactrdquo Psychologi-cal Review vol 97 no 3 pp 362ndash376 1990

[5] G Deffuant S Huet and F Amblard ldquoAn individual-basedmodel of innovation diffusion mixing social value and individ-ual benefitrdquo American Journal of Sociology vol 110 no 4 pp1041ndash1069 2005

[6] G Deffuant D Neau F Amblard and G Weisbuch ldquoMixingbeliefs among interacting agentsrdquoAdvances in Complex Systems(ACS) vol 3 no 01n04 pp 87ndash98 2000

[7] J Rouchier P Tubaro and C Emery ldquoOpinion transmission inorganizations an agent-based modeling approachrdquo Computa-tional and Mathematical OrganizationTheory vol 20 no 3 pp252ndash277 2014

[8] M Meadows and D Cliff ldquoThe relative disagreement model ofopinion dynamics where do extremists come fromrdquo in Pro-ceedings of the International Workshop on Self-Organizing Sys-tems Lecture Notes in Computer Science pp 66ndash77 Springer2013

[9] W Jager and F Amblard ldquoUniformity bipolarization and plu-riformity captured as generic stylized behavior with an agent-based simulation model of attitude changerdquo Computational ampMathematical Organization Theory vol 10 no 4 pp 295ndash3032005

[10] Y-F Zhang H-Y Duan and Z-L Geng ldquoEvolutionary mech-anism of frangibility in social consensus system based onnegative emotions spreadrdquo Complexity vol 2017 Article ID4037049 8 pages 2017

[11] A Flache andMWMacy ldquoSmall worlds and cultural polariza-tionrdquoThe Journal of Mathematical Sociology vol 35 no 1-3 pp146ndash176 2011

[12] A Tsang and K Larson ldquoOpinion dynamics of skepticalagentsrdquo in Proceedings of the 13th International Conference onAutonomous Agents and Multiagent Systems AAMAS 2014 pp277ndash284 International Foundation forAutonomousAgents andMultiagent Systems France May 2014

[13] K M Hansen Deliberative democracy and opinion formationUniversity Press of Southern Denmark Odense 2004

[14] C R Sunstein ldquoThe law of group polarizationrdquo Journal ofPolitical Philosophy vol 10 no 2 pp 175ndash195 2002 WileyOnline Library

[15] S Moscovici and W Doise Dissensions et Consensus unetheorie generale des decisions collectives Presses Universitairesde France-PUF 1992

[16] J S Fishkin and R C Luskin ldquoExperimenting with a demo-cratic ideal deliberative polling and public opinionrdquo ActaPolitica vol 40 no 3 pp 284ndash298 2005

[17] R C Luskin J S Fishkin and S Iyengar Considered Opinionson US Foreign Policy Face-To-Face versus Online Delibera-tive Polling International Communication Association NewOrleans La USA 2004

[18] J Barabas ldquoHowdeliberation affects policy opinionsrdquoAmericanPolitical Science Review vol 98 no 4 pp 687ndash701 2004

[19] O Pryymak A Rogers and N R Jennings ldquoEfficient opinionsharing in large decentralised teamsrdquo in Proceedings of the 11thInternational Conference on Autonomous Agents andMultiagentSystems vol 1 pp 543ndash550 Spain June 2012

[20] J Rouchier and E Tanimura ldquoWhen overconfident agents slowdown collective learningrdquo Simulation vol 88 no 1 pp 33ndash492012

[21] D Miller ldquoDeliberative democracy and social choicerdquo PoliticalStudies vol 40 S1 pp 54ndash67 1992

[22] M Sherif and C I Hovland Social Judgment Assimilation andContrast Effects in Communication and Attitude Change YaleUniversity Press 1961

[23] P M Dung ldquoOn the acceptability of arguments and its funda-mental role in nonmonotonic reasoning logic programmingand n-person gamesrdquo Artificial Intelligence vol 77 no 2 pp321ndash357 1995

[24] M Caminada and G Pigozzi ldquoOn judgment aggregation inabstract argumentationrdquo Autonomous Agents and Multi-AgentSystems vol 22 no 1 pp 64ndash102 2011

[25] P Baroni M Caminada andM Giacomin ldquoAn introduction toargumentation semanticsrdquoThe Knowledge Engineering Reviewvol 26 no 4 pp 365ndash410 2011

[26] S Chambers ldquoDeliberative democratic theoryrdquo Annual Reviewof Political Science vol 6 no 1 pp 307ndash326 2003

[27] E Bonzon andNMaudet ldquoOn the outcomes of multiparty per-suasionrdquo in Proceedings of the 10th International Conference onAutonomous Agents and Multiagent Systems 2011 AAMAS 2011vol 1 pp 47ndash54 International Foundation for AutonomousAgents and Multiagent Systems Taiwan May 2011

[28] J Ranciere Disagreement Politics and Philosophy University ofMinnesota Press 1999

[29] T Bench-Capon and K Atkinson ldquoAbstract argumentation andvaluesrdquo in Argumentation in Artificial Intelligence pp 45ndash64Springer 2009

[30] M Caminada ldquoOn the issue of reinstatement in argumenta-tionrdquo in Proceedings of the European Workshop on Logics in

Complexity 31

Artificial Intelligence vol 4160 of Lecture Notes in ComputerScience pp 111ndash123 Springer Berlin Heidelberg

[31] GAVreeswijk ldquoAn algorithm to computeminimally groundedand admissible defence sets in argument systemsrdquo in Pro-ceedings of the 2006 conference on Computational Models ofArgument Proceedings of COMMA 2006 vol 144 pp 109ndash120IOS Press Amsterdam The Netherlands 2006

[32] C W Sherif M Kelly H L Rodgers et al ldquoPersonal involve-ment social judgment and actionrdquo Journal of Personality andSocial Psychology vol 27 no 3 pp 311ndash328 1973

[33] G Deffuant F Amblard G Weisbuch and T Faure ldquoHow canextremism prevail A study based on the relative agreementinteraction modelrdquo Journal of Artificial Societies and SocialSimulation vol 5 no 4 p 1 2002

[34] E Awad M W Caminada G Pigozzi M Podlaszewski andI Rahwan ldquoPareto optimality and strategy-proofness in groupargument evaluationrdquo Journal of Logic and Computation vol 27no 8 pp 2581ndash2609 2017

[35] D Grossi and G Pigozzi ldquoJudgment aggregation a primerrdquoSynthesis Lectures on Artificial Intelligence and Machine Learn-ing vol 8 no 2 pp 1ndash151 2014

[36] H Landemore and S E Page ldquoDeliberation and disagreementproblem solving prediction and positive dissensusrdquo PoliticsPhilosophy and Economics vol 14 no 3 pp 229ndash254 2015

[37] R Hegselmann and U Krause ldquoOpinion dynamics andbounded confidence models analysis and simulationrdquo Journalof Artificial Societies and Social Simulation vol 5 no 3 2002

[38] A Stefanelli and R Seidl ldquoModerate and polarized opinionsUsing empirical data for an agent-based simulationrdquo inProceed-ings of the Social Simulation Conference 2014

[39] S Gabbriellini and P Torroni ldquoA new framework for ABMsbased on argumentative reasoningrdquo in Advances in SocialSimulation vol 229 pp 25ndash36 Springer Berlin Germany 2014

[40] H Mercier and D Sperber ldquoWhy do humans reason Argu-ments for an argumentative theoryrdquo Behavioral and BrainSciences vol 34 no 2 pp 57ndash74 2011

[41] H V Parunak T C Belding R Hilscher and S BruecknerldquoModeling and managing collective cognitive convergencerdquoin In Proceedings of the 7th iNternational Joint Conference onAutonomous Agents and Multiagent Systems vol 3 pp 1505ndash1508 International Foundation for Autonomous Agents andMultiagent Systems 2008

[42] H Prakken ldquoFormal systems for persuasion dialoguerdquo TheKnowledge Engineering Review vol 21 no 2 pp 163ndash188 2006

[43] K Atkinson T Bench-Capon and P Mcburney ldquoA dialoguegame protocol for multi-agent argument over proposals foractionrdquoAutonomousAgents andMulti-Agent Systems vol 11 no2 pp 153ndash171 2005

[44] M Thimm and A J Garcıa ldquoClassification and strategicalissues of argumentation games on structured argumentationframeworksrdquo in Proceedings of the 9th International Conferenceon Autonomous Agents and Multiagent Systems vol 1 pp 1247ndash1254 International Foundation for Autonomous Agents andMultiagent Systems Canada 2010

[45] I Rahwan and K Larson ldquoMechanism design for abstractargumentationrdquo in Proceedings of the 7th International JointConference on Autonomous Agents and Multiagent SystemsAAMAS 2008 pp 1031ndash1038 International Foundation forAutonomous Agents and Multiagent Systems Portugal May2008

[46] H Prakken R Riveret A Rotolo and G Sartor ldquoHeuristics inArgumentation A Game-Theoretical Investigationrdquo EuropeanUniversity Institute 2008

[47] N Hadidi Y Dimopoulos and P Moraitis ldquoTactics and con-cessions for argumentation-based negotiationrdquo in Proceedingsof the COMMA vol 245 pp 285ndash296 2012

[48] N Oren and T J Norman ldquoArguing using opponent modelsrdquoin Proceedings of the International Workshop on Argumentationin Multi-Agent Systems pp 160ndash174 Springer 2009

[49] S Delre W Jager T Bijmolt and M Janssen ldquoTargeting andtiming promotional activities An agent-based model for thetakeoff of new productsrdquo Journal of Business Research vol 60no 8 pp 826ndash835 2007

[50] A Mirtabatabaei P Jia and F Bullo ldquoEulerian opinion dynam-ics with bounded confidence and exogenous inputsrdquo SIAMJournal on Applied Dynamical Systems vol 13 no 1 pp 425ndash446 2014

[51] V Barbet N Guiraud V Laperriere and J Rouchier Hagglingon Values Towards Consensus or Trouble 2019

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Complexity 7

235 Agent Behavior during Deliberation Agents maybehave in twodifferentways in deliberationTheymay behavenaively or focusedly Naive agents use deliberation to voicetheir opinions on the principle through arguments Focusedagents strategically argue in favor of proposal arguments thatsupport the principle they favor In both cases agents advancearguments in terms of the argumentsrsquo relative proximity totheir own positions [22]

Let 119886119894 isin A119894 denote the argument an agent advancesin a debate 119889 (of a deliberation process 119863119868) over a centralargument 119868 and V119886 the valence of an argument 119886 isin A Let

(i) 119891119894(119886) equiv (119900119894 times V119886 gt 0 or 119900119894 = V119886 = 0) the formula thatis satisfied if the argument 119886 is of the same valence asthe opinion of agent 119894

(ii) 119892119894(119886) equiv (119886 119887 isin A119894A119889 and forall119887(|119900119894 minus V119887| gt |119900119894 minus V119886|)) theformula that is satisfied only if the argument 119886 is in 119894rsquossack (A119894) and not in the debate and is the closest interms of adherence to the principle P to 119894rsquos opinion

(iii) ℎ119894(119886) equiv (119891119894(119886) and 119892119894(119886)) the conjunction of thepreceding formuli indicating that an argument 119886 is inagent 119894rsquos sack and not in the debate of the same signas 119900119894 and closest to 119900119894 in absolute value

Naive agents choose 119886119894 = 119886 such that ℎ119894(119886) is true given anyproposal argument 119868 Agents that behave strategically or tosay focusedly choose their moves in the debate as follows(119873119868119871 indicates no move or null argument)

(a) if agent 119894rsquos position with respect to P is of the samesign as the proposal argumentrsquos adherence to P (119891119894(119868)) sheadvances an argument 119886 such that

(i) ℎ119894(119886) and ((119886 119868) isin 119863119890119891119889119894 )) is true If 119886119894 = 119873119868119871(i1) notℎ119894(119886) and 119892119894(119886) and ((119886 119868) isin 119863119890119891119889119894 ) (chooses anargument not of the same sign and closest to heropinion and defending 119868) is true If 119886119894 = 119873119868119871(ii) ℎ119894(119886) and ((exist119888 isin A119894)((119888 119868) isin 119860119905119905119889119894 and (119886 119888) isin 119860119905119905119894))(anticipates an attack from an argument 119888 on 119868 andadvances an argument that attacks the argument 119888)6is true If 119886119894 = 119873119868119871(ii1) notℎ119894(119886)and119892119894(119886)and ((exist119888 isin A119894)((119888 119868) isin 119860119905119905119889119894 and(119886 119888) isin119860119905119905119894)) is true If 119886119894 = 119873119868119871(iii) ℎ119894(119886) and ((119886 119868) notin 119860119905119905119889119894 ) (simply does not attack 119868)is true If 119886119894 = 119873119868119871(iii1)notℎ119894(119886)and119892119894(119886)and((119886 119868) notin 119860119905119905119889119894 ) is true If 119886119894 = 119873119868119871(iv) 119886119894 = 119873119868119871 (the agent will advance no argument)

(b) if agent 119894rsquos position with respect toP is of the oppositesign of the proposal argumentrsquos adherence to P (not119891119894(119868)) sheplays the argument 119886 such that

(i) ℎ119894(119886) and ((119886 119868) isin 119860119905119905119889119894 ) 7 If 119886119894 = 119873119868119871(i1) notℎ119894(119886)and119892119894(119886)and ((119886 119868) isin 119860119905119905119889119894 ) is true If 119886119894 = 119873119868119871(ii) ℎ119894(119886) and ((exist119887 isin A119894)((119887 119868) isin 119863119890119891119889119894 and (119886 119887) isin 119860119905119905119894))(anticipates a defense from an argument 119888 to 119868 andadvances an argument that attacks 119888) is true If 119886119894 =119873119868119871

(ii1)notℎ119894(119886)and119892119894(119886)and((exist119887 isin A119894)((119887 119868) isin 119863119890119891119889119894 and(119886 119887) isin119860119905119905119894)) is true If 119886119894 = 119873119868119871(iii) ℎ119894(119886) and ((forall119887 isin A119889)((119887 119868) isin 119860119905119905119889119894 and (119886 119887) notin 119860119905119905119889119894 ))(avoids attacking any attacker of 119868) If 119886119894 = 119873119868119871(iii1)notℎ119894(119886)and119892119894(119886)and((forall119887 isin A119889)((119887 119868) isin 119860119905119905119889119894 and(119886 119887) notin119860119905119905119889119894 )) If 119886119894 = 119873119868119871(iv) 119886119894 = 119873119868119871

In English a focused agent that is opposed to the proposalattempts to either attack it directly or attack an argument thatis defending it whereas an agent that agrees with the proposalattempts to either defend it or avoid attacking it altogetherThis behavioral approach is similar to debate protocols inabstract argumentation but for the fact that agents mayadvance arguments that do not result in an ldquoadvancementrdquoof the debate In our case the debates stop for other reasons(see Section 24)

Two important points deserve to be highlighted Firstlyfocused agents with small argument sacks will regularly findthemselves applying rules (119894119894119894) or (119894V) In effect if they donot have enough arguments to infer attacks and defensestheir behavior in deliberation is likely to be similar to that ofthe naive agents Secondly the naive and focused behavioralassumptions presented here are very different in terms ofcomputational complexity The first assumes that the agentobserves the debate but what she sees has no effect on hercourse of action ie the argument she voices is uniquelydetermined by how she feels about the principle behindthe proposal argument In the second an agent debatesessentially to knock out any proposal argument she disagreeswith8

24 Tables for Deliberation Tables are the physical or virtualplaces where the exchange of arguments occurs Agentsdeliberate on these tables to determine whether the proposalargument is acceptable or not and are thus subdued bythe deliberation procedure imposed by the tablersquos centralauthority (CA) [27] The CA decides on the structure andlength of the collective decision-making process and thedeliberation procedure It controls the percentage of agentsfrom the population that actively participates in the debate(119899119863) and the labeling-based semantics used to extract accept-able arguments from the framework (120590) The percentage 119899119863denotes either the proportion of agents in the population thatgets to advance arguments in a population-scale deliberationor an independent sample of agents summoned to activelyparticipate in the debate 120590 stands for the procedure used toconclude on the epistemic status of the proposal argumentand the arguments advanced during deliberation

The CA has the ability to stop debates at will using a stoprule that inherently depends on the number of debates thatought to take place before a decision is deemed sufficientlydiscussed (119898) the maximum number of debates that cantake place before abandoning deliberation (119898) and the labelgiven to the proposal argument 119868 (119904119903(119898119898L119886119887(119868))) 119904119903is a Boolean function whose value ldquotruerdquo signals the callfor a vote andor the end of the decision-making process119898 is associated with a minimal dialectical or epistemic

8 Complexity

requirement to consider a proposal for voting and to a lowerbound of the length of the deliberation process 119898 providesan upper bound Moreover the CA controls the size of thetime interval between debates (119905119863) the collective decisionrule (eg if there is voting on the deliberated proposals) andthe majority quota rule for accepting proposal arguments(120572) in the decision-making process 119905119863 may represent thefrequency or density of pair-wise interactions in a decision-making process

241 The Construction of a Decision-Making Process Adeliberation process or debate in ourmodel is a tool to obtainlabels for proposal arguments that are as close as possible tothe ideal or consensual ones To define a deliberation processformally we introduce the notion of debate step as constituentof a deliberation process Informally a debate step is a timestep in which a debate occurs Formally and more event-oriented a debate step (1198891198731015840 1198611015840 1198781015840) of a debate 119863119868 on 119868 isa quadruplet composed of a set of agents 1198731015840 sub 119873 a set ofarguments 1198611015840 sub ⋃119894isin1198731015840 A119894 a set of attack relations 1198781015840 sub Rand a mapping

119889 (AF 2A times 2R) 997888rarr AF

(119860119865 (1198611015840 1198781015840)) 997891997888rarr 1198601198651015840 = (119861 cup 1198611015840 119878 ⊔ 1198781015840) (3)

that adds the arguments in 1198611015840 and some attacks in 1198781015840 to aframework 119860119865 = (119861 119878) and where 119878 ⊔ 1198781015840 = (119886 119887) isin 119878 cup 1198781015840 |119886 119887 isin 119861 cup 1198611015840 In the same spirit we define a deliberationprocess119863119868 on a proposal argument 119868 as a sequence of debatesteps (119889119873119904 119861119904 119878119904)119904gt0 such that1198730 = 0 and

119889((119904minus1⋃119895=0

119861119895 119904minus1⨆119895=0

119878119895) (119861119904 119878119904)) sub (AR) (4)

with

(i) 1198610 = 119868 (the initial argument is the proposal argu-ment)

(ii) 1198780 = 0 (no reflexive attack from 119868 to 119868 is allowed)(iii) (forall119904 gt 0)(⋃119895lt119904 119861119895cap119861119904 = 0) (any newly added argument

to the framework has yet to be added to it)(iv) (forall119904 gt 0)(⨆119895lt119904 119878119895 cap 119878119904 = 0) (any newly considered

attack among arguments cannot be declared amongarguments that are not in the framework)

(v) lim119904997888rarrinfin(⋃119904gt0 119861119904 ⋃119904gt0 119878119904) sube (AR) (the system isstable no arguments are created during deliberation)

Finally let L119886119887119889(119868) denote the label given to argument 119868 ata debate step 119889 during a deliberation process 119863119868 A decision-making process notedD119868 = (119899119863 119905119863 119904119903(119898119898L119886119887119889(119868)) 120572 120590)over a proposalwhose justification argument is 119868 is a sequenceof debate and nondebate steps (119911119905)119905isinN 119911119905 isin 119889 119889 such that

(i) 119889 are debate steps and 119889 nondebate steps that corre-spond to time steps at which there are no debates

(ii) If 119905(mod 119905119863) equiv 0 then 119911119905 = 119889 and 119889 otherwise

For Against

Table

CA

in out

out out in

out in

proparg

Figure 3 A representation of a table of deliberation betweentwo groups of 3 agents with 4 arguments each Each agent hasadvanced an argument Agents tacitly agree on the attacks betweenthe arguments The CA determines the labeling using the groundedlabeling-based semantics

d d d d d d d d d d d d dTime 0 1 2 3 4 5 6 7 8 9 10 11 12I

Figure 4 A decision-making processD119868 or collective decision stepstarting at 119905 = 0 composed of debate (119889) and nondebate (119889) substepsDebate steps are shaded The number of nondebate steps betweendebates is 119905119863 = 3 the number of debates in the decision-makingprocess is119898 = 4

(iii) The subsequence (119911119897119905)119897119905isinN of 119911119905 with 119897119905 = 119905 such that119905(mod 119905119863) equiv 0 is a deliberation process with |1198731015840| =119899119863 times |119873|(iv) 119904119903(sdot sdotL119886119887119889(119868)) pursues the deliberation process as

long asL119886119887119889(119868) = 119906119899119889 under the semantics 120590(v) The length of the sequence |119911119905| or equivalently the

duration of the process is somewhere in the interval⟦119898119905119863 119898119905119863⟧(vi) The final labeling for the proposal argument 119868

L119886119887(119868) is determined by a combination of its delib-erated labelL119886119887119889(119868) and a majority vote of majorityquota 120572

A decision-making process ends when a final decision hasbeen taken concerning the acceptability of the proposal Peg 119868 has been deemed unacceptable (L119886119887119889(119868) = 119900119906119905) or amajority of agents have voted againstP For a representationof a deliberative interaction on a table see Figure 3 for one ofa decision-making process see Figure 4

Please note that every collective decision can be seen asa ldquotime steprdquo in as much as it describes how agents updateopinions and make collective decisions In the model thedecision-making process and the parameters define the sub-steps or events that occur within the time step Henceforth

Complexity 9

Fix proceduralsm m α nD tD σ Init count m larr 0

Given proposal anargument I is randomly generated

nD times |N | agents arerandomly picked

m larr m + 1

Begin decision processon proposal at time t

t larr t + 1Deliberation step m

larr 0

t larr t + 1dyadic opinionupdate Eq (5)tm larr tm + 1

m ge m

tm gt tD

ℒd (I) = in

m ge m

yields ℒ(I)

ℒ(I) = in Deliberationupdate Eq (6)

End decision

yes

no

no

no

yes

yes

no

yes

yes

yes

no

no

I

Vote on ℒd(I)

ℒd (I) = und

ＪＬＩ＝ＭＭ I

Init count tmyields ℒd(I)

Figure 5 Process chart for a decision processD119868 on a proposalP 119905 corresponds to time and 119898 (119905119898) corresponds to the number of debate(non-debate) steps that have taken place in the deliberation process (in between two debate steps) over time for the mixed discussion modelWe useL119886119887(119868) andL(119868) interchangeably in the figure for presentation

comparing simulations on the basis of the different decision-making processes translates into comparing these collectivedecision steps

242 Deliberation Protocol To define the deliberation proto-col held in the table either as a consequence of the definitionprocess or as a statement we assume the following

(i) Agents may decide not to contribute to the debate(ii) All agents have the same probability conditional to

their opinions of being picked to participate in adebate or deliberation step

(iii) There is no restriction on the number of times eachagent can participate in a deliberation process

(iv) Each agent may only place one argument per debatestep

(v) Arguments that have already been advanced in thedebate may attack newly placed arguments9

The deliberation or debate protocol goes as follows

(1) The CA randomly generates and makes public acentral argument or proposal argument 119868 and informsall agents about the rules of the decision-makingprocess

(2) The CA randomly draws two sets of (119899119863 times |119873|)2agents with divergent views10 onP and P It mergesthem to create the set of debaters1198731015840

(3) Each agent 119894 advances an argument from her sackA119894 The CA makes sure that there are no repeatingarguments (agents already take argument repetitioninto account)

(4) The CA establishes the debate step 119889rsquos argumentationframework and computes its labelingL120590

119889(A119889) using120590(5) If the computed label for 119868 is L119886119887119889(119868) = 119906119899119889

or the number of debates steps is inferior or equalto 119898 at time 119905 then the CA stops the debate andresumes it at the (119905 + 119905119863 + 1)rsquoth time step by repeating(1) (2) (3) (4) and (5)

(6) Let L119886119887(119868) be the final label given to the proposalargument 119868 If voting is allowed then if more than120572 times |119873| agents agree with 119868 119868 is accepted (L119886119887(119868) =119894119899) it is rejected if strictly less than 120572times |119873| agree withit (L119886119887(119868) = 119900119906119905) When there is a tie ifL119886119887119889(119868) =119906119899119889 then L119886119887(119868) = 119900119906119905 if L119886119887119889(119868) = 119894119899 thenL119886119887(119868) = 119894119899 When voting is not part of the decision-making process (120572 = 0)L119886119887119889(119868) = L119886119887(119868)

One important point to notice about this protocol is that itinduces a stop rule 119904119903 (cf steps (5) and (6) of protocol) forthe decision process and thus it always ends Either agentsdebate and agree on the proposalrsquos acceptability throughargumentation or after 119898 debate steps they directly voteon it so that a decision concerning the proposal is alwaysreached Refer to Figure 5 for a sweeping description of thedeliberation protocol and the decision-making process

10 Complexity

-1

-1 1

1

Latitude of acceptance

Latitude of rejection Agent i

Agent j

Figure 6 A dyadic interaction between agents 119894 and 119895 as expressedin (5) Agent 119894rsquos opinion falls within agent 119895rsquos latitude of noncommit-ment and conversely Neither agent affects the opinion of the other

25 Opinion Dynamics for Social Interactions In this subsec-tion we describe the opinion diffusion model based on pair-wise interactions among agents and deliberation Sherif rsquosand Hovlandrsquos social judgment theory [22] motivates part ofour approach It describes how individualsrsquo opinions changeon the basis of their attitude structures Attitude structuresrefer to the relative scope width or latitude of categoriesused by individuals when evaluating information namelythe latitudes of acceptance rejection and noncommitment[32] The idea behind this theory is that individuals changetheir positions only in accordance to how far or close thecommunicative cues they receive are from (to) their anchorpositions It holds that if communicative cues are far (close)from (to) an agentrsquos position say over her latitude of rejection(acceptance) then the agent shifts her position away from(towards) the position defended by the cues In the casewherethe cues fall within the agentrsquos latitude of noncommitmenther position does not change (see Figure 6 and Equation (5))

251 Pair-Wise or Dyadic Opinion Dynamics As agentsmay communicate and deliberate collectively they may alsoengage in one-to-one conversations with other agents toponder on their positions We loosely associate this type ofcommunication with dyadic nonargumentative exchange ordiscussion based on fallacious arguments and persuasionIn the light of social judgment theory and the descriptionof agents in the system we model pair-wise symmetricinteractions following the opinion dynamicsmodel in [9] Anagent 119895rsquos influence on an agent 119894rsquos opinion at time 119905 is governedby the ensuing difference equation

119900119905+1119894 =

119900119905119894 + 120583119894 (119900119905119895 minus 119900119905119894) 119894119891 10038161003816100381610038161003816119900119905119894 minus 11990011990511989510038161003816100381610038161003816 lt 119880119894119900119905119894 minus 120583119894 (119900119905119895 minus 119900119905119894) 119894119891 1003816100381610038161003816119900119905119894 minus 119900119905119894 1003816100381610038161003816 gt 119879119894119900119905119894 119900119905ℎ119890119903119908119894119904119890(5)

where the parameter 120583119894 isin [0 05] controls for the strengthof the attraction and repulsion in social influence and 119879119894and 119880119894 are the latitudes of rejection and acceptance for agent119894 respectively The parameter 120583119894 may be thought of as therelative importance agent 119894 gives to the opinions of her peersor analogously the weight she gives to her own opinionwhen updating 120583 lt 05 means that agents will never givemore weight to the opinions of other agents than to their

own (egocentric bias) When two agents 119894 and 119895 discuss iffor instance 119895 advances an informational cue (argumentpersuasion tactic) and it happens to be close enough to119894rsquos opinion anchor then 119894 shifts her opinion towards thedirection of 119895rsquos informational cue The symmetric influencefrom agent 119894 to agent 119895 takes effect in the same way (seeFigure 6)

We add to the dyadic dynamics a rule of encounters ateach step of pair-wise interactions each agent meets exactlyone other agent at random This rule may be associated withrandom day-to-day encounters among agents Steps of socialinfluence correspond to nondebate steps (119889) in decision-making processes

252 Deliberative Opinion Dynamics We define an opinionupdate equation that links the proposal arguments that areadvanced during deliberation and opinions We combine theuncertain and probabilistic nature of the effect of delibera-tion have it a moderating [15 16] or polarizing [14] one witha mechanism similar to the one in social judgment theorybased on the distance between different informational cuesand opinion anchors [32] The probabilistic modeling of theopinion update can be related to deliberation encouragingopen-mindedness11 during collective discussion [18] Fromthis choice it follows that deliberated cues can affect even themost extreme of agents as opposed to some classic opiniondiffusion models (eg [5 8]) where once agents becomeextremists they may no longer become moderate

LetP be a proposal V119905119868 the proposal argument 119868rsquos level ofsupport for a principle P and 119900119905119894 an agent 119894rsquos opinion at time119905 Then given the difference 120575119905119894 = (V119905119868 minus 119900119905119894)2 we define 119894rsquosprobability of being attracted to a decisionrsquos informational cue119868 by 119909119905119894 = 119901|120575119905119894 |119886 where 119901119886 denotes a general probability param-eter that characterizes how important deliberated results arefor the group The parameter may also be interpreted asthe grouprsquos tendency to be swayed by a decisional majoritySimilarly we define 119894rsquos probability of being repulsed from adecision informational cue 119868 by 119910119905119894 = 1199011|120575119905119894 |119903 where 119901119903 denotesa general probability parameter that characterizes the grouprsquosdislike of deliberated results 119901119903 can also be thought of as thegrouprsquos distrust of the system symbolized by deliberation anddemocracy

Let L119886119887(119868) denote the epistemic status of the proposalargument 119868 at the end of a decision process over P everyagent 119894 updates her opinion as follows

(i) IfL119886119887(119868) = 119894119899

119900119905+1119894 =

119900119905119894 + 2120574119894120575119905119894 119908119894119905ℎ 119901119903119900119887119886119887119894119897119894119905119910 119909119905119894119900119905119894 minus 2120574119894120575t119894 119908119894119905ℎ 119901119903119900119887119886119887119894119897119894119905119910 119910119905119894119900119905119894 119908119894119905ℎ 119901119903119900119887119886119887119894119897119894119905119910 1 minus 119909119905119894 minus 119910119905119894

(6)

(ii) IfL119886119887(119868) = 119900119906119905 119900119905+1119894 = 119900119905119894where 120574119894 isin [0 05] is the strength of repulsion and attractionin the dynamicThemeaning of 120574119894 is analogous to the one of120583119894in the dyadic interactionsmodel but for the observation that 119894

Complexity 11

weights her opinion to an argument rather than to an opinionThe interpretation of this dynamics is straightforward Ifthe deliberated proposal argument is close to an agentrsquosopinion then it is very likely that the agent shifts its opiniontowards it Please note that the probability 119909119905119894 (119910119905119894 ) that anagent is attracted (repelled) to (from) an accepted deliberatedproposal argument is bounded below (above) by 119901119886 (119901119903)(equivalently 119909119905119894 isin [119901119886 1] 119910119905119894 isin [0 119901119903]) Steps of deliberativeopinion dynamics are associated with debate steps in which aconcluding collective decision is made

253 Mixed Opinion Dynamics Opinion dynamics in thecontext of decision-making processes can be best described asa combination of the two preceding dynamics In this opiniondynamics model agents are engaged in a democratic systemin which they deliberate vote for proposals and occasionallydiscuss with one another on the principle supported by theproposalsThe opinion dynamics for an agent 119894 can be writtenas a sequence of dyadic and deliberative opinion updates

119900119905+1119894 =

119864119902119906119886119905119894119900119899 (5) 119894119891 119905 equiv 0 (119898119900119889119905119863) and 119904119903 (sdot) = 119891119886119897119904119890119864119902119906119886119905119894119900119899 (6) 119894119891 119905 equiv 0 (119898119900119889119905119863) and 119904119903 (sdot) = 119905119903119906119890119900119905119894 119900119905ℎ119890119903119908119894119904119890

(7)

where 119904119903(sdot) is the decision processrsquos stop rule 119905119863 the numberof time steps between each deliberation step Equation (5)describes the dynamics for the dyadic interactions and Equa-tion (6) posits the changes in opinions due to the deliberatedcues Equation (5) applies when agents are not deliberatingand when each agent encounters another agent to exchangeinformation that may lead to local opinion updates Whenthere is deliberation a handful of agents deliberate and ifthey reach the stop condition imposed by the table all agentsvote for the proposal They update (see Equation (6)) theiropinions on the basis of the proposal argumentrsquos support forthe principle and the result of the scrutiny Otherwise thereis no change in the opinions of the agents A vote indicatesthe end of the decision-making process

3 Experiments and Calibration

In this section we introduce the metrics that enable us toobserve the simulations and characterize the calibration ofthe model

31 Observations and Initialization In this section wedescribe the simulations and the protocol used to test andobserve the results of our model We introduce the metricsof interest explain the calibration of the model describesome results obtained from the simulation data used forcalibration and conclude with a brief discussion on theexpected outcomes of the mixed opinion dynamics model

311 Metrics or Statistics of Interest Let 119878 denote the end ofa simulation We are interested in the effect of deliberationand of the modelrsquos procedural parameters on the followingmetrics or statistics

(i) Variance of opinions (119881119886119903(119900)) the variance ofopinions at time 119878 Since opinions live in the unionof the positive and negative unit intervals 119881119886119903(119900) isin[0 1] The higher the variance of the distribution isthe more ldquodiverserdquo the opinions are(ii) Proportion of extremists in the population(119901119903119900119901119890119909) the proportion of agents in the populationwith opinions such that |119900119894| ge 075 A high proportionof extremists makes ldquohealthyrdquo consensus12 difficult toreach and deliberation more or less informative(iii) Shifts of opinions (119878ℎ) [33] statistic that measuresthe aggregated change in individual opinion at time 119878with respect to aggregated individual opinion at thebeginning of the simulation

119878ℎ = 2sum119894isin119873 10038161003816100381610038161199000119894 minus 1199001198781198941003816100381610038161003816max119894isin1198731199000119894 minusmin119894isin1198731199000119894 (8)

119878ℎ is positive and bounded above by 2|119873| since 119900119894 isin[minus1 1] A low shift statistic implies that the processhas a small impact on opinions(iv) Judgment or consensual inaccuracy (119890119888) consen-sual accuracy of a group consists of an ad hoc statisticmeasuring a grouprsquos ability to infer correct labels forproposal arguments from a decision-making processgiven the ideal consensual labeling based on full-informationL120590

120576 (A) We use a Hamming-based dis-tance on labelings [34] to define the statistic

119890119888 = 1|I| sum119868isinI1198861198681003816100381610038161003816L119886119887120576 (119868) = L119886119887 (119868)1003816100381610038161003816 (9)

whereI is the set of all discussed proposal argumentsand 119886119868 = 12 ifL119886119887120576(119868) = 119906119899119889 or ifL119886119887(119868) = 119906119899119889119886119868 = 1 otherwise119890119888 lives in the interval [0 1] An inaccuracy statisticclose to 1 indicates that agents subdued to a particulardecision-making process make many mistakes injudging the labels of the proposal arguments Notethat when there is voting all proposal arguments arelabeled either 119894119899 or 119900119906119905 after deliberation(v) Coherence (119894119903) letL119886119887119863(119868) be the label obtainedfor 119868 from the deliberation process without votingThe coherence statistic measures how well votingresults adjust to results obtained during deliberationonly We use the proportion of arguments that havebeen labeled 119894119899 in the debate and that agents havevoted favorably for

119894119903 = 1003816100381610038161003816119868 isin I | L119886119887119863 (119868) = 119894119899 andL119886119887 (119868) = 11989411989910038161003816100381610038161003816100381610038161003816119868 isin I | L119886119887119863 (119868) = 1198941198991003816100381610038161003816 (10)

The coherence statisticrsquos domain is [0 1] If after 119872debates no deliberated central argument has beenlabeled 119894119899 or voting is not part of the procedure ofcollective decision-making then the statistic equals

12 Complexity

1 or said differently agents are perfectly coherentThis comes for the fact that if the central argumentis labeled 119900119906119905 then it is not even eligible for voting iflabeled 119906119899119889 then the debate will always be coherentwith the preferences of the agents since the result willbe a simple aggregation of their votes If the statisticequals 0 then none of the proposal arguments labeled119894119899 are voted in favorably by the agents It follows that ahigh coherence statistic implies that when agents votefor acceptable proposal arguments the results of thescrutiny reflect the consensual rationality expressedin the deliberation process

312 Parameters of Interest We recall the parameters ofinterest in our study that are linked to the structure of thedecision-making processes

(i) 119898 the minimum number of time steps in which adebate or a scrutiny occurs in a decision-making pro-cess before a final deliberated decision is submitted toa vote

(ii) 119899119863 the number of agents that deliberate as a propor-tion of the population

(iii) 119905119863 the number of time steps between debates inwhich pair-wise interactions among agents mayoccur

(iv) 120572 the proportional majority requirement for theacceptance of a proposal When 120572 = 0 it stands forno voting ldquoany proposal argument that is labeled 119894119899during deliberation is acceptedrdquo

Please recall that in terms of the definition of a decision-making process each parameter controls either the length orthe content of the sequence D119868 (119905119863 119898) or the rules that areapplied during the debate steps (120572 119899119863) (refer to Figure 4)313 Initialization All |119873| = 400 agents start off with anopinion 119900119894 drawn from a uniform distribution U(minus1 1) 13Given 119900119894 every agent 119894 randomly draws a setA119894 (|A119894| = 119896) ofarguments from a balanced14 argument pool A of119872 = 600nonneutral (V119886 = 0) arguments on the basis of 119900119894 if |119900119894| lt 025(the opinion is moderate) then agent 119894 randomly fills halfof her argument sack with arguments such that V119886 lt 0 andthe other half with arguments such that V119886 gt 0 (120601119898119894 = 05)Otherwise she randomly fills 120601119890119894 ge 05 of her sack witharguments such that V119886 and 119900119894 are of the same sign and theremaining (1 minus 120601119890119894 ) with arguments of the opposite sign15

Like for the opinion of agents every argument 119886rsquosadherence to the principle P V119886 is drawn from a uniformdistribution U(minus1 1) The attack relation that gives birth tothe ideal argumentation framework (119860119865120576) is established onthe basis of the V119886rsquos and the argumentsrsquo epistemic reach 119890119903

Let 120579(119886 119887 119903) = (|V119886 + V119887|2)119903 be an auxiliary positivereal-valued function that takes two arguments (119886 119887) and apositive real number (119903 gt 0) as inputThe probability that anyargument 119886 isin A creates a link to any other argument 119887 isin A

is given by the equation 119901 = 119901120579(1198861198872)119890 and thus argumentssupporting opposing views of the principle always attack each

other We fix 119901119890 the epistemic correlation parameter to 015If 119901119890 was greater than 015 then arguments of the same signwould attack each other too often (and focused agents wouldbe rarely incited to advance favorable arguments duringdeliberation) If 119901119890 was lower then the arguments of thesame type would induce an almost empty graph which isunrealistic The number of arguments that any argument 119886can attack is bounded by 119886rsquos epistemic reach that we fix to15 A higher value of epistemic reach makes the argumentlattice too conflicting and nearly bipartite A lower epistemicreachmakes the lattice not sufficiently conflicting in the sensethat too many arguments can attack the proposal argumentrelatively to the few that can defend it If any argument attacksthe proposal argument the chances that another argumentattacks the attacker are low Hence the proposal argumentis almost never accepted and deliberative opinion updatesbecome rare

The arguments in the resulting argumentation frame-work 119860119865120576 are given a permanent labelingL120590

120576 (A) (in shortL120576) using120590 = grounded semanticsWe choose the groundedlabeling-based semantics because it provides a unique admis-sible labeling respects minimal rationality constraints whilesimplifying the model and models a skeptic approach toaccepting arguments (see [23 25 31]) as it maximizes thecardinality of the set 119906119899119889 L120576 To some extent if we considerproposals as committing because they may guide collectiveaction choosing a skeptic semantics seems reasonable insofaras it labels an argument 119894119899 or 119900119906119905 only if it has no reasonto label it 119906119899119889 For instance if a proposal is a public policythat requires large amounts of resources and engages tofuture course of action then it may also determine policycycles and heavily burden a group and its future decisionsIn important situations like these it seems reasonable tolengthen deliberative cues and ask for more demanding andldquogroundedrdquo criteria for policy argument acceptability

On the proposalrsquos side we create an argument 119868 whoseadherence to P is also drawn from aU(minus1 1) and we label itL119886119887(119868) = 119906119899119889 119868 is interpreted as being the main argumentjustifying the discussed proposal P and V119905119868 as its supportfor P at time 119905 119868 cannot attack other arguments but otherarguments can attack it For each argument 119886 isin A a directedarc or attack from 119886 to 119868 is activated with probability 119901119901 =003120579(11988611986802) When 119901119901rsquos lower bound is too high the proposalargument is almost always defeated and thus deliberativeopinion updates do not happen When 119901119901rsquos lower boundis lower than 003 then the opposite occursmdashthe proposalargument is always accepted due to an absence of attackstowards it Finally we set the maximum number of debatesto119898 = 7 for the decision-making processes stop rules1632 Calibration and Simulation Protocol In this subsectionwe discuss the calibration of the models the terminationconditions for runs in the deliberative and mixed model andthe expected outcomes in terms of the observations

321 Calibrating Dyadic Opinion Dynamics Dyadic opiniondynamics correspond to a space of parameters in whichargumentation and deliberation spaces are not taken into

Complexity 13

Table 1 Parameters and parameter types and their domains used to compare dyadic collective and mixed opinion dynamics

Length ofD119868 Decisional in119863119868 Social influence Deliberation influence Cognitive119905119863 119898 119898 119899119863 120572 120583 119879 119880 119901119886 119901119903 120574 119896 119861119890ℎ119886V1198941199001199031 3 6 1 3 6 7 001 002 005 0 05 066 01 14 18 02 06 03 05 01 005 01 02 4 8 16 119873119886119894V119890 119865119900119888119906119904119890119889account Deliberated arguments and informational cues haveno effect on agent opinion but agents still vote for theproposal argument17 We use and calibrate this model forcomparability in terms of all our metrics since they arenot explicitly observed in [9] a reference opinion dynamicsmodel Furthermore for simplicity and comparability againwe suppose that agents are homogeneous in terms of theirattitude structures and opinion weightings We fix 120583119894 =120583 119879119894 = 119879 and 119880119894 = 119880 for all 119894 isin 119873 and for some triplet(120583 119880 119879) isin [0 05] times [0 2] times [119880 2]

Experiments suggest that the strength of the dyadic inter-actions is an explanatory factor of the time of convergence(120583 lt 05) and of the dyamicsrsquo steady states Bigger values of 120583speed up convergence towards a stable set of opinion clustersand affect the size and the relative position of these clustersin the opinion distribution 120583 however does not play a majorrole in the number of opinion clusters observed at the steadystate of the dynamics

We set 120583 to the value given to it in Jagger et alrsquos opiniondynamics model (120583 = 01) [9] For one it is the smallestreal number for which most results found in [9] hold and thedynamics are smooth secondly it seems to be a reasonableldquostrengthrdquo of influence considering that we would not wantpair-wise social interactions to completely shadow or maskan eventual effect of deliberation in the opinion dynamics

For different values of attitude structures 119879 and 119880 weget exactly the same results as in [9] in terms of the numberand density of the opinion clusters The different couplesof values of 119880 and 119879 model the size of the latitude ofnoncommitment (119879 minus 119880) and thus the agentsrsquo propensityto update opinions Whenever 119879 minus 119880 le 02 regularitiesin convergence and opinion clustering appear If 119880 gt 1there is always central convergence if 119879 lt 1 there is alwaysbipolar convergence with clusters of similar sizes For 119879 minus119880 gt 02 relatively low values of 119879 (119879 lt 1) and relativelyhigh values of 119880 always yield extreme bipolar convergencewith a relatively small cluster of moderate agents (bipolar-central convergence) For 119879 sufficiently high (119879 ge 12) and119880 sufficiently low (119880 le 08) in other words not very ego-involved agents there is a bigger spectrum of steady statesFor this reason we studied the metrics of interest for thesevalue differences and kept the couples of (119880 119879) depicted inTable 1 For these couples central bipolar bipolar-centralconvergence (3 clusters) and multicluster convergence (4 or5 clusters) are possible and result frommeaningfully differentattitude structures Judgment accuracy varies significantlyacross the four different attitude structures we retain and soit happens with the variance of opinions These scenariosprovides us with reference results from which to build upon

322 Calibrating Deliberative Social Interactions Delibera-tive social interactions correspond to social situations in

which individuals are not influenced by pair-wise discussionwith peers but are sensitive to collectively deliberated infor-mational cues On these terms parameters such as the sizeof agentsrsquo argument sacks (119896119894) the distribution of argumentsin them (120601119894) the minimum number of debates (119898) andthe proportion of debaters from the population (119899119863) are nolonger mute We assume for simplicity that the formulaexist120601119890exist119896forall119894(119896119894 = 119896 and 120601119890119894 = 120601119890 and 120601119898119894 = 05) is true with 119894 isin 119873for some value 120601119890 isin [05 1] and for 119896 isin N

The chosen domains for these parameters (see Table 1)are justified by the size of the population and the size of theargument pool (119872 = 600) which we set from the start If 119896is too big (119896 ge 32) andor too many agents are allowed inthe debate (119899119863 ge 02) then individuals always disprove thecentral argument and deliberation is never or rarely takeninto account in the update of opinions Similar effects occurwhen 119898 is high yet we only calibrate it with respect to themean running time of a simulation and the values of 119905119863Different distributions of 120601119890119894 seem to significantly or directlyexplain none of the statistics we analyze so we convenientlyset 120601119890119894 = 075 We chose the parameter 119896 to be always divisibleby 4 for it is convenient given the initial conditions regardingthe argument distribution in the agentsrsquo argument sacks

We fix the value space for the parameters 120574 119901119886 119901119903 as inTable 1 to account for different intensities of the effect ofdeliberative voting retroaction on the system For examplethe scenarios where the three parameters are at their lowestvalues correspond to the scenarios with the smallest fixedprior effect of deliberation in the model Increasing any ofthese parameters should mechanically be associated witha world in which agents are more sensitive to collectivedecisions The values for 119901119886 are taken from [13] wherethe author states that by means of deliberation 7 to28 of individuals changed their opinion from agreeingto disagreeing or vice versa on a referendum question forDenmarkrsquos participation in the EuroWe choose two differentvalues for 119901119886 that account for high (03) and very high (05)sensitivity to deliberation in a group 119901119903 on the other handis taken to be small (equal to 01) since we believe that thereexist agents that will always go against the reached consensusTheir numbers are meager

For simplicity we assume that for all 119894 isin 119873 120574119894 = 120574 forsome 120574 isin [0 05] In effect we posit that the heterogeneityof agents in the deliberative context only derives from theirarguments and their opinions We decide to calibrate 120574 inallusion to the strength of the social dynamics in pair-wiseinteractions In doing so we limit ourselves to consider threedifferent scenarios in which deliberation has respectivelyhalf equal and twice as much opinion-shifting power thanone-to-one social influence Finally we include the accept-ability voting quota rule 120572 whose domain is inspired fromclassical majority rules [35] observed empirically (Table 1)

14 Complexity

Table 2 Parameters and their domains used to study the effect of the procedural parameters on the metrics

Procedural parameters Other parameters119905119863 119898 119899119863 120572 119898 120583 119879 119880 119901119886 119901119903 120574 119896 119861119890ℎ119886V1198941199001199031 2 3 4 5 6 1 2 3 4 5 001 002 003 004 005 0 05 066 7 01 16 02 03 005 02 12 119873119886119894V119890 119865119900119888119906119904119890119889

323 Calibrating the Mixed Social Interactions Model Themixed interaction model ascribes to a parameter space inwhich the effect of collective choices on our metrics isnontrivial and where deliberation and voting on proposalsdetermine their acceptability Since the mixed model isequivalent to periodic iterations of pair-wise and collectivesocial interactions the calibration of the parameters in bothpreceding models holds (see Table 1 for the calibration)The first reason for this is comparability the second is thatthe previous calibration takes into account the fact thatboth dynamics are going to be combined We include thefrequency of debates by adding the parameter 119905119863 whichcontrols for the amount of pair-wise interaction amongagents between two distinct deliberation steps

324 Termination Conditions for Runs Simulations stoponce 100 proposals are deliberated on andor voted for inother words after 100 collective decision steps have occuredThe number of proposals discussed seems arbitrary yet it ishigh enough to observe the effects of deliberation on opiniondistributions and on other metrics related to coherence andjudgment accuracy We choose the number of runs as afunction of our research questions which give relevance tothe procedural parameters in the mixed model rather than tothe parameters set to describe the population and its behavior

325 Expected Outcomes of the Simulations We expect moredeliberation in terms of higher 119898 and 119899119863 to increasejudgment accuracy and at the same time to reduce thevariance of opinions In turn a smaller variance of opinionsimplies that the argument pool for deliberation is smaller andtherefore judgment accuracy should be lower Moreovervariance increasing dyadic interactions (or rejection) shouldalso increase judgment accuracy and coherence since bipo-larization and dissensus foster argument diversity in delibera-tionThe coherence statistic should be stronger in simulationsin which central convergence appears quickly and agentsare naive Attitude structures sensitivity to informationalcues and weights given to deliberated cues that makes pathsto bipolar convergence smaller or to central convergencelonger should be associated with higher judgment accuracyShifts of opinion should be more visible in scenarios inwhich extreme agents are pulled away from the extremesHence the sensitivity to deliberated cues (119901119886 119901119903) and 120574should explain the shifts When crossed with high values of119880 and 119879 the shifting power of deliberation should be at itshighest

In the end we do not know how these mechanismswill play out The results of the subsequent experimentsgive us insight on the interplay between the aforementionedeffects

4 Results

Before pointing at any result obtained from the mixedinteractions model we describe and compare the dyadicand deliberative interactions models in the parameter spaceobtained from the calibration in Section 3 (see Table 1)Primarily we require our results to describe two orthogonaltypes of opinion dynamics the pair-wise dyadic and thedeliberative The latter comprises scenarios in which indi-viduals do not influence each other by means of pair-wisediscussion and only update their opinions from delibera-tion the former scenarios where only pair-wise interactionsamong agents determine the dynamics We describe both onthe parameter space given in Table 1 by performing at least20 simulations per scenario Quantitative results from thepair-wise interaction model alone are described during thecalibration and are not treated heremdashdeliberative parametershave no effect on it Moreover we show how these differentmodel specifications yield qualitatively different opiniondistributions and compare them on the basis of the metricsof interest To this end we perform independent two-sampleStudentrsquos t-tests18 or Welch t-tests19 and compute confidenceintervals at a 095 level of confidence Last but not least wecomment ordinary least squares (OLS) regression estimatesto account for the direction and magnitude20 of the effectsof the parameters on the metrics Estimates are declaredsignificant at a 5 level of risk

We perform the same analysis on the mixed interactionsmodel as we compare it to the pair-wise dyadic and deliber-ative interactions models and discuss the marginal effects ofthe parameters on the metrics More precisely we obtain twodifferent types of results for the mixed interactions modelthe first compares the mixed scenarios to their monolithiccounterparts with respect to each metric and with respect tothe regression estimates the second gives a clear idea of themarginal effect of each governance parameter (or parameterof interest) on our observations In the first we allow controland procedural parameters to vary on a parameter spacesimilar to those of the dyadic and deliberative opiniondynamics (see Table 1) In the second we fix all of ourcontrol parameters execute 36000 balanced21 runs andfocus only on the one-way pooled effects of our proceduraland behavioral (all agents are focused or naive) parametersWe generate comprehensive graphs and tables to account forthe obtained results

Please notice the slight change in the size of the parameterspace for the procedural parameters and in the values ofthe nonprocedural ones for the first and second type ofresults (refer to Tables 1 and 2 respectively) Since we wantto have a finer idea of the marginal effects of each of theprocedural parameters on how well a group decides onproposals we make the value jumps for each parameter (1)

Complexity 15

() (U T) = (02 18) (b) ( pa m nD) = (005 05 00 2 02) (＝) ０ＬＧＮＬＭ ＣＨ (＜) (＝) Ｈ＞ tD = 3

(d) (５ ４) = (02 14) (e) ( pa m nD) = (01 03 05 3 02) () ０ＬＧＮＬＭ ＣＨ (＞) () Ｈ＞ tD = 6

Figure 7 From left to right common opinion trajectories and distributions for dyadic deliberative and mixed opinion dynamics within thestructure of different decision-making processes

small enough to detect significant differences in the estimatesand (2) meaningful for interpretation For the fixed one-valued parameters in Table 2 we use Table 1 to set them totheir minimum mean or median values

41 Comments on Dyadic and Deliberative Opinions Dynam-ics In this subsection we present our first results regardingthe deliberative opinion dynamics described in Equation (6)and its differences with the dyadic dynamics (Equation (5))in terms of our observations

411 Qualitative Analysis of Pair-Wise Opinion DynamicsPair-wise opinion dynamics alone produces multiclusterstable convergencewith variable cluster sizes (see Figures 7(a)and 7(d)) Insight on the complexity of these dynamics wasdiscussed in Section 321

In Figures 7(a) and 7(d) agents discuss randomly withone another in pairs Clusters in the extremes form quicklysince either agents that are close to the extremes attractone another or are convinced to stay close to the extremesby interacting with other agents they disagree with Otheragents with near-extreme positions may be simply attractedto one of the several moderate foci of agents in the opiniondistribution On the other hand moderate agents are eitherattracted to the closest opinion focus or pushed towards theextremes of the distribution Over time agents in the centralfocus either attract other agents into it or ignore the opinions

of the extreme agents that interact with them The result ofthese dynamics yields amulticlustered convergencewhere theopinion foci are at least at distance 119880 from one another andonly the extreme foci (clusters of agents with |119900119894| ge 075) areat a distance greater than or equal to 119879 from one anotherThefoci at the bounds of the opinion distribution are very stableand in analogy to the (12119889)-rule in assimilation bounded-confidence models [33] the number of clusters that form isroughly equal to 119880minus1412 Qualitative Analysis of Deliberative Opinion DynamicsQualitatively deliberative opinion dynamics yield a looseunipolar convergence of opinions near the center (centralconvergence see Figure 7(b)) near the center of either the leftor right portions of the distribution (group polarization seeFigure 7(e)) or to a sparse bipolar opinion distribution withclusters at the center and at the extremes (see Figure 7(c)) ifmixed with pair-wise interactions This is probably a reasonto expect the variance of opinions and the proportion ofextremists (almost perfectly correlated 120588 = 089 119901 lt 0001)to be low for these scenarios The side towards which theopinion cluster skews depends essentially on the valence ofthe first argument that is collectively accepted

In Figures 7(e) and 7(b) agents that update their opin-ions do so at the same time and only when deliberationis successful In these scenarios every time a decision iscollectively accepted agents update their opinions towards

16 Complexity

Table 3 Ordinary least squares estimates for pair-wise interactions scenarios for most of the observed metrics

Dependent variable119881119886119903(119900) 119878ℎ 119890119888 119894119903(1) (2) (3) (4)Constant 0610lowastlowastlowast 4429lowastlowastlowast 0472lowastlowastlowast 0000

(0002) (0206) (0003) (0000)

119880 = 06 (119880 = 02) minus0012lowastlowastlowast 2981lowastlowastlowast 0061lowastlowastlowast 0000(0002) (0260) (0004) (0000)

119879 = 18 (ref 119879 = 14) minus0264lowastlowastlowast minus3261lowastlowastlowast 0004 0000(0002) (0260) (0004) (0000)

120572 = 066 (ref 120572 = 05) 0002 minus0059 minus0100lowastlowastlowast 0000(0002) (0184) (0003) (0000)

(119880 119879) = (06 18) (ref (119880 119879) = (02 14)) minus0324lowastlowastlowast minus3728lowastlowastlowast minus0033lowastlowastlowast 0000(0003) (0368) (0006) (0000)

Observations 1440 1440 1440 1440R2 0983 0389 0529Adjusted R2 0983 0388 0527Residual Std Error (df = 1435) 0032 3492 0053 0000F Statistic (df = 4 1435) 21 272740lowastlowastlowast 228686lowastlowastlowast 402326lowastlowastlowastNote lowastplt01 lowastlowastplt005 lowastlowastlowastplt001

the argumentrsquos defended position Hence if there is votingand if agents happen to obtain 119894119899 proposal arguments theyinexorably cluster towards one or the other side of the opiniondistribution resulting in opinion convergence remeniscent ofgroup polarization (see Figure 7(e)) Furthermore as agentscluster in the same side of the opinion spectrum the argu-ments that may be advanced in deliberation are fewer andmore skewed towards that one side of the opinion spectrumAgents become more easily persuaded in deliberation andonly by arguments supporting one side of the spectrumConvergence of opinions towards one loose opinion focusbecomes faster and more certain If there is no vote (seeFigure 7(b)) the dynamics are similar but for the fact thatit is the uniform randomness of the proposal argument thatguarantees a central convergence of opinions

413 Quantitative Analysis of Pair-Wise Opinion DynamicsThe three parameters that allow for quantitative analysis are119880 and119879 the agentsrsquo attitude structure and themajority quotavoting rule 120572 Table 3 supports the claim that going from 120572 =05 to 120572 = 066 does not affect the variance nor the shifts ofopinions However judgment accuracy is significantly higherwhen accepting proposals becomes more difficult This israther surprising but also intuitive since decisions that aretaken using more restrictive methods of scrutiny should bemore accurate

Concerning the attitude structures the interaction termbetween 119880 and 119879 shows that 119880 has a small effect on thevariance of opinions given that 119879 is low This shows thateven when the assimilation threshold is high a sufficientlysmall rejection threshold can make its one-way effect vanishHowever high levels of both 119880 and 119879 reduce the variance ofopinions dramatically In contrast when it comes to overallchanges in opinion the magnitude of 119880rsquos one-way marginal

effect increases the shifts of opinions whereas 119879 lowers itThis can be interpreted as having the rejection dynamics pushmoderate-to-extreme agents to positions that are close tothe extremes and very moderate agents to the center of thedistribution Not too many quantitative changes (on average)can therefore be observedThe fact that higher119880 is associatedwith faster convergence of opinions seems to imply lowerjudgment accuracy At some point voting for the same type ofproposals for too longmay result inmany collectivemistakesThis is an interesting conclusion to consider in the analysis ofthe mixed model Finally the interaction term shows that 119879has no strong impact on the metric

414 Quantitative Analysis of Deliberative OpinionDynamicsSeven parameters can be considered for quantitative anal-ysis in this dynamics the parameters of interest recalledin Section 312 the size of the argument sacks (119896) agentbehavior during deliberation the strength of the deliberationdynamics (120574) and the measure sensitivity to deliberatedresults (119901119886) The most ldquounexpectedrdquo result obtained fromthe deliberation dynamics is that it shows how deliberationcontributes to the proportion of extremists and the varianceof opinions It happens that having more agents discussproposals and having more of these discussions increasesthe variance of opinions and the proportion of extremistswhile lowering judgment inaccuracy and coherence For theformer the fact that far too many arguments are advancedduring deliberation plummets the chances that the centralargument is accepted hence deliberation does not moveopinions too often For the latter we observe that only ldquobigrdquodifferences (going from the lowest value of a parameter to thehighest) in protocol actually prove to have a preponderantimpact on the metrics These observations hint at a possibletrade-off between judgment accuracy and the variance of

Complexity 17

Table 4 Ordinary least squares estimates for the deliberative interactions scenarios for most of the observed metrics

Dependent variable119881119886119903(119900) 119878ℎ 119890119888 119894119903(1) (2) (3) (4)Constant 0032lowastlowastlowast 13092lowastlowastlowast 0370lowastlowastlowast 1007lowastlowastlowast

(0001) (0406) (0001) (0001)

119898 = 3 (ref119898 = 1) 0012lowastlowastlowast minus2062lowastlowastlowast minus0012lowastlowastlowast minus0001(0001) (0276) (0001) (0001)

119898 = 6 (ref119898 = 1) 0023lowastlowastlowast minus3285lowastlowastlowast minus0018lowastlowastlowast minus0004lowastlowastlowast(0001) (0276) (0001) (0001)

119899119863 = 002 (ref 119899119863 = 001) 0007lowastlowastlowast minus0965lowastlowastlowast minus0006lowastlowastlowast minus0002lowastlowast(0001) (0276) (0001) (0001)

119899119863 = 005 (ref 119899119863 = 001) 0020lowastlowastlowast minus3316lowastlowastlowast minus0015lowastlowastlowast minus0004lowastlowastlowast(0001) (0276) (0001) (0001)

120572 = 05 (ref 120572 = 0) 0146lowastlowastlowast 68932lowastlowastlowast 0068lowastlowastlowast minus0507lowastlowastlowast(0001) (0276) (0001) (0001)

120572 = 066 (ref 120572 = 0) 0291lowastlowastlowast minus13842lowastlowastlowast 00004 minus0000(0001) (0276) (0001) (0001)

119896 = 8 (ref 119896 = 4) 0004lowastlowastlowast minus0815lowastlowastlowast minus0002lowastlowastlowast minus0001(0001) (0276) (0001) (0001)

119896 = 16 (ref 119896 = 4) 0008lowastlowastlowast minus1118lowastlowastlowast minus0004lowastlowastlowast minus0003lowastlowastlowast(0001) (0276) (0001) (0001)

119861119890ℎ119886V119894119900119903 = ldquoFocusedrdquo (ref ldquoNaiverdquo) 0015lowastlowastlowast minus1980lowastlowastlowast minus0012lowastlowastlowast minus0004lowastlowastlowast(00005) (0225) (00004) (0001)

120574 = 01 (ref 120574 = 005) minus0020lowastlowastlowast 5762lowastlowastlowast minus00003 minus0001(0001) (0276) (0001) (0001)

120574 = 02 (ref 120574 = 005) minus0025lowastlowastlowast 8648lowastlowastlowast minus00003 minus0001(0001) (0276) (0001) (0001)

119901119886 = 05 (ref 119901119886 = 03) minus0013lowastlowastlowast 1581lowastlowastlowast 00004 00003(00005) (0225) (00004) (0001)

Observations 19440 19440 19440 19440R2 0931 0844 0562 0958Adjusted R2 0931 0844 0562 0958Residual Std Error (df = 19427) 0033 15701 0030 0050F Statistic (df = 12 19427) 21 844790lowastlowastlowast 8 733563lowastlowastlowast 2 079585lowastlowastlowast 36 761250lowastlowastlowastNote lowastplt01 lowastlowastplt005 lowastlowastlowastplt001

opinions that may be of interest in the mixed discussionmodel

Group coherence shifts of opinions and the varianceof opinions are highly affected by the voting thresholdthe coherence metric is reduced by half when allowing forclassic majority voting shifts explode and the variance ofopinions increases dramatically This is not surprising sincethe majority quota rule works as a buffer to the success ofdecision-making processes and in consequence it hampersthe deliberative opinion updates For the shifts of opinions itis clear that the two-third majority voting rule results in littleopinion change and that the simple majority rule inducesgroup polarization As expected there are less extremists inscenarios where there is no voting and the shifts of opinionare more likely to happen when agents are more sensitiveto deliberation When we include voting in the system

the grouprsquos judgment accuracy decreases due to the correctdeliberated decisions that are vetoed by a salient opinion-agent majority Strangely the effect only holds for the simplemajority decision rule Lastly the proportion of individualsthat participate in the debate and the minimal amount ofdebates have similar marginal effects on all the metrics (seeTable 4)

415 Comparing Deliberative and Pair-Wise Opinion Dynam-ics in a Nuthshell Qualitatively both models yield similaroutcomes but for the fact that deliberation never producesopinion polarization with both types of extremists (seeFigure 7) Furthermore clusters in the deliberative interac-tions model are less homogeneous in size and multiclusterconvergence occurs only when deliberation is rarely suc-cessful which happens sporadically given the calibration of

18 Complexity

Table 5 Mean difference 095 confidence intervals for metrics by model comparison

Model Metric119881119886119903(119900) 119901119903119900119901119890119909 119878ℎ 119890119888 119894119903Dyadic vs Deliberative [0191 0216] [0170 0197] [-2948 -2828] [0074 0082] [0165 0172]Mixed vs Deliberative [0142 0147] [0150 0155] [-2085 -1976] [0003 0002] [-0038 -0030]Mixed vs Dyadic [-0072 -0046] [-0045 -0018] [8332 8859] [-0079 -0071] [-0205 -0201]the model On the other hand pair-wise opinion dynamicsproduces multicluster convergence most of the time andcentral convergence only in the infrequent cases where thelatitude of acceptance is large

The comparison of these two scenarios in terms ofthe metrics is straightforward Whilst dyadic interactionsallow for more variability in opinions through pair-wiseinteractions and more mistakes in judging proposals delib-erative updates simply act as an antagonist force On aver-age the deliberative interaction model produces opiniondistributions with smaller variance and yields steady stateswith higher judgment accuracy a higher shift statistic andnaturally lower group coherence since only proposals that aredeliberated 119894119899 are eligible for voting (see Table 5)

42 Articulating Dyadic and Deliberative Social Interactionsin Opinion Dynamics So far we have considered scenarioswhere argumentation had no place and where individualsmade decisions according to opinions that were purelyconstructed from pair-wise discussions We also observedcases in which agents formed their opinions only by inte-grating deliberated proposals into their opinion updates Thescenarios we present in this section combine the two pre-ceding scenarios to account for both the effects of pair-wisediscussion and the effects of deliberation on opinions judg-ment accuracy and group coherence From the precedingobservations we conclude the following deliberative opiniondynamics reduces the variance and the proportion of extrem-ists because it shifts opinions greatly and polarizes groupsby eliminating at least one type of extremist the one whoseopinions are initially opposed to the deliberated resultsor it brings opinions close to neutrality (see Figure 7(b))This may not be desired in a group since the diversity andstability of opinions are necessary for deliberation to makesense and lead to the collective acceptance of controversialproposals Dyadic opinion dynamics on the other handtend to increase the variance of opinions and the numberof extremists while having a lesser aggregated effect on theshifts of opinion This is an immediate result of the pair-wiseinteraction dynamics inasmuch as it polarizes individualsvery quickly

To sum up in the mixed interactions model deliberationhinders the effect of voting on opinions by making someproposals that would normally be submitted for voting noteligible for voting It may also undo opinion changes dueto pair-wise interactions The mixed model can be seen asthe deliberation opinion dynamics model in which pair-wiseinteractions occur in between collective discussion and mayaffect which arguments are advanced during deliberationEquivalently it can be seen as the dyadic opinion dynamics

model in which deliberation accounts for endogenousldquoshocksrdquo on the opinion distribution

421 Qualitative Analysis of the Mixed Dynamics In Figures7(c) and 7(f) agents update their opinions through delibera-tion and pair-wise interactions Opinion trajectories visualsshow a combination of both previously described opiniontrajectories Over time as agents discuss with one anotheropinion clusters form as in the pair-wise or dyadic opiniondynamics Deliberation may disrupt the formation of theseclusters and three different situations may occur

(i) Deliberation disrupts the formation of the opinionclusters but not sufficiently to jeopardize pair-wiseopinion cluster formation (eg for high values 119905119863)Two phenomena may be responsible for it (1) thevariability of the proposal argumentrsquos support for theprinciple P and (2) too few (successive) deliberationsteps that result in opinion updates Once the opinionclusters are formed successful deliberation foreshad-ows the merging of the previously formed clustersmay bring extremists closer to moderate agents andthanks to the assimilation component of the pair-wisedynamics allow bigger and more moderate opinionclusters to form (refer to Figure 7(c))

(ii) Deliberation significantly disrupts the pair-wisedynamic formation of clusters This may happenwhen the requirements for accepting proposals areweak in the decision-making process (eg low 120572) Inthis case one has either central convergence sinceall agents follow the deliberated results in the samemanner or 3-cluster convergence with two extremeopinion foci In the latter situation the extremegroups form quickly (like in the pair-wise model) anddeliberation cannot pull extreme agents away fromthe extreme foci they are in The main reason for thisis that immediately after an extreme agent attempts toleave her foci through deliberation updates she getspulled or pushed back into it by means of a dyadicinteraction with another extreme agent

(iii) Deliberation does not disrupt the formation of theopinions cluster This is the rare case scenario forvery demanding decision-making processes in termsof inclusion (eg 119899119863) and argument acceptability (eg120572) Opinions converge as in the pair-wise opiniondynamics model (refer to Figures 7(a) and 7(d))

Last when there is voting (120572 = 0) the position ofthe moderate opinion cluster is determined by the first

Complexity 19

minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

Var(o)

sh

ec

ir

Var(o)

propex

sh

ec

ir

1

098

minus02

minus004

001

1

minus014

minus001

minus003

1

026

minus028

1

minus057 1

ＪＬＩＪ

Ｒ

Figure 8 Significant correlation estimates for themetrics of interestin the mixed opinion dyamics scenarios

accepted proposal argument andor the opinionmajority thatis formed from previous dyadic interactions (see Figure 7(f))

422 Relating the Metrics of Interest in Mixed OpinionDynamics The first thing to notice is that for all of thesescenarios all the metrics are significantly correlated (seeFigure 8) The variance of opinions and the proportion ofextremists are almost perfectly correlated To some extentthis means that a high variance of opinions known thatthe mean of opinions is statistically null equates to havingmany extremists in the population be towards one or theother side of the opinion distribution (eg extreme bipolar-ization) Similarly we find a negative and strong correlationbetween the judgment inaccuracymeasure and the coherencecoefficient (see Figure 8)mdashwhen judgment inaccuracy is high(low) coherence to deliberation is low (high) Cases likethese are telling since they may be related to sequences ofcollective decisions in which agents do not make too manymistakes when judging proposal arguments and still makedeliberated decisions that are on average in line with theirprinciples

The shift statistic interestingly is negatively correlatedwith the variance of opinions with judgment accuracy andwith the coherence coefficient In a way this means thathaving more moderate agents conduces on average to moremistakes in judging arguments and to less coherent collectivedecisions An explanation for this phenomenon could be thatas agents reach consensus in opinion important (possiblyextreme) arguments for correctly judging argument propos-als are not advanced in the deliberation arenawhich results inmore mistakes and in possibly accepting proposal argumentsthat after voting are not meant to be accepted Indeed the

negative correlation between the variance of opinions andjudgment inaccuracy points towards this direction since onaverage the more shifts there are the smaller the variance ofopinions is (see Figure 8)

423 Comparing Mixed Dynamics with Other MonolithicDynamics of Interactions Qualitatively opinion trajectoriesin the mixed model are more chaotic than in the pair-wise and deliberative interaction models (see Figure 7) Asanalyzed previously deliberation dynamics disrupt the pair-wise dynamics and create situations in which central andmulticlustered convergence (with two clusters at the extremesof the distribution) are possible at the expense of more(pair-wise dynamics) or less (deliberative dynamics) clustersIt tolerates the survival of very small isolated group ofagents that the previous models rarely tolerate in a hundredcollective decision-making steps (see Figure 7(c))

One of the most interesting observations one can makeof this family of models regards how groups change (in termsof our metrics) when dyadic social interactions are addedto deliberative interactions and vice-versa When comparingthe mixed interactions with only deliberative ones shiftsagentsrsquo aggregated change in opinions are on average lesspronounced The variance of opinions and the proportionof extremists are significantly higher for the former than forthe latter group coherence in contrast is lower with respectto the scenarios where opinions are formed only throughdeliberation Judgment accuracy follows that same trendA reason for this might be that mixed dynamics are fasterto yield semistable clusters andor opinion convergence Inconsequence the potential argument pool for deliberativeexchange shrinks faster and deliberation becomes less effec-tive over time which results in more collective mistakes Theoutcomeon coherence is predictable inasmuch as deliberativeinteractions produce group polarization If a group agrees inopinion debates are sterile because themembers of the groupwill most likely refrain from attacking the proposal argu-ments they all support and will work together to disparagethe proposal arguments they do not support (see Table 5)

In the same manner we report that the proportion ofextremists and the variance of opinions are barely lower inthe pair-wise interaction model when including deliberationin the decision-making process Shifts of opinions are alsobarely lower when there is no deliberation This can indicatethat deliberation does not undo the changes in opiniondue to pair-wise interactions nor does it amplifies them Incontrast when we consider the labeling-based metrics themixed interaction model ensures lower judgment inaccuracybecause of deliberation and lower group coherence (seeTable 5)

In sum the mixed discussion model is a compromisebetween the pair-wise interaction and deliberative opiniondynamic models in that it offers some guarantees in termsof judgment accuracy and coherence while allowing forldquoreasonablerdquo variance in the opinion pool It establishes somekind of trade-off between labeling-based metrics and thevariance of opinions that may be of interest for decision-making process design

20 Complexity

Table 6 Summary of sign or relative marginal effects for all varying parameters on dyadic deliberative and mixed interactions models inand for all experiments

Metrics Parameters119905119863 119898 119899119863 120572 119879 119880 119901119886 120574 119896 119861119890ℎ119886V119894119900119903119881119886119903(119900) XX X X O X X X X X X119901119903119900119901119890119909 XX X X O X X X X X X119878ℎ XX X X O X X X X X119890119888 XX X X XO X XOO XOO X X119894119903 XX X X O XO O X OOX XOO X XNota gray symbols correspond to dyadic dynamics light gray to deliberative dynamics and bold black to mixed dynamics Given a scenario the symbol 119883stands for constant parameter and 119874 for insignificant marginal effect and as parameters grow for positive (negative) marginal effect for the parameterFrom left to right at the parameter level the leftmost arrow corresponds to the scenario with the strongest (in absolute value) marginal effect of the parameter(with respect to to its factors) Example Let 119901 be a parameter and 119874 stands for ldquo119901 has a significant positive effect in dyadic discussion scenario negativein the deliberative one and insignificant in mixed discussion The positive effect in dyadic discussion is greater than the absolute value of the negative one inthe purely deliberative scenariordquo

424 Comments on the Differences between the Three Modelsfor OLS Regression Estimates The significance of most of theregression parameters estimated for the mixed interactionsscenarios are similar to those estimated for the pure dyadicanddeliberative scenariosNevertheless themagnitude of theeffects is different insomuch that the dynamics themselvesare different So instead of commenting the linear regressionestimates of the mixed model we compare when possiblethe standardized significant coefficients of the regressors ofthe mixed model with those of the pair-wise and deliberativeinteractionmodels (see Table 6 for a summary of the results)

In mixed interactions for instance the influence of119898 onthe opinion distribution is much more preponderant than inthe deliberative dynamics scenarios Going from 119898 = 1 to119898 = 3 increases the standard variation units of the variance ofopinions by twice as much as the one in the deliberative sce-nariosThis in particular is due to the design of the decision-making process If 119898 is higher then the number of socialinfluence steps (119905119863) in the decision process is also higher(compared to 119905119863 = 0 in the deliberative case) If we decideto consider these steps as contributing to the variance ofopinions then the effect of119898 on the variance of opinions hasto be stronger in the mixed interactions model In contrastthe number of agents participating in the debate has a weakereffect on the variance of opinions in the mixed interactionsmodel than in the deliberative dynamics specially for 119899119863 =005 (more than twice as much) The difference may comefrom the fact that in mixed discussions the part of theargument pool used in deliberation is more restrictive thanin the deliberative case We can argue that since pair-wisedynamics may get individuals closer to one another aftersome deliberation the same happens to the arguments thatindividuals will most likely use in deliberation Taking thisinto account means that deliberation will at many instancesbe unfruitful and have no effect on opinions thus failing tomoderate extreme opinions andor polarize the group In thesame direction a high number of participants in deliberationcan make the deliberation dynamics stiff In this case thedistribution of opinion is close to the uniform distributionwhich has a relatively high-to-moderate variancewith respectto the reference results

The strength of the deliberative dynamics has a positivestronger effect on the variance of opinions in themixedmodelthan in the deliberative model when 120574 = 02 and a negativeweaker one when 120574 = 01 A sound interpretation of thisresult is that when deliberation is successful and its effect onopinions does not get agents to move sufficiently away fromtheir current positions agents are pulled or pushed back totheir previously held opinions the variance of opinions doesnot changemuchOn the other hand if the effect of successfuldeliberation is strong opinions can change sufficiently tomake agents much more prone to assimilation into moderateopinion foci the variance of opinions falls

The deliberative opinion dynamics model is much moresensitive to 120572 than the mixed discussion model Differ-ences in marginal effects can double The probability ofattraction to deliberated results (119901119886) for instance has astrongmoderating effect (negative) on agentsrsquo opinions in thedeliberative opinion dynamics but a meager one (two-fold)in the mixed and dyadic dynamic models This is becauseopinion updates in the deliberative model depend only onthat parameter whereas in the mixed model more numeroustypes of interactions make opinion updates possible

In terms of judgment accuracy all parameters except thesize of the argument sacks (119896) have a significant slightlystronger effect on the deliberative dynamics than in themixedmodel In effect the difference may be attributed to theupshot of pair-wise interactions on the distribution of opin-ions and on the arguments that would be advanced duringdeliberation The result respecting parameter 119896 however iscounterintuitive Since argument sacks are static scenarioswhere shifts are higher should make 119896 more important fordeliberation to succeed and it is precisely in the scenarios ofpure deliberative dynamics that we find the biggest opinionshifts This may point to the hypothesis that in deliberativeinteractions agents change opinions substantially but do notoften change their adherence to the principle (go from anegative opinion to a positive one or vice versa) when thereis no group polarization

43 Sensitivity Analysis for the Procedural Parameters ofInterest Weperform a sensitivity analysis of the observationson the parameters of interest in themixed interactionsmodel

Complexity 21

We extend the parameter domains of the pure dyadic anddeliberative scenarios as we make it explicit in Table 2

431 Minimum Number of Debates (119898) We observe that theminimum number of debates has a significant well-observedeffect on all of our metrics excluding group coherence Forthe variance of opinions we can observe that the moredebates there are the bigger the value of the metric isalthough the higher 119899119863 and 119905119863 are the weaker the overalleffect of119898 becomes (Figures 9(d) and 9(a)) Furthermore themarginal effect of increasing minimal debates is decreasingan additional deliberation step requirement increases thevariance of opinions but less and less as it grows 119898 coupledwith 119899119863 has a shy S-shaped effect (see Figure 9(d)) on thevariance of opinions meaning that it has a stronger effectwhen 119899119863 is low which suggests a possible trade-off betweenthe number of arguments accepted in the debate and thenumber of debates required before accepting a proposal

In contrast the shifts in opinion are less and less likelyas 119898 grows and this is independent of the variations of theother parameters Again the effect is marginally decreasingand is only truly significant when 120572 = 05 An explanationto these effects may be linked to the design of the systemFirst the variance of opinions is higher when deliberation isasked for because the more deliberation steps there are thehigher the chances are that the central proposal argumentis deemed unacceptable specially when agents are focusedFurthermoremechanically speaking increasing theminimalamount of debates implies that whenever a decision is to betaken at least 119898 times 119905119863 time steps have to take place and if atany moment a proposal argument is considered undecided119905119863 time steps are added to the process So unless the debatedoes not yield 119906119899119889 labels for proposal arguments (highlyunlikely considering that 120590 = grounded semantics) the morenondeliberation steps there are in the decision process thehigher the variance of opinions is Concerning the shiftswhen 120572 = 05 either the system is too stiff to accept anyproposal argument and opinions do not changemuch or theeffects of pair-wise discussion and deliberation cancel out insuch a way that individual shifts are minimal (Figure 9(b))

On the side of labeling-based metrics the more debatesare asked for the more accurate a group is in its judgmentthe effect being smaller as 119898 grows When agents are naivethe effect is quasilinear while when they are focused thestrongest effects of acquiring more deliberation are foundwhen levels of deliberation are already low (Figure 9(f))Thiscan be explained by the fact that the more debates thereare in the decision process the closer one gets to the idealargumentation framework

432 Proportion of the Population in Deliberation Steps (119899119863)Like for 119898 119899119863 has a significant effect on the proportionof extremists and on the shifts and variance of opinions(Figure 9(e)) This may result from the fact that being ableto put more arguments in play at the same debate step mayincrease the odds of revealing cycles around the proposalargument in119860119865120576 Given that we use grounded semantics thearguments in the cycles are labeled 119906119899119889 and in consequence

debates are more often postponed when 119899119863 is higher Post-poning debates in turn increases the number of nondebatessteps in the decision process which increases the varianceof opinions and limits over time the moderating effect ofdeliberation Moreover the effect of this parameter is verydependent on the value of 120572 (Figure 9(e)) For shifts forinstance 120572 = 05 makes the effect of 119899119863 negative while120572 = 0 makes it positive but to a lesser extent Furthermorefor higher requirements of deliberation (119898) adding moreindividuals to the deliberation process has weaker effectson the variance of opinions and on the other metrics It isalso quite interesting to notice that it has the same effect onagents independently of agent behavior during deliberationThis is a surprising result as one would have expected morefocused agents in a deliberation arena to cause a raise inthe proportion of extremists as they play to knock outopposing proposal arguments and thereof reducing the effectof deliberation (lowers variance) on the whole population

Finally similar to119898 addingmore people into the deliber-ation process increases judgment accuracy (see Figure 9(c))and has no clear effect if any on group coherence (seeFigure 9(i))

433 Steps between Deliberation Steps (119905119863) In all configu-rations 119905119863 increases the proportion of extremists (variance)and decreases the shifts of opinions among agents Theshifts and the effect on the variance of opinions are presentfor 120572 = 05 insignificant otherwise (Figure 9(b)) 119905119863 ishighly linked to 119898 by construction When 119898 = 1 thecurve linking the variance of opinions and 119905119863 is convex As119898 increases the curve becomes more and more concavewhich means that 119905119863 has a more important effect on theopinion distribution as collective decisions take longer to beachieved This seems counterintuitive but in reality it reflectsthemultiplicative relationship between deliberation and pair-wise interactions and the semantics in the model If 119898 is lowand 119905119863 is high the effective number of pair-wise interactionsis on average smaller in the deliberation process Hencethe structure of the decision-making process hinders anyincrease in the variance of opinions If119898 is high the oppositeeffect is observed Additionally since the grounded semanticsyields few 119894119899 arguments with respect to other admissibility-based semantics getting closer to the ideal argumentationframeworkmaymake it difficult to obtain the expected effectsof deliberative discussion on agents Lower 119898 constrains thevariance of opinions by making deliberation more influentialon opinions

In terms of judgment accuracy and coherence we observethat 119905119863 does not explain coherence yet it tends to decreasejudgment inaccuracy This may be due to a more variedargument pool in the debate which in turn may be aconsequence of a higher variance of opinions or vice versa

434 Acceptability Voting Quota (120572) By far it is the mostinfluential parameter in our study It changes the directionand the intensity of the effect of all the other proceduralparameters and by construction heavily constrains the roadto accepting a proposal (see Figure 9)

22 Complexity

015

020

025

030

035

040

1 2 3 4 5 6

12345

Values of Ｇ

(a) 119905119863 times 119901119903119900119901119890119909 by119898

0

10

20

30

40

50

1 2 3 4 5 6

005066

Values of

(b) 119905119863 times 119878ℎ by 120572

040

045

050

055

060

1 2 3 4 5

001002003004005

Values of Ｈ＄

(c) 119898 times 119890119888 by 119899119863

015

020

025

030

035

040

1 2 3 4 5

001002003004005

Values of Ｈ＄

(d) 119898 times 119901119903119900119901119890119909 by 119899119863

0

10

20

30

40

50

001 002 003 004 005

00005066

Values of

(e) 119899119863 times 119878ℎ by 120572

040

045

050

055

060

1 2 3 4 5

BehaviorNaiveFocused

(f) 119898 times 119890119888 by 119887119890ℎ119886V119894119900119903

015

020

025

030

035

040

0 05 066Behavior

NaiveFocused

(g) 120572 times 119901119903119900119901119890119909 by 119887119890ℎ119886V119894119900119903

05

06

07

08

09

10

1 2 3 4 5 6

00005066

Values of

(h) 119905119863 times 119894119903 by 120572

08700

08725

08750

08775

08800

001 002 003 004 005

BehaviorNaiveFocused

(i) 119899119863 times 119894119903 by 119887119890ℎ119886V119894119900119903

Figure 9 Curves of mean observations for different parameters and metrics for 36000 runs Bars are 095 confidence intervals Plots withthe same observables (ordinate) except for 119894119903 are on the same scale

Complexity 23

The higher the requirement for accepting the proposal isthe higher the proportion of extremists and the variance ofopinions are and the lower the shift statistic is In few words120572 constrains the world to dyadic discussion or throws it intoa process in which deliberation is much more importantthan dyadic interactions This is why for any 120572 = 05 oneeither gives too much weight to deliberated results whichshadows pair-wise interactions or too little weight in thesense that no deliberated result is ever accepted and thusnever integrated into the opinions of the agents that havevoted for it The reason for the latter is that the latitudeacceptance is too low and thereafter pair-wise interactionsare not enough to unevenly polarize the population in such away that deliberated proposal arguments are voted favorablyby a non-negligeable majority Shifts for 120572 = 0 are stable fordifferent levels of 119905119863 which implies that deliberation keeps incheck all the shifts of opinions related to an increase in thenumber of pair-wise interactions For 120572 = 05 deliberationis less likely to be successful and shifts tend towards the shiftlevels for 120572 = 0 (see Figure 9(b))

Concerning labeling-basedmetrics120572 entirely determinesthe coherence statistic For 120572 = 05 there is no difference incoherence because of how coherence is defined it is maximalwhen 120572 = 0 and strangely maximal when 120572 = 066(see Figure 9(h)) Surprisingly 120572 has no effect on judgmentaccuracy One would have expected a higher 120572 to increasejudgment accuracy since agents would hardly collectivelyaccept a proposal argument discussed during deliberationProposal arguments have given how the argument lattice isgenerated a higher probability of being rejected than of beingaccepted in the ideal argumentation framework

435 A Word on Focused and Naive Agents Focused andnaive specifications for agent behavior are an importantparameter in the decision process since they model howagents choose arguments in the deliberation arena In the sce-narios where agents are focused agents show equal or highervariance of opinions and are therefore more extreme (seeFigure 9(g)) independently of the voting requirements Shiftsof opinion are less likely as well We can see this happeningbecause focused agents knock out proposal arguments moreoften than naive agents provided that 119899119863 is low

One could have expected naive agents to yield morecoherent decisions since they argue sincerely However thisseems not to be the case (see Figure 9(i)) Agents are unable todeliberate in away thatmakes deliberation reflect their votingintentions either because they do not have the necessaryarguments to do so or because such arguments do not exist

In terms of judgment accuracy focused agents do betterthan naive agents in finding the ideal labels for the proposalarguments This is likely because when reconstructing theframework focused agents take into account the deliberatedproposal argument and choose themost pertinent argumentsin their sacks that relate to the state of affairs at the deliber-ation arena In fine debates result in a better approximationofL119886119887120576(119868) even though one could have believed the oppositethat is focused agents are ready to sacrifice a correct collectivelabel of a proposal argument in the goal of defending theirpositions

44 Points of Discussion In this subsection we discuss theresults obtained from the simulations We attempt to makesense of them from a social sciences perspective

441 Opinion Consensus and Dissensus The deliberativeopinion dynamics corresponds to the simulated scenarios inwhich the variance of opinions is at its lowest and judgmentaccuracy at its highest Dyadic opinion dynamics are associ-ated with the scenarios in which the variance of opinions is atits highest judgment accuracy at its lowest and coherence atitsmaximumpossible levelThemixed interactions dynamicsis a combination of these aspects for all the metrics exceptfor coherence at which it is the worsemdashdeliberation leadsto some kind of loose opinion consensus that is reinforcedby pair-wise interactions Likewise the mixed discussionmodel confirms some results exposed in [16] expressingthat opinions and voting intentions in a deliberative contextldquooften changerdquo That being said and more particularly indeliberative opinion dynamics outputs align well with twointeresting results on consensus formation given by aninterpretation of Moscovici and Doisersquos [15] work Sunsteinrsquos[14] account on deliberation and group polarization andsocial impact theory [4] The first states that mild consensusis reached in deliberation as procedure disengages agentsthe second is that within discussion groups majorities tendto become larger and opinions more extrememdashnot on thebasis of competing arguments but of the initial balance ofopinions After several decision processes we find opiniondistributions that reflect these observations even though weare not able to precisely coin the set of parameters that yieldssuch distributions (see Figure 7)

Although differences between mixed and dyadic opiniondynamics in terms of the variance of opinions and shiftsare slim the former does much better in terms of judg-ment accuracy In this sense deliberation does moderateopinions but not as much when non-deliberation steps arerelatively frequent throughout the decision-making processIn terms of Moscovici and Doise theory of consensus [15]this makes sense since pair-wise discussion engages or ego-involve agents in the discussion and pushes to convergenceof opinion at the extremes of the opinion distributionHowever what remains unclear is the reason why proceduralparameters that constrain deliberation do not disengageagents from discussion and thus result in a mild consensusas also expressed by Moscovici and Doise [15] A possibleexplanation of this phenomenon may stem from the factthat deliberation in the model is always accompanied by aseries of ego-involving discussions that shadowor counter themoderating effect of deliberation Not only this deliberationcould also be responsible for crystallizing the opinions ofextremists for it favors one or the other part of the distributionof opinions as explained in [14] and to some extent in [13]Furthermore the mixed discussion model gives an exampleof polarization that results from a minority of dissenterson one side of an issue acceding to views of the majorityrsquosside [4] as opposed to the strong polarization observed indyadic interactions Situations in which the status quo orthe undecidability of the proposal argument lingers cannotpossibly yield situations of moderation In cognitive science

24 Complexity

one can be inclined to believe that the longer a deliberationprocess is the more likely individuals will take an immutablestance on the situation In view of the assumption that longdecision-making processes can be more costly for a group orsociety as a whole agents may feel forced to take a stancejust for the sake of ending the process The stance that isadopted would therefore depend on pair-wise interactionsamong individuals and not on deliberation since deliberationtakes too long to be conclusive

442 For Hypothetical Recommendations in Decision-MakingProcess Elaboration On another tune results from themodelsuggest that procedural parameters have marginally decreas-ing positive or negative effects on the metrics of interest Ina hypothetical policy analysis perspective this may indicatethat after choosing a correct number of individuals andimposing the right amount of debates one can attain rea-sonable levels of judgment accuracy coherence and varianceof opinions As accounted in [18] deliberation increasesknowledge (collective knowledge as well) and therefore itheightens the chances of making correct decisions If onebelieves that social welfare is linked to deliberation andeventually to the metrics studied previously then proceduralparameters can both be used as an instrument to attaindesired levels of social welfare and reinforce the notionof legitimacy of collective decisions It can also provide ajustification for choosing a deliberation regime rather thananother on the basis of how important (or urgent) a topic ofdiscussion is Indeed if we adopt the claim that deliberationis a forerunner of welfare then any state in which deliberativecues are at their maximum (highest number of participantsmany deliberation steps before making a decision and asfrequent as possible) has to be mapped to the highest attain-able social welfare In this case is a situation where judgmentaccuracy is at its highest yet the variance of opinions or theproportion of extremists at its lowest an ideal situation Wecannot say for sure since strong opinion consensus and longdecision-making processes in many situations may result ina loss of welfare rather than in a gain of it

Another interesting result in procedure for deliberationis the very similar estimates we find for the effects of theminimum number (119898) of debates necessary to make adecision on a proposal and the proportion (119899119863) of individualsparticipating in deliberation on judgment accuracy In thescope of social welfare this raises the question of whetheradvocating for longer decision processes by demandingmoredeliberation steps (higher 119898) rather than bigger debates(higher 119899119863) may increase welfare Depending on the objec-tives of the decider of the deliberation procedure he or shemay choose to concentrate on one or the other A recom-mendation we can assert from the mixed social interactionsmodel is that if one prefers low to high variance of opinionsandor high to low coherence the decider is better off ifshe focuses on increasing the size of the debate rather thanorganizing debates very frequently One of the reasons forthis might be that when making bigger debates all agents aretruly considered as equals in the decision-making processIn organizing many debates individuals that are chosen toparticipate in many of these will be more influential and will

somehow bias the set of arguments in the deliberation arenaThemajority rulemodeled through120572 also determines the sizeof the shifts and the distribution of opinions Because a bigger120572 is associated with higher levels of extremism and opinionvariance deliberated cues are harder to accept and thereforeit becomesmore difficult for agents to reach concensus andormoderate their opinions

On the other hand and further into the idea of legitimacythe goal of deliberative processes is the agreement amongall participants This agreement stands for the right or bestdecision from a formal or procedural perspective and it is alsothe best from a substantive point of view In other words thelegitimacy of decisions not only derives fromwell-establishedprocedural requirements (respect of the protocol) but alsobegs for the fulfillment of two essential conditions (1) satisfy-ing the procedural requirements for a correct procedure (for-mal legitimacy) and (2) the rational acceptability of the resultsof this procedure (substantive acceptability) Legitimacy canbe then obtained from the coherence statistic scrutinizedin the model insofar as it measures how well agents acceptthe results of a correct procedure (eg deliberation) So fora proposal to be legitimate one has to choose parametersthat would maximize group coherence in decision-makingprocesses

As deliberation may give rise to consensus in the delib-eration arena it may also sow the seeds of dissensus inthe group Our model clearly illustrates this phenomenonthrough the procedural parameters of deliberation and themetrics observed From another perspective in deliberativedemocracy dissensus in collective decision-making may bedesirable According to Landermore and Pagersquos accounts [36]lack of consensus can be perceived as having agents withalternative ideas that combined with other ideas can providea better approximation of the ideal status of an argumentTheauthors in [36] call this dissensus ldquopositive dissensusrdquo andit makes normative sense when solving complex collectivedecision-making problems Our model captures the ideaof having normative requirements to define a good andlegitimate collective decision requirements of high consen-sual deliberative accuracy (judgment accuracy) of legitimacy(coherence) and of diversity (variance of opinions) are alltaken into account and for such we can conclude that ourmodel is successful in describing a sound process of collectivedecision-making with interpretable outcomes

Of course the present simulation results and analysis donot prove that all opinion dynamics can be accounted forby such simple processes like the one presented here Theydo however shed light on the expressiveness of combiningdifferent paradigms to account formore interpretablemodelsand pave the way to creating original models of the sortThe simulations suggest the desirability of discovering theconsequences of relatively simple laws of communication atdifferent levels (micro meso and macro) to determine whatstill needs to be explained

5 Related Work

We see our model as a contribution to the influence andopinion dynamics field in ABM and a pragmatic application

Complexity 25

of abstract argumentation theory To our knowledge we areunaware of existing literature on agent-based modeling thatexplicitly relates collective choice and the notions of deliber-ation and opinion diffusion through abstract argumentationas we have done it

Most models in the literature of opinion diffusion areinterested in opinions because these have an influence incollective decision and in questions of social order Forinstance in [3] the authors are interested in consensus andin how a group collectively chooses among two alternativesIn other models authors are interested in the emergenceof extremism [8] and on the distribution of opinions whenextremists are introduced in the population [5 33 37] whileother authors coin the notion of opinion polarization as anemergent property of the system [8 12] They show usingmodels of ldquobounded confidencerdquo and opinion diffusion withtrust that three different kinds of steady states (unipolarbipolar and central) are possible depending on whetheragents are sufficiently uncertain about their opinions andsufficiently connected andor a certain proportion of indi-viduals are already extreme Recent articles on informationand opinion dynamics stress the importance of governanceand government intervention in the spread of emotions(opinions) in the frangibility of social consensus systemIn [10] the authors show that government intervention inthe spread of negative emotions can lead to an even fasterspread of negative emotions (single-peeked convergence) andin a faster collapse of the social consensus system Opinionas a function of trust is studied in [12] and relates to howindividuals form their opinions on the basis of how muchthey trust agents in their networks

Another stream of discrete opinion dynamics modelsthat explain the emergence of opinion can be found in theliterature Computer multiagent models of attitude changebased on Latanersquos social impact theory are presented in [4] Inthe theory an agent changes her opinion on the basis of theinformational impact she is subject to which depends on thepersuasiveness (strength) supportiveness and immediacy(group structure) of the environment and information shereceives The simulations predict two emergent groups ofphenomena the shifting of attitudes towards incompletelypolarized equilibria and the formation of coherent clusteringof subgroups with deviant attitudes Similarly but in acontinuous more argumentative fashion Mas and Flache[11] present a model of opinion diffusion using argumentsArguments are considered to be for or against a proposal andare given an agent-dependant relevance in dyadic persuasiveinteractions Arguments also determine the agentsrsquo opinionsand their dissimilarities which posits the rules of pair-wiseencounters in theirmodel as agents are assumed to form theiropinions according to the arguments they own They showthat in an argumentative opinion diffusion model the onlyobservable steady states are bipolarization and consensusand that bipolarization can emerge from interactions amongsimilar agents In other words bipolarization of opinionsis possible with homophily and without negative influenceif there is argumentation They subsequently compare theirresults to experimental data In [38] a similar approach toargumentation and opinion formation is taken but for the fact

that they introduce different types of arguments explicitlyapply social judgment theory [22] and use survey data toestablish model-to-real-world comparisons

Closer to opinion formation and abstract argumentationGabbriellini and Torroni [39] are the first to have origi-nally and soundly merged opinion diffusion and abstractargumentation They define an agentrsquos opinion as a functionof the arguments she holds and the attack relation amongthem They devised a focused peer-to-peer dialogue systemof persuasion (NetArg) inspired from Mercier and Sperberrsquosargumentative theory of reasoning [40] which used onlyabstract argumentation to study opinion polarization andopinion dynamics Moreover they considered networks andthe notions of trust and epistemic vigilance to define adynamics for knowledge revision and trust itself whichwe clearly lack in our model They showed that if a con-servative belief operator in argumentation was applied byagents when they reasoned then their dialogue protocoldid not increase polarization among agents That said theyused the model to study the effect of Groenvetterrsquos weaklink theory in the spreading of arguments and were notparticularly interested in notions like judgment accuracy orcoherence nor in tackling questions related to collectivedecision-making procedures Their system is very expressiveand helps position our work in the litterature our work is atthe frontier of pure argumentative opinion diffusion modelsopinion diffusion with arguments and continuous bounded-confidence models of opinion diffusion a la Deffuant

On another note work on collective cognitive conver-gence [41] and opinion sharing [19] shows that consensustowards a certain ldquocorrectrdquo opinion or cognitive state is alwayspossible yet dependent on noise variability and awarenessof agents In [20] the authors show that learning about anexogenous correct state of the world (represented by bits)under confidence was possible but only if the agents werenot too confident In a homogeneous population they showthat the higher the confidence the worse the learningmdashthevery confident agents do not learn the properties of the truestate of the world and disrupt the learning process of the lessconfident ones Collective cognitive convergence can be seenas a result of deliberation in truth-seeking models

When it comes to abstract argumentation theory we takean approach that wires two types of dialogues that are well-studied in the literature persuasion dialogues [42] and delib-eration dialogues [43] Another interesting line of work is theone on mechanism design [44] or the problem of devisingan argumentation protocol where strategic argumentation isnot a liability for success in debates We tackle mechanismdesign in a different way though Instead of consideringstrategy-proofness we are interested in how differences inprotocol can result in epistemologically ldquobetterrdquo collectivechoices and guarantee that opinion distributions are favorablefor deliberation For a survey on persuasion dialogue see[42]

Work on agent-based argumentation usually assumes thatthe semantic relationship between arguments is fixed Inother words if two individuals were to put two argumentsin the public arena such that one logically attacked the otherthen everyone would agree that such attack exists [45 46]

26 Complexity

Other models which do not make this restrictive assumptioncan also be found in the literature [27 47] and derive fromthe class or family of opponent models [27 48] in which twoopposing sides attempt to win a dialogue Ourmodel is in theintersection of these two but the frame combining opiniondiffusion of the kind and argumentation remains new

The idea of mixing interpersonal influence and verticalcommunication however is not original It is described andimplemented in innovation diffusion models such as [5 49]In both vertical communication is modeled as exogenoustransparent informationmdashagents are aware of the existenceof an innovation thus triggering several processes of choiceand stabilization of opinions Also in [50] an Eulerian modelis implemented to show the effect of media and exogenousinformation on opinion distributions With respect to thispoint the originality of our work is in that the informationemitted as vertical communication is endogenous It is issuedfrom a deliberation model that agents shape on the basisof their opinions arguments and behavior In the spirit of[51] where the authors control for the design of verticalcommunication (to whom the vertical communication isaddressed its timing how it influences agents opinions) wecontrol for the process generating the information for ourown set of variables length of debates majority quotas andthe frequency at which discussions take place

6 Conclusion and Perspectives

The main objective of this article was to build a bridgewhere decision-making argumentation and opinion dif-fusion could come together We proposed a model thatcombined abstract argumentation theory and a boundedconfidence opinion diffusion model and showed to whatextent it could explain variance of opinions extremismcoherence in collective decisions and judgment accuracyon arguments given an ideal state of full information Thesecond objective was to show in what way governance pro-viding agents with arenas of discussion and deliberation wasimportant in addressing the success of collective decision-making processes and the quality of their outcomes

The model revealed that allowing for more deliberationtime though in lower frequency and allowing for widerparticipation in deliberation increased the variance of opin-ions and the proportion of extremists in a group Theseobservations are consistent with the combination of resultsfound in [9] and in [13 14] which stress that deliberationmay polarize groups and may have a meager effect on theshifts of opinions and inconsistent with [16] where it isargued that opinions tend to moderate after deliberationIt happens that deliberation alone did moderate opinionyet when integrated into a more complex system in whichindividuals were allowed to interact with one another andnot all that was deliberated was accepted its influence wasshadowed by other more individualistic dynamics

Undeniably the grounded semantics played an importantrole in the weak effect of deliberation since it models askeptical way of reasoning over arguments For agentsaccepting arguments thus updating their opinions accord-ingly happened not too often Nevertheless asking for more

deliberation did increase judgment accuracy as observed in[18] yet in a marginally decreasing fashion We showed thatvoting within the deliberation protocol not only increasedthe proportion of extremists and the variance of opinions ina group but also determined how coherent deliberation andvoting procedures were with each other Lastly we showedthat agents that are focused judged arguments better hadmore stable opinions and constituted a group with higherproportion of extremists than their naive counterparts

In terms of governance the model says that there is per-haps no trade-off between extremismand judgment accuracyInstead it asserts that the higher the variance of opinions thecloser the group gets to the correct decisions This situationmay be interpreted as Landermore and Pagersquos notion of ldquoposi-tive dissensusrdquo (in complex collective decision-making tasks)[36] as opposed to consensus as a normative requirementfor correct collective decision-making If a decider has toorganize deliberation to legitimize a public policy dependingon what kind of world he or she wants different set ofparameters may be chosen to account for different levels oflegitimacy and correctness For instance one may ask for adecision to be legitimate to respect a certain level of coherenceor of judgment accuracy and to be discussed by a sundryof different-opinionated peoples If so deliberation has tobe made more often more participants have to be includedin debates or in deliberation instances and not too manytime steps should be introduced in between two deliberationinstances So the model may capture normative ideas todefine correct and legitimate collective decisions and tomake recommendations on those grounds Requirements ofhigh consensual deliberative accuracy (judgment accuracy)legitimacy (coherence) and diversity (variance of opinions)are all taken into account and for such a reason the model israther successful in describing a sound process of collectivedecision-making with interpretable outcomes

61 Extensions of the Model and Other Ideas Our model canbe extended in many ways Value-based abstract argumen-tation frameworks (VAFs) for example as an extension ofDungrsquos argumentation framework provide a formal descrip-tion of the process of decision-making in which argumentsare given values and audiences (set of agents) ignore attacksbetween arguments on the basis of their preferences over val-ues Social abstract argumentation frameworks (SAFs) alsoprovide an interesting and seemingly convenient formalismfor our model In SAFs agents vote on the acceptability ofarguments and the resulting labeling is a combination of thearguments that are voted for and labeling-based semantics Inour model we apply the same rule yet only for one argumentand given precise agent-based votingmechanisms and votingrules which are eventually parametrized

To consider our model as a specification of a mixed VAFand SAF model may be of great interest for future workThis formalism can provide solid foundations to describeexplain interpret and eventually extend the results foundhere from an argument-based perspective On those groundsfuture work will focus on creating a hybrid argumentationframeworkmdashwith its semanticsmdashthat takes heed of bothvalues or principles and voting

Complexity 27

Concerning other possible extensions of the model anatural reflex would be to argue on the validity and therobustness of the mix between the bounded confidenceopinion diffusion models and deliberation Consequentiallyadding deliberation to different bounded confidence modelsof opinion diffusion such as the Hegelsmann-Krause model[37] or to models of social impact theory like in [4] canbe of great interest for future work We acknowledge that afiner understanding of argumentation frameworks randomlattices and their respective applications for this modelare necessary Testing for different labeling-based semanticsor even trading labeling-based semantics for scoring-basedsemantics (semantics that defines acceptability on scoresgiven to arguments) such as debate semantics and ranking-based semantics (semantics that yield a preorder) may beinteresting to consider in future work

More into modeling deliberation it is not inadequateto believe that agents may also advance proposals duringdeliberation and replace to some extent the proposal forwhich the deliberation is taking place Endogenizing theprocess that produces the type of argument to be deliberatedon may be interesting to explore either by having agentsstrategically replace proposals at certain moments duringthe debate or having the central authority attempt to chooseproposals such that the probability of having them acceptedis high Agents may also be thought to be autoorganizing(independent of a central authority) and concede on the sizeof the instances of deliberation To consider differences insemantics across agents or even endogenizing the semanticson how urgent or important the proposal that is beingdiscussed is can also be an interesting direction to take

Last but not least many nontrivial modifications of ourmodel are possible and most should include better-thoughtdeliberation protocols It may be interesting to design andobserve protocols in which deliberation only affects agentsthat are actually debating for example Trust network effectsmultidimensionality of opinions new processes of argumentexchange or learning among agents are notions to furtherdevelop in order to make the model more realistic and relaxsome unreliable assumptions In sum the model is to berefined and extended with the objective of either studyingconcrete cases of deliberative polling and opinion dynamicsor implementing more intuitive thought experiments

Appendix

We present below some of the pseudocode we used toimplement the model

Algorithms

(a) Algorithm 1 implements the dyadic influence in themodel It describes how two agents interact andupdate their opinions

(b) Algorithm 2 implements the deliberative influence inthe model which describes how agents update theiropinions in sight of the accepted deliberated proposalarguments

(c) Algorithm 3 is used to obtain the grounded labelingof an argumentation framework

(d) Algorithm 4 provides an overview of a run in thesimulations119872 is the number of proposal argumentsto discuss which we set to 100 in the model

Each object in the system is indexed by a119908ℎ119900 number Hencea set of agents is annotated by a set of natural numbers Weuse this fact in the implementation of the algorithm 119874119887119895119890119888119905119894denotes the object of 119908ℎ119900 number 119894 Agentsrsquo opinions (119900) arecalibrated so that 119900 isin [minus1 1]Data Availability

The program and data used to support the findings of thisstudy are available from the corresponding author uponrequest

Disclosure

A preliminary version of this work was presented anddiscussed at the conference Bridging the Gap between FormalArgumentation and Actual Human Reasoning on October 4-5 2018 at BochumGermanyThe authors thank the audiencefor their valuable comments and suggestions

Conflicts of Interest

The authors declare that they have no conflicts of interestregarding the publication of this article

Acknowledgments

The contribution of Gabriella Pigozzi was supported by theDeutsche Forschungsgemeinschaft (DFG) and the CzechScience Foundation (GACR) as part of the joint project FromShared Evidence to Group Attitudes (RO 45486-1)

Endnotes

1 In other words the superiority of the ldquoright answerrdquo orlabeling in our case will appear as such to all

2 All agents hold the same opinion3 By sensitivity we mean how much an agent is ready to

change her opinion on the sole fact that an informationalcue has been debated on

4 The assumption is strong but this trust among agentsmay be rooted from a genuine individual motivation toreach a democratic consensus

5 We adopted a rule of thumbs that says that if an agentis neutral with respect to the principle then she willvote for any proposal that is presented to her duringdeliberation

6 If the argument is not unique she chooses one atrandom

7 If there are more than one candidate the agent will favorthe argument at minimal distance from 119868

28 Complexity

Require Vector of values for parameters (119881 = (120583 119880 119879))Ensure Successful interactions among agents119896 larr997888 0119873119890 larr997888 0

while 119896 le |119873|2 do ⊳Fill119873119890 with |119873|2 agents119896 larr997888 119896 + 1119873119890 larr997888 119877119886119899119889119900119898(|119873| + 1) ⊳Pseudo-random generator Generates integer between 0 and |119873|119873119903 larr997888 0 120575119894119895 larr997888 0 ⊳Variable to store differences in opinionfor all 119894 isin 119873119890 doRandomly pick a number 119895 isin 119873(119873119890 cup 119873119903)119873119903 larr997888 119873119903 cup 119895 120575119894119895 larr997888 |119900119894 minus 119900119895|procedure dyadic-update(119894 119895) ⊳Two agents 119894 and 119895 discuss and update opinions1199001015840 larr997888 119900119894 (120583119895 119880119895 119879119895) larr997888 (120583119894 119880119894 119879119894) larr997888 (120583 119880 119879) ⊳Save agent irsquos opinion before updating and homogenize agents

if 120575119894119895 lt 119880119894 then119900119894 larr997888 120583119894119900119895 + (1 minus 120583119894)119900119894if 120575119894119895 gt 119879119894 then119900119894 larr997888 (1 + 120583119894)119900119894 minus 120583119894119900119895if 120575119894119895 lt 119880119895 then119900119895 larr997888 1205831198951199001015840 + (1 minus 120583119895)119900119895if 120575119894119895 gt 119879119895 then119900119895 larr997888 (1 + 120583119895)119900119895 minus 1205831198951199001015840

Algorithm 1 Dyadic social influence

Require Vector of values for parameters (119881 = (120574 119901119886 119901119903))Ensure Successful interaction between proposal arguments and agents

for all 119894 isin 119873 doprocedure Probability-Update(V119886 119901119886 119901119903) ⊳Update individual probabilities of change due to deliberated cues119909119894 larr997888 119901|V119886minus119900119894 |2119886 119910119894 larr997888 1199012|V119886minus119900119894 |119903119902 larr997888 0if L119886119887(119868) = 119894119899 then

procedure deliberative-update(119894 119886) ⊳Agent i argument 119886120574119894 larr997888 120574 119902 larr997888 119906 sim U(0 1) ⊳ 119906 is a realization of a uniform random variableif 119902 lt 119909119894 then119900119894 larr997888 120574119894119900119894 + (1 minus 120574119894)V119886break

if 119902 lt 119910119894(1 minus 119909119894) then119900119894 larr997888 (1 + 120574119894)119900119894 minus 120574119894V119886Algorithm 2 Deliberation influence

8 In terms of computational complexity naive agentsmaketheirmoves in119874(|A119894| log(|A119894|)) while focused agents doat least as bad

9 Tables have memory Therefore debates are not neces-sarily trees

10 If there are less than (119899119863 times |119873|)2 agents for any of bothgroups the CA summons all of them to the deliberationtable

11 Or careful consideration of the viewpoints of otherswhich implies that citizens keep an open mind and donot reject arguments outright

12 A ldquohealthyrdquo consensus refers to the case in which manyconsensual decisions are taken but opinion-wise agentsdo not agree with each other Quotes are used on theword healthy because some political science theorists

(eg [36]) believe that in certain situations agreement inopinions for deliberation is epistemologically damaging

13 Simulations on groups of |119873| = 800 and |119873| = 400agents shows that scaling the population has an effect onall themetrics but it is confounding depending onwhichcouples of attitudes structures are considered (119880 119879) Westick to |119873| = 400 as in [9] for comparability andsimulation running time

14 By balanced wemean with as many arguments with V119886 lt0 as with V119886 gt 0

15 Experiments on the system suggest that a bigger(smaller) argument pool 119872 = 1000 (119872 = 200) isassociated with a higher (lower) variance of opinions(and extremism) and higher (lower) judgment accuracyFor the rest of the observations 119872 seem to have noeffect We choose a middle value for 119872 Similarly theargument pool chosen to be balanced or uniformly

Complexity 29

Require Argumentation framework 119860119865 = (AR)Ensure Assignment of labels to arguments from the grounded labeling of 119860119865L(A)

L0(A) larr997888 (0 0 0)119896 larr997888 0do119894119899 L119896+1(A) larr997888 119894119899 L119896(A) cup 119886 | 119886 is not labeled 119894119899 L119896(A) 119886119899119889 forall119887 isin A 119904119905 119887 R 119886 then 119887 isin 119900119906119905 L119896(A)119900119906119905 L119896+1(A) larr997888 119900119906119905 L119896(A) cup 119886 | 119886 is not labeled 119894119899 L119896(A) 119886119899119889 exist119887 isin A 119904119905 119887 R 119886 and 119887 isin 119894119899 L119896+1(A)

while L119896+1(A) = L119896(A) = 0Algorithm 3 Solve argumentation frameworks (with grounded semantics)

Require Vector of values for parameters (119881)Ensure Vector of statistics (119880)

procedure Set-parameters(119881) ⊳Initialize parameters for scenarioprocedure Init-objs(119899119898 119896 ) ⊳Create119898 arguments 119899 agents and 119860119865120576procedure Solve-Argumentation-Framework(119860119865) ⊳Assigns the epistemic labelingL120576 toA119894 larr997888 1119905 larr997888 1while 119894 le 119872 do ⊳ 119872 the number of arguments to discuss119868119894 larr997888 119866119890119899 119886119903119892(1) ⊳Generates proposal argument 119868119860 119894119905 larr997888 (119868119894 0)L119886119887(119868119894) larr997888 119906119899119889 ⊳Initialize label for 119868119895 larr997888 1 ⊳Initialize debate counter 119895while ((L119886119887(119868) = 119906119899119889 or 119898 le 119895) and 119895 le 119898) doif 119905 equiv 0(119898119900119889 119905119863) then

procedure build-argumentation-framework(119899119863 119887119890ℎ119886V119894119900119903) ⊳Build the argument graph agents deliberate(119861119894119905 119878119894119905) larr997888 (119883 119884) sub (⋃119897isin119873119899119863 A119897119894119905 ⋃119897isin119873119899119863 R119897119894119905) ⊳ 119873119899119863 sub 119873 are agents sampled by the CA119860 119894119905 larr997888 119889(119860 119894119905 (119861119894119905 119878119894119905))procedure Solve-Argumentation-Framework(A119889) ⊳FindL(A119889)L119886119887(119868119894) larr997888 L119886119887119863(119868119894)119895 larr997888 119895 + 1

else119896 larr997888 1while 119896 le 119905119863 doprocedure dyadic-social-influence(120583 119879 119880) ⊳Agents discuss one-to-one119896 larr997888 119896 + 1119905 larr997888 119905 + 1

if 120572 gt 0 andL119886119887(119868119894) = 119900119906119905 then ⊳Agents vote for the proposalif (|119897 isin 119873|119900119897 times V119868119894 ge 0| ge 120572 times 119899) then

L119886119887(119868119894) larr997888 119894119899else

L119886119887(119868119894) larr997888 119900119906119905if L119886119887(119868119894) = 119894119899 thenprocedure Deliberation-influence(120574 119901119886 119901119903) ⊳Agents are influenced by the result of the decision process119894 larr997888 119894 + 1119905 larr997888 119905 + 1119880 larr997888 119862119900119898119901119906119905119890 119878119905119886119905119894119904119905119894119888119904(119881)

return 119880 ⊳Get the statisticsAlgorithm 4 Run for one scenario

generated does not seem to play a significant explanatoryrole on any of the metrics

16 Experiments show that themaximumnumber of debatesis not an important parameter even if it is an intuitiveone when considering decision-making processes Inpractice debates end after at most 119898 + 1 deliberation

steps and even then it happens rarely However to avoidunpleasant surprises like infinite debates on a completeargument graph containing the proposal argument weadded the constraint 119898 Its value is precisely 7 becausethe biggest minimum number of debates we studyis 6

30 Complexity

17 To ensure comparability with the deliberative andmixedopinion dynamics models on judgment accuracy andcoherence agents vote on proposal arguments every 119905119863times(119898+1) 119905119863 and119898does not explain variability in judgmentaccuracy nor in coherence

18 Student t-tests for independent samples are robust tothe violation of two hypothesis on the distribution ofthe samples normality and homogeneity of variancesas long as the sample size is big enough (central limittheorem) and if the difference in size of the comparedsamples is smallrespectively

19 Studentrsquos test version forwhen variances are not assumedequal and the difference in size of the compared samplesis big

20 We do not correct for homoskedasticity nor normality oferrors in the estimates

21 Balanced as the same number of runs per scenario orsuch that no significant correlations are found betweenparameters

References

[1] R Axelrod ldquoThe dissemination of culture a model withlocal convergence and global polarizationrdquo Journal of ConflictResolution vol 41 no 2 pp 203ndash226 1997

[2] D M Rousseau Reinforcing the MicroMacro Bridge Organiza-tionalThinking and Pluralistic Vehicles SAGEPublications SageLos Angeles Calif USA 2011

[3] S Galam and SMoscovici ldquoTowards a theory of collective phe-nomena consensus and attitude changes in groupsrdquo EuropeanJournal of Social Psychology vol 21 no 1 pp 49ndash74 1991

[4] A Nowak J Szamrej and B Latane ldquoFrom private attitude topublic opinion a dynamic theory of social impactrdquo Psychologi-cal Review vol 97 no 3 pp 362ndash376 1990

[5] G Deffuant S Huet and F Amblard ldquoAn individual-basedmodel of innovation diffusion mixing social value and individ-ual benefitrdquo American Journal of Sociology vol 110 no 4 pp1041ndash1069 2005

[6] G Deffuant D Neau F Amblard and G Weisbuch ldquoMixingbeliefs among interacting agentsrdquoAdvances in Complex Systems(ACS) vol 3 no 01n04 pp 87ndash98 2000

[7] J Rouchier P Tubaro and C Emery ldquoOpinion transmission inorganizations an agent-based modeling approachrdquo Computa-tional and Mathematical OrganizationTheory vol 20 no 3 pp252ndash277 2014

[8] M Meadows and D Cliff ldquoThe relative disagreement model ofopinion dynamics where do extremists come fromrdquo in Pro-ceedings of the International Workshop on Self-Organizing Sys-tems Lecture Notes in Computer Science pp 66ndash77 Springer2013

[9] W Jager and F Amblard ldquoUniformity bipolarization and plu-riformity captured as generic stylized behavior with an agent-based simulation model of attitude changerdquo Computational ampMathematical Organization Theory vol 10 no 4 pp 295ndash3032005

[10] Y-F Zhang H-Y Duan and Z-L Geng ldquoEvolutionary mech-anism of frangibility in social consensus system based onnegative emotions spreadrdquo Complexity vol 2017 Article ID4037049 8 pages 2017

[11] A Flache andMWMacy ldquoSmall worlds and cultural polariza-tionrdquoThe Journal of Mathematical Sociology vol 35 no 1-3 pp146ndash176 2011

[12] A Tsang and K Larson ldquoOpinion dynamics of skepticalagentsrdquo in Proceedings of the 13th International Conference onAutonomous Agents and Multiagent Systems AAMAS 2014 pp277ndash284 International Foundation forAutonomousAgents andMultiagent Systems France May 2014

[13] K M Hansen Deliberative democracy and opinion formationUniversity Press of Southern Denmark Odense 2004

[14] C R Sunstein ldquoThe law of group polarizationrdquo Journal ofPolitical Philosophy vol 10 no 2 pp 175ndash195 2002 WileyOnline Library

[15] S Moscovici and W Doise Dissensions et Consensus unetheorie generale des decisions collectives Presses Universitairesde France-PUF 1992

[16] J S Fishkin and R C Luskin ldquoExperimenting with a demo-cratic ideal deliberative polling and public opinionrdquo ActaPolitica vol 40 no 3 pp 284ndash298 2005

[17] R C Luskin J S Fishkin and S Iyengar Considered Opinionson US Foreign Policy Face-To-Face versus Online Delibera-tive Polling International Communication Association NewOrleans La USA 2004

[18] J Barabas ldquoHowdeliberation affects policy opinionsrdquoAmericanPolitical Science Review vol 98 no 4 pp 687ndash701 2004

[19] O Pryymak A Rogers and N R Jennings ldquoEfficient opinionsharing in large decentralised teamsrdquo in Proceedings of the 11thInternational Conference on Autonomous Agents andMultiagentSystems vol 1 pp 543ndash550 Spain June 2012

[20] J Rouchier and E Tanimura ldquoWhen overconfident agents slowdown collective learningrdquo Simulation vol 88 no 1 pp 33ndash492012

[21] D Miller ldquoDeliberative democracy and social choicerdquo PoliticalStudies vol 40 S1 pp 54ndash67 1992

[22] M Sherif and C I Hovland Social Judgment Assimilation andContrast Effects in Communication and Attitude Change YaleUniversity Press 1961

[23] P M Dung ldquoOn the acceptability of arguments and its funda-mental role in nonmonotonic reasoning logic programmingand n-person gamesrdquo Artificial Intelligence vol 77 no 2 pp321ndash357 1995

[24] M Caminada and G Pigozzi ldquoOn judgment aggregation inabstract argumentationrdquo Autonomous Agents and Multi-AgentSystems vol 22 no 1 pp 64ndash102 2011

[25] P Baroni M Caminada andM Giacomin ldquoAn introduction toargumentation semanticsrdquoThe Knowledge Engineering Reviewvol 26 no 4 pp 365ndash410 2011

[26] S Chambers ldquoDeliberative democratic theoryrdquo Annual Reviewof Political Science vol 6 no 1 pp 307ndash326 2003

[27] E Bonzon andNMaudet ldquoOn the outcomes of multiparty per-suasionrdquo in Proceedings of the 10th International Conference onAutonomous Agents and Multiagent Systems 2011 AAMAS 2011vol 1 pp 47ndash54 International Foundation for AutonomousAgents and Multiagent Systems Taiwan May 2011

[28] J Ranciere Disagreement Politics and Philosophy University ofMinnesota Press 1999

[29] T Bench-Capon and K Atkinson ldquoAbstract argumentation andvaluesrdquo in Argumentation in Artificial Intelligence pp 45ndash64Springer 2009

[30] M Caminada ldquoOn the issue of reinstatement in argumenta-tionrdquo in Proceedings of the European Workshop on Logics in

Complexity 31

Artificial Intelligence vol 4160 of Lecture Notes in ComputerScience pp 111ndash123 Springer Berlin Heidelberg

[31] GAVreeswijk ldquoAn algorithm to computeminimally groundedand admissible defence sets in argument systemsrdquo in Pro-ceedings of the 2006 conference on Computational Models ofArgument Proceedings of COMMA 2006 vol 144 pp 109ndash120IOS Press Amsterdam The Netherlands 2006

[32] C W Sherif M Kelly H L Rodgers et al ldquoPersonal involve-ment social judgment and actionrdquo Journal of Personality andSocial Psychology vol 27 no 3 pp 311ndash328 1973

[33] G Deffuant F Amblard G Weisbuch and T Faure ldquoHow canextremism prevail A study based on the relative agreementinteraction modelrdquo Journal of Artificial Societies and SocialSimulation vol 5 no 4 p 1 2002

[34] E Awad M W Caminada G Pigozzi M Podlaszewski andI Rahwan ldquoPareto optimality and strategy-proofness in groupargument evaluationrdquo Journal of Logic and Computation vol 27no 8 pp 2581ndash2609 2017

[35] D Grossi and G Pigozzi ldquoJudgment aggregation a primerrdquoSynthesis Lectures on Artificial Intelligence and Machine Learn-ing vol 8 no 2 pp 1ndash151 2014

[36] H Landemore and S E Page ldquoDeliberation and disagreementproblem solving prediction and positive dissensusrdquo PoliticsPhilosophy and Economics vol 14 no 3 pp 229ndash254 2015

[37] R Hegselmann and U Krause ldquoOpinion dynamics andbounded confidence models analysis and simulationrdquo Journalof Artificial Societies and Social Simulation vol 5 no 3 2002

[38] A Stefanelli and R Seidl ldquoModerate and polarized opinionsUsing empirical data for an agent-based simulationrdquo inProceed-ings of the Social Simulation Conference 2014

[39] S Gabbriellini and P Torroni ldquoA new framework for ABMsbased on argumentative reasoningrdquo in Advances in SocialSimulation vol 229 pp 25ndash36 Springer Berlin Germany 2014

[40] H Mercier and D Sperber ldquoWhy do humans reason Argu-ments for an argumentative theoryrdquo Behavioral and BrainSciences vol 34 no 2 pp 57ndash74 2011

[41] H V Parunak T C Belding R Hilscher and S BruecknerldquoModeling and managing collective cognitive convergencerdquoin In Proceedings of the 7th iNternational Joint Conference onAutonomous Agents and Multiagent Systems vol 3 pp 1505ndash1508 International Foundation for Autonomous Agents andMultiagent Systems 2008

[42] H Prakken ldquoFormal systems for persuasion dialoguerdquo TheKnowledge Engineering Review vol 21 no 2 pp 163ndash188 2006

[43] K Atkinson T Bench-Capon and P Mcburney ldquoA dialoguegame protocol for multi-agent argument over proposals foractionrdquoAutonomousAgents andMulti-Agent Systems vol 11 no2 pp 153ndash171 2005

[44] M Thimm and A J Garcıa ldquoClassification and strategicalissues of argumentation games on structured argumentationframeworksrdquo in Proceedings of the 9th International Conferenceon Autonomous Agents and Multiagent Systems vol 1 pp 1247ndash1254 International Foundation for Autonomous Agents andMultiagent Systems Canada 2010

[45] I Rahwan and K Larson ldquoMechanism design for abstractargumentationrdquo in Proceedings of the 7th International JointConference on Autonomous Agents and Multiagent SystemsAAMAS 2008 pp 1031ndash1038 International Foundation forAutonomous Agents and Multiagent Systems Portugal May2008

[46] H Prakken R Riveret A Rotolo and G Sartor ldquoHeuristics inArgumentation A Game-Theoretical Investigationrdquo EuropeanUniversity Institute 2008

[47] N Hadidi Y Dimopoulos and P Moraitis ldquoTactics and con-cessions for argumentation-based negotiationrdquo in Proceedingsof the COMMA vol 245 pp 285ndash296 2012

[48] N Oren and T J Norman ldquoArguing using opponent modelsrdquoin Proceedings of the International Workshop on Argumentationin Multi-Agent Systems pp 160ndash174 Springer 2009

[49] S Delre W Jager T Bijmolt and M Janssen ldquoTargeting andtiming promotional activities An agent-based model for thetakeoff of new productsrdquo Journal of Business Research vol 60no 8 pp 826ndash835 2007

[50] A Mirtabatabaei P Jia and F Bullo ldquoEulerian opinion dynam-ics with bounded confidence and exogenous inputsrdquo SIAMJournal on Applied Dynamical Systems vol 13 no 1 pp 425ndash446 2014

[51] V Barbet N Guiraud V Laperriere and J Rouchier Hagglingon Values Towards Consensus or Trouble 2019

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8 Complexity

requirement to consider a proposal for voting and to a lowerbound of the length of the deliberation process 119898 providesan upper bound Moreover the CA controls the size of thetime interval between debates (119905119863) the collective decisionrule (eg if there is voting on the deliberated proposals) andthe majority quota rule for accepting proposal arguments(120572) in the decision-making process 119905119863 may represent thefrequency or density of pair-wise interactions in a decision-making process

241 The Construction of a Decision-Making Process Adeliberation process or debate in ourmodel is a tool to obtainlabels for proposal arguments that are as close as possible tothe ideal or consensual ones To define a deliberation processformally we introduce the notion of debate step as constituentof a deliberation process Informally a debate step is a timestep in which a debate occurs Formally and more event-oriented a debate step (1198891198731015840 1198611015840 1198781015840) of a debate 119863119868 on 119868 isa quadruplet composed of a set of agents 1198731015840 sub 119873 a set ofarguments 1198611015840 sub ⋃119894isin1198731015840 A119894 a set of attack relations 1198781015840 sub Rand a mapping

119889 (AF 2A times 2R) 997888rarr AF

(119860119865 (1198611015840 1198781015840)) 997891997888rarr 1198601198651015840 = (119861 cup 1198611015840 119878 ⊔ 1198781015840) (3)

that adds the arguments in 1198611015840 and some attacks in 1198781015840 to aframework 119860119865 = (119861 119878) and where 119878 ⊔ 1198781015840 = (119886 119887) isin 119878 cup 1198781015840 |119886 119887 isin 119861 cup 1198611015840 In the same spirit we define a deliberationprocess119863119868 on a proposal argument 119868 as a sequence of debatesteps (119889119873119904 119861119904 119878119904)119904gt0 such that1198730 = 0 and

119889((119904minus1⋃119895=0

119861119895 119904minus1⨆119895=0

119878119895) (119861119904 119878119904)) sub (AR) (4)

with

(i) 1198610 = 119868 (the initial argument is the proposal argu-ment)

(ii) 1198780 = 0 (no reflexive attack from 119868 to 119868 is allowed)(iii) (forall119904 gt 0)(⋃119895lt119904 119861119895cap119861119904 = 0) (any newly added argument

to the framework has yet to be added to it)(iv) (forall119904 gt 0)(⨆119895lt119904 119878119895 cap 119878119904 = 0) (any newly considered

attack among arguments cannot be declared amongarguments that are not in the framework)

(v) lim119904997888rarrinfin(⋃119904gt0 119861119904 ⋃119904gt0 119878119904) sube (AR) (the system isstable no arguments are created during deliberation)

Finally let L119886119887119889(119868) denote the label given to argument 119868 ata debate step 119889 during a deliberation process 119863119868 A decision-making process notedD119868 = (119899119863 119905119863 119904119903(119898119898L119886119887119889(119868)) 120572 120590)over a proposalwhose justification argument is 119868 is a sequenceof debate and nondebate steps (119911119905)119905isinN 119911119905 isin 119889 119889 such that

(i) 119889 are debate steps and 119889 nondebate steps that corre-spond to time steps at which there are no debates

(ii) If 119905(mod 119905119863) equiv 0 then 119911119905 = 119889 and 119889 otherwise

For Against

Table

CA

in out

out out in

out in

proparg

Figure 3 A representation of a table of deliberation betweentwo groups of 3 agents with 4 arguments each Each agent hasadvanced an argument Agents tacitly agree on the attacks betweenthe arguments The CA determines the labeling using the groundedlabeling-based semantics

d d d d d d d d d d d d dTime 0 1 2 3 4 5 6 7 8 9 10 11 12I

Figure 4 A decision-making processD119868 or collective decision stepstarting at 119905 = 0 composed of debate (119889) and nondebate (119889) substepsDebate steps are shaded The number of nondebate steps betweendebates is 119905119863 = 3 the number of debates in the decision-makingprocess is119898 = 4

(iii) The subsequence (119911119897119905)119897119905isinN of 119911119905 with 119897119905 = 119905 such that119905(mod 119905119863) equiv 0 is a deliberation process with |1198731015840| =119899119863 times |119873|(iv) 119904119903(sdot sdotL119886119887119889(119868)) pursues the deliberation process as

long asL119886119887119889(119868) = 119906119899119889 under the semantics 120590(v) The length of the sequence |119911119905| or equivalently the

duration of the process is somewhere in the interval⟦119898119905119863 119898119905119863⟧(vi) The final labeling for the proposal argument 119868

L119886119887(119868) is determined by a combination of its delib-erated labelL119886119887119889(119868) and a majority vote of majorityquota 120572

A decision-making process ends when a final decision hasbeen taken concerning the acceptability of the proposal Peg 119868 has been deemed unacceptable (L119886119887119889(119868) = 119900119906119905) or amajority of agents have voted againstP For a representationof a deliberative interaction on a table see Figure 3 for one ofa decision-making process see Figure 4

Please note that every collective decision can be seen asa ldquotime steprdquo in as much as it describes how agents updateopinions and make collective decisions In the model thedecision-making process and the parameters define the sub-steps or events that occur within the time step Henceforth

Complexity 9

Fix proceduralsm m α nD tD σ Init count m larr 0

Given proposal anargument I is randomly generated

nD times |N | agents arerandomly picked

m larr m + 1

Begin decision processon proposal at time t

t larr t + 1Deliberation step m

larr 0

t larr t + 1dyadic opinionupdate Eq (5)tm larr tm + 1

m ge m

tm gt tD

ℒd (I) = in

m ge m

yields ℒ(I)

ℒ(I) = in Deliberationupdate Eq (6)

End decision

yes

no

no

no

yes

yes

no

yes

yes

yes

no

no

I

Vote on ℒd(I)

ℒd (I) = und

ＪＬＩ＝ＭＭ I

Init count tmyields ℒd(I)

Figure 5 Process chart for a decision processD119868 on a proposalP 119905 corresponds to time and 119898 (119905119898) corresponds to the number of debate(non-debate) steps that have taken place in the deliberation process (in between two debate steps) over time for the mixed discussion modelWe useL119886119887(119868) andL(119868) interchangeably in the figure for presentation

comparing simulations on the basis of the different decision-making processes translates into comparing these collectivedecision steps

242 Deliberation Protocol To define the deliberation proto-col held in the table either as a consequence of the definitionprocess or as a statement we assume the following

(i) Agents may decide not to contribute to the debate(ii) All agents have the same probability conditional to

their opinions of being picked to participate in adebate or deliberation step

(iii) There is no restriction on the number of times eachagent can participate in a deliberation process

(iv) Each agent may only place one argument per debatestep

(v) Arguments that have already been advanced in thedebate may attack newly placed arguments9

The deliberation or debate protocol goes as follows

(1) The CA randomly generates and makes public acentral argument or proposal argument 119868 and informsall agents about the rules of the decision-makingprocess

(2) The CA randomly draws two sets of (119899119863 times |119873|)2agents with divergent views10 onP and P It mergesthem to create the set of debaters1198731015840

(3) Each agent 119894 advances an argument from her sackA119894 The CA makes sure that there are no repeatingarguments (agents already take argument repetitioninto account)

(4) The CA establishes the debate step 119889rsquos argumentationframework and computes its labelingL120590

119889(A119889) using120590(5) If the computed label for 119868 is L119886119887119889(119868) = 119906119899119889

or the number of debates steps is inferior or equalto 119898 at time 119905 then the CA stops the debate andresumes it at the (119905 + 119905119863 + 1)rsquoth time step by repeating(1) (2) (3) (4) and (5)

(6) Let L119886119887(119868) be the final label given to the proposalargument 119868 If voting is allowed then if more than120572 times |119873| agents agree with 119868 119868 is accepted (L119886119887(119868) =119894119899) it is rejected if strictly less than 120572times |119873| agree withit (L119886119887(119868) = 119900119906119905) When there is a tie ifL119886119887119889(119868) =119906119899119889 then L119886119887(119868) = 119900119906119905 if L119886119887119889(119868) = 119894119899 thenL119886119887(119868) = 119894119899 When voting is not part of the decision-making process (120572 = 0)L119886119887119889(119868) = L119886119887(119868)

One important point to notice about this protocol is that itinduces a stop rule 119904119903 (cf steps (5) and (6) of protocol) forthe decision process and thus it always ends Either agentsdebate and agree on the proposalrsquos acceptability throughargumentation or after 119898 debate steps they directly voteon it so that a decision concerning the proposal is alwaysreached Refer to Figure 5 for a sweeping description of thedeliberation protocol and the decision-making process

10 Complexity

-1

-1 1

1

Latitude of acceptance

Latitude of rejection Agent i

Agent j

Figure 6 A dyadic interaction between agents 119894 and 119895 as expressedin (5) Agent 119894rsquos opinion falls within agent 119895rsquos latitude of noncommit-ment and conversely Neither agent affects the opinion of the other

25 Opinion Dynamics for Social Interactions In this subsec-tion we describe the opinion diffusion model based on pair-wise interactions among agents and deliberation Sherif rsquosand Hovlandrsquos social judgment theory [22] motivates part ofour approach It describes how individualsrsquo opinions changeon the basis of their attitude structures Attitude structuresrefer to the relative scope width or latitude of categoriesused by individuals when evaluating information namelythe latitudes of acceptance rejection and noncommitment[32] The idea behind this theory is that individuals changetheir positions only in accordance to how far or close thecommunicative cues they receive are from (to) their anchorpositions It holds that if communicative cues are far (close)from (to) an agentrsquos position say over her latitude of rejection(acceptance) then the agent shifts her position away from(towards) the position defended by the cues In the casewherethe cues fall within the agentrsquos latitude of noncommitmenther position does not change (see Figure 6 and Equation (5))

251 Pair-Wise or Dyadic Opinion Dynamics As agentsmay communicate and deliberate collectively they may alsoengage in one-to-one conversations with other agents toponder on their positions We loosely associate this type ofcommunication with dyadic nonargumentative exchange ordiscussion based on fallacious arguments and persuasionIn the light of social judgment theory and the descriptionof agents in the system we model pair-wise symmetricinteractions following the opinion dynamicsmodel in [9] Anagent 119895rsquos influence on an agent 119894rsquos opinion at time 119905 is governedby the ensuing difference equation

119900119905+1119894 =

119900119905119894 + 120583119894 (119900119905119895 minus 119900119905119894) 119894119891 10038161003816100381610038161003816119900119905119894 minus 11990011990511989510038161003816100381610038161003816 lt 119880119894119900119905119894 minus 120583119894 (119900119905119895 minus 119900119905119894) 119894119891 1003816100381610038161003816119900119905119894 minus 119900119905119894 1003816100381610038161003816 gt 119879119894119900119905119894 119900119905ℎ119890119903119908119894119904119890(5)

where the parameter 120583119894 isin [0 05] controls for the strengthof the attraction and repulsion in social influence and 119879119894and 119880119894 are the latitudes of rejection and acceptance for agent119894 respectively The parameter 120583119894 may be thought of as therelative importance agent 119894 gives to the opinions of her peersor analogously the weight she gives to her own opinionwhen updating 120583 lt 05 means that agents will never givemore weight to the opinions of other agents than to their

own (egocentric bias) When two agents 119894 and 119895 discuss iffor instance 119895 advances an informational cue (argumentpersuasion tactic) and it happens to be close enough to119894rsquos opinion anchor then 119894 shifts her opinion towards thedirection of 119895rsquos informational cue The symmetric influencefrom agent 119894 to agent 119895 takes effect in the same way (seeFigure 6)

We add to the dyadic dynamics a rule of encounters ateach step of pair-wise interactions each agent meets exactlyone other agent at random This rule may be associated withrandom day-to-day encounters among agents Steps of socialinfluence correspond to nondebate steps (119889) in decision-making processes

252 Deliberative Opinion Dynamics We define an opinionupdate equation that links the proposal arguments that areadvanced during deliberation and opinions We combine theuncertain and probabilistic nature of the effect of delibera-tion have it a moderating [15 16] or polarizing [14] one witha mechanism similar to the one in social judgment theorybased on the distance between different informational cuesand opinion anchors [32] The probabilistic modeling of theopinion update can be related to deliberation encouragingopen-mindedness11 during collective discussion [18] Fromthis choice it follows that deliberated cues can affect even themost extreme of agents as opposed to some classic opiniondiffusion models (eg [5 8]) where once agents becomeextremists they may no longer become moderate

LetP be a proposal V119905119868 the proposal argument 119868rsquos level ofsupport for a principle P and 119900119905119894 an agent 119894rsquos opinion at time119905 Then given the difference 120575119905119894 = (V119905119868 minus 119900119905119894)2 we define 119894rsquosprobability of being attracted to a decisionrsquos informational cue119868 by 119909119905119894 = 119901|120575119905119894 |119886 where 119901119886 denotes a general probability param-eter that characterizes how important deliberated results arefor the group The parameter may also be interpreted asthe grouprsquos tendency to be swayed by a decisional majoritySimilarly we define 119894rsquos probability of being repulsed from adecision informational cue 119868 by 119910119905119894 = 1199011|120575119905119894 |119903 where 119901119903 denotesa general probability parameter that characterizes the grouprsquosdislike of deliberated results 119901119903 can also be thought of as thegrouprsquos distrust of the system symbolized by deliberation anddemocracy

Let L119886119887(119868) denote the epistemic status of the proposalargument 119868 at the end of a decision process over P everyagent 119894 updates her opinion as follows

(i) IfL119886119887(119868) = 119894119899

119900119905+1119894 =

119900119905119894 + 2120574119894120575119905119894 119908119894119905ℎ 119901119903119900119887119886119887119894119897119894119905119910 119909119905119894119900119905119894 minus 2120574119894120575t119894 119908119894119905ℎ 119901119903119900119887119886119887119894119897119894119905119910 119910119905119894119900119905119894 119908119894119905ℎ 119901119903119900119887119886119887119894119897119894119905119910 1 minus 119909119905119894 minus 119910119905119894

(6)

(ii) IfL119886119887(119868) = 119900119906119905 119900119905+1119894 = 119900119905119894where 120574119894 isin [0 05] is the strength of repulsion and attractionin the dynamicThemeaning of 120574119894 is analogous to the one of120583119894in the dyadic interactionsmodel but for the observation that 119894

Complexity 11

weights her opinion to an argument rather than to an opinionThe interpretation of this dynamics is straightforward Ifthe deliberated proposal argument is close to an agentrsquosopinion then it is very likely that the agent shifts its opiniontowards it Please note that the probability 119909119905119894 (119910119905119894 ) that anagent is attracted (repelled) to (from) an accepted deliberatedproposal argument is bounded below (above) by 119901119886 (119901119903)(equivalently 119909119905119894 isin [119901119886 1] 119910119905119894 isin [0 119901119903]) Steps of deliberativeopinion dynamics are associated with debate steps in which aconcluding collective decision is made

253 Mixed Opinion Dynamics Opinion dynamics in thecontext of decision-making processes can be best described asa combination of the two preceding dynamics In this opiniondynamics model agents are engaged in a democratic systemin which they deliberate vote for proposals and occasionallydiscuss with one another on the principle supported by theproposalsThe opinion dynamics for an agent 119894 can be writtenas a sequence of dyadic and deliberative opinion updates

119900119905+1119894 =

119864119902119906119886119905119894119900119899 (5) 119894119891 119905 equiv 0 (119898119900119889119905119863) and 119904119903 (sdot) = 119891119886119897119904119890119864119902119906119886119905119894119900119899 (6) 119894119891 119905 equiv 0 (119898119900119889119905119863) and 119904119903 (sdot) = 119905119903119906119890119900119905119894 119900119905ℎ119890119903119908119894119904119890

(7)

where 119904119903(sdot) is the decision processrsquos stop rule 119905119863 the numberof time steps between each deliberation step Equation (5)describes the dynamics for the dyadic interactions and Equa-tion (6) posits the changes in opinions due to the deliberatedcues Equation (5) applies when agents are not deliberatingand when each agent encounters another agent to exchangeinformation that may lead to local opinion updates Whenthere is deliberation a handful of agents deliberate and ifthey reach the stop condition imposed by the table all agentsvote for the proposal They update (see Equation (6)) theiropinions on the basis of the proposal argumentrsquos support forthe principle and the result of the scrutiny Otherwise thereis no change in the opinions of the agents A vote indicatesthe end of the decision-making process

3 Experiments and Calibration

In this section we introduce the metrics that enable us toobserve the simulations and characterize the calibration ofthe model

31 Observations and Initialization In this section wedescribe the simulations and the protocol used to test andobserve the results of our model We introduce the metricsof interest explain the calibration of the model describesome results obtained from the simulation data used forcalibration and conclude with a brief discussion on theexpected outcomes of the mixed opinion dynamics model

311 Metrics or Statistics of Interest Let 119878 denote the end ofa simulation We are interested in the effect of deliberationand of the modelrsquos procedural parameters on the followingmetrics or statistics

(i) Variance of opinions (119881119886119903(119900)) the variance ofopinions at time 119878 Since opinions live in the unionof the positive and negative unit intervals 119881119886119903(119900) isin[0 1] The higher the variance of the distribution isthe more ldquodiverserdquo the opinions are(ii) Proportion of extremists in the population(119901119903119900119901119890119909) the proportion of agents in the populationwith opinions such that |119900119894| ge 075 A high proportionof extremists makes ldquohealthyrdquo consensus12 difficult toreach and deliberation more or less informative(iii) Shifts of opinions (119878ℎ) [33] statistic that measuresthe aggregated change in individual opinion at time 119878with respect to aggregated individual opinion at thebeginning of the simulation

119878ℎ = 2sum119894isin119873 10038161003816100381610038161199000119894 minus 1199001198781198941003816100381610038161003816max119894isin1198731199000119894 minusmin119894isin1198731199000119894 (8)

119878ℎ is positive and bounded above by 2|119873| since 119900119894 isin[minus1 1] A low shift statistic implies that the processhas a small impact on opinions(iv) Judgment or consensual inaccuracy (119890119888) consen-sual accuracy of a group consists of an ad hoc statisticmeasuring a grouprsquos ability to infer correct labels forproposal arguments from a decision-making processgiven the ideal consensual labeling based on full-informationL120590

120576 (A) We use a Hamming-based dis-tance on labelings [34] to define the statistic

119890119888 = 1|I| sum119868isinI1198861198681003816100381610038161003816L119886119887120576 (119868) = L119886119887 (119868)1003816100381610038161003816 (9)

whereI is the set of all discussed proposal argumentsand 119886119868 = 12 ifL119886119887120576(119868) = 119906119899119889 or ifL119886119887(119868) = 119906119899119889119886119868 = 1 otherwise119890119888 lives in the interval [0 1] An inaccuracy statisticclose to 1 indicates that agents subdued to a particulardecision-making process make many mistakes injudging the labels of the proposal arguments Notethat when there is voting all proposal arguments arelabeled either 119894119899 or 119900119906119905 after deliberation(v) Coherence (119894119903) letL119886119887119863(119868) be the label obtainedfor 119868 from the deliberation process without votingThe coherence statistic measures how well votingresults adjust to results obtained during deliberationonly We use the proportion of arguments that havebeen labeled 119894119899 in the debate and that agents havevoted favorably for

119894119903 = 1003816100381610038161003816119868 isin I | L119886119887119863 (119868) = 119894119899 andL119886119887 (119868) = 11989411989910038161003816100381610038161003816100381610038161003816119868 isin I | L119886119887119863 (119868) = 1198941198991003816100381610038161003816 (10)

The coherence statisticrsquos domain is [0 1] If after 119872debates no deliberated central argument has beenlabeled 119894119899 or voting is not part of the procedure ofcollective decision-making then the statistic equals

12 Complexity

1 or said differently agents are perfectly coherentThis comes for the fact that if the central argumentis labeled 119900119906119905 then it is not even eligible for voting iflabeled 119906119899119889 then the debate will always be coherentwith the preferences of the agents since the result willbe a simple aggregation of their votes If the statisticequals 0 then none of the proposal arguments labeled119894119899 are voted in favorably by the agents It follows that ahigh coherence statistic implies that when agents votefor acceptable proposal arguments the results of thescrutiny reflect the consensual rationality expressedin the deliberation process

312 Parameters of Interest We recall the parameters ofinterest in our study that are linked to the structure of thedecision-making processes

(i) 119898 the minimum number of time steps in which adebate or a scrutiny occurs in a decision-making pro-cess before a final deliberated decision is submitted toa vote

(ii) 119899119863 the number of agents that deliberate as a propor-tion of the population

(iii) 119905119863 the number of time steps between debates inwhich pair-wise interactions among agents mayoccur

(iv) 120572 the proportional majority requirement for theacceptance of a proposal When 120572 = 0 it stands forno voting ldquoany proposal argument that is labeled 119894119899during deliberation is acceptedrdquo

Please recall that in terms of the definition of a decision-making process each parameter controls either the length orthe content of the sequence D119868 (119905119863 119898) or the rules that areapplied during the debate steps (120572 119899119863) (refer to Figure 4)313 Initialization All |119873| = 400 agents start off with anopinion 119900119894 drawn from a uniform distribution U(minus1 1) 13Given 119900119894 every agent 119894 randomly draws a setA119894 (|A119894| = 119896) ofarguments from a balanced14 argument pool A of119872 = 600nonneutral (V119886 = 0) arguments on the basis of 119900119894 if |119900119894| lt 025(the opinion is moderate) then agent 119894 randomly fills halfof her argument sack with arguments such that V119886 lt 0 andthe other half with arguments such that V119886 gt 0 (120601119898119894 = 05)Otherwise she randomly fills 120601119890119894 ge 05 of her sack witharguments such that V119886 and 119900119894 are of the same sign and theremaining (1 minus 120601119890119894 ) with arguments of the opposite sign15

Like for the opinion of agents every argument 119886rsquosadherence to the principle P V119886 is drawn from a uniformdistribution U(minus1 1) The attack relation that gives birth tothe ideal argumentation framework (119860119865120576) is established onthe basis of the V119886rsquos and the argumentsrsquo epistemic reach 119890119903

Let 120579(119886 119887 119903) = (|V119886 + V119887|2)119903 be an auxiliary positivereal-valued function that takes two arguments (119886 119887) and apositive real number (119903 gt 0) as inputThe probability that anyargument 119886 isin A creates a link to any other argument 119887 isin A

is given by the equation 119901 = 119901120579(1198861198872)119890 and thus argumentssupporting opposing views of the principle always attack each

other We fix 119901119890 the epistemic correlation parameter to 015If 119901119890 was greater than 015 then arguments of the same signwould attack each other too often (and focused agents wouldbe rarely incited to advance favorable arguments duringdeliberation) If 119901119890 was lower then the arguments of thesame type would induce an almost empty graph which isunrealistic The number of arguments that any argument 119886can attack is bounded by 119886rsquos epistemic reach that we fix to15 A higher value of epistemic reach makes the argumentlattice too conflicting and nearly bipartite A lower epistemicreachmakes the lattice not sufficiently conflicting in the sensethat too many arguments can attack the proposal argumentrelatively to the few that can defend it If any argument attacksthe proposal argument the chances that another argumentattacks the attacker are low Hence the proposal argumentis almost never accepted and deliberative opinion updatesbecome rare

The arguments in the resulting argumentation frame-work 119860119865120576 are given a permanent labelingL120590

120576 (A) (in shortL120576) using120590 = grounded semanticsWe choose the groundedlabeling-based semantics because it provides a unique admis-sible labeling respects minimal rationality constraints whilesimplifying the model and models a skeptic approach toaccepting arguments (see [23 25 31]) as it maximizes thecardinality of the set 119906119899119889 L120576 To some extent if we considerproposals as committing because they may guide collectiveaction choosing a skeptic semantics seems reasonable insofaras it labels an argument 119894119899 or 119900119906119905 only if it has no reasonto label it 119906119899119889 For instance if a proposal is a public policythat requires large amounts of resources and engages tofuture course of action then it may also determine policycycles and heavily burden a group and its future decisionsIn important situations like these it seems reasonable tolengthen deliberative cues and ask for more demanding andldquogroundedrdquo criteria for policy argument acceptability

On the proposalrsquos side we create an argument 119868 whoseadherence to P is also drawn from aU(minus1 1) and we label itL119886119887(119868) = 119906119899119889 119868 is interpreted as being the main argumentjustifying the discussed proposal P and V119905119868 as its supportfor P at time 119905 119868 cannot attack other arguments but otherarguments can attack it For each argument 119886 isin A a directedarc or attack from 119886 to 119868 is activated with probability 119901119901 =003120579(11988611986802) When 119901119901rsquos lower bound is too high the proposalargument is almost always defeated and thus deliberativeopinion updates do not happen When 119901119901rsquos lower boundis lower than 003 then the opposite occursmdashthe proposalargument is always accepted due to an absence of attackstowards it Finally we set the maximum number of debatesto119898 = 7 for the decision-making processes stop rules1632 Calibration and Simulation Protocol In this subsectionwe discuss the calibration of the models the terminationconditions for runs in the deliberative and mixed model andthe expected outcomes in terms of the observations

321 Calibrating Dyadic Opinion Dynamics Dyadic opiniondynamics correspond to a space of parameters in whichargumentation and deliberation spaces are not taken into

Complexity 13

Table 1 Parameters and parameter types and their domains used to compare dyadic collective and mixed opinion dynamics

Length ofD119868 Decisional in119863119868 Social influence Deliberation influence Cognitive119905119863 119898 119898 119899119863 120572 120583 119879 119880 119901119886 119901119903 120574 119896 119861119890ℎ119886V1198941199001199031 3 6 1 3 6 7 001 002 005 0 05 066 01 14 18 02 06 03 05 01 005 01 02 4 8 16 119873119886119894V119890 119865119900119888119906119904119890119889account Deliberated arguments and informational cues haveno effect on agent opinion but agents still vote for theproposal argument17 We use and calibrate this model forcomparability in terms of all our metrics since they arenot explicitly observed in [9] a reference opinion dynamicsmodel Furthermore for simplicity and comparability againwe suppose that agents are homogeneous in terms of theirattitude structures and opinion weightings We fix 120583119894 =120583 119879119894 = 119879 and 119880119894 = 119880 for all 119894 isin 119873 and for some triplet(120583 119880 119879) isin [0 05] times [0 2] times [119880 2]

Experiments suggest that the strength of the dyadic inter-actions is an explanatory factor of the time of convergence(120583 lt 05) and of the dyamicsrsquo steady states Bigger values of 120583speed up convergence towards a stable set of opinion clustersand affect the size and the relative position of these clustersin the opinion distribution 120583 however does not play a majorrole in the number of opinion clusters observed at the steadystate of the dynamics

We set 120583 to the value given to it in Jagger et alrsquos opiniondynamics model (120583 = 01) [9] For one it is the smallestreal number for which most results found in [9] hold and thedynamics are smooth secondly it seems to be a reasonableldquostrengthrdquo of influence considering that we would not wantpair-wise social interactions to completely shadow or maskan eventual effect of deliberation in the opinion dynamics

For different values of attitude structures 119879 and 119880 weget exactly the same results as in [9] in terms of the numberand density of the opinion clusters The different couplesof values of 119880 and 119879 model the size of the latitude ofnoncommitment (119879 minus 119880) and thus the agentsrsquo propensityto update opinions Whenever 119879 minus 119880 le 02 regularitiesin convergence and opinion clustering appear If 119880 gt 1there is always central convergence if 119879 lt 1 there is alwaysbipolar convergence with clusters of similar sizes For 119879 minus119880 gt 02 relatively low values of 119879 (119879 lt 1) and relativelyhigh values of 119880 always yield extreme bipolar convergencewith a relatively small cluster of moderate agents (bipolar-central convergence) For 119879 sufficiently high (119879 ge 12) and119880 sufficiently low (119880 le 08) in other words not very ego-involved agents there is a bigger spectrum of steady statesFor this reason we studied the metrics of interest for thesevalue differences and kept the couples of (119880 119879) depicted inTable 1 For these couples central bipolar bipolar-centralconvergence (3 clusters) and multicluster convergence (4 or5 clusters) are possible and result frommeaningfully differentattitude structures Judgment accuracy varies significantlyacross the four different attitude structures we retain and soit happens with the variance of opinions These scenariosprovides us with reference results from which to build upon

322 Calibrating Deliberative Social Interactions Delibera-tive social interactions correspond to social situations in

which individuals are not influenced by pair-wise discussionwith peers but are sensitive to collectively deliberated infor-mational cues On these terms parameters such as the sizeof agentsrsquo argument sacks (119896119894) the distribution of argumentsin them (120601119894) the minimum number of debates (119898) andthe proportion of debaters from the population (119899119863) are nolonger mute We assume for simplicity that the formulaexist120601119890exist119896forall119894(119896119894 = 119896 and 120601119890119894 = 120601119890 and 120601119898119894 = 05) is true with 119894 isin 119873for some value 120601119890 isin [05 1] and for 119896 isin N

The chosen domains for these parameters (see Table 1)are justified by the size of the population and the size of theargument pool (119872 = 600) which we set from the start If 119896is too big (119896 ge 32) andor too many agents are allowed inthe debate (119899119863 ge 02) then individuals always disprove thecentral argument and deliberation is never or rarely takeninto account in the update of opinions Similar effects occurwhen 119898 is high yet we only calibrate it with respect to themean running time of a simulation and the values of 119905119863Different distributions of 120601119890119894 seem to significantly or directlyexplain none of the statistics we analyze so we convenientlyset 120601119890119894 = 075 We chose the parameter 119896 to be always divisibleby 4 for it is convenient given the initial conditions regardingthe argument distribution in the agentsrsquo argument sacks

We fix the value space for the parameters 120574 119901119886 119901119903 as inTable 1 to account for different intensities of the effect ofdeliberative voting retroaction on the system For examplethe scenarios where the three parameters are at their lowestvalues correspond to the scenarios with the smallest fixedprior effect of deliberation in the model Increasing any ofthese parameters should mechanically be associated witha world in which agents are more sensitive to collectivedecisions The values for 119901119886 are taken from [13] wherethe author states that by means of deliberation 7 to28 of individuals changed their opinion from agreeingto disagreeing or vice versa on a referendum question forDenmarkrsquos participation in the EuroWe choose two differentvalues for 119901119886 that account for high (03) and very high (05)sensitivity to deliberation in a group 119901119903 on the other handis taken to be small (equal to 01) since we believe that thereexist agents that will always go against the reached consensusTheir numbers are meager

For simplicity we assume that for all 119894 isin 119873 120574119894 = 120574 forsome 120574 isin [0 05] In effect we posit that the heterogeneityof agents in the deliberative context only derives from theirarguments and their opinions We decide to calibrate 120574 inallusion to the strength of the social dynamics in pair-wiseinteractions In doing so we limit ourselves to consider threedifferent scenarios in which deliberation has respectivelyhalf equal and twice as much opinion-shifting power thanone-to-one social influence Finally we include the accept-ability voting quota rule 120572 whose domain is inspired fromclassical majority rules [35] observed empirically (Table 1)

14 Complexity

Table 2 Parameters and their domains used to study the effect of the procedural parameters on the metrics

Procedural parameters Other parameters119905119863 119898 119899119863 120572 119898 120583 119879 119880 119901119886 119901119903 120574 119896 119861119890ℎ119886V1198941199001199031 2 3 4 5 6 1 2 3 4 5 001 002 003 004 005 0 05 066 7 01 16 02 03 005 02 12 119873119886119894V119890 119865119900119888119906119904119890119889

323 Calibrating the Mixed Social Interactions Model Themixed interaction model ascribes to a parameter space inwhich the effect of collective choices on our metrics isnontrivial and where deliberation and voting on proposalsdetermine their acceptability Since the mixed model isequivalent to periodic iterations of pair-wise and collectivesocial interactions the calibration of the parameters in bothpreceding models holds (see Table 1 for the calibration)The first reason for this is comparability the second is thatthe previous calibration takes into account the fact thatboth dynamics are going to be combined We include thefrequency of debates by adding the parameter 119905119863 whichcontrols for the amount of pair-wise interaction amongagents between two distinct deliberation steps

324 Termination Conditions for Runs Simulations stoponce 100 proposals are deliberated on andor voted for inother words after 100 collective decision steps have occuredThe number of proposals discussed seems arbitrary yet it ishigh enough to observe the effects of deliberation on opiniondistributions and on other metrics related to coherence andjudgment accuracy We choose the number of runs as afunction of our research questions which give relevance tothe procedural parameters in the mixed model rather than tothe parameters set to describe the population and its behavior

325 Expected Outcomes of the Simulations We expect moredeliberation in terms of higher 119898 and 119899119863 to increasejudgment accuracy and at the same time to reduce thevariance of opinions In turn a smaller variance of opinionsimplies that the argument pool for deliberation is smaller andtherefore judgment accuracy should be lower Moreovervariance increasing dyadic interactions (or rejection) shouldalso increase judgment accuracy and coherence since bipo-larization and dissensus foster argument diversity in delibera-tionThe coherence statistic should be stronger in simulationsin which central convergence appears quickly and agentsare naive Attitude structures sensitivity to informationalcues and weights given to deliberated cues that makes pathsto bipolar convergence smaller or to central convergencelonger should be associated with higher judgment accuracyShifts of opinion should be more visible in scenarios inwhich extreme agents are pulled away from the extremesHence the sensitivity to deliberated cues (119901119886 119901119903) and 120574should explain the shifts When crossed with high values of119880 and 119879 the shifting power of deliberation should be at itshighest

In the end we do not know how these mechanismswill play out The results of the subsequent experimentsgive us insight on the interplay between the aforementionedeffects

4 Results

Before pointing at any result obtained from the mixedinteractions model we describe and compare the dyadicand deliberative interactions models in the parameter spaceobtained from the calibration in Section 3 (see Table 1)Primarily we require our results to describe two orthogonaltypes of opinion dynamics the pair-wise dyadic and thedeliberative The latter comprises scenarios in which indi-viduals do not influence each other by means of pair-wisediscussion and only update their opinions from delibera-tion the former scenarios where only pair-wise interactionsamong agents determine the dynamics We describe both onthe parameter space given in Table 1 by performing at least20 simulations per scenario Quantitative results from thepair-wise interaction model alone are described during thecalibration and are not treated heremdashdeliberative parametershave no effect on it Moreover we show how these differentmodel specifications yield qualitatively different opiniondistributions and compare them on the basis of the metricsof interest To this end we perform independent two-sampleStudentrsquos t-tests18 or Welch t-tests19 and compute confidenceintervals at a 095 level of confidence Last but not least wecomment ordinary least squares (OLS) regression estimatesto account for the direction and magnitude20 of the effectsof the parameters on the metrics Estimates are declaredsignificant at a 5 level of risk

We perform the same analysis on the mixed interactionsmodel as we compare it to the pair-wise dyadic and deliber-ative interactions models and discuss the marginal effects ofthe parameters on the metrics More precisely we obtain twodifferent types of results for the mixed interactions modelthe first compares the mixed scenarios to their monolithiccounterparts with respect to each metric and with respect tothe regression estimates the second gives a clear idea of themarginal effect of each governance parameter (or parameterof interest) on our observations In the first we allow controland procedural parameters to vary on a parameter spacesimilar to those of the dyadic and deliberative opiniondynamics (see Table 1) In the second we fix all of ourcontrol parameters execute 36000 balanced21 runs andfocus only on the one-way pooled effects of our proceduraland behavioral (all agents are focused or naive) parametersWe generate comprehensive graphs and tables to account forthe obtained results

Please notice the slight change in the size of the parameterspace for the procedural parameters and in the values ofthe nonprocedural ones for the first and second type ofresults (refer to Tables 1 and 2 respectively) Since we wantto have a finer idea of the marginal effects of each of theprocedural parameters on how well a group decides onproposals we make the value jumps for each parameter (1)

Complexity 15

() (U T) = (02 18) (b) ( pa m nD) = (005 05 00 2 02) (＝) ０ＬＧＮＬＭ ＣＨ (＜) (＝) Ｈ＞ tD = 3

(d) (５ ４) = (02 14) (e) ( pa m nD) = (01 03 05 3 02) () ０ＬＧＮＬＭ ＣＨ (＞) () Ｈ＞ tD = 6

Figure 7 From left to right common opinion trajectories and distributions for dyadic deliberative and mixed opinion dynamics within thestructure of different decision-making processes

small enough to detect significant differences in the estimatesand (2) meaningful for interpretation For the fixed one-valued parameters in Table 2 we use Table 1 to set them totheir minimum mean or median values

41 Comments on Dyadic and Deliberative Opinions Dynam-ics In this subsection we present our first results regardingthe deliberative opinion dynamics described in Equation (6)and its differences with the dyadic dynamics (Equation (5))in terms of our observations

411 Qualitative Analysis of Pair-Wise Opinion DynamicsPair-wise opinion dynamics alone produces multiclusterstable convergencewith variable cluster sizes (see Figures 7(a)and 7(d)) Insight on the complexity of these dynamics wasdiscussed in Section 321

In Figures 7(a) and 7(d) agents discuss randomly withone another in pairs Clusters in the extremes form quicklysince either agents that are close to the extremes attractone another or are convinced to stay close to the extremesby interacting with other agents they disagree with Otheragents with near-extreme positions may be simply attractedto one of the several moderate foci of agents in the opiniondistribution On the other hand moderate agents are eitherattracted to the closest opinion focus or pushed towards theextremes of the distribution Over time agents in the centralfocus either attract other agents into it or ignore the opinions

of the extreme agents that interact with them The result ofthese dynamics yields amulticlustered convergencewhere theopinion foci are at least at distance 119880 from one another andonly the extreme foci (clusters of agents with |119900119894| ge 075) areat a distance greater than or equal to 119879 from one anotherThefoci at the bounds of the opinion distribution are very stableand in analogy to the (12119889)-rule in assimilation bounded-confidence models [33] the number of clusters that form isroughly equal to 119880minus1412 Qualitative Analysis of Deliberative Opinion DynamicsQualitatively deliberative opinion dynamics yield a looseunipolar convergence of opinions near the center (centralconvergence see Figure 7(b)) near the center of either the leftor right portions of the distribution (group polarization seeFigure 7(e)) or to a sparse bipolar opinion distribution withclusters at the center and at the extremes (see Figure 7(c)) ifmixed with pair-wise interactions This is probably a reasonto expect the variance of opinions and the proportion ofextremists (almost perfectly correlated 120588 = 089 119901 lt 0001)to be low for these scenarios The side towards which theopinion cluster skews depends essentially on the valence ofthe first argument that is collectively accepted

In Figures 7(e) and 7(b) agents that update their opin-ions do so at the same time and only when deliberationis successful In these scenarios every time a decision iscollectively accepted agents update their opinions towards

16 Complexity

Table 3 Ordinary least squares estimates for pair-wise interactions scenarios for most of the observed metrics

Dependent variable119881119886119903(119900) 119878ℎ 119890119888 119894119903(1) (2) (3) (4)Constant 0610lowastlowastlowast 4429lowastlowastlowast 0472lowastlowastlowast 0000

(0002) (0206) (0003) (0000)

119880 = 06 (119880 = 02) minus0012lowastlowastlowast 2981lowastlowastlowast 0061lowastlowastlowast 0000(0002) (0260) (0004) (0000)

119879 = 18 (ref 119879 = 14) minus0264lowastlowastlowast minus3261lowastlowastlowast 0004 0000(0002) (0260) (0004) (0000)

120572 = 066 (ref 120572 = 05) 0002 minus0059 minus0100lowastlowastlowast 0000(0002) (0184) (0003) (0000)

(119880 119879) = (06 18) (ref (119880 119879) = (02 14)) minus0324lowastlowastlowast minus3728lowastlowastlowast minus0033lowastlowastlowast 0000(0003) (0368) (0006) (0000)

Observations 1440 1440 1440 1440R2 0983 0389 0529Adjusted R2 0983 0388 0527Residual Std Error (df = 1435) 0032 3492 0053 0000F Statistic (df = 4 1435) 21 272740lowastlowastlowast 228686lowastlowastlowast 402326lowastlowastlowastNote lowastplt01 lowastlowastplt005 lowastlowastlowastplt001

the argumentrsquos defended position Hence if there is votingand if agents happen to obtain 119894119899 proposal arguments theyinexorably cluster towards one or the other side of the opiniondistribution resulting in opinion convergence remeniscent ofgroup polarization (see Figure 7(e)) Furthermore as agentscluster in the same side of the opinion spectrum the argu-ments that may be advanced in deliberation are fewer andmore skewed towards that one side of the opinion spectrumAgents become more easily persuaded in deliberation andonly by arguments supporting one side of the spectrumConvergence of opinions towards one loose opinion focusbecomes faster and more certain If there is no vote (seeFigure 7(b)) the dynamics are similar but for the fact thatit is the uniform randomness of the proposal argument thatguarantees a central convergence of opinions

413 Quantitative Analysis of Pair-Wise Opinion DynamicsThe three parameters that allow for quantitative analysis are119880 and119879 the agentsrsquo attitude structure and themajority quotavoting rule 120572 Table 3 supports the claim that going from 120572 =05 to 120572 = 066 does not affect the variance nor the shifts ofopinions However judgment accuracy is significantly higherwhen accepting proposals becomes more difficult This israther surprising but also intuitive since decisions that aretaken using more restrictive methods of scrutiny should bemore accurate

Concerning the attitude structures the interaction termbetween 119880 and 119879 shows that 119880 has a small effect on thevariance of opinions given that 119879 is low This shows thateven when the assimilation threshold is high a sufficientlysmall rejection threshold can make its one-way effect vanishHowever high levels of both 119880 and 119879 reduce the variance ofopinions dramatically In contrast when it comes to overallchanges in opinion the magnitude of 119880rsquos one-way marginal

effect increases the shifts of opinions whereas 119879 lowers itThis can be interpreted as having the rejection dynamics pushmoderate-to-extreme agents to positions that are close tothe extremes and very moderate agents to the center of thedistribution Not too many quantitative changes (on average)can therefore be observedThe fact that higher119880 is associatedwith faster convergence of opinions seems to imply lowerjudgment accuracy At some point voting for the same type ofproposals for too longmay result inmany collectivemistakesThis is an interesting conclusion to consider in the analysis ofthe mixed model Finally the interaction term shows that 119879has no strong impact on the metric

414 Quantitative Analysis of Deliberative OpinionDynamicsSeven parameters can be considered for quantitative anal-ysis in this dynamics the parameters of interest recalledin Section 312 the size of the argument sacks (119896) agentbehavior during deliberation the strength of the deliberationdynamics (120574) and the measure sensitivity to deliberatedresults (119901119886) The most ldquounexpectedrdquo result obtained fromthe deliberation dynamics is that it shows how deliberationcontributes to the proportion of extremists and the varianceof opinions It happens that having more agents discussproposals and having more of these discussions increasesthe variance of opinions and the proportion of extremistswhile lowering judgment inaccuracy and coherence For theformer the fact that far too many arguments are advancedduring deliberation plummets the chances that the centralargument is accepted hence deliberation does not moveopinions too often For the latter we observe that only ldquobigrdquodifferences (going from the lowest value of a parameter to thehighest) in protocol actually prove to have a preponderantimpact on the metrics These observations hint at a possibletrade-off between judgment accuracy and the variance of

Complexity 17

Table 4 Ordinary least squares estimates for the deliberative interactions scenarios for most of the observed metrics

Dependent variable119881119886119903(119900) 119878ℎ 119890119888 119894119903(1) (2) (3) (4)Constant 0032lowastlowastlowast 13092lowastlowastlowast 0370lowastlowastlowast 1007lowastlowastlowast

(0001) (0406) (0001) (0001)

119898 = 3 (ref119898 = 1) 0012lowastlowastlowast minus2062lowastlowastlowast minus0012lowastlowastlowast minus0001(0001) (0276) (0001) (0001)

119898 = 6 (ref119898 = 1) 0023lowastlowastlowast minus3285lowastlowastlowast minus0018lowastlowastlowast minus0004lowastlowastlowast(0001) (0276) (0001) (0001)

119899119863 = 002 (ref 119899119863 = 001) 0007lowastlowastlowast minus0965lowastlowastlowast minus0006lowastlowastlowast minus0002lowastlowast(0001) (0276) (0001) (0001)

119899119863 = 005 (ref 119899119863 = 001) 0020lowastlowastlowast minus3316lowastlowastlowast minus0015lowastlowastlowast minus0004lowastlowastlowast(0001) (0276) (0001) (0001)

120572 = 05 (ref 120572 = 0) 0146lowastlowastlowast 68932lowastlowastlowast 0068lowastlowastlowast minus0507lowastlowastlowast(0001) (0276) (0001) (0001)

120572 = 066 (ref 120572 = 0) 0291lowastlowastlowast minus13842lowastlowastlowast 00004 minus0000(0001) (0276) (0001) (0001)

119896 = 8 (ref 119896 = 4) 0004lowastlowastlowast minus0815lowastlowastlowast minus0002lowastlowastlowast minus0001(0001) (0276) (0001) (0001)

119896 = 16 (ref 119896 = 4) 0008lowastlowastlowast minus1118lowastlowastlowast minus0004lowastlowastlowast minus0003lowastlowastlowast(0001) (0276) (0001) (0001)

119861119890ℎ119886V119894119900119903 = ldquoFocusedrdquo (ref ldquoNaiverdquo) 0015lowastlowastlowast minus1980lowastlowastlowast minus0012lowastlowastlowast minus0004lowastlowastlowast(00005) (0225) (00004) (0001)

120574 = 01 (ref 120574 = 005) minus0020lowastlowastlowast 5762lowastlowastlowast minus00003 minus0001(0001) (0276) (0001) (0001)

120574 = 02 (ref 120574 = 005) minus0025lowastlowastlowast 8648lowastlowastlowast minus00003 minus0001(0001) (0276) (0001) (0001)

119901119886 = 05 (ref 119901119886 = 03) minus0013lowastlowastlowast 1581lowastlowastlowast 00004 00003(00005) (0225) (00004) (0001)

Observations 19440 19440 19440 19440R2 0931 0844 0562 0958Adjusted R2 0931 0844 0562 0958Residual Std Error (df = 19427) 0033 15701 0030 0050F Statistic (df = 12 19427) 21 844790lowastlowastlowast 8 733563lowastlowastlowast 2 079585lowastlowastlowast 36 761250lowastlowastlowastNote lowastplt01 lowastlowastplt005 lowastlowastlowastplt001

opinions that may be of interest in the mixed discussionmodel

Group coherence shifts of opinions and the varianceof opinions are highly affected by the voting thresholdthe coherence metric is reduced by half when allowing forclassic majority voting shifts explode and the variance ofopinions increases dramatically This is not surprising sincethe majority quota rule works as a buffer to the success ofdecision-making processes and in consequence it hampersthe deliberative opinion updates For the shifts of opinions itis clear that the two-third majority voting rule results in littleopinion change and that the simple majority rule inducesgroup polarization As expected there are less extremists inscenarios where there is no voting and the shifts of opinionare more likely to happen when agents are more sensitiveto deliberation When we include voting in the system

the grouprsquos judgment accuracy decreases due to the correctdeliberated decisions that are vetoed by a salient opinion-agent majority Strangely the effect only holds for the simplemajority decision rule Lastly the proportion of individualsthat participate in the debate and the minimal amount ofdebates have similar marginal effects on all the metrics (seeTable 4)

415 Comparing Deliberative and Pair-Wise Opinion Dynam-ics in a Nuthshell Qualitatively both models yield similaroutcomes but for the fact that deliberation never producesopinion polarization with both types of extremists (seeFigure 7) Furthermore clusters in the deliberative interac-tions model are less homogeneous in size and multiclusterconvergence occurs only when deliberation is rarely suc-cessful which happens sporadically given the calibration of

18 Complexity

Table 5 Mean difference 095 confidence intervals for metrics by model comparison

Model Metric119881119886119903(119900) 119901119903119900119901119890119909 119878ℎ 119890119888 119894119903Dyadic vs Deliberative [0191 0216] [0170 0197] [-2948 -2828] [0074 0082] [0165 0172]Mixed vs Deliberative [0142 0147] [0150 0155] [-2085 -1976] [0003 0002] [-0038 -0030]Mixed vs Dyadic [-0072 -0046] [-0045 -0018] [8332 8859] [-0079 -0071] [-0205 -0201]the model On the other hand pair-wise opinion dynamicsproduces multicluster convergence most of the time andcentral convergence only in the infrequent cases where thelatitude of acceptance is large

The comparison of these two scenarios in terms ofthe metrics is straightforward Whilst dyadic interactionsallow for more variability in opinions through pair-wiseinteractions and more mistakes in judging proposals delib-erative updates simply act as an antagonist force On aver-age the deliberative interaction model produces opiniondistributions with smaller variance and yields steady stateswith higher judgment accuracy a higher shift statistic andnaturally lower group coherence since only proposals that aredeliberated 119894119899 are eligible for voting (see Table 5)

42 Articulating Dyadic and Deliberative Social Interactionsin Opinion Dynamics So far we have considered scenarioswhere argumentation had no place and where individualsmade decisions according to opinions that were purelyconstructed from pair-wise discussions We also observedcases in which agents formed their opinions only by inte-grating deliberated proposals into their opinion updates Thescenarios we present in this section combine the two pre-ceding scenarios to account for both the effects of pair-wisediscussion and the effects of deliberation on opinions judg-ment accuracy and group coherence From the precedingobservations we conclude the following deliberative opiniondynamics reduces the variance and the proportion of extrem-ists because it shifts opinions greatly and polarizes groupsby eliminating at least one type of extremist the one whoseopinions are initially opposed to the deliberated resultsor it brings opinions close to neutrality (see Figure 7(b))This may not be desired in a group since the diversity andstability of opinions are necessary for deliberation to makesense and lead to the collective acceptance of controversialproposals Dyadic opinion dynamics on the other handtend to increase the variance of opinions and the numberof extremists while having a lesser aggregated effect on theshifts of opinion This is an immediate result of the pair-wiseinteraction dynamics inasmuch as it polarizes individualsvery quickly

To sum up in the mixed interactions model deliberationhinders the effect of voting on opinions by making someproposals that would normally be submitted for voting noteligible for voting It may also undo opinion changes dueto pair-wise interactions The mixed model can be seen asthe deliberation opinion dynamics model in which pair-wiseinteractions occur in between collective discussion and mayaffect which arguments are advanced during deliberationEquivalently it can be seen as the dyadic opinion dynamics

model in which deliberation accounts for endogenousldquoshocksrdquo on the opinion distribution

421 Qualitative Analysis of the Mixed Dynamics In Figures7(c) and 7(f) agents update their opinions through delibera-tion and pair-wise interactions Opinion trajectories visualsshow a combination of both previously described opiniontrajectories Over time as agents discuss with one anotheropinion clusters form as in the pair-wise or dyadic opiniondynamics Deliberation may disrupt the formation of theseclusters and three different situations may occur

(i) Deliberation disrupts the formation of the opinionclusters but not sufficiently to jeopardize pair-wiseopinion cluster formation (eg for high values 119905119863)Two phenomena may be responsible for it (1) thevariability of the proposal argumentrsquos support for theprinciple P and (2) too few (successive) deliberationsteps that result in opinion updates Once the opinionclusters are formed successful deliberation foreshad-ows the merging of the previously formed clustersmay bring extremists closer to moderate agents andthanks to the assimilation component of the pair-wisedynamics allow bigger and more moderate opinionclusters to form (refer to Figure 7(c))

(ii) Deliberation significantly disrupts the pair-wisedynamic formation of clusters This may happenwhen the requirements for accepting proposals areweak in the decision-making process (eg low 120572) Inthis case one has either central convergence sinceall agents follow the deliberated results in the samemanner or 3-cluster convergence with two extremeopinion foci In the latter situation the extremegroups form quickly (like in the pair-wise model) anddeliberation cannot pull extreme agents away fromthe extreme foci they are in The main reason for thisis that immediately after an extreme agent attempts toleave her foci through deliberation updates she getspulled or pushed back into it by means of a dyadicinteraction with another extreme agent

(iii) Deliberation does not disrupt the formation of theopinions cluster This is the rare case scenario forvery demanding decision-making processes in termsof inclusion (eg 119899119863) and argument acceptability (eg120572) Opinions converge as in the pair-wise opiniondynamics model (refer to Figures 7(a) and 7(d))

Last when there is voting (120572 = 0) the position ofthe moderate opinion cluster is determined by the first

Complexity 19

minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

Var(o)

sh

ec

ir

Var(o)

propex

sh

ec

ir

1

098

minus02

minus004

001

1

minus014

minus001

minus003

1

026

minus028

1

minus057 1

ＪＬＩＪ

Ｒ

Figure 8 Significant correlation estimates for themetrics of interestin the mixed opinion dyamics scenarios

accepted proposal argument andor the opinionmajority thatis formed from previous dyadic interactions (see Figure 7(f))

422 Relating the Metrics of Interest in Mixed OpinionDynamics The first thing to notice is that for all of thesescenarios all the metrics are significantly correlated (seeFigure 8) The variance of opinions and the proportion ofextremists are almost perfectly correlated To some extentthis means that a high variance of opinions known thatthe mean of opinions is statistically null equates to havingmany extremists in the population be towards one or theother side of the opinion distribution (eg extreme bipolar-ization) Similarly we find a negative and strong correlationbetween the judgment inaccuracymeasure and the coherencecoefficient (see Figure 8)mdashwhen judgment inaccuracy is high(low) coherence to deliberation is low (high) Cases likethese are telling since they may be related to sequences ofcollective decisions in which agents do not make too manymistakes when judging proposal arguments and still makedeliberated decisions that are on average in line with theirprinciples

The shift statistic interestingly is negatively correlatedwith the variance of opinions with judgment accuracy andwith the coherence coefficient In a way this means thathaving more moderate agents conduces on average to moremistakes in judging arguments and to less coherent collectivedecisions An explanation for this phenomenon could be thatas agents reach consensus in opinion important (possiblyextreme) arguments for correctly judging argument propos-als are not advanced in the deliberation arenawhich results inmore mistakes and in possibly accepting proposal argumentsthat after voting are not meant to be accepted Indeed the

negative correlation between the variance of opinions andjudgment inaccuracy points towards this direction since onaverage the more shifts there are the smaller the variance ofopinions is (see Figure 8)

423 Comparing Mixed Dynamics with Other MonolithicDynamics of Interactions Qualitatively opinion trajectoriesin the mixed model are more chaotic than in the pair-wise and deliberative interaction models (see Figure 7) Asanalyzed previously deliberation dynamics disrupt the pair-wise dynamics and create situations in which central andmulticlustered convergence (with two clusters at the extremesof the distribution) are possible at the expense of more(pair-wise dynamics) or less (deliberative dynamics) clustersIt tolerates the survival of very small isolated group ofagents that the previous models rarely tolerate in a hundredcollective decision-making steps (see Figure 7(c))

One of the most interesting observations one can makeof this family of models regards how groups change (in termsof our metrics) when dyadic social interactions are addedto deliberative interactions and vice-versa When comparingthe mixed interactions with only deliberative ones shiftsagentsrsquo aggregated change in opinions are on average lesspronounced The variance of opinions and the proportionof extremists are significantly higher for the former than forthe latter group coherence in contrast is lower with respectto the scenarios where opinions are formed only throughdeliberation Judgment accuracy follows that same trendA reason for this might be that mixed dynamics are fasterto yield semistable clusters andor opinion convergence Inconsequence the potential argument pool for deliberativeexchange shrinks faster and deliberation becomes less effec-tive over time which results in more collective mistakes Theoutcomeon coherence is predictable inasmuch as deliberativeinteractions produce group polarization If a group agrees inopinion debates are sterile because themembers of the groupwill most likely refrain from attacking the proposal argu-ments they all support and will work together to disparagethe proposal arguments they do not support (see Table 5)

In the same manner we report that the proportion ofextremists and the variance of opinions are barely lower inthe pair-wise interaction model when including deliberationin the decision-making process Shifts of opinions are alsobarely lower when there is no deliberation This can indicatethat deliberation does not undo the changes in opiniondue to pair-wise interactions nor does it amplifies them Incontrast when we consider the labeling-based metrics themixed interaction model ensures lower judgment inaccuracybecause of deliberation and lower group coherence (seeTable 5)

In sum the mixed discussion model is a compromisebetween the pair-wise interaction and deliberative opiniondynamic models in that it offers some guarantees in termsof judgment accuracy and coherence while allowing forldquoreasonablerdquo variance in the opinion pool It establishes somekind of trade-off between labeling-based metrics and thevariance of opinions that may be of interest for decision-making process design

20 Complexity

Table 6 Summary of sign or relative marginal effects for all varying parameters on dyadic deliberative and mixed interactions models inand for all experiments

Metrics Parameters119905119863 119898 119899119863 120572 119879 119880 119901119886 120574 119896 119861119890ℎ119886V119894119900119903119881119886119903(119900) XX X X O X X X X X X119901119903119900119901119890119909 XX X X O X X X X X X119878ℎ XX X X O X X X X X119890119888 XX X X XO X XOO XOO X X119894119903 XX X X O XO O X OOX XOO X XNota gray symbols correspond to dyadic dynamics light gray to deliberative dynamics and bold black to mixed dynamics Given a scenario the symbol 119883stands for constant parameter and 119874 for insignificant marginal effect and as parameters grow for positive (negative) marginal effect for the parameterFrom left to right at the parameter level the leftmost arrow corresponds to the scenario with the strongest (in absolute value) marginal effect of the parameter(with respect to to its factors) Example Let 119901 be a parameter and 119874 stands for ldquo119901 has a significant positive effect in dyadic discussion scenario negativein the deliberative one and insignificant in mixed discussion The positive effect in dyadic discussion is greater than the absolute value of the negative one inthe purely deliberative scenariordquo

424 Comments on the Differences between the Three Modelsfor OLS Regression Estimates The significance of most of theregression parameters estimated for the mixed interactionsscenarios are similar to those estimated for the pure dyadicanddeliberative scenariosNevertheless themagnitude of theeffects is different insomuch that the dynamics themselvesare different So instead of commenting the linear regressionestimates of the mixed model we compare when possiblethe standardized significant coefficients of the regressors ofthe mixed model with those of the pair-wise and deliberativeinteractionmodels (see Table 6 for a summary of the results)

In mixed interactions for instance the influence of119898 onthe opinion distribution is much more preponderant than inthe deliberative dynamics scenarios Going from 119898 = 1 to119898 = 3 increases the standard variation units of the variance ofopinions by twice as much as the one in the deliberative sce-nariosThis in particular is due to the design of the decision-making process If 119898 is higher then the number of socialinfluence steps (119905119863) in the decision process is also higher(compared to 119905119863 = 0 in the deliberative case) If we decideto consider these steps as contributing to the variance ofopinions then the effect of119898 on the variance of opinions hasto be stronger in the mixed interactions model In contrastthe number of agents participating in the debate has a weakereffect on the variance of opinions in the mixed interactionsmodel than in the deliberative dynamics specially for 119899119863 =005 (more than twice as much) The difference may comefrom the fact that in mixed discussions the part of theargument pool used in deliberation is more restrictive thanin the deliberative case We can argue that since pair-wisedynamics may get individuals closer to one another aftersome deliberation the same happens to the arguments thatindividuals will most likely use in deliberation Taking thisinto account means that deliberation will at many instancesbe unfruitful and have no effect on opinions thus failing tomoderate extreme opinions andor polarize the group In thesame direction a high number of participants in deliberationcan make the deliberation dynamics stiff In this case thedistribution of opinion is close to the uniform distributionwhich has a relatively high-to-moderate variancewith respectto the reference results

The strength of the deliberative dynamics has a positivestronger effect on the variance of opinions in themixedmodelthan in the deliberative model when 120574 = 02 and a negativeweaker one when 120574 = 01 A sound interpretation of thisresult is that when deliberation is successful and its effect onopinions does not get agents to move sufficiently away fromtheir current positions agents are pulled or pushed back totheir previously held opinions the variance of opinions doesnot changemuchOn the other hand if the effect of successfuldeliberation is strong opinions can change sufficiently tomake agents much more prone to assimilation into moderateopinion foci the variance of opinions falls

The deliberative opinion dynamics model is much moresensitive to 120572 than the mixed discussion model Differ-ences in marginal effects can double The probability ofattraction to deliberated results (119901119886) for instance has astrongmoderating effect (negative) on agentsrsquo opinions in thedeliberative opinion dynamics but a meager one (two-fold)in the mixed and dyadic dynamic models This is becauseopinion updates in the deliberative model depend only onthat parameter whereas in the mixed model more numeroustypes of interactions make opinion updates possible

In terms of judgment accuracy all parameters except thesize of the argument sacks (119896) have a significant slightlystronger effect on the deliberative dynamics than in themixedmodel In effect the difference may be attributed to theupshot of pair-wise interactions on the distribution of opin-ions and on the arguments that would be advanced duringdeliberation The result respecting parameter 119896 however iscounterintuitive Since argument sacks are static scenarioswhere shifts are higher should make 119896 more important fordeliberation to succeed and it is precisely in the scenarios ofpure deliberative dynamics that we find the biggest opinionshifts This may point to the hypothesis that in deliberativeinteractions agents change opinions substantially but do notoften change their adherence to the principle (go from anegative opinion to a positive one or vice versa) when thereis no group polarization

43 Sensitivity Analysis for the Procedural Parameters ofInterest Weperform a sensitivity analysis of the observationson the parameters of interest in themixed interactionsmodel

Complexity 21

We extend the parameter domains of the pure dyadic anddeliberative scenarios as we make it explicit in Table 2

431 Minimum Number of Debates (119898) We observe that theminimum number of debates has a significant well-observedeffect on all of our metrics excluding group coherence Forthe variance of opinions we can observe that the moredebates there are the bigger the value of the metric isalthough the higher 119899119863 and 119905119863 are the weaker the overalleffect of119898 becomes (Figures 9(d) and 9(a)) Furthermore themarginal effect of increasing minimal debates is decreasingan additional deliberation step requirement increases thevariance of opinions but less and less as it grows 119898 coupledwith 119899119863 has a shy S-shaped effect (see Figure 9(d)) on thevariance of opinions meaning that it has a stronger effectwhen 119899119863 is low which suggests a possible trade-off betweenthe number of arguments accepted in the debate and thenumber of debates required before accepting a proposal

In contrast the shifts in opinion are less and less likelyas 119898 grows and this is independent of the variations of theother parameters Again the effect is marginally decreasingand is only truly significant when 120572 = 05 An explanationto these effects may be linked to the design of the systemFirst the variance of opinions is higher when deliberation isasked for because the more deliberation steps there are thehigher the chances are that the central proposal argumentis deemed unacceptable specially when agents are focusedFurthermoremechanically speaking increasing theminimalamount of debates implies that whenever a decision is to betaken at least 119898 times 119905119863 time steps have to take place and if atany moment a proposal argument is considered undecided119905119863 time steps are added to the process So unless the debatedoes not yield 119906119899119889 labels for proposal arguments (highlyunlikely considering that 120590 = grounded semantics) the morenondeliberation steps there are in the decision process thehigher the variance of opinions is Concerning the shiftswhen 120572 = 05 either the system is too stiff to accept anyproposal argument and opinions do not changemuch or theeffects of pair-wise discussion and deliberation cancel out insuch a way that individual shifts are minimal (Figure 9(b))

On the side of labeling-based metrics the more debatesare asked for the more accurate a group is in its judgmentthe effect being smaller as 119898 grows When agents are naivethe effect is quasilinear while when they are focused thestrongest effects of acquiring more deliberation are foundwhen levels of deliberation are already low (Figure 9(f))Thiscan be explained by the fact that the more debates thereare in the decision process the closer one gets to the idealargumentation framework

432 Proportion of the Population in Deliberation Steps (119899119863)Like for 119898 119899119863 has a significant effect on the proportionof extremists and on the shifts and variance of opinions(Figure 9(e)) This may result from the fact that being ableto put more arguments in play at the same debate step mayincrease the odds of revealing cycles around the proposalargument in119860119865120576 Given that we use grounded semantics thearguments in the cycles are labeled 119906119899119889 and in consequence

debates are more often postponed when 119899119863 is higher Post-poning debates in turn increases the number of nondebatessteps in the decision process which increases the varianceof opinions and limits over time the moderating effect ofdeliberation Moreover the effect of this parameter is verydependent on the value of 120572 (Figure 9(e)) For shifts forinstance 120572 = 05 makes the effect of 119899119863 negative while120572 = 0 makes it positive but to a lesser extent Furthermorefor higher requirements of deliberation (119898) adding moreindividuals to the deliberation process has weaker effectson the variance of opinions and on the other metrics It isalso quite interesting to notice that it has the same effect onagents independently of agent behavior during deliberationThis is a surprising result as one would have expected morefocused agents in a deliberation arena to cause a raise inthe proportion of extremists as they play to knock outopposing proposal arguments and thereof reducing the effectof deliberation (lowers variance) on the whole population

Finally similar to119898 addingmore people into the deliber-ation process increases judgment accuracy (see Figure 9(c))and has no clear effect if any on group coherence (seeFigure 9(i))

433 Steps between Deliberation Steps (119905119863) In all configu-rations 119905119863 increases the proportion of extremists (variance)and decreases the shifts of opinions among agents Theshifts and the effect on the variance of opinions are presentfor 120572 = 05 insignificant otherwise (Figure 9(b)) 119905119863 ishighly linked to 119898 by construction When 119898 = 1 thecurve linking the variance of opinions and 119905119863 is convex As119898 increases the curve becomes more and more concavewhich means that 119905119863 has a more important effect on theopinion distribution as collective decisions take longer to beachieved This seems counterintuitive but in reality it reflectsthemultiplicative relationship between deliberation and pair-wise interactions and the semantics in the model If 119898 is lowand 119905119863 is high the effective number of pair-wise interactionsis on average smaller in the deliberation process Hencethe structure of the decision-making process hinders anyincrease in the variance of opinions If119898 is high the oppositeeffect is observed Additionally since the grounded semanticsyields few 119894119899 arguments with respect to other admissibility-based semantics getting closer to the ideal argumentationframeworkmaymake it difficult to obtain the expected effectsof deliberative discussion on agents Lower 119898 constrains thevariance of opinions by making deliberation more influentialon opinions

In terms of judgment accuracy and coherence we observethat 119905119863 does not explain coherence yet it tends to decreasejudgment inaccuracy This may be due to a more variedargument pool in the debate which in turn may be aconsequence of a higher variance of opinions or vice versa

434 Acceptability Voting Quota (120572) By far it is the mostinfluential parameter in our study It changes the directionand the intensity of the effect of all the other proceduralparameters and by construction heavily constrains the roadto accepting a proposal (see Figure 9)

22 Complexity

015

020

025

030

035

040

1 2 3 4 5 6

12345

Values of Ｇ

(a) 119905119863 times 119901119903119900119901119890119909 by119898

0

10

20

30

40

50

1 2 3 4 5 6

005066

Values of

(b) 119905119863 times 119878ℎ by 120572

040

045

050

055

060

1 2 3 4 5

001002003004005

Values of Ｈ＄

(c) 119898 times 119890119888 by 119899119863

015

020

025

030

035

040

1 2 3 4 5

001002003004005

Values of Ｈ＄

(d) 119898 times 119901119903119900119901119890119909 by 119899119863

0

10

20

30

40

50

001 002 003 004 005

00005066

Values of

(e) 119899119863 times 119878ℎ by 120572

040

045

050

055

060

1 2 3 4 5

BehaviorNaiveFocused

(f) 119898 times 119890119888 by 119887119890ℎ119886V119894119900119903

015

020

025

030

035

040

0 05 066Behavior

NaiveFocused

(g) 120572 times 119901119903119900119901119890119909 by 119887119890ℎ119886V119894119900119903

05

06

07

08

09

10

1 2 3 4 5 6

00005066

Values of

(h) 119905119863 times 119894119903 by 120572

08700

08725

08750

08775

08800

001 002 003 004 005

BehaviorNaiveFocused

(i) 119899119863 times 119894119903 by 119887119890ℎ119886V119894119900119903

Figure 9 Curves of mean observations for different parameters and metrics for 36000 runs Bars are 095 confidence intervals Plots withthe same observables (ordinate) except for 119894119903 are on the same scale

Complexity 23

The higher the requirement for accepting the proposal isthe higher the proportion of extremists and the variance ofopinions are and the lower the shift statistic is In few words120572 constrains the world to dyadic discussion or throws it intoa process in which deliberation is much more importantthan dyadic interactions This is why for any 120572 = 05 oneeither gives too much weight to deliberated results whichshadows pair-wise interactions or too little weight in thesense that no deliberated result is ever accepted and thusnever integrated into the opinions of the agents that havevoted for it The reason for the latter is that the latitudeacceptance is too low and thereafter pair-wise interactionsare not enough to unevenly polarize the population in such away that deliberated proposal arguments are voted favorablyby a non-negligeable majority Shifts for 120572 = 0 are stable fordifferent levels of 119905119863 which implies that deliberation keeps incheck all the shifts of opinions related to an increase in thenumber of pair-wise interactions For 120572 = 05 deliberationis less likely to be successful and shifts tend towards the shiftlevels for 120572 = 0 (see Figure 9(b))

Concerning labeling-basedmetrics120572 entirely determinesthe coherence statistic For 120572 = 05 there is no difference incoherence because of how coherence is defined it is maximalwhen 120572 = 0 and strangely maximal when 120572 = 066(see Figure 9(h)) Surprisingly 120572 has no effect on judgmentaccuracy One would have expected a higher 120572 to increasejudgment accuracy since agents would hardly collectivelyaccept a proposal argument discussed during deliberationProposal arguments have given how the argument lattice isgenerated a higher probability of being rejected than of beingaccepted in the ideal argumentation framework

435 A Word on Focused and Naive Agents Focused andnaive specifications for agent behavior are an importantparameter in the decision process since they model howagents choose arguments in the deliberation arena In the sce-narios where agents are focused agents show equal or highervariance of opinions and are therefore more extreme (seeFigure 9(g)) independently of the voting requirements Shiftsof opinion are less likely as well We can see this happeningbecause focused agents knock out proposal arguments moreoften than naive agents provided that 119899119863 is low

One could have expected naive agents to yield morecoherent decisions since they argue sincerely However thisseems not to be the case (see Figure 9(i)) Agents are unable todeliberate in away thatmakes deliberation reflect their votingintentions either because they do not have the necessaryarguments to do so or because such arguments do not exist

In terms of judgment accuracy focused agents do betterthan naive agents in finding the ideal labels for the proposalarguments This is likely because when reconstructing theframework focused agents take into account the deliberatedproposal argument and choose themost pertinent argumentsin their sacks that relate to the state of affairs at the deliber-ation arena In fine debates result in a better approximationofL119886119887120576(119868) even though one could have believed the oppositethat is focused agents are ready to sacrifice a correct collectivelabel of a proposal argument in the goal of defending theirpositions

44 Points of Discussion In this subsection we discuss theresults obtained from the simulations We attempt to makesense of them from a social sciences perspective

441 Opinion Consensus and Dissensus The deliberativeopinion dynamics corresponds to the simulated scenarios inwhich the variance of opinions is at its lowest and judgmentaccuracy at its highest Dyadic opinion dynamics are associ-ated with the scenarios in which the variance of opinions is atits highest judgment accuracy at its lowest and coherence atitsmaximumpossible levelThemixed interactions dynamicsis a combination of these aspects for all the metrics exceptfor coherence at which it is the worsemdashdeliberation leadsto some kind of loose opinion consensus that is reinforcedby pair-wise interactions Likewise the mixed discussionmodel confirms some results exposed in [16] expressingthat opinions and voting intentions in a deliberative contextldquooften changerdquo That being said and more particularly indeliberative opinion dynamics outputs align well with twointeresting results on consensus formation given by aninterpretation of Moscovici and Doisersquos [15] work Sunsteinrsquos[14] account on deliberation and group polarization andsocial impact theory [4] The first states that mild consensusis reached in deliberation as procedure disengages agentsthe second is that within discussion groups majorities tendto become larger and opinions more extrememdashnot on thebasis of competing arguments but of the initial balance ofopinions After several decision processes we find opiniondistributions that reflect these observations even though weare not able to precisely coin the set of parameters that yieldssuch distributions (see Figure 7)

Although differences between mixed and dyadic opiniondynamics in terms of the variance of opinions and shiftsare slim the former does much better in terms of judg-ment accuracy In this sense deliberation does moderateopinions but not as much when non-deliberation steps arerelatively frequent throughout the decision-making processIn terms of Moscovici and Doise theory of consensus [15]this makes sense since pair-wise discussion engages or ego-involve agents in the discussion and pushes to convergenceof opinion at the extremes of the opinion distributionHowever what remains unclear is the reason why proceduralparameters that constrain deliberation do not disengageagents from discussion and thus result in a mild consensusas also expressed by Moscovici and Doise [15] A possibleexplanation of this phenomenon may stem from the factthat deliberation in the model is always accompanied by aseries of ego-involving discussions that shadowor counter themoderating effect of deliberation Not only this deliberationcould also be responsible for crystallizing the opinions ofextremists for it favors one or the other part of the distributionof opinions as explained in [14] and to some extent in [13]Furthermore the mixed discussion model gives an exampleof polarization that results from a minority of dissenterson one side of an issue acceding to views of the majorityrsquosside [4] as opposed to the strong polarization observed indyadic interactions Situations in which the status quo orthe undecidability of the proposal argument lingers cannotpossibly yield situations of moderation In cognitive science

24 Complexity

one can be inclined to believe that the longer a deliberationprocess is the more likely individuals will take an immutablestance on the situation In view of the assumption that longdecision-making processes can be more costly for a group orsociety as a whole agents may feel forced to take a stancejust for the sake of ending the process The stance that isadopted would therefore depend on pair-wise interactionsamong individuals and not on deliberation since deliberationtakes too long to be conclusive

442 For Hypothetical Recommendations in Decision-MakingProcess Elaboration On another tune results from themodelsuggest that procedural parameters have marginally decreas-ing positive or negative effects on the metrics of interest Ina hypothetical policy analysis perspective this may indicatethat after choosing a correct number of individuals andimposing the right amount of debates one can attain rea-sonable levels of judgment accuracy coherence and varianceof opinions As accounted in [18] deliberation increasesknowledge (collective knowledge as well) and therefore itheightens the chances of making correct decisions If onebelieves that social welfare is linked to deliberation andeventually to the metrics studied previously then proceduralparameters can both be used as an instrument to attaindesired levels of social welfare and reinforce the notionof legitimacy of collective decisions It can also provide ajustification for choosing a deliberation regime rather thananother on the basis of how important (or urgent) a topic ofdiscussion is Indeed if we adopt the claim that deliberationis a forerunner of welfare then any state in which deliberativecues are at their maximum (highest number of participantsmany deliberation steps before making a decision and asfrequent as possible) has to be mapped to the highest attain-able social welfare In this case is a situation where judgmentaccuracy is at its highest yet the variance of opinions or theproportion of extremists at its lowest an ideal situation Wecannot say for sure since strong opinion consensus and longdecision-making processes in many situations may result ina loss of welfare rather than in a gain of it

Another interesting result in procedure for deliberationis the very similar estimates we find for the effects of theminimum number (119898) of debates necessary to make adecision on a proposal and the proportion (119899119863) of individualsparticipating in deliberation on judgment accuracy In thescope of social welfare this raises the question of whetheradvocating for longer decision processes by demandingmoredeliberation steps (higher 119898) rather than bigger debates(higher 119899119863) may increase welfare Depending on the objec-tives of the decider of the deliberation procedure he or shemay choose to concentrate on one or the other A recom-mendation we can assert from the mixed social interactionsmodel is that if one prefers low to high variance of opinionsandor high to low coherence the decider is better off ifshe focuses on increasing the size of the debate rather thanorganizing debates very frequently One of the reasons forthis might be that when making bigger debates all agents aretruly considered as equals in the decision-making processIn organizing many debates individuals that are chosen toparticipate in many of these will be more influential and will

somehow bias the set of arguments in the deliberation arenaThemajority rulemodeled through120572 also determines the sizeof the shifts and the distribution of opinions Because a bigger120572 is associated with higher levels of extremism and opinionvariance deliberated cues are harder to accept and thereforeit becomesmore difficult for agents to reach concensus andormoderate their opinions

On the other hand and further into the idea of legitimacythe goal of deliberative processes is the agreement amongall participants This agreement stands for the right or bestdecision from a formal or procedural perspective and it is alsothe best from a substantive point of view In other words thelegitimacy of decisions not only derives fromwell-establishedprocedural requirements (respect of the protocol) but alsobegs for the fulfillment of two essential conditions (1) satisfy-ing the procedural requirements for a correct procedure (for-mal legitimacy) and (2) the rational acceptability of the resultsof this procedure (substantive acceptability) Legitimacy canbe then obtained from the coherence statistic scrutinizedin the model insofar as it measures how well agents acceptthe results of a correct procedure (eg deliberation) So fora proposal to be legitimate one has to choose parametersthat would maximize group coherence in decision-makingprocesses

As deliberation may give rise to consensus in the delib-eration arena it may also sow the seeds of dissensus inthe group Our model clearly illustrates this phenomenonthrough the procedural parameters of deliberation and themetrics observed From another perspective in deliberativedemocracy dissensus in collective decision-making may bedesirable According to Landermore and Pagersquos accounts [36]lack of consensus can be perceived as having agents withalternative ideas that combined with other ideas can providea better approximation of the ideal status of an argumentTheauthors in [36] call this dissensus ldquopositive dissensusrdquo andit makes normative sense when solving complex collectivedecision-making problems Our model captures the ideaof having normative requirements to define a good andlegitimate collective decision requirements of high consen-sual deliberative accuracy (judgment accuracy) of legitimacy(coherence) and of diversity (variance of opinions) are alltaken into account and for such we can conclude that ourmodel is successful in describing a sound process of collectivedecision-making with interpretable outcomes

Of course the present simulation results and analysis donot prove that all opinion dynamics can be accounted forby such simple processes like the one presented here Theydo however shed light on the expressiveness of combiningdifferent paradigms to account formore interpretablemodelsand pave the way to creating original models of the sortThe simulations suggest the desirability of discovering theconsequences of relatively simple laws of communication atdifferent levels (micro meso and macro) to determine whatstill needs to be explained

5 Related Work

We see our model as a contribution to the influence andopinion dynamics field in ABM and a pragmatic application

Complexity 25

of abstract argumentation theory To our knowledge we areunaware of existing literature on agent-based modeling thatexplicitly relates collective choice and the notions of deliber-ation and opinion diffusion through abstract argumentationas we have done it

Most models in the literature of opinion diffusion areinterested in opinions because these have an influence incollective decision and in questions of social order Forinstance in [3] the authors are interested in consensus andin how a group collectively chooses among two alternativesIn other models authors are interested in the emergenceof extremism [8] and on the distribution of opinions whenextremists are introduced in the population [5 33 37] whileother authors coin the notion of opinion polarization as anemergent property of the system [8 12] They show usingmodels of ldquobounded confidencerdquo and opinion diffusion withtrust that three different kinds of steady states (unipolarbipolar and central) are possible depending on whetheragents are sufficiently uncertain about their opinions andsufficiently connected andor a certain proportion of indi-viduals are already extreme Recent articles on informationand opinion dynamics stress the importance of governanceand government intervention in the spread of emotions(opinions) in the frangibility of social consensus systemIn [10] the authors show that government intervention inthe spread of negative emotions can lead to an even fasterspread of negative emotions (single-peeked convergence) andin a faster collapse of the social consensus system Opinionas a function of trust is studied in [12] and relates to howindividuals form their opinions on the basis of how muchthey trust agents in their networks

Another stream of discrete opinion dynamics modelsthat explain the emergence of opinion can be found in theliterature Computer multiagent models of attitude changebased on Latanersquos social impact theory are presented in [4] Inthe theory an agent changes her opinion on the basis of theinformational impact she is subject to which depends on thepersuasiveness (strength) supportiveness and immediacy(group structure) of the environment and information shereceives The simulations predict two emergent groups ofphenomena the shifting of attitudes towards incompletelypolarized equilibria and the formation of coherent clusteringof subgroups with deviant attitudes Similarly but in acontinuous more argumentative fashion Mas and Flache[11] present a model of opinion diffusion using argumentsArguments are considered to be for or against a proposal andare given an agent-dependant relevance in dyadic persuasiveinteractions Arguments also determine the agentsrsquo opinionsand their dissimilarities which posits the rules of pair-wiseencounters in theirmodel as agents are assumed to form theiropinions according to the arguments they own They showthat in an argumentative opinion diffusion model the onlyobservable steady states are bipolarization and consensusand that bipolarization can emerge from interactions amongsimilar agents In other words bipolarization of opinionsis possible with homophily and without negative influenceif there is argumentation They subsequently compare theirresults to experimental data In [38] a similar approach toargumentation and opinion formation is taken but for the fact

that they introduce different types of arguments explicitlyapply social judgment theory [22] and use survey data toestablish model-to-real-world comparisons

Closer to opinion formation and abstract argumentationGabbriellini and Torroni [39] are the first to have origi-nally and soundly merged opinion diffusion and abstractargumentation They define an agentrsquos opinion as a functionof the arguments she holds and the attack relation amongthem They devised a focused peer-to-peer dialogue systemof persuasion (NetArg) inspired from Mercier and Sperberrsquosargumentative theory of reasoning [40] which used onlyabstract argumentation to study opinion polarization andopinion dynamics Moreover they considered networks andthe notions of trust and epistemic vigilance to define adynamics for knowledge revision and trust itself whichwe clearly lack in our model They showed that if a con-servative belief operator in argumentation was applied byagents when they reasoned then their dialogue protocoldid not increase polarization among agents That said theyused the model to study the effect of Groenvetterrsquos weaklink theory in the spreading of arguments and were notparticularly interested in notions like judgment accuracy orcoherence nor in tackling questions related to collectivedecision-making procedures Their system is very expressiveand helps position our work in the litterature our work is atthe frontier of pure argumentative opinion diffusion modelsopinion diffusion with arguments and continuous bounded-confidence models of opinion diffusion a la Deffuant

On another note work on collective cognitive conver-gence [41] and opinion sharing [19] shows that consensustowards a certain ldquocorrectrdquo opinion or cognitive state is alwayspossible yet dependent on noise variability and awarenessof agents In [20] the authors show that learning about anexogenous correct state of the world (represented by bits)under confidence was possible but only if the agents werenot too confident In a homogeneous population they showthat the higher the confidence the worse the learningmdashthevery confident agents do not learn the properties of the truestate of the world and disrupt the learning process of the lessconfident ones Collective cognitive convergence can be seenas a result of deliberation in truth-seeking models

When it comes to abstract argumentation theory we takean approach that wires two types of dialogues that are well-studied in the literature persuasion dialogues [42] and delib-eration dialogues [43] Another interesting line of work is theone on mechanism design [44] or the problem of devisingan argumentation protocol where strategic argumentation isnot a liability for success in debates We tackle mechanismdesign in a different way though Instead of consideringstrategy-proofness we are interested in how differences inprotocol can result in epistemologically ldquobetterrdquo collectivechoices and guarantee that opinion distributions are favorablefor deliberation For a survey on persuasion dialogue see[42]

Work on agent-based argumentation usually assumes thatthe semantic relationship between arguments is fixed Inother words if two individuals were to put two argumentsin the public arena such that one logically attacked the otherthen everyone would agree that such attack exists [45 46]

26 Complexity

Other models which do not make this restrictive assumptioncan also be found in the literature [27 47] and derive fromthe class or family of opponent models [27 48] in which twoopposing sides attempt to win a dialogue Ourmodel is in theintersection of these two but the frame combining opiniondiffusion of the kind and argumentation remains new

The idea of mixing interpersonal influence and verticalcommunication however is not original It is described andimplemented in innovation diffusion models such as [5 49]In both vertical communication is modeled as exogenoustransparent informationmdashagents are aware of the existenceof an innovation thus triggering several processes of choiceand stabilization of opinions Also in [50] an Eulerian modelis implemented to show the effect of media and exogenousinformation on opinion distributions With respect to thispoint the originality of our work is in that the informationemitted as vertical communication is endogenous It is issuedfrom a deliberation model that agents shape on the basisof their opinions arguments and behavior In the spirit of[51] where the authors control for the design of verticalcommunication (to whom the vertical communication isaddressed its timing how it influences agents opinions) wecontrol for the process generating the information for ourown set of variables length of debates majority quotas andthe frequency at which discussions take place

6 Conclusion and Perspectives

The main objective of this article was to build a bridgewhere decision-making argumentation and opinion dif-fusion could come together We proposed a model thatcombined abstract argumentation theory and a boundedconfidence opinion diffusion model and showed to whatextent it could explain variance of opinions extremismcoherence in collective decisions and judgment accuracyon arguments given an ideal state of full information Thesecond objective was to show in what way governance pro-viding agents with arenas of discussion and deliberation wasimportant in addressing the success of collective decision-making processes and the quality of their outcomes

The model revealed that allowing for more deliberationtime though in lower frequency and allowing for widerparticipation in deliberation increased the variance of opin-ions and the proportion of extremists in a group Theseobservations are consistent with the combination of resultsfound in [9] and in [13 14] which stress that deliberationmay polarize groups and may have a meager effect on theshifts of opinions and inconsistent with [16] where it isargued that opinions tend to moderate after deliberationIt happens that deliberation alone did moderate opinionyet when integrated into a more complex system in whichindividuals were allowed to interact with one another andnot all that was deliberated was accepted its influence wasshadowed by other more individualistic dynamics

Undeniably the grounded semantics played an importantrole in the weak effect of deliberation since it models askeptical way of reasoning over arguments For agentsaccepting arguments thus updating their opinions accord-ingly happened not too often Nevertheless asking for more

deliberation did increase judgment accuracy as observed in[18] yet in a marginally decreasing fashion We showed thatvoting within the deliberation protocol not only increasedthe proportion of extremists and the variance of opinions ina group but also determined how coherent deliberation andvoting procedures were with each other Lastly we showedthat agents that are focused judged arguments better hadmore stable opinions and constituted a group with higherproportion of extremists than their naive counterparts

In terms of governance the model says that there is per-haps no trade-off between extremismand judgment accuracyInstead it asserts that the higher the variance of opinions thecloser the group gets to the correct decisions This situationmay be interpreted as Landermore and Pagersquos notion of ldquoposi-tive dissensusrdquo (in complex collective decision-making tasks)[36] as opposed to consensus as a normative requirementfor correct collective decision-making If a decider has toorganize deliberation to legitimize a public policy dependingon what kind of world he or she wants different set ofparameters may be chosen to account for different levels oflegitimacy and correctness For instance one may ask for adecision to be legitimate to respect a certain level of coherenceor of judgment accuracy and to be discussed by a sundryof different-opinionated peoples If so deliberation has tobe made more often more participants have to be includedin debates or in deliberation instances and not too manytime steps should be introduced in between two deliberationinstances So the model may capture normative ideas todefine correct and legitimate collective decisions and tomake recommendations on those grounds Requirements ofhigh consensual deliberative accuracy (judgment accuracy)legitimacy (coherence) and diversity (variance of opinions)are all taken into account and for such a reason the model israther successful in describing a sound process of collectivedecision-making with interpretable outcomes

61 Extensions of the Model and Other Ideas Our model canbe extended in many ways Value-based abstract argumen-tation frameworks (VAFs) for example as an extension ofDungrsquos argumentation framework provide a formal descrip-tion of the process of decision-making in which argumentsare given values and audiences (set of agents) ignore attacksbetween arguments on the basis of their preferences over val-ues Social abstract argumentation frameworks (SAFs) alsoprovide an interesting and seemingly convenient formalismfor our model In SAFs agents vote on the acceptability ofarguments and the resulting labeling is a combination of thearguments that are voted for and labeling-based semantics Inour model we apply the same rule yet only for one argumentand given precise agent-based votingmechanisms and votingrules which are eventually parametrized

To consider our model as a specification of a mixed VAFand SAF model may be of great interest for future workThis formalism can provide solid foundations to describeexplain interpret and eventually extend the results foundhere from an argument-based perspective On those groundsfuture work will focus on creating a hybrid argumentationframeworkmdashwith its semanticsmdashthat takes heed of bothvalues or principles and voting

Complexity 27

Concerning other possible extensions of the model anatural reflex would be to argue on the validity and therobustness of the mix between the bounded confidenceopinion diffusion models and deliberation Consequentiallyadding deliberation to different bounded confidence modelsof opinion diffusion such as the Hegelsmann-Krause model[37] or to models of social impact theory like in [4] canbe of great interest for future work We acknowledge that afiner understanding of argumentation frameworks randomlattices and their respective applications for this modelare necessary Testing for different labeling-based semanticsor even trading labeling-based semantics for scoring-basedsemantics (semantics that defines acceptability on scoresgiven to arguments) such as debate semantics and ranking-based semantics (semantics that yield a preorder) may beinteresting to consider in future work

More into modeling deliberation it is not inadequateto believe that agents may also advance proposals duringdeliberation and replace to some extent the proposal forwhich the deliberation is taking place Endogenizing theprocess that produces the type of argument to be deliberatedon may be interesting to explore either by having agentsstrategically replace proposals at certain moments duringthe debate or having the central authority attempt to chooseproposals such that the probability of having them acceptedis high Agents may also be thought to be autoorganizing(independent of a central authority) and concede on the sizeof the instances of deliberation To consider differences insemantics across agents or even endogenizing the semanticson how urgent or important the proposal that is beingdiscussed is can also be an interesting direction to take

Last but not least many nontrivial modifications of ourmodel are possible and most should include better-thoughtdeliberation protocols It may be interesting to design andobserve protocols in which deliberation only affects agentsthat are actually debating for example Trust network effectsmultidimensionality of opinions new processes of argumentexchange or learning among agents are notions to furtherdevelop in order to make the model more realistic and relaxsome unreliable assumptions In sum the model is to berefined and extended with the objective of either studyingconcrete cases of deliberative polling and opinion dynamicsor implementing more intuitive thought experiments

Appendix

We present below some of the pseudocode we used toimplement the model

Algorithms

(a) Algorithm 1 implements the dyadic influence in themodel It describes how two agents interact andupdate their opinions

(b) Algorithm 2 implements the deliberative influence inthe model which describes how agents update theiropinions in sight of the accepted deliberated proposalarguments

(c) Algorithm 3 is used to obtain the grounded labelingof an argumentation framework

(d) Algorithm 4 provides an overview of a run in thesimulations119872 is the number of proposal argumentsto discuss which we set to 100 in the model

Each object in the system is indexed by a119908ℎ119900 number Hencea set of agents is annotated by a set of natural numbers Weuse this fact in the implementation of the algorithm 119874119887119895119890119888119905119894denotes the object of 119908ℎ119900 number 119894 Agentsrsquo opinions (119900) arecalibrated so that 119900 isin [minus1 1]Data Availability

The program and data used to support the findings of thisstudy are available from the corresponding author uponrequest

Disclosure

A preliminary version of this work was presented anddiscussed at the conference Bridging the Gap between FormalArgumentation and Actual Human Reasoning on October 4-5 2018 at BochumGermanyThe authors thank the audiencefor their valuable comments and suggestions

Conflicts of Interest

The authors declare that they have no conflicts of interestregarding the publication of this article

Acknowledgments

The contribution of Gabriella Pigozzi was supported by theDeutsche Forschungsgemeinschaft (DFG) and the CzechScience Foundation (GACR) as part of the joint project FromShared Evidence to Group Attitudes (RO 45486-1)

Endnotes

1 In other words the superiority of the ldquoright answerrdquo orlabeling in our case will appear as such to all

2 All agents hold the same opinion3 By sensitivity we mean how much an agent is ready to

change her opinion on the sole fact that an informationalcue has been debated on

4 The assumption is strong but this trust among agentsmay be rooted from a genuine individual motivation toreach a democratic consensus

5 We adopted a rule of thumbs that says that if an agentis neutral with respect to the principle then she willvote for any proposal that is presented to her duringdeliberation

6 If the argument is not unique she chooses one atrandom

7 If there are more than one candidate the agent will favorthe argument at minimal distance from 119868

28 Complexity

Require Vector of values for parameters (119881 = (120583 119880 119879))Ensure Successful interactions among agents119896 larr997888 0119873119890 larr997888 0

while 119896 le |119873|2 do ⊳Fill119873119890 with |119873|2 agents119896 larr997888 119896 + 1119873119890 larr997888 119877119886119899119889119900119898(|119873| + 1) ⊳Pseudo-random generator Generates integer between 0 and |119873|119873119903 larr997888 0 120575119894119895 larr997888 0 ⊳Variable to store differences in opinionfor all 119894 isin 119873119890 doRandomly pick a number 119895 isin 119873(119873119890 cup 119873119903)119873119903 larr997888 119873119903 cup 119895 120575119894119895 larr997888 |119900119894 minus 119900119895|procedure dyadic-update(119894 119895) ⊳Two agents 119894 and 119895 discuss and update opinions1199001015840 larr997888 119900119894 (120583119895 119880119895 119879119895) larr997888 (120583119894 119880119894 119879119894) larr997888 (120583 119880 119879) ⊳Save agent irsquos opinion before updating and homogenize agents

if 120575119894119895 lt 119880119894 then119900119894 larr997888 120583119894119900119895 + (1 minus 120583119894)119900119894if 120575119894119895 gt 119879119894 then119900119894 larr997888 (1 + 120583119894)119900119894 minus 120583119894119900119895if 120575119894119895 lt 119880119895 then119900119895 larr997888 1205831198951199001015840 + (1 minus 120583119895)119900119895if 120575119894119895 gt 119879119895 then119900119895 larr997888 (1 + 120583119895)119900119895 minus 1205831198951199001015840

Algorithm 1 Dyadic social influence

Require Vector of values for parameters (119881 = (120574 119901119886 119901119903))Ensure Successful interaction between proposal arguments and agents

for all 119894 isin 119873 doprocedure Probability-Update(V119886 119901119886 119901119903) ⊳Update individual probabilities of change due to deliberated cues119909119894 larr997888 119901|V119886minus119900119894 |2119886 119910119894 larr997888 1199012|V119886minus119900119894 |119903119902 larr997888 0if L119886119887(119868) = 119894119899 then

procedure deliberative-update(119894 119886) ⊳Agent i argument 119886120574119894 larr997888 120574 119902 larr997888 119906 sim U(0 1) ⊳ 119906 is a realization of a uniform random variableif 119902 lt 119909119894 then119900119894 larr997888 120574119894119900119894 + (1 minus 120574119894)V119886break

if 119902 lt 119910119894(1 minus 119909119894) then119900119894 larr997888 (1 + 120574119894)119900119894 minus 120574119894V119886Algorithm 2 Deliberation influence

8 In terms of computational complexity naive agentsmaketheirmoves in119874(|A119894| log(|A119894|)) while focused agents doat least as bad

9 Tables have memory Therefore debates are not neces-sarily trees

10 If there are less than (119899119863 times |119873|)2 agents for any of bothgroups the CA summons all of them to the deliberationtable

11 Or careful consideration of the viewpoints of otherswhich implies that citizens keep an open mind and donot reject arguments outright

12 A ldquohealthyrdquo consensus refers to the case in which manyconsensual decisions are taken but opinion-wise agentsdo not agree with each other Quotes are used on theword healthy because some political science theorists

(eg [36]) believe that in certain situations agreement inopinions for deliberation is epistemologically damaging

13 Simulations on groups of |119873| = 800 and |119873| = 400agents shows that scaling the population has an effect onall themetrics but it is confounding depending onwhichcouples of attitudes structures are considered (119880 119879) Westick to |119873| = 400 as in [9] for comparability andsimulation running time

14 By balanced wemean with as many arguments with V119886 lt0 as with V119886 gt 0

15 Experiments on the system suggest that a bigger(smaller) argument pool 119872 = 1000 (119872 = 200) isassociated with a higher (lower) variance of opinions(and extremism) and higher (lower) judgment accuracyFor the rest of the observations 119872 seem to have noeffect We choose a middle value for 119872 Similarly theargument pool chosen to be balanced or uniformly

Complexity 29

Require Argumentation framework 119860119865 = (AR)Ensure Assignment of labels to arguments from the grounded labeling of 119860119865L(A)

L0(A) larr997888 (0 0 0)119896 larr997888 0do119894119899 L119896+1(A) larr997888 119894119899 L119896(A) cup 119886 | 119886 is not labeled 119894119899 L119896(A) 119886119899119889 forall119887 isin A 119904119905 119887 R 119886 then 119887 isin 119900119906119905 L119896(A)119900119906119905 L119896+1(A) larr997888 119900119906119905 L119896(A) cup 119886 | 119886 is not labeled 119894119899 L119896(A) 119886119899119889 exist119887 isin A 119904119905 119887 R 119886 and 119887 isin 119894119899 L119896+1(A)

while L119896+1(A) = L119896(A) = 0Algorithm 3 Solve argumentation frameworks (with grounded semantics)

Require Vector of values for parameters (119881)Ensure Vector of statistics (119880)

procedure Set-parameters(119881) ⊳Initialize parameters for scenarioprocedure Init-objs(119899119898 119896 ) ⊳Create119898 arguments 119899 agents and 119860119865120576procedure Solve-Argumentation-Framework(119860119865) ⊳Assigns the epistemic labelingL120576 toA119894 larr997888 1119905 larr997888 1while 119894 le 119872 do ⊳ 119872 the number of arguments to discuss119868119894 larr997888 119866119890119899 119886119903119892(1) ⊳Generates proposal argument 119868119860 119894119905 larr997888 (119868119894 0)L119886119887(119868119894) larr997888 119906119899119889 ⊳Initialize label for 119868119895 larr997888 1 ⊳Initialize debate counter 119895while ((L119886119887(119868) = 119906119899119889 or 119898 le 119895) and 119895 le 119898) doif 119905 equiv 0(119898119900119889 119905119863) then

procedure build-argumentation-framework(119899119863 119887119890ℎ119886V119894119900119903) ⊳Build the argument graph agents deliberate(119861119894119905 119878119894119905) larr997888 (119883 119884) sub (⋃119897isin119873119899119863 A119897119894119905 ⋃119897isin119873119899119863 R119897119894119905) ⊳ 119873119899119863 sub 119873 are agents sampled by the CA119860 119894119905 larr997888 119889(119860 119894119905 (119861119894119905 119878119894119905))procedure Solve-Argumentation-Framework(A119889) ⊳FindL(A119889)L119886119887(119868119894) larr997888 L119886119887119863(119868119894)119895 larr997888 119895 + 1

else119896 larr997888 1while 119896 le 119905119863 doprocedure dyadic-social-influence(120583 119879 119880) ⊳Agents discuss one-to-one119896 larr997888 119896 + 1119905 larr997888 119905 + 1

if 120572 gt 0 andL119886119887(119868119894) = 119900119906119905 then ⊳Agents vote for the proposalif (|119897 isin 119873|119900119897 times V119868119894 ge 0| ge 120572 times 119899) then

L119886119887(119868119894) larr997888 119894119899else

L119886119887(119868119894) larr997888 119900119906119905if L119886119887(119868119894) = 119894119899 thenprocedure Deliberation-influence(120574 119901119886 119901119903) ⊳Agents are influenced by the result of the decision process119894 larr997888 119894 + 1119905 larr997888 119905 + 1119880 larr997888 119862119900119898119901119906119905119890 119878119905119886119905119894119904119905119894119888119904(119881)

return 119880 ⊳Get the statisticsAlgorithm 4 Run for one scenario

generated does not seem to play a significant explanatoryrole on any of the metrics

16 Experiments show that themaximumnumber of debatesis not an important parameter even if it is an intuitiveone when considering decision-making processes Inpractice debates end after at most 119898 + 1 deliberation

steps and even then it happens rarely However to avoidunpleasant surprises like infinite debates on a completeargument graph containing the proposal argument weadded the constraint 119898 Its value is precisely 7 becausethe biggest minimum number of debates we studyis 6

30 Complexity

17 To ensure comparability with the deliberative andmixedopinion dynamics models on judgment accuracy andcoherence agents vote on proposal arguments every 119905119863times(119898+1) 119905119863 and119898does not explain variability in judgmentaccuracy nor in coherence

18 Student t-tests for independent samples are robust tothe violation of two hypothesis on the distribution ofthe samples normality and homogeneity of variancesas long as the sample size is big enough (central limittheorem) and if the difference in size of the comparedsamples is smallrespectively

19 Studentrsquos test version forwhen variances are not assumedequal and the difference in size of the compared samplesis big

20 We do not correct for homoskedasticity nor normality oferrors in the estimates

21 Balanced as the same number of runs per scenario orsuch that no significant correlations are found betweenparameters

References

[1] R Axelrod ldquoThe dissemination of culture a model withlocal convergence and global polarizationrdquo Journal of ConflictResolution vol 41 no 2 pp 203ndash226 1997

[2] D M Rousseau Reinforcing the MicroMacro Bridge Organiza-tionalThinking and Pluralistic Vehicles SAGEPublications SageLos Angeles Calif USA 2011

[3] S Galam and SMoscovici ldquoTowards a theory of collective phe-nomena consensus and attitude changes in groupsrdquo EuropeanJournal of Social Psychology vol 21 no 1 pp 49ndash74 1991

[4] A Nowak J Szamrej and B Latane ldquoFrom private attitude topublic opinion a dynamic theory of social impactrdquo Psychologi-cal Review vol 97 no 3 pp 362ndash376 1990

[5] G Deffuant S Huet and F Amblard ldquoAn individual-basedmodel of innovation diffusion mixing social value and individ-ual benefitrdquo American Journal of Sociology vol 110 no 4 pp1041ndash1069 2005

[6] G Deffuant D Neau F Amblard and G Weisbuch ldquoMixingbeliefs among interacting agentsrdquoAdvances in Complex Systems(ACS) vol 3 no 01n04 pp 87ndash98 2000

[7] J Rouchier P Tubaro and C Emery ldquoOpinion transmission inorganizations an agent-based modeling approachrdquo Computa-tional and Mathematical OrganizationTheory vol 20 no 3 pp252ndash277 2014

[8] M Meadows and D Cliff ldquoThe relative disagreement model ofopinion dynamics where do extremists come fromrdquo in Pro-ceedings of the International Workshop on Self-Organizing Sys-tems Lecture Notes in Computer Science pp 66ndash77 Springer2013

[9] W Jager and F Amblard ldquoUniformity bipolarization and plu-riformity captured as generic stylized behavior with an agent-based simulation model of attitude changerdquo Computational ampMathematical Organization Theory vol 10 no 4 pp 295ndash3032005

[10] Y-F Zhang H-Y Duan and Z-L Geng ldquoEvolutionary mech-anism of frangibility in social consensus system based onnegative emotions spreadrdquo Complexity vol 2017 Article ID4037049 8 pages 2017

[11] A Flache andMWMacy ldquoSmall worlds and cultural polariza-tionrdquoThe Journal of Mathematical Sociology vol 35 no 1-3 pp146ndash176 2011

[12] A Tsang and K Larson ldquoOpinion dynamics of skepticalagentsrdquo in Proceedings of the 13th International Conference onAutonomous Agents and Multiagent Systems AAMAS 2014 pp277ndash284 International Foundation forAutonomousAgents andMultiagent Systems France May 2014

[13] K M Hansen Deliberative democracy and opinion formationUniversity Press of Southern Denmark Odense 2004

[14] C R Sunstein ldquoThe law of group polarizationrdquo Journal ofPolitical Philosophy vol 10 no 2 pp 175ndash195 2002 WileyOnline Library

[15] S Moscovici and W Doise Dissensions et Consensus unetheorie generale des decisions collectives Presses Universitairesde France-PUF 1992

[16] J S Fishkin and R C Luskin ldquoExperimenting with a demo-cratic ideal deliberative polling and public opinionrdquo ActaPolitica vol 40 no 3 pp 284ndash298 2005

[17] R C Luskin J S Fishkin and S Iyengar Considered Opinionson US Foreign Policy Face-To-Face versus Online Delibera-tive Polling International Communication Association NewOrleans La USA 2004

[18] J Barabas ldquoHowdeliberation affects policy opinionsrdquoAmericanPolitical Science Review vol 98 no 4 pp 687ndash701 2004

[19] O Pryymak A Rogers and N R Jennings ldquoEfficient opinionsharing in large decentralised teamsrdquo in Proceedings of the 11thInternational Conference on Autonomous Agents andMultiagentSystems vol 1 pp 543ndash550 Spain June 2012

[20] J Rouchier and E Tanimura ldquoWhen overconfident agents slowdown collective learningrdquo Simulation vol 88 no 1 pp 33ndash492012

[21] D Miller ldquoDeliberative democracy and social choicerdquo PoliticalStudies vol 40 S1 pp 54ndash67 1992

[22] M Sherif and C I Hovland Social Judgment Assimilation andContrast Effects in Communication and Attitude Change YaleUniversity Press 1961

[23] P M Dung ldquoOn the acceptability of arguments and its funda-mental role in nonmonotonic reasoning logic programmingand n-person gamesrdquo Artificial Intelligence vol 77 no 2 pp321ndash357 1995

[24] M Caminada and G Pigozzi ldquoOn judgment aggregation inabstract argumentationrdquo Autonomous Agents and Multi-AgentSystems vol 22 no 1 pp 64ndash102 2011

[25] P Baroni M Caminada andM Giacomin ldquoAn introduction toargumentation semanticsrdquoThe Knowledge Engineering Reviewvol 26 no 4 pp 365ndash410 2011

[26] S Chambers ldquoDeliberative democratic theoryrdquo Annual Reviewof Political Science vol 6 no 1 pp 307ndash326 2003

[27] E Bonzon andNMaudet ldquoOn the outcomes of multiparty per-suasionrdquo in Proceedings of the 10th International Conference onAutonomous Agents and Multiagent Systems 2011 AAMAS 2011vol 1 pp 47ndash54 International Foundation for AutonomousAgents and Multiagent Systems Taiwan May 2011

[28] J Ranciere Disagreement Politics and Philosophy University ofMinnesota Press 1999

[29] T Bench-Capon and K Atkinson ldquoAbstract argumentation andvaluesrdquo in Argumentation in Artificial Intelligence pp 45ndash64Springer 2009

[30] M Caminada ldquoOn the issue of reinstatement in argumenta-tionrdquo in Proceedings of the European Workshop on Logics in

Complexity 31

Artificial Intelligence vol 4160 of Lecture Notes in ComputerScience pp 111ndash123 Springer Berlin Heidelberg

[31] GAVreeswijk ldquoAn algorithm to computeminimally groundedand admissible defence sets in argument systemsrdquo in Pro-ceedings of the 2006 conference on Computational Models ofArgument Proceedings of COMMA 2006 vol 144 pp 109ndash120IOS Press Amsterdam The Netherlands 2006

[32] C W Sherif M Kelly H L Rodgers et al ldquoPersonal involve-ment social judgment and actionrdquo Journal of Personality andSocial Psychology vol 27 no 3 pp 311ndash328 1973

[33] G Deffuant F Amblard G Weisbuch and T Faure ldquoHow canextremism prevail A study based on the relative agreementinteraction modelrdquo Journal of Artificial Societies and SocialSimulation vol 5 no 4 p 1 2002

[34] E Awad M W Caminada G Pigozzi M Podlaszewski andI Rahwan ldquoPareto optimality and strategy-proofness in groupargument evaluationrdquo Journal of Logic and Computation vol 27no 8 pp 2581ndash2609 2017

[35] D Grossi and G Pigozzi ldquoJudgment aggregation a primerrdquoSynthesis Lectures on Artificial Intelligence and Machine Learn-ing vol 8 no 2 pp 1ndash151 2014

[36] H Landemore and S E Page ldquoDeliberation and disagreementproblem solving prediction and positive dissensusrdquo PoliticsPhilosophy and Economics vol 14 no 3 pp 229ndash254 2015

[37] R Hegselmann and U Krause ldquoOpinion dynamics andbounded confidence models analysis and simulationrdquo Journalof Artificial Societies and Social Simulation vol 5 no 3 2002

[38] A Stefanelli and R Seidl ldquoModerate and polarized opinionsUsing empirical data for an agent-based simulationrdquo inProceed-ings of the Social Simulation Conference 2014

[39] S Gabbriellini and P Torroni ldquoA new framework for ABMsbased on argumentative reasoningrdquo in Advances in SocialSimulation vol 229 pp 25ndash36 Springer Berlin Germany 2014

[40] H Mercier and D Sperber ldquoWhy do humans reason Argu-ments for an argumentative theoryrdquo Behavioral and BrainSciences vol 34 no 2 pp 57ndash74 2011

[41] H V Parunak T C Belding R Hilscher and S BruecknerldquoModeling and managing collective cognitive convergencerdquoin In Proceedings of the 7th iNternational Joint Conference onAutonomous Agents and Multiagent Systems vol 3 pp 1505ndash1508 International Foundation for Autonomous Agents andMultiagent Systems 2008

[42] H Prakken ldquoFormal systems for persuasion dialoguerdquo TheKnowledge Engineering Review vol 21 no 2 pp 163ndash188 2006

[43] K Atkinson T Bench-Capon and P Mcburney ldquoA dialoguegame protocol for multi-agent argument over proposals foractionrdquoAutonomousAgents andMulti-Agent Systems vol 11 no2 pp 153ndash171 2005

[44] M Thimm and A J Garcıa ldquoClassification and strategicalissues of argumentation games on structured argumentationframeworksrdquo in Proceedings of the 9th International Conferenceon Autonomous Agents and Multiagent Systems vol 1 pp 1247ndash1254 International Foundation for Autonomous Agents andMultiagent Systems Canada 2010

[45] I Rahwan and K Larson ldquoMechanism design for abstractargumentationrdquo in Proceedings of the 7th International JointConference on Autonomous Agents and Multiagent SystemsAAMAS 2008 pp 1031ndash1038 International Foundation forAutonomous Agents and Multiagent Systems Portugal May2008

[46] H Prakken R Riveret A Rotolo and G Sartor ldquoHeuristics inArgumentation A Game-Theoretical Investigationrdquo EuropeanUniversity Institute 2008

[47] N Hadidi Y Dimopoulos and P Moraitis ldquoTactics and con-cessions for argumentation-based negotiationrdquo in Proceedingsof the COMMA vol 245 pp 285ndash296 2012

[48] N Oren and T J Norman ldquoArguing using opponent modelsrdquoin Proceedings of the International Workshop on Argumentationin Multi-Agent Systems pp 160ndash174 Springer 2009

[49] S Delre W Jager T Bijmolt and M Janssen ldquoTargeting andtiming promotional activities An agent-based model for thetakeoff of new productsrdquo Journal of Business Research vol 60no 8 pp 826ndash835 2007

[50] A Mirtabatabaei P Jia and F Bullo ldquoEulerian opinion dynam-ics with bounded confidence and exogenous inputsrdquo SIAMJournal on Applied Dynamical Systems vol 13 no 1 pp 425ndash446 2014

[51] V Barbet N Guiraud V Laperriere and J Rouchier Hagglingon Values Towards Consensus or Trouble 2019

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Submit your manuscripts atwwwhindawicom

Complexity 9

Fix proceduralsm m α nD tD σ Init count m larr 0

Given proposal anargument I is randomly generated

nD times |N | agents arerandomly picked

m larr m + 1

Begin decision processon proposal at time t

t larr t + 1Deliberation step m

larr 0

t larr t + 1dyadic opinionupdate Eq (5)tm larr tm + 1

m ge m

tm gt tD

ℒd (I) = in

m ge m

yields ℒ(I)

ℒ(I) = in Deliberationupdate Eq (6)

End decision

yes

no

no

no

yes

yes

no

yes

yes

yes

no

no

I

Vote on ℒd(I)

ℒd (I) = und

ＪＬＩ＝ＭＭ I

Init count tmyields ℒd(I)

Figure 5 Process chart for a decision processD119868 on a proposalP 119905 corresponds to time and 119898 (119905119898) corresponds to the number of debate(non-debate) steps that have taken place in the deliberation process (in between two debate steps) over time for the mixed discussion modelWe useL119886119887(119868) andL(119868) interchangeably in the figure for presentation

comparing simulations on the basis of the different decision-making processes translates into comparing these collectivedecision steps

242 Deliberation Protocol To define the deliberation proto-col held in the table either as a consequence of the definitionprocess or as a statement we assume the following

(i) Agents may decide not to contribute to the debate(ii) All agents have the same probability conditional to

their opinions of being picked to participate in adebate or deliberation step

(iii) There is no restriction on the number of times eachagent can participate in a deliberation process

(iv) Each agent may only place one argument per debatestep

(v) Arguments that have already been advanced in thedebate may attack newly placed arguments9

The deliberation or debate protocol goes as follows

(1) The CA randomly generates and makes public acentral argument or proposal argument 119868 and informsall agents about the rules of the decision-makingprocess

(2) The CA randomly draws two sets of (119899119863 times |119873|)2agents with divergent views10 onP and P It mergesthem to create the set of debaters1198731015840

(3) Each agent 119894 advances an argument from her sackA119894 The CA makes sure that there are no repeatingarguments (agents already take argument repetitioninto account)

(4) The CA establishes the debate step 119889rsquos argumentationframework and computes its labelingL120590

119889(A119889) using120590(5) If the computed label for 119868 is L119886119887119889(119868) = 119906119899119889

or the number of debates steps is inferior or equalto 119898 at time 119905 then the CA stops the debate andresumes it at the (119905 + 119905119863 + 1)rsquoth time step by repeating(1) (2) (3) (4) and (5)

(6) Let L119886119887(119868) be the final label given to the proposalargument 119868 If voting is allowed then if more than120572 times |119873| agents agree with 119868 119868 is accepted (L119886119887(119868) =119894119899) it is rejected if strictly less than 120572times |119873| agree withit (L119886119887(119868) = 119900119906119905) When there is a tie ifL119886119887119889(119868) =119906119899119889 then L119886119887(119868) = 119900119906119905 if L119886119887119889(119868) = 119894119899 thenL119886119887(119868) = 119894119899 When voting is not part of the decision-making process (120572 = 0)L119886119887119889(119868) = L119886119887(119868)

One important point to notice about this protocol is that itinduces a stop rule 119904119903 (cf steps (5) and (6) of protocol) forthe decision process and thus it always ends Either agentsdebate and agree on the proposalrsquos acceptability throughargumentation or after 119898 debate steps they directly voteon it so that a decision concerning the proposal is alwaysreached Refer to Figure 5 for a sweeping description of thedeliberation protocol and the decision-making process

10 Complexity

-1

-1 1

1

Latitude of acceptance

Latitude of rejection Agent i

Agent j

Figure 6 A dyadic interaction between agents 119894 and 119895 as expressedin (5) Agent 119894rsquos opinion falls within agent 119895rsquos latitude of noncommit-ment and conversely Neither agent affects the opinion of the other

25 Opinion Dynamics for Social Interactions In this subsec-tion we describe the opinion diffusion model based on pair-wise interactions among agents and deliberation Sherif rsquosand Hovlandrsquos social judgment theory [22] motivates part ofour approach It describes how individualsrsquo opinions changeon the basis of their attitude structures Attitude structuresrefer to the relative scope width or latitude of categoriesused by individuals when evaluating information namelythe latitudes of acceptance rejection and noncommitment[32] The idea behind this theory is that individuals changetheir positions only in accordance to how far or close thecommunicative cues they receive are from (to) their anchorpositions It holds that if communicative cues are far (close)from (to) an agentrsquos position say over her latitude of rejection(acceptance) then the agent shifts her position away from(towards) the position defended by the cues In the casewherethe cues fall within the agentrsquos latitude of noncommitmenther position does not change (see Figure 6 and Equation (5))

251 Pair-Wise or Dyadic Opinion Dynamics As agentsmay communicate and deliberate collectively they may alsoengage in one-to-one conversations with other agents toponder on their positions We loosely associate this type ofcommunication with dyadic nonargumentative exchange ordiscussion based on fallacious arguments and persuasionIn the light of social judgment theory and the descriptionof agents in the system we model pair-wise symmetricinteractions following the opinion dynamicsmodel in [9] Anagent 119895rsquos influence on an agent 119894rsquos opinion at time 119905 is governedby the ensuing difference equation

119900119905+1119894 =

119900119905119894 + 120583119894 (119900119905119895 minus 119900119905119894) 119894119891 10038161003816100381610038161003816119900119905119894 minus 11990011990511989510038161003816100381610038161003816 lt 119880119894119900119905119894 minus 120583119894 (119900119905119895 minus 119900119905119894) 119894119891 1003816100381610038161003816119900119905119894 minus 119900119905119894 1003816100381610038161003816 gt 119879119894119900119905119894 119900119905ℎ119890119903119908119894119904119890(5)

where the parameter 120583119894 isin [0 05] controls for the strengthof the attraction and repulsion in social influence and 119879119894and 119880119894 are the latitudes of rejection and acceptance for agent119894 respectively The parameter 120583119894 may be thought of as therelative importance agent 119894 gives to the opinions of her peersor analogously the weight she gives to her own opinionwhen updating 120583 lt 05 means that agents will never givemore weight to the opinions of other agents than to their

own (egocentric bias) When two agents 119894 and 119895 discuss iffor instance 119895 advances an informational cue (argumentpersuasion tactic) and it happens to be close enough to119894rsquos opinion anchor then 119894 shifts her opinion towards thedirection of 119895rsquos informational cue The symmetric influencefrom agent 119894 to agent 119895 takes effect in the same way (seeFigure 6)

We add to the dyadic dynamics a rule of encounters ateach step of pair-wise interactions each agent meets exactlyone other agent at random This rule may be associated withrandom day-to-day encounters among agents Steps of socialinfluence correspond to nondebate steps (119889) in decision-making processes

252 Deliberative Opinion Dynamics We define an opinionupdate equation that links the proposal arguments that areadvanced during deliberation and opinions We combine theuncertain and probabilistic nature of the effect of delibera-tion have it a moderating [15 16] or polarizing [14] one witha mechanism similar to the one in social judgment theorybased on the distance between different informational cuesand opinion anchors [32] The probabilistic modeling of theopinion update can be related to deliberation encouragingopen-mindedness11 during collective discussion [18] Fromthis choice it follows that deliberated cues can affect even themost extreme of agents as opposed to some classic opiniondiffusion models (eg [5 8]) where once agents becomeextremists they may no longer become moderate

LetP be a proposal V119905119868 the proposal argument 119868rsquos level ofsupport for a principle P and 119900119905119894 an agent 119894rsquos opinion at time119905 Then given the difference 120575119905119894 = (V119905119868 minus 119900119905119894)2 we define 119894rsquosprobability of being attracted to a decisionrsquos informational cue119868 by 119909119905119894 = 119901|120575119905119894 |119886 where 119901119886 denotes a general probability param-eter that characterizes how important deliberated results arefor the group The parameter may also be interpreted asthe grouprsquos tendency to be swayed by a decisional majoritySimilarly we define 119894rsquos probability of being repulsed from adecision informational cue 119868 by 119910119905119894 = 1199011|120575119905119894 |119903 where 119901119903 denotesa general probability parameter that characterizes the grouprsquosdislike of deliberated results 119901119903 can also be thought of as thegrouprsquos distrust of the system symbolized by deliberation anddemocracy

Let L119886119887(119868) denote the epistemic status of the proposalargument 119868 at the end of a decision process over P everyagent 119894 updates her opinion as follows

(i) IfL119886119887(119868) = 119894119899

119900119905+1119894 =

119900119905119894 + 2120574119894120575119905119894 119908119894119905ℎ 119901119903119900119887119886119887119894119897119894119905119910 119909119905119894119900119905119894 minus 2120574119894120575t119894 119908119894119905ℎ 119901119903119900119887119886119887119894119897119894119905119910 119910119905119894119900119905119894 119908119894119905ℎ 119901119903119900119887119886119887119894119897119894119905119910 1 minus 119909119905119894 minus 119910119905119894

(6)

(ii) IfL119886119887(119868) = 119900119906119905 119900119905+1119894 = 119900119905119894where 120574119894 isin [0 05] is the strength of repulsion and attractionin the dynamicThemeaning of 120574119894 is analogous to the one of120583119894in the dyadic interactionsmodel but for the observation that 119894

Complexity 11

weights her opinion to an argument rather than to an opinionThe interpretation of this dynamics is straightforward Ifthe deliberated proposal argument is close to an agentrsquosopinion then it is very likely that the agent shifts its opiniontowards it Please note that the probability 119909119905119894 (119910119905119894 ) that anagent is attracted (repelled) to (from) an accepted deliberatedproposal argument is bounded below (above) by 119901119886 (119901119903)(equivalently 119909119905119894 isin [119901119886 1] 119910119905119894 isin [0 119901119903]) Steps of deliberativeopinion dynamics are associated with debate steps in which aconcluding collective decision is made

253 Mixed Opinion Dynamics Opinion dynamics in thecontext of decision-making processes can be best described asa combination of the two preceding dynamics In this opiniondynamics model agents are engaged in a democratic systemin which they deliberate vote for proposals and occasionallydiscuss with one another on the principle supported by theproposalsThe opinion dynamics for an agent 119894 can be writtenas a sequence of dyadic and deliberative opinion updates

119900119905+1119894 =

119864119902119906119886119905119894119900119899 (5) 119894119891 119905 equiv 0 (119898119900119889119905119863) and 119904119903 (sdot) = 119891119886119897119904119890119864119902119906119886119905119894119900119899 (6) 119894119891 119905 equiv 0 (119898119900119889119905119863) and 119904119903 (sdot) = 119905119903119906119890119900119905119894 119900119905ℎ119890119903119908119894119904119890

(7)

where 119904119903(sdot) is the decision processrsquos stop rule 119905119863 the numberof time steps between each deliberation step Equation (5)describes the dynamics for the dyadic interactions and Equa-tion (6) posits the changes in opinions due to the deliberatedcues Equation (5) applies when agents are not deliberatingand when each agent encounters another agent to exchangeinformation that may lead to local opinion updates Whenthere is deliberation a handful of agents deliberate and ifthey reach the stop condition imposed by the table all agentsvote for the proposal They update (see Equation (6)) theiropinions on the basis of the proposal argumentrsquos support forthe principle and the result of the scrutiny Otherwise thereis no change in the opinions of the agents A vote indicatesthe end of the decision-making process

3 Experiments and Calibration

In this section we introduce the metrics that enable us toobserve the simulations and characterize the calibration ofthe model

31 Observations and Initialization In this section wedescribe the simulations and the protocol used to test andobserve the results of our model We introduce the metricsof interest explain the calibration of the model describesome results obtained from the simulation data used forcalibration and conclude with a brief discussion on theexpected outcomes of the mixed opinion dynamics model

311 Metrics or Statistics of Interest Let 119878 denote the end ofa simulation We are interested in the effect of deliberationand of the modelrsquos procedural parameters on the followingmetrics or statistics

(i) Variance of opinions (119881119886119903(119900)) the variance ofopinions at time 119878 Since opinions live in the unionof the positive and negative unit intervals 119881119886119903(119900) isin[0 1] The higher the variance of the distribution isthe more ldquodiverserdquo the opinions are(ii) Proportion of extremists in the population(119901119903119900119901119890119909) the proportion of agents in the populationwith opinions such that |119900119894| ge 075 A high proportionof extremists makes ldquohealthyrdquo consensus12 difficult toreach and deliberation more or less informative(iii) Shifts of opinions (119878ℎ) [33] statistic that measuresthe aggregated change in individual opinion at time 119878with respect to aggregated individual opinion at thebeginning of the simulation

119878ℎ = 2sum119894isin119873 10038161003816100381610038161199000119894 minus 1199001198781198941003816100381610038161003816max119894isin1198731199000119894 minusmin119894isin1198731199000119894 (8)

119878ℎ is positive and bounded above by 2|119873| since 119900119894 isin[minus1 1] A low shift statistic implies that the processhas a small impact on opinions(iv) Judgment or consensual inaccuracy (119890119888) consen-sual accuracy of a group consists of an ad hoc statisticmeasuring a grouprsquos ability to infer correct labels forproposal arguments from a decision-making processgiven the ideal consensual labeling based on full-informationL120590

120576 (A) We use a Hamming-based dis-tance on labelings [34] to define the statistic

119890119888 = 1|I| sum119868isinI1198861198681003816100381610038161003816L119886119887120576 (119868) = L119886119887 (119868)1003816100381610038161003816 (9)

whereI is the set of all discussed proposal argumentsand 119886119868 = 12 ifL119886119887120576(119868) = 119906119899119889 or ifL119886119887(119868) = 119906119899119889119886119868 = 1 otherwise119890119888 lives in the interval [0 1] An inaccuracy statisticclose to 1 indicates that agents subdued to a particulardecision-making process make many mistakes injudging the labels of the proposal arguments Notethat when there is voting all proposal arguments arelabeled either 119894119899 or 119900119906119905 after deliberation(v) Coherence (119894119903) letL119886119887119863(119868) be the label obtainedfor 119868 from the deliberation process without votingThe coherence statistic measures how well votingresults adjust to results obtained during deliberationonly We use the proportion of arguments that havebeen labeled 119894119899 in the debate and that agents havevoted favorably for

119894119903 = 1003816100381610038161003816119868 isin I | L119886119887119863 (119868) = 119894119899 andL119886119887 (119868) = 11989411989910038161003816100381610038161003816100381610038161003816119868 isin I | L119886119887119863 (119868) = 1198941198991003816100381610038161003816 (10)

The coherence statisticrsquos domain is [0 1] If after 119872debates no deliberated central argument has beenlabeled 119894119899 or voting is not part of the procedure ofcollective decision-making then the statistic equals

12 Complexity

1 or said differently agents are perfectly coherentThis comes for the fact that if the central argumentis labeled 119900119906119905 then it is not even eligible for voting iflabeled 119906119899119889 then the debate will always be coherentwith the preferences of the agents since the result willbe a simple aggregation of their votes If the statisticequals 0 then none of the proposal arguments labeled119894119899 are voted in favorably by the agents It follows that ahigh coherence statistic implies that when agents votefor acceptable proposal arguments the results of thescrutiny reflect the consensual rationality expressedin the deliberation process

312 Parameters of Interest We recall the parameters ofinterest in our study that are linked to the structure of thedecision-making processes

(i) 119898 the minimum number of time steps in which adebate or a scrutiny occurs in a decision-making pro-cess before a final deliberated decision is submitted toa vote

(ii) 119899119863 the number of agents that deliberate as a propor-tion of the population

(iii) 119905119863 the number of time steps between debates inwhich pair-wise interactions among agents mayoccur

(iv) 120572 the proportional majority requirement for theacceptance of a proposal When 120572 = 0 it stands forno voting ldquoany proposal argument that is labeled 119894119899during deliberation is acceptedrdquo

Please recall that in terms of the definition of a decision-making process each parameter controls either the length orthe content of the sequence D119868 (119905119863 119898) or the rules that areapplied during the debate steps (120572 119899119863) (refer to Figure 4)313 Initialization All |119873| = 400 agents start off with anopinion 119900119894 drawn from a uniform distribution U(minus1 1) 13Given 119900119894 every agent 119894 randomly draws a setA119894 (|A119894| = 119896) ofarguments from a balanced14 argument pool A of119872 = 600nonneutral (V119886 = 0) arguments on the basis of 119900119894 if |119900119894| lt 025(the opinion is moderate) then agent 119894 randomly fills halfof her argument sack with arguments such that V119886 lt 0 andthe other half with arguments such that V119886 gt 0 (120601119898119894 = 05)Otherwise she randomly fills 120601119890119894 ge 05 of her sack witharguments such that V119886 and 119900119894 are of the same sign and theremaining (1 minus 120601119890119894 ) with arguments of the opposite sign15

Like for the opinion of agents every argument 119886rsquosadherence to the principle P V119886 is drawn from a uniformdistribution U(minus1 1) The attack relation that gives birth tothe ideal argumentation framework (119860119865120576) is established onthe basis of the V119886rsquos and the argumentsrsquo epistemic reach 119890119903

Let 120579(119886 119887 119903) = (|V119886 + V119887|2)119903 be an auxiliary positivereal-valued function that takes two arguments (119886 119887) and apositive real number (119903 gt 0) as inputThe probability that anyargument 119886 isin A creates a link to any other argument 119887 isin A

is given by the equation 119901 = 119901120579(1198861198872)119890 and thus argumentssupporting opposing views of the principle always attack each

other We fix 119901119890 the epistemic correlation parameter to 015If 119901119890 was greater than 015 then arguments of the same signwould attack each other too often (and focused agents wouldbe rarely incited to advance favorable arguments duringdeliberation) If 119901119890 was lower then the arguments of thesame type would induce an almost empty graph which isunrealistic The number of arguments that any argument 119886can attack is bounded by 119886rsquos epistemic reach that we fix to15 A higher value of epistemic reach makes the argumentlattice too conflicting and nearly bipartite A lower epistemicreachmakes the lattice not sufficiently conflicting in the sensethat too many arguments can attack the proposal argumentrelatively to the few that can defend it If any argument attacksthe proposal argument the chances that another argumentattacks the attacker are low Hence the proposal argumentis almost never accepted and deliberative opinion updatesbecome rare

The arguments in the resulting argumentation frame-work 119860119865120576 are given a permanent labelingL120590

120576 (A) (in shortL120576) using120590 = grounded semanticsWe choose the groundedlabeling-based semantics because it provides a unique admis-sible labeling respects minimal rationality constraints whilesimplifying the model and models a skeptic approach toaccepting arguments (see [23 25 31]) as it maximizes thecardinality of the set 119906119899119889 L120576 To some extent if we considerproposals as committing because they may guide collectiveaction choosing a skeptic semantics seems reasonable insofaras it labels an argument 119894119899 or 119900119906119905 only if it has no reasonto label it 119906119899119889 For instance if a proposal is a public policythat requires large amounts of resources and engages tofuture course of action then it may also determine policycycles and heavily burden a group and its future decisionsIn important situations like these it seems reasonable tolengthen deliberative cues and ask for more demanding andldquogroundedrdquo criteria for policy argument acceptability

On the proposalrsquos side we create an argument 119868 whoseadherence to P is also drawn from aU(minus1 1) and we label itL119886119887(119868) = 119906119899119889 119868 is interpreted as being the main argumentjustifying the discussed proposal P and V119905119868 as its supportfor P at time 119905 119868 cannot attack other arguments but otherarguments can attack it For each argument 119886 isin A a directedarc or attack from 119886 to 119868 is activated with probability 119901119901 =003120579(11988611986802) When 119901119901rsquos lower bound is too high the proposalargument is almost always defeated and thus deliberativeopinion updates do not happen When 119901119901rsquos lower boundis lower than 003 then the opposite occursmdashthe proposalargument is always accepted due to an absence of attackstowards it Finally we set the maximum number of debatesto119898 = 7 for the decision-making processes stop rules1632 Calibration and Simulation Protocol In this subsectionwe discuss the calibration of the models the terminationconditions for runs in the deliberative and mixed model andthe expected outcomes in terms of the observations

321 Calibrating Dyadic Opinion Dynamics Dyadic opiniondynamics correspond to a space of parameters in whichargumentation and deliberation spaces are not taken into

Complexity 13

Table 1 Parameters and parameter types and their domains used to compare dyadic collective and mixed opinion dynamics

Length ofD119868 Decisional in119863119868 Social influence Deliberation influence Cognitive119905119863 119898 119898 119899119863 120572 120583 119879 119880 119901119886 119901119903 120574 119896 119861119890ℎ119886V1198941199001199031 3 6 1 3 6 7 001 002 005 0 05 066 01 14 18 02 06 03 05 01 005 01 02 4 8 16 119873119886119894V119890 119865119900119888119906119904119890119889account Deliberated arguments and informational cues haveno effect on agent opinion but agents still vote for theproposal argument17 We use and calibrate this model forcomparability in terms of all our metrics since they arenot explicitly observed in [9] a reference opinion dynamicsmodel Furthermore for simplicity and comparability againwe suppose that agents are homogeneous in terms of theirattitude structures and opinion weightings We fix 120583119894 =120583 119879119894 = 119879 and 119880119894 = 119880 for all 119894 isin 119873 and for some triplet(120583 119880 119879) isin [0 05] times [0 2] times [119880 2]

Experiments suggest that the strength of the dyadic inter-actions is an explanatory factor of the time of convergence(120583 lt 05) and of the dyamicsrsquo steady states Bigger values of 120583speed up convergence towards a stable set of opinion clustersand affect the size and the relative position of these clustersin the opinion distribution 120583 however does not play a majorrole in the number of opinion clusters observed at the steadystate of the dynamics

We set 120583 to the value given to it in Jagger et alrsquos opiniondynamics model (120583 = 01) [9] For one it is the smallestreal number for which most results found in [9] hold and thedynamics are smooth secondly it seems to be a reasonableldquostrengthrdquo of influence considering that we would not wantpair-wise social interactions to completely shadow or maskan eventual effect of deliberation in the opinion dynamics

For different values of attitude structures 119879 and 119880 weget exactly the same results as in [9] in terms of the numberand density of the opinion clusters The different couplesof values of 119880 and 119879 model the size of the latitude ofnoncommitment (119879 minus 119880) and thus the agentsrsquo propensityto update opinions Whenever 119879 minus 119880 le 02 regularitiesin convergence and opinion clustering appear If 119880 gt 1there is always central convergence if 119879 lt 1 there is alwaysbipolar convergence with clusters of similar sizes For 119879 minus119880 gt 02 relatively low values of 119879 (119879 lt 1) and relativelyhigh values of 119880 always yield extreme bipolar convergencewith a relatively small cluster of moderate agents (bipolar-central convergence) For 119879 sufficiently high (119879 ge 12) and119880 sufficiently low (119880 le 08) in other words not very ego-involved agents there is a bigger spectrum of steady statesFor this reason we studied the metrics of interest for thesevalue differences and kept the couples of (119880 119879) depicted inTable 1 For these couples central bipolar bipolar-centralconvergence (3 clusters) and multicluster convergence (4 or5 clusters) are possible and result frommeaningfully differentattitude structures Judgment accuracy varies significantlyacross the four different attitude structures we retain and soit happens with the variance of opinions These scenariosprovides us with reference results from which to build upon

322 Calibrating Deliberative Social Interactions Delibera-tive social interactions correspond to social situations in

which individuals are not influenced by pair-wise discussionwith peers but are sensitive to collectively deliberated infor-mational cues On these terms parameters such as the sizeof agentsrsquo argument sacks (119896119894) the distribution of argumentsin them (120601119894) the minimum number of debates (119898) andthe proportion of debaters from the population (119899119863) are nolonger mute We assume for simplicity that the formulaexist120601119890exist119896forall119894(119896119894 = 119896 and 120601119890119894 = 120601119890 and 120601119898119894 = 05) is true with 119894 isin 119873for some value 120601119890 isin [05 1] and for 119896 isin N

The chosen domains for these parameters (see Table 1)are justified by the size of the population and the size of theargument pool (119872 = 600) which we set from the start If 119896is too big (119896 ge 32) andor too many agents are allowed inthe debate (119899119863 ge 02) then individuals always disprove thecentral argument and deliberation is never or rarely takeninto account in the update of opinions Similar effects occurwhen 119898 is high yet we only calibrate it with respect to themean running time of a simulation and the values of 119905119863Different distributions of 120601119890119894 seem to significantly or directlyexplain none of the statistics we analyze so we convenientlyset 120601119890119894 = 075 We chose the parameter 119896 to be always divisibleby 4 for it is convenient given the initial conditions regardingthe argument distribution in the agentsrsquo argument sacks

We fix the value space for the parameters 120574 119901119886 119901119903 as inTable 1 to account for different intensities of the effect ofdeliberative voting retroaction on the system For examplethe scenarios where the three parameters are at their lowestvalues correspond to the scenarios with the smallest fixedprior effect of deliberation in the model Increasing any ofthese parameters should mechanically be associated witha world in which agents are more sensitive to collectivedecisions The values for 119901119886 are taken from [13] wherethe author states that by means of deliberation 7 to28 of individuals changed their opinion from agreeingto disagreeing or vice versa on a referendum question forDenmarkrsquos participation in the EuroWe choose two differentvalues for 119901119886 that account for high (03) and very high (05)sensitivity to deliberation in a group 119901119903 on the other handis taken to be small (equal to 01) since we believe that thereexist agents that will always go against the reached consensusTheir numbers are meager

For simplicity we assume that for all 119894 isin 119873 120574119894 = 120574 forsome 120574 isin [0 05] In effect we posit that the heterogeneityof agents in the deliberative context only derives from theirarguments and their opinions We decide to calibrate 120574 inallusion to the strength of the social dynamics in pair-wiseinteractions In doing so we limit ourselves to consider threedifferent scenarios in which deliberation has respectivelyhalf equal and twice as much opinion-shifting power thanone-to-one social influence Finally we include the accept-ability voting quota rule 120572 whose domain is inspired fromclassical majority rules [35] observed empirically (Table 1)

14 Complexity

Table 2 Parameters and their domains used to study the effect of the procedural parameters on the metrics

Procedural parameters Other parameters119905119863 119898 119899119863 120572 119898 120583 119879 119880 119901119886 119901119903 120574 119896 119861119890ℎ119886V1198941199001199031 2 3 4 5 6 1 2 3 4 5 001 002 003 004 005 0 05 066 7 01 16 02 03 005 02 12 119873119886119894V119890 119865119900119888119906119904119890119889

323 Calibrating the Mixed Social Interactions Model Themixed interaction model ascribes to a parameter space inwhich the effect of collective choices on our metrics isnontrivial and where deliberation and voting on proposalsdetermine their acceptability Since the mixed model isequivalent to periodic iterations of pair-wise and collectivesocial interactions the calibration of the parameters in bothpreceding models holds (see Table 1 for the calibration)The first reason for this is comparability the second is thatthe previous calibration takes into account the fact thatboth dynamics are going to be combined We include thefrequency of debates by adding the parameter 119905119863 whichcontrols for the amount of pair-wise interaction amongagents between two distinct deliberation steps

324 Termination Conditions for Runs Simulations stoponce 100 proposals are deliberated on andor voted for inother words after 100 collective decision steps have occuredThe number of proposals discussed seems arbitrary yet it ishigh enough to observe the effects of deliberation on opiniondistributions and on other metrics related to coherence andjudgment accuracy We choose the number of runs as afunction of our research questions which give relevance tothe procedural parameters in the mixed model rather than tothe parameters set to describe the population and its behavior

325 Expected Outcomes of the Simulations We expect moredeliberation in terms of higher 119898 and 119899119863 to increasejudgment accuracy and at the same time to reduce thevariance of opinions In turn a smaller variance of opinionsimplies that the argument pool for deliberation is smaller andtherefore judgment accuracy should be lower Moreovervariance increasing dyadic interactions (or rejection) shouldalso increase judgment accuracy and coherence since bipo-larization and dissensus foster argument diversity in delibera-tionThe coherence statistic should be stronger in simulationsin which central convergence appears quickly and agentsare naive Attitude structures sensitivity to informationalcues and weights given to deliberated cues that makes pathsto bipolar convergence smaller or to central convergencelonger should be associated with higher judgment accuracyShifts of opinion should be more visible in scenarios inwhich extreme agents are pulled away from the extremesHence the sensitivity to deliberated cues (119901119886 119901119903) and 120574should explain the shifts When crossed with high values of119880 and 119879 the shifting power of deliberation should be at itshighest

In the end we do not know how these mechanismswill play out The results of the subsequent experimentsgive us insight on the interplay between the aforementionedeffects

4 Results

Before pointing at any result obtained from the mixedinteractions model we describe and compare the dyadicand deliberative interactions models in the parameter spaceobtained from the calibration in Section 3 (see Table 1)Primarily we require our results to describe two orthogonaltypes of opinion dynamics the pair-wise dyadic and thedeliberative The latter comprises scenarios in which indi-viduals do not influence each other by means of pair-wisediscussion and only update their opinions from delibera-tion the former scenarios where only pair-wise interactionsamong agents determine the dynamics We describe both onthe parameter space given in Table 1 by performing at least20 simulations per scenario Quantitative results from thepair-wise interaction model alone are described during thecalibration and are not treated heremdashdeliberative parametershave no effect on it Moreover we show how these differentmodel specifications yield qualitatively different opiniondistributions and compare them on the basis of the metricsof interest To this end we perform independent two-sampleStudentrsquos t-tests18 or Welch t-tests19 and compute confidenceintervals at a 095 level of confidence Last but not least wecomment ordinary least squares (OLS) regression estimatesto account for the direction and magnitude20 of the effectsof the parameters on the metrics Estimates are declaredsignificant at a 5 level of risk

We perform the same analysis on the mixed interactionsmodel as we compare it to the pair-wise dyadic and deliber-ative interactions models and discuss the marginal effects ofthe parameters on the metrics More precisely we obtain twodifferent types of results for the mixed interactions modelthe first compares the mixed scenarios to their monolithiccounterparts with respect to each metric and with respect tothe regression estimates the second gives a clear idea of themarginal effect of each governance parameter (or parameterof interest) on our observations In the first we allow controland procedural parameters to vary on a parameter spacesimilar to those of the dyadic and deliberative opiniondynamics (see Table 1) In the second we fix all of ourcontrol parameters execute 36000 balanced21 runs andfocus only on the one-way pooled effects of our proceduraland behavioral (all agents are focused or naive) parametersWe generate comprehensive graphs and tables to account forthe obtained results

Please notice the slight change in the size of the parameterspace for the procedural parameters and in the values ofthe nonprocedural ones for the first and second type ofresults (refer to Tables 1 and 2 respectively) Since we wantto have a finer idea of the marginal effects of each of theprocedural parameters on how well a group decides onproposals we make the value jumps for each parameter (1)

Complexity 15

() (U T) = (02 18) (b) ( pa m nD) = (005 05 00 2 02) (＝) ０ＬＧＮＬＭ ＣＨ (＜) (＝) Ｈ＞ tD = 3

(d) (５ ４) = (02 14) (e) ( pa m nD) = (01 03 05 3 02) () ０ＬＧＮＬＭ ＣＨ (＞) () Ｈ＞ tD = 6

Figure 7 From left to right common opinion trajectories and distributions for dyadic deliberative and mixed opinion dynamics within thestructure of different decision-making processes

small enough to detect significant differences in the estimatesand (2) meaningful for interpretation For the fixed one-valued parameters in Table 2 we use Table 1 to set them totheir minimum mean or median values

41 Comments on Dyadic and Deliberative Opinions Dynam-ics In this subsection we present our first results regardingthe deliberative opinion dynamics described in Equation (6)and its differences with the dyadic dynamics (Equation (5))in terms of our observations

411 Qualitative Analysis of Pair-Wise Opinion DynamicsPair-wise opinion dynamics alone produces multiclusterstable convergencewith variable cluster sizes (see Figures 7(a)and 7(d)) Insight on the complexity of these dynamics wasdiscussed in Section 321

In Figures 7(a) and 7(d) agents discuss randomly withone another in pairs Clusters in the extremes form quicklysince either agents that are close to the extremes attractone another or are convinced to stay close to the extremesby interacting with other agents they disagree with Otheragents with near-extreme positions may be simply attractedto one of the several moderate foci of agents in the opiniondistribution On the other hand moderate agents are eitherattracted to the closest opinion focus or pushed towards theextremes of the distribution Over time agents in the centralfocus either attract other agents into it or ignore the opinions

of the extreme agents that interact with them The result ofthese dynamics yields amulticlustered convergencewhere theopinion foci are at least at distance 119880 from one another andonly the extreme foci (clusters of agents with |119900119894| ge 075) areat a distance greater than or equal to 119879 from one anotherThefoci at the bounds of the opinion distribution are very stableand in analogy to the (12119889)-rule in assimilation bounded-confidence models [33] the number of clusters that form isroughly equal to 119880minus1412 Qualitative Analysis of Deliberative Opinion DynamicsQualitatively deliberative opinion dynamics yield a looseunipolar convergence of opinions near the center (centralconvergence see Figure 7(b)) near the center of either the leftor right portions of the distribution (group polarization seeFigure 7(e)) or to a sparse bipolar opinion distribution withclusters at the center and at the extremes (see Figure 7(c)) ifmixed with pair-wise interactions This is probably a reasonto expect the variance of opinions and the proportion ofextremists (almost perfectly correlated 120588 = 089 119901 lt 0001)to be low for these scenarios The side towards which theopinion cluster skews depends essentially on the valence ofthe first argument that is collectively accepted

In Figures 7(e) and 7(b) agents that update their opin-ions do so at the same time and only when deliberationis successful In these scenarios every time a decision iscollectively accepted agents update their opinions towards

16 Complexity

Table 3 Ordinary least squares estimates for pair-wise interactions scenarios for most of the observed metrics

Dependent variable119881119886119903(119900) 119878ℎ 119890119888 119894119903(1) (2) (3) (4)Constant 0610lowastlowastlowast 4429lowastlowastlowast 0472lowastlowastlowast 0000

(0002) (0206) (0003) (0000)

119880 = 06 (119880 = 02) minus0012lowastlowastlowast 2981lowastlowastlowast 0061lowastlowastlowast 0000(0002) (0260) (0004) (0000)

119879 = 18 (ref 119879 = 14) minus0264lowastlowastlowast minus3261lowastlowastlowast 0004 0000(0002) (0260) (0004) (0000)

120572 = 066 (ref 120572 = 05) 0002 minus0059 minus0100lowastlowastlowast 0000(0002) (0184) (0003) (0000)

(119880 119879) = (06 18) (ref (119880 119879) = (02 14)) minus0324lowastlowastlowast minus3728lowastlowastlowast minus0033lowastlowastlowast 0000(0003) (0368) (0006) (0000)

Observations 1440 1440 1440 1440R2 0983 0389 0529Adjusted R2 0983 0388 0527Residual Std Error (df = 1435) 0032 3492 0053 0000F Statistic (df = 4 1435) 21 272740lowastlowastlowast 228686lowastlowastlowast 402326lowastlowastlowastNote lowastplt01 lowastlowastplt005 lowastlowastlowastplt001

the argumentrsquos defended position Hence if there is votingand if agents happen to obtain 119894119899 proposal arguments theyinexorably cluster towards one or the other side of the opiniondistribution resulting in opinion convergence remeniscent ofgroup polarization (see Figure 7(e)) Furthermore as agentscluster in the same side of the opinion spectrum the argu-ments that may be advanced in deliberation are fewer andmore skewed towards that one side of the opinion spectrumAgents become more easily persuaded in deliberation andonly by arguments supporting one side of the spectrumConvergence of opinions towards one loose opinion focusbecomes faster and more certain If there is no vote (seeFigure 7(b)) the dynamics are similar but for the fact thatit is the uniform randomness of the proposal argument thatguarantees a central convergence of opinions

413 Quantitative Analysis of Pair-Wise Opinion DynamicsThe three parameters that allow for quantitative analysis are119880 and119879 the agentsrsquo attitude structure and themajority quotavoting rule 120572 Table 3 supports the claim that going from 120572 =05 to 120572 = 066 does not affect the variance nor the shifts ofopinions However judgment accuracy is significantly higherwhen accepting proposals becomes more difficult This israther surprising but also intuitive since decisions that aretaken using more restrictive methods of scrutiny should bemore accurate

Concerning the attitude structures the interaction termbetween 119880 and 119879 shows that 119880 has a small effect on thevariance of opinions given that 119879 is low This shows thateven when the assimilation threshold is high a sufficientlysmall rejection threshold can make its one-way effect vanishHowever high levels of both 119880 and 119879 reduce the variance ofopinions dramatically In contrast when it comes to overallchanges in opinion the magnitude of 119880rsquos one-way marginal

effect increases the shifts of opinions whereas 119879 lowers itThis can be interpreted as having the rejection dynamics pushmoderate-to-extreme agents to positions that are close tothe extremes and very moderate agents to the center of thedistribution Not too many quantitative changes (on average)can therefore be observedThe fact that higher119880 is associatedwith faster convergence of opinions seems to imply lowerjudgment accuracy At some point voting for the same type ofproposals for too longmay result inmany collectivemistakesThis is an interesting conclusion to consider in the analysis ofthe mixed model Finally the interaction term shows that 119879has no strong impact on the metric

414 Quantitative Analysis of Deliberative OpinionDynamicsSeven parameters can be considered for quantitative anal-ysis in this dynamics the parameters of interest recalledin Section 312 the size of the argument sacks (119896) agentbehavior during deliberation the strength of the deliberationdynamics (120574) and the measure sensitivity to deliberatedresults (119901119886) The most ldquounexpectedrdquo result obtained fromthe deliberation dynamics is that it shows how deliberationcontributes to the proportion of extremists and the varianceof opinions It happens that having more agents discussproposals and having more of these discussions increasesthe variance of opinions and the proportion of extremistswhile lowering judgment inaccuracy and coherence For theformer the fact that far too many arguments are advancedduring deliberation plummets the chances that the centralargument is accepted hence deliberation does not moveopinions too often For the latter we observe that only ldquobigrdquodifferences (going from the lowest value of a parameter to thehighest) in protocol actually prove to have a preponderantimpact on the metrics These observations hint at a possibletrade-off between judgment accuracy and the variance of

Complexity 17

Table 4 Ordinary least squares estimates for the deliberative interactions scenarios for most of the observed metrics

Dependent variable119881119886119903(119900) 119878ℎ 119890119888 119894119903(1) (2) (3) (4)Constant 0032lowastlowastlowast 13092lowastlowastlowast 0370lowastlowastlowast 1007lowastlowastlowast

(0001) (0406) (0001) (0001)

119898 = 3 (ref119898 = 1) 0012lowastlowastlowast minus2062lowastlowastlowast minus0012lowastlowastlowast minus0001(0001) (0276) (0001) (0001)

119898 = 6 (ref119898 = 1) 0023lowastlowastlowast minus3285lowastlowastlowast minus0018lowastlowastlowast minus0004lowastlowastlowast(0001) (0276) (0001) (0001)

119899119863 = 002 (ref 119899119863 = 001) 0007lowastlowastlowast minus0965lowastlowastlowast minus0006lowastlowastlowast minus0002lowastlowast(0001) (0276) (0001) (0001)

119899119863 = 005 (ref 119899119863 = 001) 0020lowastlowastlowast minus3316lowastlowastlowast minus0015lowastlowastlowast minus0004lowastlowastlowast(0001) (0276) (0001) (0001)

120572 = 05 (ref 120572 = 0) 0146lowastlowastlowast 68932lowastlowastlowast 0068lowastlowastlowast minus0507lowastlowastlowast(0001) (0276) (0001) (0001)

120572 = 066 (ref 120572 = 0) 0291lowastlowastlowast minus13842lowastlowastlowast 00004 minus0000(0001) (0276) (0001) (0001)

119896 = 8 (ref 119896 = 4) 0004lowastlowastlowast minus0815lowastlowastlowast minus0002lowastlowastlowast minus0001(0001) (0276) (0001) (0001)

119896 = 16 (ref 119896 = 4) 0008lowastlowastlowast minus1118lowastlowastlowast minus0004lowastlowastlowast minus0003lowastlowastlowast(0001) (0276) (0001) (0001)

119861119890ℎ119886V119894119900119903 = ldquoFocusedrdquo (ref ldquoNaiverdquo) 0015lowastlowastlowast minus1980lowastlowastlowast minus0012lowastlowastlowast minus0004lowastlowastlowast(00005) (0225) (00004) (0001)

120574 = 01 (ref 120574 = 005) minus0020lowastlowastlowast 5762lowastlowastlowast minus00003 minus0001(0001) (0276) (0001) (0001)

120574 = 02 (ref 120574 = 005) minus0025lowastlowastlowast 8648lowastlowastlowast minus00003 minus0001(0001) (0276) (0001) (0001)

119901119886 = 05 (ref 119901119886 = 03) minus0013lowastlowastlowast 1581lowastlowastlowast 00004 00003(00005) (0225) (00004) (0001)

Observations 19440 19440 19440 19440R2 0931 0844 0562 0958Adjusted R2 0931 0844 0562 0958Residual Std Error (df = 19427) 0033 15701 0030 0050F Statistic (df = 12 19427) 21 844790lowastlowastlowast 8 733563lowastlowastlowast 2 079585lowastlowastlowast 36 761250lowastlowastlowastNote lowastplt01 lowastlowastplt005 lowastlowastlowastplt001

opinions that may be of interest in the mixed discussionmodel

Group coherence shifts of opinions and the varianceof opinions are highly affected by the voting thresholdthe coherence metric is reduced by half when allowing forclassic majority voting shifts explode and the variance ofopinions increases dramatically This is not surprising sincethe majority quota rule works as a buffer to the success ofdecision-making processes and in consequence it hampersthe deliberative opinion updates For the shifts of opinions itis clear that the two-third majority voting rule results in littleopinion change and that the simple majority rule inducesgroup polarization As expected there are less extremists inscenarios where there is no voting and the shifts of opinionare more likely to happen when agents are more sensitiveto deliberation When we include voting in the system

the grouprsquos judgment accuracy decreases due to the correctdeliberated decisions that are vetoed by a salient opinion-agent majority Strangely the effect only holds for the simplemajority decision rule Lastly the proportion of individualsthat participate in the debate and the minimal amount ofdebates have similar marginal effects on all the metrics (seeTable 4)

415 Comparing Deliberative and Pair-Wise Opinion Dynam-ics in a Nuthshell Qualitatively both models yield similaroutcomes but for the fact that deliberation never producesopinion polarization with both types of extremists (seeFigure 7) Furthermore clusters in the deliberative interac-tions model are less homogeneous in size and multiclusterconvergence occurs only when deliberation is rarely suc-cessful which happens sporadically given the calibration of

18 Complexity

Table 5 Mean difference 095 confidence intervals for metrics by model comparison

Model Metric119881119886119903(119900) 119901119903119900119901119890119909 119878ℎ 119890119888 119894119903Dyadic vs Deliberative [0191 0216] [0170 0197] [-2948 -2828] [0074 0082] [0165 0172]Mixed vs Deliberative [0142 0147] [0150 0155] [-2085 -1976] [0003 0002] [-0038 -0030]Mixed vs Dyadic [-0072 -0046] [-0045 -0018] [8332 8859] [-0079 -0071] [-0205 -0201]the model On the other hand pair-wise opinion dynamicsproduces multicluster convergence most of the time andcentral convergence only in the infrequent cases where thelatitude of acceptance is large

The comparison of these two scenarios in terms ofthe metrics is straightforward Whilst dyadic interactionsallow for more variability in opinions through pair-wiseinteractions and more mistakes in judging proposals delib-erative updates simply act as an antagonist force On aver-age the deliberative interaction model produces opiniondistributions with smaller variance and yields steady stateswith higher judgment accuracy a higher shift statistic andnaturally lower group coherence since only proposals that aredeliberated 119894119899 are eligible for voting (see Table 5)

42 Articulating Dyadic and Deliberative Social Interactionsin Opinion Dynamics So far we have considered scenarioswhere argumentation had no place and where individualsmade decisions according to opinions that were purelyconstructed from pair-wise discussions We also observedcases in which agents formed their opinions only by inte-grating deliberated proposals into their opinion updates Thescenarios we present in this section combine the two pre-ceding scenarios to account for both the effects of pair-wisediscussion and the effects of deliberation on opinions judg-ment accuracy and group coherence From the precedingobservations we conclude the following deliberative opiniondynamics reduces the variance and the proportion of extrem-ists because it shifts opinions greatly and polarizes groupsby eliminating at least one type of extremist the one whoseopinions are initially opposed to the deliberated resultsor it brings opinions close to neutrality (see Figure 7(b))This may not be desired in a group since the diversity andstability of opinions are necessary for deliberation to makesense and lead to the collective acceptance of controversialproposals Dyadic opinion dynamics on the other handtend to increase the variance of opinions and the numberof extremists while having a lesser aggregated effect on theshifts of opinion This is an immediate result of the pair-wiseinteraction dynamics inasmuch as it polarizes individualsvery quickly

To sum up in the mixed interactions model deliberationhinders the effect of voting on opinions by making someproposals that would normally be submitted for voting noteligible for voting It may also undo opinion changes dueto pair-wise interactions The mixed model can be seen asthe deliberation opinion dynamics model in which pair-wiseinteractions occur in between collective discussion and mayaffect which arguments are advanced during deliberationEquivalently it can be seen as the dyadic opinion dynamics

model in which deliberation accounts for endogenousldquoshocksrdquo on the opinion distribution

421 Qualitative Analysis of the Mixed Dynamics In Figures7(c) and 7(f) agents update their opinions through delibera-tion and pair-wise interactions Opinion trajectories visualsshow a combination of both previously described opiniontrajectories Over time as agents discuss with one anotheropinion clusters form as in the pair-wise or dyadic opiniondynamics Deliberation may disrupt the formation of theseclusters and three different situations may occur

(i) Deliberation disrupts the formation of the opinionclusters but not sufficiently to jeopardize pair-wiseopinion cluster formation (eg for high values 119905119863)Two phenomena may be responsible for it (1) thevariability of the proposal argumentrsquos support for theprinciple P and (2) too few (successive) deliberationsteps that result in opinion updates Once the opinionclusters are formed successful deliberation foreshad-ows the merging of the previously formed clustersmay bring extremists closer to moderate agents andthanks to the assimilation component of the pair-wisedynamics allow bigger and more moderate opinionclusters to form (refer to Figure 7(c))

(ii) Deliberation significantly disrupts the pair-wisedynamic formation of clusters This may happenwhen the requirements for accepting proposals areweak in the decision-making process (eg low 120572) Inthis case one has either central convergence sinceall agents follow the deliberated results in the samemanner or 3-cluster convergence with two extremeopinion foci In the latter situation the extremegroups form quickly (like in the pair-wise model) anddeliberation cannot pull extreme agents away fromthe extreme foci they are in The main reason for thisis that immediately after an extreme agent attempts toleave her foci through deliberation updates she getspulled or pushed back into it by means of a dyadicinteraction with another extreme agent

(iii) Deliberation does not disrupt the formation of theopinions cluster This is the rare case scenario forvery demanding decision-making processes in termsof inclusion (eg 119899119863) and argument acceptability (eg120572) Opinions converge as in the pair-wise opiniondynamics model (refer to Figures 7(a) and 7(d))

Last when there is voting (120572 = 0) the position ofthe moderate opinion cluster is determined by the first

Complexity 19

minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

Var(o)

sh

ec

ir

Var(o)

propex

sh

ec

ir

1

098

minus02

minus004

001

1

minus014

minus001

minus003

1

026

minus028

1

minus057 1

ＪＬＩＪ

Ｒ

Figure 8 Significant correlation estimates for themetrics of interestin the mixed opinion dyamics scenarios

accepted proposal argument andor the opinionmajority thatis formed from previous dyadic interactions (see Figure 7(f))

422 Relating the Metrics of Interest in Mixed OpinionDynamics The first thing to notice is that for all of thesescenarios all the metrics are significantly correlated (seeFigure 8) The variance of opinions and the proportion ofextremists are almost perfectly correlated To some extentthis means that a high variance of opinions known thatthe mean of opinions is statistically null equates to havingmany extremists in the population be towards one or theother side of the opinion distribution (eg extreme bipolar-ization) Similarly we find a negative and strong correlationbetween the judgment inaccuracymeasure and the coherencecoefficient (see Figure 8)mdashwhen judgment inaccuracy is high(low) coherence to deliberation is low (high) Cases likethese are telling since they may be related to sequences ofcollective decisions in which agents do not make too manymistakes when judging proposal arguments and still makedeliberated decisions that are on average in line with theirprinciples

The shift statistic interestingly is negatively correlatedwith the variance of opinions with judgment accuracy andwith the coherence coefficient In a way this means thathaving more moderate agents conduces on average to moremistakes in judging arguments and to less coherent collectivedecisions An explanation for this phenomenon could be thatas agents reach consensus in opinion important (possiblyextreme) arguments for correctly judging argument propos-als are not advanced in the deliberation arenawhich results inmore mistakes and in possibly accepting proposal argumentsthat after voting are not meant to be accepted Indeed the

negative correlation between the variance of opinions andjudgment inaccuracy points towards this direction since onaverage the more shifts there are the smaller the variance ofopinions is (see Figure 8)

423 Comparing Mixed Dynamics with Other MonolithicDynamics of Interactions Qualitatively opinion trajectoriesin the mixed model are more chaotic than in the pair-wise and deliberative interaction models (see Figure 7) Asanalyzed previously deliberation dynamics disrupt the pair-wise dynamics and create situations in which central andmulticlustered convergence (with two clusters at the extremesof the distribution) are possible at the expense of more(pair-wise dynamics) or less (deliberative dynamics) clustersIt tolerates the survival of very small isolated group ofagents that the previous models rarely tolerate in a hundredcollective decision-making steps (see Figure 7(c))

One of the most interesting observations one can makeof this family of models regards how groups change (in termsof our metrics) when dyadic social interactions are addedto deliberative interactions and vice-versa When comparingthe mixed interactions with only deliberative ones shiftsagentsrsquo aggregated change in opinions are on average lesspronounced The variance of opinions and the proportionof extremists are significantly higher for the former than forthe latter group coherence in contrast is lower with respectto the scenarios where opinions are formed only throughdeliberation Judgment accuracy follows that same trendA reason for this might be that mixed dynamics are fasterto yield semistable clusters andor opinion convergence Inconsequence the potential argument pool for deliberativeexchange shrinks faster and deliberation becomes less effec-tive over time which results in more collective mistakes Theoutcomeon coherence is predictable inasmuch as deliberativeinteractions produce group polarization If a group agrees inopinion debates are sterile because themembers of the groupwill most likely refrain from attacking the proposal argu-ments they all support and will work together to disparagethe proposal arguments they do not support (see Table 5)

In the same manner we report that the proportion ofextremists and the variance of opinions are barely lower inthe pair-wise interaction model when including deliberationin the decision-making process Shifts of opinions are alsobarely lower when there is no deliberation This can indicatethat deliberation does not undo the changes in opiniondue to pair-wise interactions nor does it amplifies them Incontrast when we consider the labeling-based metrics themixed interaction model ensures lower judgment inaccuracybecause of deliberation and lower group coherence (seeTable 5)

In sum the mixed discussion model is a compromisebetween the pair-wise interaction and deliberative opiniondynamic models in that it offers some guarantees in termsof judgment accuracy and coherence while allowing forldquoreasonablerdquo variance in the opinion pool It establishes somekind of trade-off between labeling-based metrics and thevariance of opinions that may be of interest for decision-making process design

20 Complexity

Table 6 Summary of sign or relative marginal effects for all varying parameters on dyadic deliberative and mixed interactions models inand for all experiments

Metrics Parameters119905119863 119898 119899119863 120572 119879 119880 119901119886 120574 119896 119861119890ℎ119886V119894119900119903119881119886119903(119900) XX X X O X X X X X X119901119903119900119901119890119909 XX X X O X X X X X X119878ℎ XX X X O X X X X X119890119888 XX X X XO X XOO XOO X X119894119903 XX X X O XO O X OOX XOO X XNota gray symbols correspond to dyadic dynamics light gray to deliberative dynamics and bold black to mixed dynamics Given a scenario the symbol 119883stands for constant parameter and 119874 for insignificant marginal effect and as parameters grow for positive (negative) marginal effect for the parameterFrom left to right at the parameter level the leftmost arrow corresponds to the scenario with the strongest (in absolute value) marginal effect of the parameter(with respect to to its factors) Example Let 119901 be a parameter and 119874 stands for ldquo119901 has a significant positive effect in dyadic discussion scenario negativein the deliberative one and insignificant in mixed discussion The positive effect in dyadic discussion is greater than the absolute value of the negative one inthe purely deliberative scenariordquo

424 Comments on the Differences between the Three Modelsfor OLS Regression Estimates The significance of most of theregression parameters estimated for the mixed interactionsscenarios are similar to those estimated for the pure dyadicanddeliberative scenariosNevertheless themagnitude of theeffects is different insomuch that the dynamics themselvesare different So instead of commenting the linear regressionestimates of the mixed model we compare when possiblethe standardized significant coefficients of the regressors ofthe mixed model with those of the pair-wise and deliberativeinteractionmodels (see Table 6 for a summary of the results)

In mixed interactions for instance the influence of119898 onthe opinion distribution is much more preponderant than inthe deliberative dynamics scenarios Going from 119898 = 1 to119898 = 3 increases the standard variation units of the variance ofopinions by twice as much as the one in the deliberative sce-nariosThis in particular is due to the design of the decision-making process If 119898 is higher then the number of socialinfluence steps (119905119863) in the decision process is also higher(compared to 119905119863 = 0 in the deliberative case) If we decideto consider these steps as contributing to the variance ofopinions then the effect of119898 on the variance of opinions hasto be stronger in the mixed interactions model In contrastthe number of agents participating in the debate has a weakereffect on the variance of opinions in the mixed interactionsmodel than in the deliberative dynamics specially for 119899119863 =005 (more than twice as much) The difference may comefrom the fact that in mixed discussions the part of theargument pool used in deliberation is more restrictive thanin the deliberative case We can argue that since pair-wisedynamics may get individuals closer to one another aftersome deliberation the same happens to the arguments thatindividuals will most likely use in deliberation Taking thisinto account means that deliberation will at many instancesbe unfruitful and have no effect on opinions thus failing tomoderate extreme opinions andor polarize the group In thesame direction a high number of participants in deliberationcan make the deliberation dynamics stiff In this case thedistribution of opinion is close to the uniform distributionwhich has a relatively high-to-moderate variancewith respectto the reference results

The strength of the deliberative dynamics has a positivestronger effect on the variance of opinions in themixedmodelthan in the deliberative model when 120574 = 02 and a negativeweaker one when 120574 = 01 A sound interpretation of thisresult is that when deliberation is successful and its effect onopinions does not get agents to move sufficiently away fromtheir current positions agents are pulled or pushed back totheir previously held opinions the variance of opinions doesnot changemuchOn the other hand if the effect of successfuldeliberation is strong opinions can change sufficiently tomake agents much more prone to assimilation into moderateopinion foci the variance of opinions falls

The deliberative opinion dynamics model is much moresensitive to 120572 than the mixed discussion model Differ-ences in marginal effects can double The probability ofattraction to deliberated results (119901119886) for instance has astrongmoderating effect (negative) on agentsrsquo opinions in thedeliberative opinion dynamics but a meager one (two-fold)in the mixed and dyadic dynamic models This is becauseopinion updates in the deliberative model depend only onthat parameter whereas in the mixed model more numeroustypes of interactions make opinion updates possible

In terms of judgment accuracy all parameters except thesize of the argument sacks (119896) have a significant slightlystronger effect on the deliberative dynamics than in themixedmodel In effect the difference may be attributed to theupshot of pair-wise interactions on the distribution of opin-ions and on the arguments that would be advanced duringdeliberation The result respecting parameter 119896 however iscounterintuitive Since argument sacks are static scenarioswhere shifts are higher should make 119896 more important fordeliberation to succeed and it is precisely in the scenarios ofpure deliberative dynamics that we find the biggest opinionshifts This may point to the hypothesis that in deliberativeinteractions agents change opinions substantially but do notoften change their adherence to the principle (go from anegative opinion to a positive one or vice versa) when thereis no group polarization

43 Sensitivity Analysis for the Procedural Parameters ofInterest Weperform a sensitivity analysis of the observationson the parameters of interest in themixed interactionsmodel

Complexity 21

We extend the parameter domains of the pure dyadic anddeliberative scenarios as we make it explicit in Table 2

431 Minimum Number of Debates (119898) We observe that theminimum number of debates has a significant well-observedeffect on all of our metrics excluding group coherence Forthe variance of opinions we can observe that the moredebates there are the bigger the value of the metric isalthough the higher 119899119863 and 119905119863 are the weaker the overalleffect of119898 becomes (Figures 9(d) and 9(a)) Furthermore themarginal effect of increasing minimal debates is decreasingan additional deliberation step requirement increases thevariance of opinions but less and less as it grows 119898 coupledwith 119899119863 has a shy S-shaped effect (see Figure 9(d)) on thevariance of opinions meaning that it has a stronger effectwhen 119899119863 is low which suggests a possible trade-off betweenthe number of arguments accepted in the debate and thenumber of debates required before accepting a proposal

In contrast the shifts in opinion are less and less likelyas 119898 grows and this is independent of the variations of theother parameters Again the effect is marginally decreasingand is only truly significant when 120572 = 05 An explanationto these effects may be linked to the design of the systemFirst the variance of opinions is higher when deliberation isasked for because the more deliberation steps there are thehigher the chances are that the central proposal argumentis deemed unacceptable specially when agents are focusedFurthermoremechanically speaking increasing theminimalamount of debates implies that whenever a decision is to betaken at least 119898 times 119905119863 time steps have to take place and if atany moment a proposal argument is considered undecided119905119863 time steps are added to the process So unless the debatedoes not yield 119906119899119889 labels for proposal arguments (highlyunlikely considering that 120590 = grounded semantics) the morenondeliberation steps there are in the decision process thehigher the variance of opinions is Concerning the shiftswhen 120572 = 05 either the system is too stiff to accept anyproposal argument and opinions do not changemuch or theeffects of pair-wise discussion and deliberation cancel out insuch a way that individual shifts are minimal (Figure 9(b))

On the side of labeling-based metrics the more debatesare asked for the more accurate a group is in its judgmentthe effect being smaller as 119898 grows When agents are naivethe effect is quasilinear while when they are focused thestrongest effects of acquiring more deliberation are foundwhen levels of deliberation are already low (Figure 9(f))Thiscan be explained by the fact that the more debates thereare in the decision process the closer one gets to the idealargumentation framework

432 Proportion of the Population in Deliberation Steps (119899119863)Like for 119898 119899119863 has a significant effect on the proportionof extremists and on the shifts and variance of opinions(Figure 9(e)) This may result from the fact that being ableto put more arguments in play at the same debate step mayincrease the odds of revealing cycles around the proposalargument in119860119865120576 Given that we use grounded semantics thearguments in the cycles are labeled 119906119899119889 and in consequence

debates are more often postponed when 119899119863 is higher Post-poning debates in turn increases the number of nondebatessteps in the decision process which increases the varianceof opinions and limits over time the moderating effect ofdeliberation Moreover the effect of this parameter is verydependent on the value of 120572 (Figure 9(e)) For shifts forinstance 120572 = 05 makes the effect of 119899119863 negative while120572 = 0 makes it positive but to a lesser extent Furthermorefor higher requirements of deliberation (119898) adding moreindividuals to the deliberation process has weaker effectson the variance of opinions and on the other metrics It isalso quite interesting to notice that it has the same effect onagents independently of agent behavior during deliberationThis is a surprising result as one would have expected morefocused agents in a deliberation arena to cause a raise inthe proportion of extremists as they play to knock outopposing proposal arguments and thereof reducing the effectof deliberation (lowers variance) on the whole population

Finally similar to119898 addingmore people into the deliber-ation process increases judgment accuracy (see Figure 9(c))and has no clear effect if any on group coherence (seeFigure 9(i))

433 Steps between Deliberation Steps (119905119863) In all configu-rations 119905119863 increases the proportion of extremists (variance)and decreases the shifts of opinions among agents Theshifts and the effect on the variance of opinions are presentfor 120572 = 05 insignificant otherwise (Figure 9(b)) 119905119863 ishighly linked to 119898 by construction When 119898 = 1 thecurve linking the variance of opinions and 119905119863 is convex As119898 increases the curve becomes more and more concavewhich means that 119905119863 has a more important effect on theopinion distribution as collective decisions take longer to beachieved This seems counterintuitive but in reality it reflectsthemultiplicative relationship between deliberation and pair-wise interactions and the semantics in the model If 119898 is lowand 119905119863 is high the effective number of pair-wise interactionsis on average smaller in the deliberation process Hencethe structure of the decision-making process hinders anyincrease in the variance of opinions If119898 is high the oppositeeffect is observed Additionally since the grounded semanticsyields few 119894119899 arguments with respect to other admissibility-based semantics getting closer to the ideal argumentationframeworkmaymake it difficult to obtain the expected effectsof deliberative discussion on agents Lower 119898 constrains thevariance of opinions by making deliberation more influentialon opinions

In terms of judgment accuracy and coherence we observethat 119905119863 does not explain coherence yet it tends to decreasejudgment inaccuracy This may be due to a more variedargument pool in the debate which in turn may be aconsequence of a higher variance of opinions or vice versa

434 Acceptability Voting Quota (120572) By far it is the mostinfluential parameter in our study It changes the directionand the intensity of the effect of all the other proceduralparameters and by construction heavily constrains the roadto accepting a proposal (see Figure 9)

22 Complexity

015

020

025

030

035

040

1 2 3 4 5 6

12345

Values of Ｇ

(a) 119905119863 times 119901119903119900119901119890119909 by119898

0

10

20

30

40

50

1 2 3 4 5 6

005066

Values of

(b) 119905119863 times 119878ℎ by 120572

040

045

050

055

060

1 2 3 4 5

001002003004005

Values of Ｈ＄

(c) 119898 times 119890119888 by 119899119863

015

020

025

030

035

040

1 2 3 4 5

001002003004005

Values of Ｈ＄

(d) 119898 times 119901119903119900119901119890119909 by 119899119863

0

10

20

30

40

50

001 002 003 004 005

00005066

Values of

(e) 119899119863 times 119878ℎ by 120572

040

045

050

055

060

1 2 3 4 5

BehaviorNaiveFocused

(f) 119898 times 119890119888 by 119887119890ℎ119886V119894119900119903

015

020

025

030

035

040

0 05 066Behavior

NaiveFocused

(g) 120572 times 119901119903119900119901119890119909 by 119887119890ℎ119886V119894119900119903

05

06

07

08

09

10

1 2 3 4 5 6

00005066

Values of

(h) 119905119863 times 119894119903 by 120572

08700

08725

08750

08775

08800

001 002 003 004 005

BehaviorNaiveFocused

(i) 119899119863 times 119894119903 by 119887119890ℎ119886V119894119900119903

Figure 9 Curves of mean observations for different parameters and metrics for 36000 runs Bars are 095 confidence intervals Plots withthe same observables (ordinate) except for 119894119903 are on the same scale

Complexity 23

The higher the requirement for accepting the proposal isthe higher the proportion of extremists and the variance ofopinions are and the lower the shift statistic is In few words120572 constrains the world to dyadic discussion or throws it intoa process in which deliberation is much more importantthan dyadic interactions This is why for any 120572 = 05 oneeither gives too much weight to deliberated results whichshadows pair-wise interactions or too little weight in thesense that no deliberated result is ever accepted and thusnever integrated into the opinions of the agents that havevoted for it The reason for the latter is that the latitudeacceptance is too low and thereafter pair-wise interactionsare not enough to unevenly polarize the population in such away that deliberated proposal arguments are voted favorablyby a non-negligeable majority Shifts for 120572 = 0 are stable fordifferent levels of 119905119863 which implies that deliberation keeps incheck all the shifts of opinions related to an increase in thenumber of pair-wise interactions For 120572 = 05 deliberationis less likely to be successful and shifts tend towards the shiftlevels for 120572 = 0 (see Figure 9(b))

Concerning labeling-basedmetrics120572 entirely determinesthe coherence statistic For 120572 = 05 there is no difference incoherence because of how coherence is defined it is maximalwhen 120572 = 0 and strangely maximal when 120572 = 066(see Figure 9(h)) Surprisingly 120572 has no effect on judgmentaccuracy One would have expected a higher 120572 to increasejudgment accuracy since agents would hardly collectivelyaccept a proposal argument discussed during deliberationProposal arguments have given how the argument lattice isgenerated a higher probability of being rejected than of beingaccepted in the ideal argumentation framework

435 A Word on Focused and Naive Agents Focused andnaive specifications for agent behavior are an importantparameter in the decision process since they model howagents choose arguments in the deliberation arena In the sce-narios where agents are focused agents show equal or highervariance of opinions and are therefore more extreme (seeFigure 9(g)) independently of the voting requirements Shiftsof opinion are less likely as well We can see this happeningbecause focused agents knock out proposal arguments moreoften than naive agents provided that 119899119863 is low

One could have expected naive agents to yield morecoherent decisions since they argue sincerely However thisseems not to be the case (see Figure 9(i)) Agents are unable todeliberate in away thatmakes deliberation reflect their votingintentions either because they do not have the necessaryarguments to do so or because such arguments do not exist

In terms of judgment accuracy focused agents do betterthan naive agents in finding the ideal labels for the proposalarguments This is likely because when reconstructing theframework focused agents take into account the deliberatedproposal argument and choose themost pertinent argumentsin their sacks that relate to the state of affairs at the deliber-ation arena In fine debates result in a better approximationofL119886119887120576(119868) even though one could have believed the oppositethat is focused agents are ready to sacrifice a correct collectivelabel of a proposal argument in the goal of defending theirpositions

44 Points of Discussion In this subsection we discuss theresults obtained from the simulations We attempt to makesense of them from a social sciences perspective

441 Opinion Consensus and Dissensus The deliberativeopinion dynamics corresponds to the simulated scenarios inwhich the variance of opinions is at its lowest and judgmentaccuracy at its highest Dyadic opinion dynamics are associ-ated with the scenarios in which the variance of opinions is atits highest judgment accuracy at its lowest and coherence atitsmaximumpossible levelThemixed interactions dynamicsis a combination of these aspects for all the metrics exceptfor coherence at which it is the worsemdashdeliberation leadsto some kind of loose opinion consensus that is reinforcedby pair-wise interactions Likewise the mixed discussionmodel confirms some results exposed in [16] expressingthat opinions and voting intentions in a deliberative contextldquooften changerdquo That being said and more particularly indeliberative opinion dynamics outputs align well with twointeresting results on consensus formation given by aninterpretation of Moscovici and Doisersquos [15] work Sunsteinrsquos[14] account on deliberation and group polarization andsocial impact theory [4] The first states that mild consensusis reached in deliberation as procedure disengages agentsthe second is that within discussion groups majorities tendto become larger and opinions more extrememdashnot on thebasis of competing arguments but of the initial balance ofopinions After several decision processes we find opiniondistributions that reflect these observations even though weare not able to precisely coin the set of parameters that yieldssuch distributions (see Figure 7)

Although differences between mixed and dyadic opiniondynamics in terms of the variance of opinions and shiftsare slim the former does much better in terms of judg-ment accuracy In this sense deliberation does moderateopinions but not as much when non-deliberation steps arerelatively frequent throughout the decision-making processIn terms of Moscovici and Doise theory of consensus [15]this makes sense since pair-wise discussion engages or ego-involve agents in the discussion and pushes to convergenceof opinion at the extremes of the opinion distributionHowever what remains unclear is the reason why proceduralparameters that constrain deliberation do not disengageagents from discussion and thus result in a mild consensusas also expressed by Moscovici and Doise [15] A possibleexplanation of this phenomenon may stem from the factthat deliberation in the model is always accompanied by aseries of ego-involving discussions that shadowor counter themoderating effect of deliberation Not only this deliberationcould also be responsible for crystallizing the opinions ofextremists for it favors one or the other part of the distributionof opinions as explained in [14] and to some extent in [13]Furthermore the mixed discussion model gives an exampleof polarization that results from a minority of dissenterson one side of an issue acceding to views of the majorityrsquosside [4] as opposed to the strong polarization observed indyadic interactions Situations in which the status quo orthe undecidability of the proposal argument lingers cannotpossibly yield situations of moderation In cognitive science

24 Complexity

one can be inclined to believe that the longer a deliberationprocess is the more likely individuals will take an immutablestance on the situation In view of the assumption that longdecision-making processes can be more costly for a group orsociety as a whole agents may feel forced to take a stancejust for the sake of ending the process The stance that isadopted would therefore depend on pair-wise interactionsamong individuals and not on deliberation since deliberationtakes too long to be conclusive

442 For Hypothetical Recommendations in Decision-MakingProcess Elaboration On another tune results from themodelsuggest that procedural parameters have marginally decreas-ing positive or negative effects on the metrics of interest Ina hypothetical policy analysis perspective this may indicatethat after choosing a correct number of individuals andimposing the right amount of debates one can attain rea-sonable levels of judgment accuracy coherence and varianceof opinions As accounted in [18] deliberation increasesknowledge (collective knowledge as well) and therefore itheightens the chances of making correct decisions If onebelieves that social welfare is linked to deliberation andeventually to the metrics studied previously then proceduralparameters can both be used as an instrument to attaindesired levels of social welfare and reinforce the notionof legitimacy of collective decisions It can also provide ajustification for choosing a deliberation regime rather thananother on the basis of how important (or urgent) a topic ofdiscussion is Indeed if we adopt the claim that deliberationis a forerunner of welfare then any state in which deliberativecues are at their maximum (highest number of participantsmany deliberation steps before making a decision and asfrequent as possible) has to be mapped to the highest attain-able social welfare In this case is a situation where judgmentaccuracy is at its highest yet the variance of opinions or theproportion of extremists at its lowest an ideal situation Wecannot say for sure since strong opinion consensus and longdecision-making processes in many situations may result ina loss of welfare rather than in a gain of it

Another interesting result in procedure for deliberationis the very similar estimates we find for the effects of theminimum number (119898) of debates necessary to make adecision on a proposal and the proportion (119899119863) of individualsparticipating in deliberation on judgment accuracy In thescope of social welfare this raises the question of whetheradvocating for longer decision processes by demandingmoredeliberation steps (higher 119898) rather than bigger debates(higher 119899119863) may increase welfare Depending on the objec-tives of the decider of the deliberation procedure he or shemay choose to concentrate on one or the other A recom-mendation we can assert from the mixed social interactionsmodel is that if one prefers low to high variance of opinionsandor high to low coherence the decider is better off ifshe focuses on increasing the size of the debate rather thanorganizing debates very frequently One of the reasons forthis might be that when making bigger debates all agents aretruly considered as equals in the decision-making processIn organizing many debates individuals that are chosen toparticipate in many of these will be more influential and will

somehow bias the set of arguments in the deliberation arenaThemajority rulemodeled through120572 also determines the sizeof the shifts and the distribution of opinions Because a bigger120572 is associated with higher levels of extremism and opinionvariance deliberated cues are harder to accept and thereforeit becomesmore difficult for agents to reach concensus andormoderate their opinions

On the other hand and further into the idea of legitimacythe goal of deliberative processes is the agreement amongall participants This agreement stands for the right or bestdecision from a formal or procedural perspective and it is alsothe best from a substantive point of view In other words thelegitimacy of decisions not only derives fromwell-establishedprocedural requirements (respect of the protocol) but alsobegs for the fulfillment of two essential conditions (1) satisfy-ing the procedural requirements for a correct procedure (for-mal legitimacy) and (2) the rational acceptability of the resultsof this procedure (substantive acceptability) Legitimacy canbe then obtained from the coherence statistic scrutinizedin the model insofar as it measures how well agents acceptthe results of a correct procedure (eg deliberation) So fora proposal to be legitimate one has to choose parametersthat would maximize group coherence in decision-makingprocesses

As deliberation may give rise to consensus in the delib-eration arena it may also sow the seeds of dissensus inthe group Our model clearly illustrates this phenomenonthrough the procedural parameters of deliberation and themetrics observed From another perspective in deliberativedemocracy dissensus in collective decision-making may bedesirable According to Landermore and Pagersquos accounts [36]lack of consensus can be perceived as having agents withalternative ideas that combined with other ideas can providea better approximation of the ideal status of an argumentTheauthors in [36] call this dissensus ldquopositive dissensusrdquo andit makes normative sense when solving complex collectivedecision-making problems Our model captures the ideaof having normative requirements to define a good andlegitimate collective decision requirements of high consen-sual deliberative accuracy (judgment accuracy) of legitimacy(coherence) and of diversity (variance of opinions) are alltaken into account and for such we can conclude that ourmodel is successful in describing a sound process of collectivedecision-making with interpretable outcomes

Of course the present simulation results and analysis donot prove that all opinion dynamics can be accounted forby such simple processes like the one presented here Theydo however shed light on the expressiveness of combiningdifferent paradigms to account formore interpretablemodelsand pave the way to creating original models of the sortThe simulations suggest the desirability of discovering theconsequences of relatively simple laws of communication atdifferent levels (micro meso and macro) to determine whatstill needs to be explained

5 Related Work

We see our model as a contribution to the influence andopinion dynamics field in ABM and a pragmatic application

Complexity 25

of abstract argumentation theory To our knowledge we areunaware of existing literature on agent-based modeling thatexplicitly relates collective choice and the notions of deliber-ation and opinion diffusion through abstract argumentationas we have done it

Most models in the literature of opinion diffusion areinterested in opinions because these have an influence incollective decision and in questions of social order Forinstance in [3] the authors are interested in consensus andin how a group collectively chooses among two alternativesIn other models authors are interested in the emergenceof extremism [8] and on the distribution of opinions whenextremists are introduced in the population [5 33 37] whileother authors coin the notion of opinion polarization as anemergent property of the system [8 12] They show usingmodels of ldquobounded confidencerdquo and opinion diffusion withtrust that three different kinds of steady states (unipolarbipolar and central) are possible depending on whetheragents are sufficiently uncertain about their opinions andsufficiently connected andor a certain proportion of indi-viduals are already extreme Recent articles on informationand opinion dynamics stress the importance of governanceand government intervention in the spread of emotions(opinions) in the frangibility of social consensus systemIn [10] the authors show that government intervention inthe spread of negative emotions can lead to an even fasterspread of negative emotions (single-peeked convergence) andin a faster collapse of the social consensus system Opinionas a function of trust is studied in [12] and relates to howindividuals form their opinions on the basis of how muchthey trust agents in their networks

Another stream of discrete opinion dynamics modelsthat explain the emergence of opinion can be found in theliterature Computer multiagent models of attitude changebased on Latanersquos social impact theory are presented in [4] Inthe theory an agent changes her opinion on the basis of theinformational impact she is subject to which depends on thepersuasiveness (strength) supportiveness and immediacy(group structure) of the environment and information shereceives The simulations predict two emergent groups ofphenomena the shifting of attitudes towards incompletelypolarized equilibria and the formation of coherent clusteringof subgroups with deviant attitudes Similarly but in acontinuous more argumentative fashion Mas and Flache[11] present a model of opinion diffusion using argumentsArguments are considered to be for or against a proposal andare given an agent-dependant relevance in dyadic persuasiveinteractions Arguments also determine the agentsrsquo opinionsand their dissimilarities which posits the rules of pair-wiseencounters in theirmodel as agents are assumed to form theiropinions according to the arguments they own They showthat in an argumentative opinion diffusion model the onlyobservable steady states are bipolarization and consensusand that bipolarization can emerge from interactions amongsimilar agents In other words bipolarization of opinionsis possible with homophily and without negative influenceif there is argumentation They subsequently compare theirresults to experimental data In [38] a similar approach toargumentation and opinion formation is taken but for the fact

that they introduce different types of arguments explicitlyapply social judgment theory [22] and use survey data toestablish model-to-real-world comparisons

Closer to opinion formation and abstract argumentationGabbriellini and Torroni [39] are the first to have origi-nally and soundly merged opinion diffusion and abstractargumentation They define an agentrsquos opinion as a functionof the arguments she holds and the attack relation amongthem They devised a focused peer-to-peer dialogue systemof persuasion (NetArg) inspired from Mercier and Sperberrsquosargumentative theory of reasoning [40] which used onlyabstract argumentation to study opinion polarization andopinion dynamics Moreover they considered networks andthe notions of trust and epistemic vigilance to define adynamics for knowledge revision and trust itself whichwe clearly lack in our model They showed that if a con-servative belief operator in argumentation was applied byagents when they reasoned then their dialogue protocoldid not increase polarization among agents That said theyused the model to study the effect of Groenvetterrsquos weaklink theory in the spreading of arguments and were notparticularly interested in notions like judgment accuracy orcoherence nor in tackling questions related to collectivedecision-making procedures Their system is very expressiveand helps position our work in the litterature our work is atthe frontier of pure argumentative opinion diffusion modelsopinion diffusion with arguments and continuous bounded-confidence models of opinion diffusion a la Deffuant

On another note work on collective cognitive conver-gence [41] and opinion sharing [19] shows that consensustowards a certain ldquocorrectrdquo opinion or cognitive state is alwayspossible yet dependent on noise variability and awarenessof agents In [20] the authors show that learning about anexogenous correct state of the world (represented by bits)under confidence was possible but only if the agents werenot too confident In a homogeneous population they showthat the higher the confidence the worse the learningmdashthevery confident agents do not learn the properties of the truestate of the world and disrupt the learning process of the lessconfident ones Collective cognitive convergence can be seenas a result of deliberation in truth-seeking models

When it comes to abstract argumentation theory we takean approach that wires two types of dialogues that are well-studied in the literature persuasion dialogues [42] and delib-eration dialogues [43] Another interesting line of work is theone on mechanism design [44] or the problem of devisingan argumentation protocol where strategic argumentation isnot a liability for success in debates We tackle mechanismdesign in a different way though Instead of consideringstrategy-proofness we are interested in how differences inprotocol can result in epistemologically ldquobetterrdquo collectivechoices and guarantee that opinion distributions are favorablefor deliberation For a survey on persuasion dialogue see[42]

Work on agent-based argumentation usually assumes thatthe semantic relationship between arguments is fixed Inother words if two individuals were to put two argumentsin the public arena such that one logically attacked the otherthen everyone would agree that such attack exists [45 46]

26 Complexity

Other models which do not make this restrictive assumptioncan also be found in the literature [27 47] and derive fromthe class or family of opponent models [27 48] in which twoopposing sides attempt to win a dialogue Ourmodel is in theintersection of these two but the frame combining opiniondiffusion of the kind and argumentation remains new

The idea of mixing interpersonal influence and verticalcommunication however is not original It is described andimplemented in innovation diffusion models such as [5 49]In both vertical communication is modeled as exogenoustransparent informationmdashagents are aware of the existenceof an innovation thus triggering several processes of choiceand stabilization of opinions Also in [50] an Eulerian modelis implemented to show the effect of media and exogenousinformation on opinion distributions With respect to thispoint the originality of our work is in that the informationemitted as vertical communication is endogenous It is issuedfrom a deliberation model that agents shape on the basisof their opinions arguments and behavior In the spirit of[51] where the authors control for the design of verticalcommunication (to whom the vertical communication isaddressed its timing how it influences agents opinions) wecontrol for the process generating the information for ourown set of variables length of debates majority quotas andthe frequency at which discussions take place

6 Conclusion and Perspectives

The main objective of this article was to build a bridgewhere decision-making argumentation and opinion dif-fusion could come together We proposed a model thatcombined abstract argumentation theory and a boundedconfidence opinion diffusion model and showed to whatextent it could explain variance of opinions extremismcoherence in collective decisions and judgment accuracyon arguments given an ideal state of full information Thesecond objective was to show in what way governance pro-viding agents with arenas of discussion and deliberation wasimportant in addressing the success of collective decision-making processes and the quality of their outcomes

The model revealed that allowing for more deliberationtime though in lower frequency and allowing for widerparticipation in deliberation increased the variance of opin-ions and the proportion of extremists in a group Theseobservations are consistent with the combination of resultsfound in [9] and in [13 14] which stress that deliberationmay polarize groups and may have a meager effect on theshifts of opinions and inconsistent with [16] where it isargued that opinions tend to moderate after deliberationIt happens that deliberation alone did moderate opinionyet when integrated into a more complex system in whichindividuals were allowed to interact with one another andnot all that was deliberated was accepted its influence wasshadowed by other more individualistic dynamics

Undeniably the grounded semantics played an importantrole in the weak effect of deliberation since it models askeptical way of reasoning over arguments For agentsaccepting arguments thus updating their opinions accord-ingly happened not too often Nevertheless asking for more

deliberation did increase judgment accuracy as observed in[18] yet in a marginally decreasing fashion We showed thatvoting within the deliberation protocol not only increasedthe proportion of extremists and the variance of opinions ina group but also determined how coherent deliberation andvoting procedures were with each other Lastly we showedthat agents that are focused judged arguments better hadmore stable opinions and constituted a group with higherproportion of extremists than their naive counterparts

In terms of governance the model says that there is per-haps no trade-off between extremismand judgment accuracyInstead it asserts that the higher the variance of opinions thecloser the group gets to the correct decisions This situationmay be interpreted as Landermore and Pagersquos notion of ldquoposi-tive dissensusrdquo (in complex collective decision-making tasks)[36] as opposed to consensus as a normative requirementfor correct collective decision-making If a decider has toorganize deliberation to legitimize a public policy dependingon what kind of world he or she wants different set ofparameters may be chosen to account for different levels oflegitimacy and correctness For instance one may ask for adecision to be legitimate to respect a certain level of coherenceor of judgment accuracy and to be discussed by a sundryof different-opinionated peoples If so deliberation has tobe made more often more participants have to be includedin debates or in deliberation instances and not too manytime steps should be introduced in between two deliberationinstances So the model may capture normative ideas todefine correct and legitimate collective decisions and tomake recommendations on those grounds Requirements ofhigh consensual deliberative accuracy (judgment accuracy)legitimacy (coherence) and diversity (variance of opinions)are all taken into account and for such a reason the model israther successful in describing a sound process of collectivedecision-making with interpretable outcomes

61 Extensions of the Model and Other Ideas Our model canbe extended in many ways Value-based abstract argumen-tation frameworks (VAFs) for example as an extension ofDungrsquos argumentation framework provide a formal descrip-tion of the process of decision-making in which argumentsare given values and audiences (set of agents) ignore attacksbetween arguments on the basis of their preferences over val-ues Social abstract argumentation frameworks (SAFs) alsoprovide an interesting and seemingly convenient formalismfor our model In SAFs agents vote on the acceptability ofarguments and the resulting labeling is a combination of thearguments that are voted for and labeling-based semantics Inour model we apply the same rule yet only for one argumentand given precise agent-based votingmechanisms and votingrules which are eventually parametrized

To consider our model as a specification of a mixed VAFand SAF model may be of great interest for future workThis formalism can provide solid foundations to describeexplain interpret and eventually extend the results foundhere from an argument-based perspective On those groundsfuture work will focus on creating a hybrid argumentationframeworkmdashwith its semanticsmdashthat takes heed of bothvalues or principles and voting

Complexity 27

Concerning other possible extensions of the model anatural reflex would be to argue on the validity and therobustness of the mix between the bounded confidenceopinion diffusion models and deliberation Consequentiallyadding deliberation to different bounded confidence modelsof opinion diffusion such as the Hegelsmann-Krause model[37] or to models of social impact theory like in [4] canbe of great interest for future work We acknowledge that afiner understanding of argumentation frameworks randomlattices and their respective applications for this modelare necessary Testing for different labeling-based semanticsor even trading labeling-based semantics for scoring-basedsemantics (semantics that defines acceptability on scoresgiven to arguments) such as debate semantics and ranking-based semantics (semantics that yield a preorder) may beinteresting to consider in future work

More into modeling deliberation it is not inadequateto believe that agents may also advance proposals duringdeliberation and replace to some extent the proposal forwhich the deliberation is taking place Endogenizing theprocess that produces the type of argument to be deliberatedon may be interesting to explore either by having agentsstrategically replace proposals at certain moments duringthe debate or having the central authority attempt to chooseproposals such that the probability of having them acceptedis high Agents may also be thought to be autoorganizing(independent of a central authority) and concede on the sizeof the instances of deliberation To consider differences insemantics across agents or even endogenizing the semanticson how urgent or important the proposal that is beingdiscussed is can also be an interesting direction to take

Last but not least many nontrivial modifications of ourmodel are possible and most should include better-thoughtdeliberation protocols It may be interesting to design andobserve protocols in which deliberation only affects agentsthat are actually debating for example Trust network effectsmultidimensionality of opinions new processes of argumentexchange or learning among agents are notions to furtherdevelop in order to make the model more realistic and relaxsome unreliable assumptions In sum the model is to berefined and extended with the objective of either studyingconcrete cases of deliberative polling and opinion dynamicsor implementing more intuitive thought experiments

Appendix

We present below some of the pseudocode we used toimplement the model

Algorithms

(a) Algorithm 1 implements the dyadic influence in themodel It describes how two agents interact andupdate their opinions

(b) Algorithm 2 implements the deliberative influence inthe model which describes how agents update theiropinions in sight of the accepted deliberated proposalarguments

(c) Algorithm 3 is used to obtain the grounded labelingof an argumentation framework

(d) Algorithm 4 provides an overview of a run in thesimulations119872 is the number of proposal argumentsto discuss which we set to 100 in the model

Each object in the system is indexed by a119908ℎ119900 number Hencea set of agents is annotated by a set of natural numbers Weuse this fact in the implementation of the algorithm 119874119887119895119890119888119905119894denotes the object of 119908ℎ119900 number 119894 Agentsrsquo opinions (119900) arecalibrated so that 119900 isin [minus1 1]Data Availability

The program and data used to support the findings of thisstudy are available from the corresponding author uponrequest

Disclosure

A preliminary version of this work was presented anddiscussed at the conference Bridging the Gap between FormalArgumentation and Actual Human Reasoning on October 4-5 2018 at BochumGermanyThe authors thank the audiencefor their valuable comments and suggestions

Conflicts of Interest

The authors declare that they have no conflicts of interestregarding the publication of this article

Acknowledgments

The contribution of Gabriella Pigozzi was supported by theDeutsche Forschungsgemeinschaft (DFG) and the CzechScience Foundation (GACR) as part of the joint project FromShared Evidence to Group Attitudes (RO 45486-1)

Endnotes

1 In other words the superiority of the ldquoright answerrdquo orlabeling in our case will appear as such to all

2 All agents hold the same opinion3 By sensitivity we mean how much an agent is ready to

change her opinion on the sole fact that an informationalcue has been debated on

4 The assumption is strong but this trust among agentsmay be rooted from a genuine individual motivation toreach a democratic consensus

5 We adopted a rule of thumbs that says that if an agentis neutral with respect to the principle then she willvote for any proposal that is presented to her duringdeliberation

6 If the argument is not unique she chooses one atrandom

7 If there are more than one candidate the agent will favorthe argument at minimal distance from 119868

28 Complexity

Require Vector of values for parameters (119881 = (120583 119880 119879))Ensure Successful interactions among agents119896 larr997888 0119873119890 larr997888 0

while 119896 le |119873|2 do ⊳Fill119873119890 with |119873|2 agents119896 larr997888 119896 + 1119873119890 larr997888 119877119886119899119889119900119898(|119873| + 1) ⊳Pseudo-random generator Generates integer between 0 and |119873|119873119903 larr997888 0 120575119894119895 larr997888 0 ⊳Variable to store differences in opinionfor all 119894 isin 119873119890 doRandomly pick a number 119895 isin 119873(119873119890 cup 119873119903)119873119903 larr997888 119873119903 cup 119895 120575119894119895 larr997888 |119900119894 minus 119900119895|procedure dyadic-update(119894 119895) ⊳Two agents 119894 and 119895 discuss and update opinions1199001015840 larr997888 119900119894 (120583119895 119880119895 119879119895) larr997888 (120583119894 119880119894 119879119894) larr997888 (120583 119880 119879) ⊳Save agent irsquos opinion before updating and homogenize agents

if 120575119894119895 lt 119880119894 then119900119894 larr997888 120583119894119900119895 + (1 minus 120583119894)119900119894if 120575119894119895 gt 119879119894 then119900119894 larr997888 (1 + 120583119894)119900119894 minus 120583119894119900119895if 120575119894119895 lt 119880119895 then119900119895 larr997888 1205831198951199001015840 + (1 minus 120583119895)119900119895if 120575119894119895 gt 119879119895 then119900119895 larr997888 (1 + 120583119895)119900119895 minus 1205831198951199001015840

Algorithm 1 Dyadic social influence

Require Vector of values for parameters (119881 = (120574 119901119886 119901119903))Ensure Successful interaction between proposal arguments and agents

for all 119894 isin 119873 doprocedure Probability-Update(V119886 119901119886 119901119903) ⊳Update individual probabilities of change due to deliberated cues119909119894 larr997888 119901|V119886minus119900119894 |2119886 119910119894 larr997888 1199012|V119886minus119900119894 |119903119902 larr997888 0if L119886119887(119868) = 119894119899 then

procedure deliberative-update(119894 119886) ⊳Agent i argument 119886120574119894 larr997888 120574 119902 larr997888 119906 sim U(0 1) ⊳ 119906 is a realization of a uniform random variableif 119902 lt 119909119894 then119900119894 larr997888 120574119894119900119894 + (1 minus 120574119894)V119886break

if 119902 lt 119910119894(1 minus 119909119894) then119900119894 larr997888 (1 + 120574119894)119900119894 minus 120574119894V119886Algorithm 2 Deliberation influence

8 In terms of computational complexity naive agentsmaketheirmoves in119874(|A119894| log(|A119894|)) while focused agents doat least as bad

9 Tables have memory Therefore debates are not neces-sarily trees

10 If there are less than (119899119863 times |119873|)2 agents for any of bothgroups the CA summons all of them to the deliberationtable

11 Or careful consideration of the viewpoints of otherswhich implies that citizens keep an open mind and donot reject arguments outright

12 A ldquohealthyrdquo consensus refers to the case in which manyconsensual decisions are taken but opinion-wise agentsdo not agree with each other Quotes are used on theword healthy because some political science theorists

(eg [36]) believe that in certain situations agreement inopinions for deliberation is epistemologically damaging

13 Simulations on groups of |119873| = 800 and |119873| = 400agents shows that scaling the population has an effect onall themetrics but it is confounding depending onwhichcouples of attitudes structures are considered (119880 119879) Westick to |119873| = 400 as in [9] for comparability andsimulation running time

14 By balanced wemean with as many arguments with V119886 lt0 as with V119886 gt 0

15 Experiments on the system suggest that a bigger(smaller) argument pool 119872 = 1000 (119872 = 200) isassociated with a higher (lower) variance of opinions(and extremism) and higher (lower) judgment accuracyFor the rest of the observations 119872 seem to have noeffect We choose a middle value for 119872 Similarly theargument pool chosen to be balanced or uniformly

Complexity 29

Require Argumentation framework 119860119865 = (AR)Ensure Assignment of labels to arguments from the grounded labeling of 119860119865L(A)

L0(A) larr997888 (0 0 0)119896 larr997888 0do119894119899 L119896+1(A) larr997888 119894119899 L119896(A) cup 119886 | 119886 is not labeled 119894119899 L119896(A) 119886119899119889 forall119887 isin A 119904119905 119887 R 119886 then 119887 isin 119900119906119905 L119896(A)119900119906119905 L119896+1(A) larr997888 119900119906119905 L119896(A) cup 119886 | 119886 is not labeled 119894119899 L119896(A) 119886119899119889 exist119887 isin A 119904119905 119887 R 119886 and 119887 isin 119894119899 L119896+1(A)

while L119896+1(A) = L119896(A) = 0Algorithm 3 Solve argumentation frameworks (with grounded semantics)

Require Vector of values for parameters (119881)Ensure Vector of statistics (119880)

procedure Set-parameters(119881) ⊳Initialize parameters for scenarioprocedure Init-objs(119899119898 119896 ) ⊳Create119898 arguments 119899 agents and 119860119865120576procedure Solve-Argumentation-Framework(119860119865) ⊳Assigns the epistemic labelingL120576 toA119894 larr997888 1119905 larr997888 1while 119894 le 119872 do ⊳ 119872 the number of arguments to discuss119868119894 larr997888 119866119890119899 119886119903119892(1) ⊳Generates proposal argument 119868119860 119894119905 larr997888 (119868119894 0)L119886119887(119868119894) larr997888 119906119899119889 ⊳Initialize label for 119868119895 larr997888 1 ⊳Initialize debate counter 119895while ((L119886119887(119868) = 119906119899119889 or 119898 le 119895) and 119895 le 119898) doif 119905 equiv 0(119898119900119889 119905119863) then

procedure build-argumentation-framework(119899119863 119887119890ℎ119886V119894119900119903) ⊳Build the argument graph agents deliberate(119861119894119905 119878119894119905) larr997888 (119883 119884) sub (⋃119897isin119873119899119863 A119897119894119905 ⋃119897isin119873119899119863 R119897119894119905) ⊳ 119873119899119863 sub 119873 are agents sampled by the CA119860 119894119905 larr997888 119889(119860 119894119905 (119861119894119905 119878119894119905))procedure Solve-Argumentation-Framework(A119889) ⊳FindL(A119889)L119886119887(119868119894) larr997888 L119886119887119863(119868119894)119895 larr997888 119895 + 1

else119896 larr997888 1while 119896 le 119905119863 doprocedure dyadic-social-influence(120583 119879 119880) ⊳Agents discuss one-to-one119896 larr997888 119896 + 1119905 larr997888 119905 + 1

if 120572 gt 0 andL119886119887(119868119894) = 119900119906119905 then ⊳Agents vote for the proposalif (|119897 isin 119873|119900119897 times V119868119894 ge 0| ge 120572 times 119899) then

L119886119887(119868119894) larr997888 119894119899else

L119886119887(119868119894) larr997888 119900119906119905if L119886119887(119868119894) = 119894119899 thenprocedure Deliberation-influence(120574 119901119886 119901119903) ⊳Agents are influenced by the result of the decision process119894 larr997888 119894 + 1119905 larr997888 119905 + 1119880 larr997888 119862119900119898119901119906119905119890 119878119905119886119905119894119904119905119894119888119904(119881)

return 119880 ⊳Get the statisticsAlgorithm 4 Run for one scenario

generated does not seem to play a significant explanatoryrole on any of the metrics

16 Experiments show that themaximumnumber of debatesis not an important parameter even if it is an intuitiveone when considering decision-making processes Inpractice debates end after at most 119898 + 1 deliberation

steps and even then it happens rarely However to avoidunpleasant surprises like infinite debates on a completeargument graph containing the proposal argument weadded the constraint 119898 Its value is precisely 7 becausethe biggest minimum number of debates we studyis 6

30 Complexity

17 To ensure comparability with the deliberative andmixedopinion dynamics models on judgment accuracy andcoherence agents vote on proposal arguments every 119905119863times(119898+1) 119905119863 and119898does not explain variability in judgmentaccuracy nor in coherence

18 Student t-tests for independent samples are robust tothe violation of two hypothesis on the distribution ofthe samples normality and homogeneity of variancesas long as the sample size is big enough (central limittheorem) and if the difference in size of the comparedsamples is smallrespectively

19 Studentrsquos test version forwhen variances are not assumedequal and the difference in size of the compared samplesis big

20 We do not correct for homoskedasticity nor normality oferrors in the estimates

21 Balanced as the same number of runs per scenario orsuch that no significant correlations are found betweenparameters

References

[1] R Axelrod ldquoThe dissemination of culture a model withlocal convergence and global polarizationrdquo Journal of ConflictResolution vol 41 no 2 pp 203ndash226 1997

[2] D M Rousseau Reinforcing the MicroMacro Bridge Organiza-tionalThinking and Pluralistic Vehicles SAGEPublications SageLos Angeles Calif USA 2011

[3] S Galam and SMoscovici ldquoTowards a theory of collective phe-nomena consensus and attitude changes in groupsrdquo EuropeanJournal of Social Psychology vol 21 no 1 pp 49ndash74 1991

[4] A Nowak J Szamrej and B Latane ldquoFrom private attitude topublic opinion a dynamic theory of social impactrdquo Psychologi-cal Review vol 97 no 3 pp 362ndash376 1990

[5] G Deffuant S Huet and F Amblard ldquoAn individual-basedmodel of innovation diffusion mixing social value and individ-ual benefitrdquo American Journal of Sociology vol 110 no 4 pp1041ndash1069 2005

[6] G Deffuant D Neau F Amblard and G Weisbuch ldquoMixingbeliefs among interacting agentsrdquoAdvances in Complex Systems(ACS) vol 3 no 01n04 pp 87ndash98 2000

[7] J Rouchier P Tubaro and C Emery ldquoOpinion transmission inorganizations an agent-based modeling approachrdquo Computa-tional and Mathematical OrganizationTheory vol 20 no 3 pp252ndash277 2014

[8] M Meadows and D Cliff ldquoThe relative disagreement model ofopinion dynamics where do extremists come fromrdquo in Pro-ceedings of the International Workshop on Self-Organizing Sys-tems Lecture Notes in Computer Science pp 66ndash77 Springer2013

[9] W Jager and F Amblard ldquoUniformity bipolarization and plu-riformity captured as generic stylized behavior with an agent-based simulation model of attitude changerdquo Computational ampMathematical Organization Theory vol 10 no 4 pp 295ndash3032005

[10] Y-F Zhang H-Y Duan and Z-L Geng ldquoEvolutionary mech-anism of frangibility in social consensus system based onnegative emotions spreadrdquo Complexity vol 2017 Article ID4037049 8 pages 2017

[11] A Flache andMWMacy ldquoSmall worlds and cultural polariza-tionrdquoThe Journal of Mathematical Sociology vol 35 no 1-3 pp146ndash176 2011

[12] A Tsang and K Larson ldquoOpinion dynamics of skepticalagentsrdquo in Proceedings of the 13th International Conference onAutonomous Agents and Multiagent Systems AAMAS 2014 pp277ndash284 International Foundation forAutonomousAgents andMultiagent Systems France May 2014

[13] K M Hansen Deliberative democracy and opinion formationUniversity Press of Southern Denmark Odense 2004

[14] C R Sunstein ldquoThe law of group polarizationrdquo Journal ofPolitical Philosophy vol 10 no 2 pp 175ndash195 2002 WileyOnline Library

[15] S Moscovici and W Doise Dissensions et Consensus unetheorie generale des decisions collectives Presses Universitairesde France-PUF 1992

[16] J S Fishkin and R C Luskin ldquoExperimenting with a demo-cratic ideal deliberative polling and public opinionrdquo ActaPolitica vol 40 no 3 pp 284ndash298 2005

[17] R C Luskin J S Fishkin and S Iyengar Considered Opinionson US Foreign Policy Face-To-Face versus Online Delibera-tive Polling International Communication Association NewOrleans La USA 2004

[18] J Barabas ldquoHowdeliberation affects policy opinionsrdquoAmericanPolitical Science Review vol 98 no 4 pp 687ndash701 2004

[19] O Pryymak A Rogers and N R Jennings ldquoEfficient opinionsharing in large decentralised teamsrdquo in Proceedings of the 11thInternational Conference on Autonomous Agents andMultiagentSystems vol 1 pp 543ndash550 Spain June 2012

[20] J Rouchier and E Tanimura ldquoWhen overconfident agents slowdown collective learningrdquo Simulation vol 88 no 1 pp 33ndash492012

[21] D Miller ldquoDeliberative democracy and social choicerdquo PoliticalStudies vol 40 S1 pp 54ndash67 1992

[22] M Sherif and C I Hovland Social Judgment Assimilation andContrast Effects in Communication and Attitude Change YaleUniversity Press 1961

[23] P M Dung ldquoOn the acceptability of arguments and its funda-mental role in nonmonotonic reasoning logic programmingand n-person gamesrdquo Artificial Intelligence vol 77 no 2 pp321ndash357 1995

[24] M Caminada and G Pigozzi ldquoOn judgment aggregation inabstract argumentationrdquo Autonomous Agents and Multi-AgentSystems vol 22 no 1 pp 64ndash102 2011

[25] P Baroni M Caminada andM Giacomin ldquoAn introduction toargumentation semanticsrdquoThe Knowledge Engineering Reviewvol 26 no 4 pp 365ndash410 2011

[26] S Chambers ldquoDeliberative democratic theoryrdquo Annual Reviewof Political Science vol 6 no 1 pp 307ndash326 2003

[27] E Bonzon andNMaudet ldquoOn the outcomes of multiparty per-suasionrdquo in Proceedings of the 10th International Conference onAutonomous Agents and Multiagent Systems 2011 AAMAS 2011vol 1 pp 47ndash54 International Foundation for AutonomousAgents and Multiagent Systems Taiwan May 2011

[28] J Ranciere Disagreement Politics and Philosophy University ofMinnesota Press 1999

[29] T Bench-Capon and K Atkinson ldquoAbstract argumentation andvaluesrdquo in Argumentation in Artificial Intelligence pp 45ndash64Springer 2009

[30] M Caminada ldquoOn the issue of reinstatement in argumenta-tionrdquo in Proceedings of the European Workshop on Logics in

Complexity 31

Artificial Intelligence vol 4160 of Lecture Notes in ComputerScience pp 111ndash123 Springer Berlin Heidelberg

[31] GAVreeswijk ldquoAn algorithm to computeminimally groundedand admissible defence sets in argument systemsrdquo in Pro-ceedings of the 2006 conference on Computational Models ofArgument Proceedings of COMMA 2006 vol 144 pp 109ndash120IOS Press Amsterdam The Netherlands 2006

[32] C W Sherif M Kelly H L Rodgers et al ldquoPersonal involve-ment social judgment and actionrdquo Journal of Personality andSocial Psychology vol 27 no 3 pp 311ndash328 1973

[33] G Deffuant F Amblard G Weisbuch and T Faure ldquoHow canextremism prevail A study based on the relative agreementinteraction modelrdquo Journal of Artificial Societies and SocialSimulation vol 5 no 4 p 1 2002

[34] E Awad M W Caminada G Pigozzi M Podlaszewski andI Rahwan ldquoPareto optimality and strategy-proofness in groupargument evaluationrdquo Journal of Logic and Computation vol 27no 8 pp 2581ndash2609 2017

[35] D Grossi and G Pigozzi ldquoJudgment aggregation a primerrdquoSynthesis Lectures on Artificial Intelligence and Machine Learn-ing vol 8 no 2 pp 1ndash151 2014

[36] H Landemore and S E Page ldquoDeliberation and disagreementproblem solving prediction and positive dissensusrdquo PoliticsPhilosophy and Economics vol 14 no 3 pp 229ndash254 2015

[37] R Hegselmann and U Krause ldquoOpinion dynamics andbounded confidence models analysis and simulationrdquo Journalof Artificial Societies and Social Simulation vol 5 no 3 2002

[38] A Stefanelli and R Seidl ldquoModerate and polarized opinionsUsing empirical data for an agent-based simulationrdquo inProceed-ings of the Social Simulation Conference 2014

[39] S Gabbriellini and P Torroni ldquoA new framework for ABMsbased on argumentative reasoningrdquo in Advances in SocialSimulation vol 229 pp 25ndash36 Springer Berlin Germany 2014

[40] H Mercier and D Sperber ldquoWhy do humans reason Argu-ments for an argumentative theoryrdquo Behavioral and BrainSciences vol 34 no 2 pp 57ndash74 2011

[41] H V Parunak T C Belding R Hilscher and S BruecknerldquoModeling and managing collective cognitive convergencerdquoin In Proceedings of the 7th iNternational Joint Conference onAutonomous Agents and Multiagent Systems vol 3 pp 1505ndash1508 International Foundation for Autonomous Agents andMultiagent Systems 2008

[42] H Prakken ldquoFormal systems for persuasion dialoguerdquo TheKnowledge Engineering Review vol 21 no 2 pp 163ndash188 2006

[43] K Atkinson T Bench-Capon and P Mcburney ldquoA dialoguegame protocol for multi-agent argument over proposals foractionrdquoAutonomousAgents andMulti-Agent Systems vol 11 no2 pp 153ndash171 2005

[44] M Thimm and A J Garcıa ldquoClassification and strategicalissues of argumentation games on structured argumentationframeworksrdquo in Proceedings of the 9th International Conferenceon Autonomous Agents and Multiagent Systems vol 1 pp 1247ndash1254 International Foundation for Autonomous Agents andMultiagent Systems Canada 2010

[45] I Rahwan and K Larson ldquoMechanism design for abstractargumentationrdquo in Proceedings of the 7th International JointConference on Autonomous Agents and Multiagent SystemsAAMAS 2008 pp 1031ndash1038 International Foundation forAutonomous Agents and Multiagent Systems Portugal May2008

[46] H Prakken R Riveret A Rotolo and G Sartor ldquoHeuristics inArgumentation A Game-Theoretical Investigationrdquo EuropeanUniversity Institute 2008

[47] N Hadidi Y Dimopoulos and P Moraitis ldquoTactics and con-cessions for argumentation-based negotiationrdquo in Proceedingsof the COMMA vol 245 pp 285ndash296 2012

[48] N Oren and T J Norman ldquoArguing using opponent modelsrdquoin Proceedings of the International Workshop on Argumentationin Multi-Agent Systems pp 160ndash174 Springer 2009

[49] S Delre W Jager T Bijmolt and M Janssen ldquoTargeting andtiming promotional activities An agent-based model for thetakeoff of new productsrdquo Journal of Business Research vol 60no 8 pp 826ndash835 2007

[50] A Mirtabatabaei P Jia and F Bullo ldquoEulerian opinion dynam-ics with bounded confidence and exogenous inputsrdquo SIAMJournal on Applied Dynamical Systems vol 13 no 1 pp 425ndash446 2014

[51] V Barbet N Guiraud V Laperriere and J Rouchier Hagglingon Values Towards Consensus or Trouble 2019

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Submit your manuscripts atwwwhindawicom

10 Complexity

-1

-1 1

1

Latitude of acceptance

Latitude of rejection Agent i

Agent j

Figure 6 A dyadic interaction between agents 119894 and 119895 as expressedin (5) Agent 119894rsquos opinion falls within agent 119895rsquos latitude of noncommit-ment and conversely Neither agent affects the opinion of the other

25 Opinion Dynamics for Social Interactions In this subsec-tion we describe the opinion diffusion model based on pair-wise interactions among agents and deliberation Sherif rsquosand Hovlandrsquos social judgment theory [22] motivates part ofour approach It describes how individualsrsquo opinions changeon the basis of their attitude structures Attitude structuresrefer to the relative scope width or latitude of categoriesused by individuals when evaluating information namelythe latitudes of acceptance rejection and noncommitment[32] The idea behind this theory is that individuals changetheir positions only in accordance to how far or close thecommunicative cues they receive are from (to) their anchorpositions It holds that if communicative cues are far (close)from (to) an agentrsquos position say over her latitude of rejection(acceptance) then the agent shifts her position away from(towards) the position defended by the cues In the casewherethe cues fall within the agentrsquos latitude of noncommitmenther position does not change (see Figure 6 and Equation (5))

251 Pair-Wise or Dyadic Opinion Dynamics As agentsmay communicate and deliberate collectively they may alsoengage in one-to-one conversations with other agents toponder on their positions We loosely associate this type ofcommunication with dyadic nonargumentative exchange ordiscussion based on fallacious arguments and persuasionIn the light of social judgment theory and the descriptionof agents in the system we model pair-wise symmetricinteractions following the opinion dynamicsmodel in [9] Anagent 119895rsquos influence on an agent 119894rsquos opinion at time 119905 is governedby the ensuing difference equation

119900119905+1119894 =

119900119905119894 + 120583119894 (119900119905119895 minus 119900119905119894) 119894119891 10038161003816100381610038161003816119900119905119894 minus 11990011990511989510038161003816100381610038161003816 lt 119880119894119900119905119894 minus 120583119894 (119900119905119895 minus 119900119905119894) 119894119891 1003816100381610038161003816119900119905119894 minus 119900119905119894 1003816100381610038161003816 gt 119879119894119900119905119894 119900119905ℎ119890119903119908119894119904119890(5)

where the parameter 120583119894 isin [0 05] controls for the strengthof the attraction and repulsion in social influence and 119879119894and 119880119894 are the latitudes of rejection and acceptance for agent119894 respectively The parameter 120583119894 may be thought of as therelative importance agent 119894 gives to the opinions of her peersor analogously the weight she gives to her own opinionwhen updating 120583 lt 05 means that agents will never givemore weight to the opinions of other agents than to their

own (egocentric bias) When two agents 119894 and 119895 discuss iffor instance 119895 advances an informational cue (argumentpersuasion tactic) and it happens to be close enough to119894rsquos opinion anchor then 119894 shifts her opinion towards thedirection of 119895rsquos informational cue The symmetric influencefrom agent 119894 to agent 119895 takes effect in the same way (seeFigure 6)

We add to the dyadic dynamics a rule of encounters ateach step of pair-wise interactions each agent meets exactlyone other agent at random This rule may be associated withrandom day-to-day encounters among agents Steps of socialinfluence correspond to nondebate steps (119889) in decision-making processes

252 Deliberative Opinion Dynamics We define an opinionupdate equation that links the proposal arguments that areadvanced during deliberation and opinions We combine theuncertain and probabilistic nature of the effect of delibera-tion have it a moderating [15 16] or polarizing [14] one witha mechanism similar to the one in social judgment theorybased on the distance between different informational cuesand opinion anchors [32] The probabilistic modeling of theopinion update can be related to deliberation encouragingopen-mindedness11 during collective discussion [18] Fromthis choice it follows that deliberated cues can affect even themost extreme of agents as opposed to some classic opiniondiffusion models (eg [5 8]) where once agents becomeextremists they may no longer become moderate

LetP be a proposal V119905119868 the proposal argument 119868rsquos level ofsupport for a principle P and 119900119905119894 an agent 119894rsquos opinion at time119905 Then given the difference 120575119905119894 = (V119905119868 minus 119900119905119894)2 we define 119894rsquosprobability of being attracted to a decisionrsquos informational cue119868 by 119909119905119894 = 119901|120575119905119894 |119886 where 119901119886 denotes a general probability param-eter that characterizes how important deliberated results arefor the group The parameter may also be interpreted asthe grouprsquos tendency to be swayed by a decisional majoritySimilarly we define 119894rsquos probability of being repulsed from adecision informational cue 119868 by 119910119905119894 = 1199011|120575119905119894 |119903 where 119901119903 denotesa general probability parameter that characterizes the grouprsquosdislike of deliberated results 119901119903 can also be thought of as thegrouprsquos distrust of the system symbolized by deliberation anddemocracy

Let L119886119887(119868) denote the epistemic status of the proposalargument 119868 at the end of a decision process over P everyagent 119894 updates her opinion as follows

(i) IfL119886119887(119868) = 119894119899

119900119905+1119894 =

119900119905119894 + 2120574119894120575119905119894 119908119894119905ℎ 119901119903119900119887119886119887119894119897119894119905119910 119909119905119894119900119905119894 minus 2120574119894120575t119894 119908119894119905ℎ 119901119903119900119887119886119887119894119897119894119905119910 119910119905119894119900119905119894 119908119894119905ℎ 119901119903119900119887119886119887119894119897119894119905119910 1 minus 119909119905119894 minus 119910119905119894

(6)

(ii) IfL119886119887(119868) = 119900119906119905 119900119905+1119894 = 119900119905119894where 120574119894 isin [0 05] is the strength of repulsion and attractionin the dynamicThemeaning of 120574119894 is analogous to the one of120583119894in the dyadic interactionsmodel but for the observation that 119894

Complexity 11

weights her opinion to an argument rather than to an opinionThe interpretation of this dynamics is straightforward Ifthe deliberated proposal argument is close to an agentrsquosopinion then it is very likely that the agent shifts its opiniontowards it Please note that the probability 119909119905119894 (119910119905119894 ) that anagent is attracted (repelled) to (from) an accepted deliberatedproposal argument is bounded below (above) by 119901119886 (119901119903)(equivalently 119909119905119894 isin [119901119886 1] 119910119905119894 isin [0 119901119903]) Steps of deliberativeopinion dynamics are associated with debate steps in which aconcluding collective decision is made

253 Mixed Opinion Dynamics Opinion dynamics in thecontext of decision-making processes can be best described asa combination of the two preceding dynamics In this opiniondynamics model agents are engaged in a democratic systemin which they deliberate vote for proposals and occasionallydiscuss with one another on the principle supported by theproposalsThe opinion dynamics for an agent 119894 can be writtenas a sequence of dyadic and deliberative opinion updates

119900119905+1119894 =

119864119902119906119886119905119894119900119899 (5) 119894119891 119905 equiv 0 (119898119900119889119905119863) and 119904119903 (sdot) = 119891119886119897119904119890119864119902119906119886119905119894119900119899 (6) 119894119891 119905 equiv 0 (119898119900119889119905119863) and 119904119903 (sdot) = 119905119903119906119890119900119905119894 119900119905ℎ119890119903119908119894119904119890

(7)

where 119904119903(sdot) is the decision processrsquos stop rule 119905119863 the numberof time steps between each deliberation step Equation (5)describes the dynamics for the dyadic interactions and Equa-tion (6) posits the changes in opinions due to the deliberatedcues Equation (5) applies when agents are not deliberatingand when each agent encounters another agent to exchangeinformation that may lead to local opinion updates Whenthere is deliberation a handful of agents deliberate and ifthey reach the stop condition imposed by the table all agentsvote for the proposal They update (see Equation (6)) theiropinions on the basis of the proposal argumentrsquos support forthe principle and the result of the scrutiny Otherwise thereis no change in the opinions of the agents A vote indicatesthe end of the decision-making process

3 Experiments and Calibration

In this section we introduce the metrics that enable us toobserve the simulations and characterize the calibration ofthe model

31 Observations and Initialization In this section wedescribe the simulations and the protocol used to test andobserve the results of our model We introduce the metricsof interest explain the calibration of the model describesome results obtained from the simulation data used forcalibration and conclude with a brief discussion on theexpected outcomes of the mixed opinion dynamics model

311 Metrics or Statistics of Interest Let 119878 denote the end ofa simulation We are interested in the effect of deliberationand of the modelrsquos procedural parameters on the followingmetrics or statistics

(i) Variance of opinions (119881119886119903(119900)) the variance ofopinions at time 119878 Since opinions live in the unionof the positive and negative unit intervals 119881119886119903(119900) isin[0 1] The higher the variance of the distribution isthe more ldquodiverserdquo the opinions are(ii) Proportion of extremists in the population(119901119903119900119901119890119909) the proportion of agents in the populationwith opinions such that |119900119894| ge 075 A high proportionof extremists makes ldquohealthyrdquo consensus12 difficult toreach and deliberation more or less informative(iii) Shifts of opinions (119878ℎ) [33] statistic that measuresthe aggregated change in individual opinion at time 119878with respect to aggregated individual opinion at thebeginning of the simulation

119878ℎ = 2sum119894isin119873 10038161003816100381610038161199000119894 minus 1199001198781198941003816100381610038161003816max119894isin1198731199000119894 minusmin119894isin1198731199000119894 (8)

119878ℎ is positive and bounded above by 2|119873| since 119900119894 isin[minus1 1] A low shift statistic implies that the processhas a small impact on opinions(iv) Judgment or consensual inaccuracy (119890119888) consen-sual accuracy of a group consists of an ad hoc statisticmeasuring a grouprsquos ability to infer correct labels forproposal arguments from a decision-making processgiven the ideal consensual labeling based on full-informationL120590

120576 (A) We use a Hamming-based dis-tance on labelings [34] to define the statistic

119890119888 = 1|I| sum119868isinI1198861198681003816100381610038161003816L119886119887120576 (119868) = L119886119887 (119868)1003816100381610038161003816 (9)

whereI is the set of all discussed proposal argumentsand 119886119868 = 12 ifL119886119887120576(119868) = 119906119899119889 or ifL119886119887(119868) = 119906119899119889119886119868 = 1 otherwise119890119888 lives in the interval [0 1] An inaccuracy statisticclose to 1 indicates that agents subdued to a particulardecision-making process make many mistakes injudging the labels of the proposal arguments Notethat when there is voting all proposal arguments arelabeled either 119894119899 or 119900119906119905 after deliberation(v) Coherence (119894119903) letL119886119887119863(119868) be the label obtainedfor 119868 from the deliberation process without votingThe coherence statistic measures how well votingresults adjust to results obtained during deliberationonly We use the proportion of arguments that havebeen labeled 119894119899 in the debate and that agents havevoted favorably for

119894119903 = 1003816100381610038161003816119868 isin I | L119886119887119863 (119868) = 119894119899 andL119886119887 (119868) = 11989411989910038161003816100381610038161003816100381610038161003816119868 isin I | L119886119887119863 (119868) = 1198941198991003816100381610038161003816 (10)

The coherence statisticrsquos domain is [0 1] If after 119872debates no deliberated central argument has beenlabeled 119894119899 or voting is not part of the procedure ofcollective decision-making then the statistic equals

12 Complexity

1 or said differently agents are perfectly coherentThis comes for the fact that if the central argumentis labeled 119900119906119905 then it is not even eligible for voting iflabeled 119906119899119889 then the debate will always be coherentwith the preferences of the agents since the result willbe a simple aggregation of their votes If the statisticequals 0 then none of the proposal arguments labeled119894119899 are voted in favorably by the agents It follows that ahigh coherence statistic implies that when agents votefor acceptable proposal arguments the results of thescrutiny reflect the consensual rationality expressedin the deliberation process

312 Parameters of Interest We recall the parameters ofinterest in our study that are linked to the structure of thedecision-making processes

(i) 119898 the minimum number of time steps in which adebate or a scrutiny occurs in a decision-making pro-cess before a final deliberated decision is submitted toa vote

(ii) 119899119863 the number of agents that deliberate as a propor-tion of the population

(iii) 119905119863 the number of time steps between debates inwhich pair-wise interactions among agents mayoccur

(iv) 120572 the proportional majority requirement for theacceptance of a proposal When 120572 = 0 it stands forno voting ldquoany proposal argument that is labeled 119894119899during deliberation is acceptedrdquo

Please recall that in terms of the definition of a decision-making process each parameter controls either the length orthe content of the sequence D119868 (119905119863 119898) or the rules that areapplied during the debate steps (120572 119899119863) (refer to Figure 4)313 Initialization All |119873| = 400 agents start off with anopinion 119900119894 drawn from a uniform distribution U(minus1 1) 13Given 119900119894 every agent 119894 randomly draws a setA119894 (|A119894| = 119896) ofarguments from a balanced14 argument pool A of119872 = 600nonneutral (V119886 = 0) arguments on the basis of 119900119894 if |119900119894| lt 025(the opinion is moderate) then agent 119894 randomly fills halfof her argument sack with arguments such that V119886 lt 0 andthe other half with arguments such that V119886 gt 0 (120601119898119894 = 05)Otherwise she randomly fills 120601119890119894 ge 05 of her sack witharguments such that V119886 and 119900119894 are of the same sign and theremaining (1 minus 120601119890119894 ) with arguments of the opposite sign15

Like for the opinion of agents every argument 119886rsquosadherence to the principle P V119886 is drawn from a uniformdistribution U(minus1 1) The attack relation that gives birth tothe ideal argumentation framework (119860119865120576) is established onthe basis of the V119886rsquos and the argumentsrsquo epistemic reach 119890119903

Let 120579(119886 119887 119903) = (|V119886 + V119887|2)119903 be an auxiliary positivereal-valued function that takes two arguments (119886 119887) and apositive real number (119903 gt 0) as inputThe probability that anyargument 119886 isin A creates a link to any other argument 119887 isin A

is given by the equation 119901 = 119901120579(1198861198872)119890 and thus argumentssupporting opposing views of the principle always attack each

other We fix 119901119890 the epistemic correlation parameter to 015If 119901119890 was greater than 015 then arguments of the same signwould attack each other too often (and focused agents wouldbe rarely incited to advance favorable arguments duringdeliberation) If 119901119890 was lower then the arguments of thesame type would induce an almost empty graph which isunrealistic The number of arguments that any argument 119886can attack is bounded by 119886rsquos epistemic reach that we fix to15 A higher value of epistemic reach makes the argumentlattice too conflicting and nearly bipartite A lower epistemicreachmakes the lattice not sufficiently conflicting in the sensethat too many arguments can attack the proposal argumentrelatively to the few that can defend it If any argument attacksthe proposal argument the chances that another argumentattacks the attacker are low Hence the proposal argumentis almost never accepted and deliberative opinion updatesbecome rare

The arguments in the resulting argumentation frame-work 119860119865120576 are given a permanent labelingL120590

120576 (A) (in shortL120576) using120590 = grounded semanticsWe choose the groundedlabeling-based semantics because it provides a unique admis-sible labeling respects minimal rationality constraints whilesimplifying the model and models a skeptic approach toaccepting arguments (see [23 25 31]) as it maximizes thecardinality of the set 119906119899119889 L120576 To some extent if we considerproposals as committing because they may guide collectiveaction choosing a skeptic semantics seems reasonable insofaras it labels an argument 119894119899 or 119900119906119905 only if it has no reasonto label it 119906119899119889 For instance if a proposal is a public policythat requires large amounts of resources and engages tofuture course of action then it may also determine policycycles and heavily burden a group and its future decisionsIn important situations like these it seems reasonable tolengthen deliberative cues and ask for more demanding andldquogroundedrdquo criteria for policy argument acceptability

On the proposalrsquos side we create an argument 119868 whoseadherence to P is also drawn from aU(minus1 1) and we label itL119886119887(119868) = 119906119899119889 119868 is interpreted as being the main argumentjustifying the discussed proposal P and V119905119868 as its supportfor P at time 119905 119868 cannot attack other arguments but otherarguments can attack it For each argument 119886 isin A a directedarc or attack from 119886 to 119868 is activated with probability 119901119901 =003120579(11988611986802) When 119901119901rsquos lower bound is too high the proposalargument is almost always defeated and thus deliberativeopinion updates do not happen When 119901119901rsquos lower boundis lower than 003 then the opposite occursmdashthe proposalargument is always accepted due to an absence of attackstowards it Finally we set the maximum number of debatesto119898 = 7 for the decision-making processes stop rules1632 Calibration and Simulation Protocol In this subsectionwe discuss the calibration of the models the terminationconditions for runs in the deliberative and mixed model andthe expected outcomes in terms of the observations

321 Calibrating Dyadic Opinion Dynamics Dyadic opiniondynamics correspond to a space of parameters in whichargumentation and deliberation spaces are not taken into

Complexity 13

Table 1 Parameters and parameter types and their domains used to compare dyadic collective and mixed opinion dynamics

Length ofD119868 Decisional in119863119868 Social influence Deliberation influence Cognitive119905119863 119898 119898 119899119863 120572 120583 119879 119880 119901119886 119901119903 120574 119896 119861119890ℎ119886V1198941199001199031 3 6 1 3 6 7 001 002 005 0 05 066 01 14 18 02 06 03 05 01 005 01 02 4 8 16 119873119886119894V119890 119865119900119888119906119904119890119889account Deliberated arguments and informational cues haveno effect on agent opinion but agents still vote for theproposal argument17 We use and calibrate this model forcomparability in terms of all our metrics since they arenot explicitly observed in [9] a reference opinion dynamicsmodel Furthermore for simplicity and comparability againwe suppose that agents are homogeneous in terms of theirattitude structures and opinion weightings We fix 120583119894 =120583 119879119894 = 119879 and 119880119894 = 119880 for all 119894 isin 119873 and for some triplet(120583 119880 119879) isin [0 05] times [0 2] times [119880 2]

Experiments suggest that the strength of the dyadic inter-actions is an explanatory factor of the time of convergence(120583 lt 05) and of the dyamicsrsquo steady states Bigger values of 120583speed up convergence towards a stable set of opinion clustersand affect the size and the relative position of these clustersin the opinion distribution 120583 however does not play a majorrole in the number of opinion clusters observed at the steadystate of the dynamics

We set 120583 to the value given to it in Jagger et alrsquos opiniondynamics model (120583 = 01) [9] For one it is the smallestreal number for which most results found in [9] hold and thedynamics are smooth secondly it seems to be a reasonableldquostrengthrdquo of influence considering that we would not wantpair-wise social interactions to completely shadow or maskan eventual effect of deliberation in the opinion dynamics

For different values of attitude structures 119879 and 119880 weget exactly the same results as in [9] in terms of the numberand density of the opinion clusters The different couplesof values of 119880 and 119879 model the size of the latitude ofnoncommitment (119879 minus 119880) and thus the agentsrsquo propensityto update opinions Whenever 119879 minus 119880 le 02 regularitiesin convergence and opinion clustering appear If 119880 gt 1there is always central convergence if 119879 lt 1 there is alwaysbipolar convergence with clusters of similar sizes For 119879 minus119880 gt 02 relatively low values of 119879 (119879 lt 1) and relativelyhigh values of 119880 always yield extreme bipolar convergencewith a relatively small cluster of moderate agents (bipolar-central convergence) For 119879 sufficiently high (119879 ge 12) and119880 sufficiently low (119880 le 08) in other words not very ego-involved agents there is a bigger spectrum of steady statesFor this reason we studied the metrics of interest for thesevalue differences and kept the couples of (119880 119879) depicted inTable 1 For these couples central bipolar bipolar-centralconvergence (3 clusters) and multicluster convergence (4 or5 clusters) are possible and result frommeaningfully differentattitude structures Judgment accuracy varies significantlyacross the four different attitude structures we retain and soit happens with the variance of opinions These scenariosprovides us with reference results from which to build upon

322 Calibrating Deliberative Social Interactions Delibera-tive social interactions correspond to social situations in

which individuals are not influenced by pair-wise discussionwith peers but are sensitive to collectively deliberated infor-mational cues On these terms parameters such as the sizeof agentsrsquo argument sacks (119896119894) the distribution of argumentsin them (120601119894) the minimum number of debates (119898) andthe proportion of debaters from the population (119899119863) are nolonger mute We assume for simplicity that the formulaexist120601119890exist119896forall119894(119896119894 = 119896 and 120601119890119894 = 120601119890 and 120601119898119894 = 05) is true with 119894 isin 119873for some value 120601119890 isin [05 1] and for 119896 isin N

The chosen domains for these parameters (see Table 1)are justified by the size of the population and the size of theargument pool (119872 = 600) which we set from the start If 119896is too big (119896 ge 32) andor too many agents are allowed inthe debate (119899119863 ge 02) then individuals always disprove thecentral argument and deliberation is never or rarely takeninto account in the update of opinions Similar effects occurwhen 119898 is high yet we only calibrate it with respect to themean running time of a simulation and the values of 119905119863Different distributions of 120601119890119894 seem to significantly or directlyexplain none of the statistics we analyze so we convenientlyset 120601119890119894 = 075 We chose the parameter 119896 to be always divisibleby 4 for it is convenient given the initial conditions regardingthe argument distribution in the agentsrsquo argument sacks

We fix the value space for the parameters 120574 119901119886 119901119903 as inTable 1 to account for different intensities of the effect ofdeliberative voting retroaction on the system For examplethe scenarios where the three parameters are at their lowestvalues correspond to the scenarios with the smallest fixedprior effect of deliberation in the model Increasing any ofthese parameters should mechanically be associated witha world in which agents are more sensitive to collectivedecisions The values for 119901119886 are taken from [13] wherethe author states that by means of deliberation 7 to28 of individuals changed their opinion from agreeingto disagreeing or vice versa on a referendum question forDenmarkrsquos participation in the EuroWe choose two differentvalues for 119901119886 that account for high (03) and very high (05)sensitivity to deliberation in a group 119901119903 on the other handis taken to be small (equal to 01) since we believe that thereexist agents that will always go against the reached consensusTheir numbers are meager

For simplicity we assume that for all 119894 isin 119873 120574119894 = 120574 forsome 120574 isin [0 05] In effect we posit that the heterogeneityof agents in the deliberative context only derives from theirarguments and their opinions We decide to calibrate 120574 inallusion to the strength of the social dynamics in pair-wiseinteractions In doing so we limit ourselves to consider threedifferent scenarios in which deliberation has respectivelyhalf equal and twice as much opinion-shifting power thanone-to-one social influence Finally we include the accept-ability voting quota rule 120572 whose domain is inspired fromclassical majority rules [35] observed empirically (Table 1)

14 Complexity

Table 2 Parameters and their domains used to study the effect of the procedural parameters on the metrics

Procedural parameters Other parameters119905119863 119898 119899119863 120572 119898 120583 119879 119880 119901119886 119901119903 120574 119896 119861119890ℎ119886V1198941199001199031 2 3 4 5 6 1 2 3 4 5 001 002 003 004 005 0 05 066 7 01 16 02 03 005 02 12 119873119886119894V119890 119865119900119888119906119904119890119889

323 Calibrating the Mixed Social Interactions Model Themixed interaction model ascribes to a parameter space inwhich the effect of collective choices on our metrics isnontrivial and where deliberation and voting on proposalsdetermine their acceptability Since the mixed model isequivalent to periodic iterations of pair-wise and collectivesocial interactions the calibration of the parameters in bothpreceding models holds (see Table 1 for the calibration)The first reason for this is comparability the second is thatthe previous calibration takes into account the fact thatboth dynamics are going to be combined We include thefrequency of debates by adding the parameter 119905119863 whichcontrols for the amount of pair-wise interaction amongagents between two distinct deliberation steps

324 Termination Conditions for Runs Simulations stoponce 100 proposals are deliberated on andor voted for inother words after 100 collective decision steps have occuredThe number of proposals discussed seems arbitrary yet it ishigh enough to observe the effects of deliberation on opiniondistributions and on other metrics related to coherence andjudgment accuracy We choose the number of runs as afunction of our research questions which give relevance tothe procedural parameters in the mixed model rather than tothe parameters set to describe the population and its behavior

325 Expected Outcomes of the Simulations We expect moredeliberation in terms of higher 119898 and 119899119863 to increasejudgment accuracy and at the same time to reduce thevariance of opinions In turn a smaller variance of opinionsimplies that the argument pool for deliberation is smaller andtherefore judgment accuracy should be lower Moreovervariance increasing dyadic interactions (or rejection) shouldalso increase judgment accuracy and coherence since bipo-larization and dissensus foster argument diversity in delibera-tionThe coherence statistic should be stronger in simulationsin which central convergence appears quickly and agentsare naive Attitude structures sensitivity to informationalcues and weights given to deliberated cues that makes pathsto bipolar convergence smaller or to central convergencelonger should be associated with higher judgment accuracyShifts of opinion should be more visible in scenarios inwhich extreme agents are pulled away from the extremesHence the sensitivity to deliberated cues (119901119886 119901119903) and 120574should explain the shifts When crossed with high values of119880 and 119879 the shifting power of deliberation should be at itshighest

In the end we do not know how these mechanismswill play out The results of the subsequent experimentsgive us insight on the interplay between the aforementionedeffects

4 Results

Before pointing at any result obtained from the mixedinteractions model we describe and compare the dyadicand deliberative interactions models in the parameter spaceobtained from the calibration in Section 3 (see Table 1)Primarily we require our results to describe two orthogonaltypes of opinion dynamics the pair-wise dyadic and thedeliberative The latter comprises scenarios in which indi-viduals do not influence each other by means of pair-wisediscussion and only update their opinions from delibera-tion the former scenarios where only pair-wise interactionsamong agents determine the dynamics We describe both onthe parameter space given in Table 1 by performing at least20 simulations per scenario Quantitative results from thepair-wise interaction model alone are described during thecalibration and are not treated heremdashdeliberative parametershave no effect on it Moreover we show how these differentmodel specifications yield qualitatively different opiniondistributions and compare them on the basis of the metricsof interest To this end we perform independent two-sampleStudentrsquos t-tests18 or Welch t-tests19 and compute confidenceintervals at a 095 level of confidence Last but not least wecomment ordinary least squares (OLS) regression estimatesto account for the direction and magnitude20 of the effectsof the parameters on the metrics Estimates are declaredsignificant at a 5 level of risk

We perform the same analysis on the mixed interactionsmodel as we compare it to the pair-wise dyadic and deliber-ative interactions models and discuss the marginal effects ofthe parameters on the metrics More precisely we obtain twodifferent types of results for the mixed interactions modelthe first compares the mixed scenarios to their monolithiccounterparts with respect to each metric and with respect tothe regression estimates the second gives a clear idea of themarginal effect of each governance parameter (or parameterof interest) on our observations In the first we allow controland procedural parameters to vary on a parameter spacesimilar to those of the dyadic and deliberative opiniondynamics (see Table 1) In the second we fix all of ourcontrol parameters execute 36000 balanced21 runs andfocus only on the one-way pooled effects of our proceduraland behavioral (all agents are focused or naive) parametersWe generate comprehensive graphs and tables to account forthe obtained results

Please notice the slight change in the size of the parameterspace for the procedural parameters and in the values ofthe nonprocedural ones for the first and second type ofresults (refer to Tables 1 and 2 respectively) Since we wantto have a finer idea of the marginal effects of each of theprocedural parameters on how well a group decides onproposals we make the value jumps for each parameter (1)

Complexity 15

() (U T) = (02 18) (b) ( pa m nD) = (005 05 00 2 02) (＝) ０ＬＧＮＬＭ ＣＨ (＜) (＝) Ｈ＞ tD = 3

(d) (５ ４) = (02 14) (e) ( pa m nD) = (01 03 05 3 02) () ０ＬＧＮＬＭ ＣＨ (＞) () Ｈ＞ tD = 6

Figure 7 From left to right common opinion trajectories and distributions for dyadic deliberative and mixed opinion dynamics within thestructure of different decision-making processes

small enough to detect significant differences in the estimatesand (2) meaningful for interpretation For the fixed one-valued parameters in Table 2 we use Table 1 to set them totheir minimum mean or median values

41 Comments on Dyadic and Deliberative Opinions Dynam-ics In this subsection we present our first results regardingthe deliberative opinion dynamics described in Equation (6)and its differences with the dyadic dynamics (Equation (5))in terms of our observations

411 Qualitative Analysis of Pair-Wise Opinion DynamicsPair-wise opinion dynamics alone produces multiclusterstable convergencewith variable cluster sizes (see Figures 7(a)and 7(d)) Insight on the complexity of these dynamics wasdiscussed in Section 321

In Figures 7(a) and 7(d) agents discuss randomly withone another in pairs Clusters in the extremes form quicklysince either agents that are close to the extremes attractone another or are convinced to stay close to the extremesby interacting with other agents they disagree with Otheragents with near-extreme positions may be simply attractedto one of the several moderate foci of agents in the opiniondistribution On the other hand moderate agents are eitherattracted to the closest opinion focus or pushed towards theextremes of the distribution Over time agents in the centralfocus either attract other agents into it or ignore the opinions

of the extreme agents that interact with them The result ofthese dynamics yields amulticlustered convergencewhere theopinion foci are at least at distance 119880 from one another andonly the extreme foci (clusters of agents with |119900119894| ge 075) areat a distance greater than or equal to 119879 from one anotherThefoci at the bounds of the opinion distribution are very stableand in analogy to the (12119889)-rule in assimilation bounded-confidence models [33] the number of clusters that form isroughly equal to 119880minus1412 Qualitative Analysis of Deliberative Opinion DynamicsQualitatively deliberative opinion dynamics yield a looseunipolar convergence of opinions near the center (centralconvergence see Figure 7(b)) near the center of either the leftor right portions of the distribution (group polarization seeFigure 7(e)) or to a sparse bipolar opinion distribution withclusters at the center and at the extremes (see Figure 7(c)) ifmixed with pair-wise interactions This is probably a reasonto expect the variance of opinions and the proportion ofextremists (almost perfectly correlated 120588 = 089 119901 lt 0001)to be low for these scenarios The side towards which theopinion cluster skews depends essentially on the valence ofthe first argument that is collectively accepted

In Figures 7(e) and 7(b) agents that update their opin-ions do so at the same time and only when deliberationis successful In these scenarios every time a decision iscollectively accepted agents update their opinions towards

16 Complexity

Table 3 Ordinary least squares estimates for pair-wise interactions scenarios for most of the observed metrics

Dependent variable119881119886119903(119900) 119878ℎ 119890119888 119894119903(1) (2) (3) (4)Constant 0610lowastlowastlowast 4429lowastlowastlowast 0472lowastlowastlowast 0000

(0002) (0206) (0003) (0000)

119880 = 06 (119880 = 02) minus0012lowastlowastlowast 2981lowastlowastlowast 0061lowastlowastlowast 0000(0002) (0260) (0004) (0000)

119879 = 18 (ref 119879 = 14) minus0264lowastlowastlowast minus3261lowastlowastlowast 0004 0000(0002) (0260) (0004) (0000)

120572 = 066 (ref 120572 = 05) 0002 minus0059 minus0100lowastlowastlowast 0000(0002) (0184) (0003) (0000)

(119880 119879) = (06 18) (ref (119880 119879) = (02 14)) minus0324lowastlowastlowast minus3728lowastlowastlowast minus0033lowastlowastlowast 0000(0003) (0368) (0006) (0000)

Observations 1440 1440 1440 1440R2 0983 0389 0529Adjusted R2 0983 0388 0527Residual Std Error (df = 1435) 0032 3492 0053 0000F Statistic (df = 4 1435) 21 272740lowastlowastlowast 228686lowastlowastlowast 402326lowastlowastlowastNote lowastplt01 lowastlowastplt005 lowastlowastlowastplt001

the argumentrsquos defended position Hence if there is votingand if agents happen to obtain 119894119899 proposal arguments theyinexorably cluster towards one or the other side of the opiniondistribution resulting in opinion convergence remeniscent ofgroup polarization (see Figure 7(e)) Furthermore as agentscluster in the same side of the opinion spectrum the argu-ments that may be advanced in deliberation are fewer andmore skewed towards that one side of the opinion spectrumAgents become more easily persuaded in deliberation andonly by arguments supporting one side of the spectrumConvergence of opinions towards one loose opinion focusbecomes faster and more certain If there is no vote (seeFigure 7(b)) the dynamics are similar but for the fact thatit is the uniform randomness of the proposal argument thatguarantees a central convergence of opinions

413 Quantitative Analysis of Pair-Wise Opinion DynamicsThe three parameters that allow for quantitative analysis are119880 and119879 the agentsrsquo attitude structure and themajority quotavoting rule 120572 Table 3 supports the claim that going from 120572 =05 to 120572 = 066 does not affect the variance nor the shifts ofopinions However judgment accuracy is significantly higherwhen accepting proposals becomes more difficult This israther surprising but also intuitive since decisions that aretaken using more restrictive methods of scrutiny should bemore accurate

Concerning the attitude structures the interaction termbetween 119880 and 119879 shows that 119880 has a small effect on thevariance of opinions given that 119879 is low This shows thateven when the assimilation threshold is high a sufficientlysmall rejection threshold can make its one-way effect vanishHowever high levels of both 119880 and 119879 reduce the variance ofopinions dramatically In contrast when it comes to overallchanges in opinion the magnitude of 119880rsquos one-way marginal

effect increases the shifts of opinions whereas 119879 lowers itThis can be interpreted as having the rejection dynamics pushmoderate-to-extreme agents to positions that are close tothe extremes and very moderate agents to the center of thedistribution Not too many quantitative changes (on average)can therefore be observedThe fact that higher119880 is associatedwith faster convergence of opinions seems to imply lowerjudgment accuracy At some point voting for the same type ofproposals for too longmay result inmany collectivemistakesThis is an interesting conclusion to consider in the analysis ofthe mixed model Finally the interaction term shows that 119879has no strong impact on the metric

414 Quantitative Analysis of Deliberative OpinionDynamicsSeven parameters can be considered for quantitative anal-ysis in this dynamics the parameters of interest recalledin Section 312 the size of the argument sacks (119896) agentbehavior during deliberation the strength of the deliberationdynamics (120574) and the measure sensitivity to deliberatedresults (119901119886) The most ldquounexpectedrdquo result obtained fromthe deliberation dynamics is that it shows how deliberationcontributes to the proportion of extremists and the varianceof opinions It happens that having more agents discussproposals and having more of these discussions increasesthe variance of opinions and the proportion of extremistswhile lowering judgment inaccuracy and coherence For theformer the fact that far too many arguments are advancedduring deliberation plummets the chances that the centralargument is accepted hence deliberation does not moveopinions too often For the latter we observe that only ldquobigrdquodifferences (going from the lowest value of a parameter to thehighest) in protocol actually prove to have a preponderantimpact on the metrics These observations hint at a possibletrade-off between judgment accuracy and the variance of

Complexity 17

Table 4 Ordinary least squares estimates for the deliberative interactions scenarios for most of the observed metrics

Dependent variable119881119886119903(119900) 119878ℎ 119890119888 119894119903(1) (2) (3) (4)Constant 0032lowastlowastlowast 13092lowastlowastlowast 0370lowastlowastlowast 1007lowastlowastlowast

(0001) (0406) (0001) (0001)

119898 = 3 (ref119898 = 1) 0012lowastlowastlowast minus2062lowastlowastlowast minus0012lowastlowastlowast minus0001(0001) (0276) (0001) (0001)

119898 = 6 (ref119898 = 1) 0023lowastlowastlowast minus3285lowastlowastlowast minus0018lowastlowastlowast minus0004lowastlowastlowast(0001) (0276) (0001) (0001)

119899119863 = 002 (ref 119899119863 = 001) 0007lowastlowastlowast minus0965lowastlowastlowast minus0006lowastlowastlowast minus0002lowastlowast(0001) (0276) (0001) (0001)

119899119863 = 005 (ref 119899119863 = 001) 0020lowastlowastlowast minus3316lowastlowastlowast minus0015lowastlowastlowast minus0004lowastlowastlowast(0001) (0276) (0001) (0001)

120572 = 05 (ref 120572 = 0) 0146lowastlowastlowast 68932lowastlowastlowast 0068lowastlowastlowast minus0507lowastlowastlowast(0001) (0276) (0001) (0001)

120572 = 066 (ref 120572 = 0) 0291lowastlowastlowast minus13842lowastlowastlowast 00004 minus0000(0001) (0276) (0001) (0001)

119896 = 8 (ref 119896 = 4) 0004lowastlowastlowast minus0815lowastlowastlowast minus0002lowastlowastlowast minus0001(0001) (0276) (0001) (0001)

119896 = 16 (ref 119896 = 4) 0008lowastlowastlowast minus1118lowastlowastlowast minus0004lowastlowastlowast minus0003lowastlowastlowast(0001) (0276) (0001) (0001)

119861119890ℎ119886V119894119900119903 = ldquoFocusedrdquo (ref ldquoNaiverdquo) 0015lowastlowastlowast minus1980lowastlowastlowast minus0012lowastlowastlowast minus0004lowastlowastlowast(00005) (0225) (00004) (0001)

120574 = 01 (ref 120574 = 005) minus0020lowastlowastlowast 5762lowastlowastlowast minus00003 minus0001(0001) (0276) (0001) (0001)

120574 = 02 (ref 120574 = 005) minus0025lowastlowastlowast 8648lowastlowastlowast minus00003 minus0001(0001) (0276) (0001) (0001)

119901119886 = 05 (ref 119901119886 = 03) minus0013lowastlowastlowast 1581lowastlowastlowast 00004 00003(00005) (0225) (00004) (0001)

Observations 19440 19440 19440 19440R2 0931 0844 0562 0958Adjusted R2 0931 0844 0562 0958Residual Std Error (df = 19427) 0033 15701 0030 0050F Statistic (df = 12 19427) 21 844790lowastlowastlowast 8 733563lowastlowastlowast 2 079585lowastlowastlowast 36 761250lowastlowastlowastNote lowastplt01 lowastlowastplt005 lowastlowastlowastplt001

opinions that may be of interest in the mixed discussionmodel

Group coherence shifts of opinions and the varianceof opinions are highly affected by the voting thresholdthe coherence metric is reduced by half when allowing forclassic majority voting shifts explode and the variance ofopinions increases dramatically This is not surprising sincethe majority quota rule works as a buffer to the success ofdecision-making processes and in consequence it hampersthe deliberative opinion updates For the shifts of opinions itis clear that the two-third majority voting rule results in littleopinion change and that the simple majority rule inducesgroup polarization As expected there are less extremists inscenarios where there is no voting and the shifts of opinionare more likely to happen when agents are more sensitiveto deliberation When we include voting in the system

the grouprsquos judgment accuracy decreases due to the correctdeliberated decisions that are vetoed by a salient opinion-agent majority Strangely the effect only holds for the simplemajority decision rule Lastly the proportion of individualsthat participate in the debate and the minimal amount ofdebates have similar marginal effects on all the metrics (seeTable 4)

415 Comparing Deliberative and Pair-Wise Opinion Dynam-ics in a Nuthshell Qualitatively both models yield similaroutcomes but for the fact that deliberation never producesopinion polarization with both types of extremists (seeFigure 7) Furthermore clusters in the deliberative interac-tions model are less homogeneous in size and multiclusterconvergence occurs only when deliberation is rarely suc-cessful which happens sporadically given the calibration of

18 Complexity

Table 5 Mean difference 095 confidence intervals for metrics by model comparison

Model Metric119881119886119903(119900) 119901119903119900119901119890119909 119878ℎ 119890119888 119894119903Dyadic vs Deliberative [0191 0216] [0170 0197] [-2948 -2828] [0074 0082] [0165 0172]Mixed vs Deliberative [0142 0147] [0150 0155] [-2085 -1976] [0003 0002] [-0038 -0030]Mixed vs Dyadic [-0072 -0046] [-0045 -0018] [8332 8859] [-0079 -0071] [-0205 -0201]the model On the other hand pair-wise opinion dynamicsproduces multicluster convergence most of the time andcentral convergence only in the infrequent cases where thelatitude of acceptance is large

The comparison of these two scenarios in terms ofthe metrics is straightforward Whilst dyadic interactionsallow for more variability in opinions through pair-wiseinteractions and more mistakes in judging proposals delib-erative updates simply act as an antagonist force On aver-age the deliberative interaction model produces opiniondistributions with smaller variance and yields steady stateswith higher judgment accuracy a higher shift statistic andnaturally lower group coherence since only proposals that aredeliberated 119894119899 are eligible for voting (see Table 5)

42 Articulating Dyadic and Deliberative Social Interactionsin Opinion Dynamics So far we have considered scenarioswhere argumentation had no place and where individualsmade decisions according to opinions that were purelyconstructed from pair-wise discussions We also observedcases in which agents formed their opinions only by inte-grating deliberated proposals into their opinion updates Thescenarios we present in this section combine the two pre-ceding scenarios to account for both the effects of pair-wisediscussion and the effects of deliberation on opinions judg-ment accuracy and group coherence From the precedingobservations we conclude the following deliberative opiniondynamics reduces the variance and the proportion of extrem-ists because it shifts opinions greatly and polarizes groupsby eliminating at least one type of extremist the one whoseopinions are initially opposed to the deliberated resultsor it brings opinions close to neutrality (see Figure 7(b))This may not be desired in a group since the diversity andstability of opinions are necessary for deliberation to makesense and lead to the collective acceptance of controversialproposals Dyadic opinion dynamics on the other handtend to increase the variance of opinions and the numberof extremists while having a lesser aggregated effect on theshifts of opinion This is an immediate result of the pair-wiseinteraction dynamics inasmuch as it polarizes individualsvery quickly

To sum up in the mixed interactions model deliberationhinders the effect of voting on opinions by making someproposals that would normally be submitted for voting noteligible for voting It may also undo opinion changes dueto pair-wise interactions The mixed model can be seen asthe deliberation opinion dynamics model in which pair-wiseinteractions occur in between collective discussion and mayaffect which arguments are advanced during deliberationEquivalently it can be seen as the dyadic opinion dynamics

model in which deliberation accounts for endogenousldquoshocksrdquo on the opinion distribution

421 Qualitative Analysis of the Mixed Dynamics In Figures7(c) and 7(f) agents update their opinions through delibera-tion and pair-wise interactions Opinion trajectories visualsshow a combination of both previously described opiniontrajectories Over time as agents discuss with one anotheropinion clusters form as in the pair-wise or dyadic opiniondynamics Deliberation may disrupt the formation of theseclusters and three different situations may occur

(i) Deliberation disrupts the formation of the opinionclusters but not sufficiently to jeopardize pair-wiseopinion cluster formation (eg for high values 119905119863)Two phenomena may be responsible for it (1) thevariability of the proposal argumentrsquos support for theprinciple P and (2) too few (successive) deliberationsteps that result in opinion updates Once the opinionclusters are formed successful deliberation foreshad-ows the merging of the previously formed clustersmay bring extremists closer to moderate agents andthanks to the assimilation component of the pair-wisedynamics allow bigger and more moderate opinionclusters to form (refer to Figure 7(c))

(ii) Deliberation significantly disrupts the pair-wisedynamic formation of clusters This may happenwhen the requirements for accepting proposals areweak in the decision-making process (eg low 120572) Inthis case one has either central convergence sinceall agents follow the deliberated results in the samemanner or 3-cluster convergence with two extremeopinion foci In the latter situation the extremegroups form quickly (like in the pair-wise model) anddeliberation cannot pull extreme agents away fromthe extreme foci they are in The main reason for thisis that immediately after an extreme agent attempts toleave her foci through deliberation updates she getspulled or pushed back into it by means of a dyadicinteraction with another extreme agent

(iii) Deliberation does not disrupt the formation of theopinions cluster This is the rare case scenario forvery demanding decision-making processes in termsof inclusion (eg 119899119863) and argument acceptability (eg120572) Opinions converge as in the pair-wise opiniondynamics model (refer to Figures 7(a) and 7(d))

Last when there is voting (120572 = 0) the position ofthe moderate opinion cluster is determined by the first

Complexity 19

minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

Var(o)

sh

ec

ir

Var(o)

propex

sh

ec

ir

1

098

minus02

minus004

001

1

minus014

minus001

minus003

1

026

minus028

1

minus057 1

ＪＬＩＪ

Ｒ

Figure 8 Significant correlation estimates for themetrics of interestin the mixed opinion dyamics scenarios

accepted proposal argument andor the opinionmajority thatis formed from previous dyadic interactions (see Figure 7(f))

422 Relating the Metrics of Interest in Mixed OpinionDynamics The first thing to notice is that for all of thesescenarios all the metrics are significantly correlated (seeFigure 8) The variance of opinions and the proportion ofextremists are almost perfectly correlated To some extentthis means that a high variance of opinions known thatthe mean of opinions is statistically null equates to havingmany extremists in the population be towards one or theother side of the opinion distribution (eg extreme bipolar-ization) Similarly we find a negative and strong correlationbetween the judgment inaccuracymeasure and the coherencecoefficient (see Figure 8)mdashwhen judgment inaccuracy is high(low) coherence to deliberation is low (high) Cases likethese are telling since they may be related to sequences ofcollective decisions in which agents do not make too manymistakes when judging proposal arguments and still makedeliberated decisions that are on average in line with theirprinciples

The shift statistic interestingly is negatively correlatedwith the variance of opinions with judgment accuracy andwith the coherence coefficient In a way this means thathaving more moderate agents conduces on average to moremistakes in judging arguments and to less coherent collectivedecisions An explanation for this phenomenon could be thatas agents reach consensus in opinion important (possiblyextreme) arguments for correctly judging argument propos-als are not advanced in the deliberation arenawhich results inmore mistakes and in possibly accepting proposal argumentsthat after voting are not meant to be accepted Indeed the

negative correlation between the variance of opinions andjudgment inaccuracy points towards this direction since onaverage the more shifts there are the smaller the variance ofopinions is (see Figure 8)

423 Comparing Mixed Dynamics with Other MonolithicDynamics of Interactions Qualitatively opinion trajectoriesin the mixed model are more chaotic than in the pair-wise and deliberative interaction models (see Figure 7) Asanalyzed previously deliberation dynamics disrupt the pair-wise dynamics and create situations in which central andmulticlustered convergence (with two clusters at the extremesof the distribution) are possible at the expense of more(pair-wise dynamics) or less (deliberative dynamics) clustersIt tolerates the survival of very small isolated group ofagents that the previous models rarely tolerate in a hundredcollective decision-making steps (see Figure 7(c))

One of the most interesting observations one can makeof this family of models regards how groups change (in termsof our metrics) when dyadic social interactions are addedto deliberative interactions and vice-versa When comparingthe mixed interactions with only deliberative ones shiftsagentsrsquo aggregated change in opinions are on average lesspronounced The variance of opinions and the proportionof extremists are significantly higher for the former than forthe latter group coherence in contrast is lower with respectto the scenarios where opinions are formed only throughdeliberation Judgment accuracy follows that same trendA reason for this might be that mixed dynamics are fasterto yield semistable clusters andor opinion convergence Inconsequence the potential argument pool for deliberativeexchange shrinks faster and deliberation becomes less effec-tive over time which results in more collective mistakes Theoutcomeon coherence is predictable inasmuch as deliberativeinteractions produce group polarization If a group agrees inopinion debates are sterile because themembers of the groupwill most likely refrain from attacking the proposal argu-ments they all support and will work together to disparagethe proposal arguments they do not support (see Table 5)

In the same manner we report that the proportion ofextremists and the variance of opinions are barely lower inthe pair-wise interaction model when including deliberationin the decision-making process Shifts of opinions are alsobarely lower when there is no deliberation This can indicatethat deliberation does not undo the changes in opiniondue to pair-wise interactions nor does it amplifies them Incontrast when we consider the labeling-based metrics themixed interaction model ensures lower judgment inaccuracybecause of deliberation and lower group coherence (seeTable 5)

In sum the mixed discussion model is a compromisebetween the pair-wise interaction and deliberative opiniondynamic models in that it offers some guarantees in termsof judgment accuracy and coherence while allowing forldquoreasonablerdquo variance in the opinion pool It establishes somekind of trade-off between labeling-based metrics and thevariance of opinions that may be of interest for decision-making process design

20 Complexity

Table 6 Summary of sign or relative marginal effects for all varying parameters on dyadic deliberative and mixed interactions models inand for all experiments

Metrics Parameters119905119863 119898 119899119863 120572 119879 119880 119901119886 120574 119896 119861119890ℎ119886V119894119900119903119881119886119903(119900) XX X X O X X X X X X119901119903119900119901119890119909 XX X X O X X X X X X119878ℎ XX X X O X X X X X119890119888 XX X X XO X XOO XOO X X119894119903 XX X X O XO O X OOX XOO X XNota gray symbols correspond to dyadic dynamics light gray to deliberative dynamics and bold black to mixed dynamics Given a scenario the symbol 119883stands for constant parameter and 119874 for insignificant marginal effect and as parameters grow for positive (negative) marginal effect for the parameterFrom left to right at the parameter level the leftmost arrow corresponds to the scenario with the strongest (in absolute value) marginal effect of the parameter(with respect to to its factors) Example Let 119901 be a parameter and 119874 stands for ldquo119901 has a significant positive effect in dyadic discussion scenario negativein the deliberative one and insignificant in mixed discussion The positive effect in dyadic discussion is greater than the absolute value of the negative one inthe purely deliberative scenariordquo

424 Comments on the Differences between the Three Modelsfor OLS Regression Estimates The significance of most of theregression parameters estimated for the mixed interactionsscenarios are similar to those estimated for the pure dyadicanddeliberative scenariosNevertheless themagnitude of theeffects is different insomuch that the dynamics themselvesare different So instead of commenting the linear regressionestimates of the mixed model we compare when possiblethe standardized significant coefficients of the regressors ofthe mixed model with those of the pair-wise and deliberativeinteractionmodels (see Table 6 for a summary of the results)

In mixed interactions for instance the influence of119898 onthe opinion distribution is much more preponderant than inthe deliberative dynamics scenarios Going from 119898 = 1 to119898 = 3 increases the standard variation units of the variance ofopinions by twice as much as the one in the deliberative sce-nariosThis in particular is due to the design of the decision-making process If 119898 is higher then the number of socialinfluence steps (119905119863) in the decision process is also higher(compared to 119905119863 = 0 in the deliberative case) If we decideto consider these steps as contributing to the variance ofopinions then the effect of119898 on the variance of opinions hasto be stronger in the mixed interactions model In contrastthe number of agents participating in the debate has a weakereffect on the variance of opinions in the mixed interactionsmodel than in the deliberative dynamics specially for 119899119863 =005 (more than twice as much) The difference may comefrom the fact that in mixed discussions the part of theargument pool used in deliberation is more restrictive thanin the deliberative case We can argue that since pair-wisedynamics may get individuals closer to one another aftersome deliberation the same happens to the arguments thatindividuals will most likely use in deliberation Taking thisinto account means that deliberation will at many instancesbe unfruitful and have no effect on opinions thus failing tomoderate extreme opinions andor polarize the group In thesame direction a high number of participants in deliberationcan make the deliberation dynamics stiff In this case thedistribution of opinion is close to the uniform distributionwhich has a relatively high-to-moderate variancewith respectto the reference results

The strength of the deliberative dynamics has a positivestronger effect on the variance of opinions in themixedmodelthan in the deliberative model when 120574 = 02 and a negativeweaker one when 120574 = 01 A sound interpretation of thisresult is that when deliberation is successful and its effect onopinions does not get agents to move sufficiently away fromtheir current positions agents are pulled or pushed back totheir previously held opinions the variance of opinions doesnot changemuchOn the other hand if the effect of successfuldeliberation is strong opinions can change sufficiently tomake agents much more prone to assimilation into moderateopinion foci the variance of opinions falls

The deliberative opinion dynamics model is much moresensitive to 120572 than the mixed discussion model Differ-ences in marginal effects can double The probability ofattraction to deliberated results (119901119886) for instance has astrongmoderating effect (negative) on agentsrsquo opinions in thedeliberative opinion dynamics but a meager one (two-fold)in the mixed and dyadic dynamic models This is becauseopinion updates in the deliberative model depend only onthat parameter whereas in the mixed model more numeroustypes of interactions make opinion updates possible

In terms of judgment accuracy all parameters except thesize of the argument sacks (119896) have a significant slightlystronger effect on the deliberative dynamics than in themixedmodel In effect the difference may be attributed to theupshot of pair-wise interactions on the distribution of opin-ions and on the arguments that would be advanced duringdeliberation The result respecting parameter 119896 however iscounterintuitive Since argument sacks are static scenarioswhere shifts are higher should make 119896 more important fordeliberation to succeed and it is precisely in the scenarios ofpure deliberative dynamics that we find the biggest opinionshifts This may point to the hypothesis that in deliberativeinteractions agents change opinions substantially but do notoften change their adherence to the principle (go from anegative opinion to a positive one or vice versa) when thereis no group polarization

43 Sensitivity Analysis for the Procedural Parameters ofInterest Weperform a sensitivity analysis of the observationson the parameters of interest in themixed interactionsmodel

Complexity 21

We extend the parameter domains of the pure dyadic anddeliberative scenarios as we make it explicit in Table 2

431 Minimum Number of Debates (119898) We observe that theminimum number of debates has a significant well-observedeffect on all of our metrics excluding group coherence Forthe variance of opinions we can observe that the moredebates there are the bigger the value of the metric isalthough the higher 119899119863 and 119905119863 are the weaker the overalleffect of119898 becomes (Figures 9(d) and 9(a)) Furthermore themarginal effect of increasing minimal debates is decreasingan additional deliberation step requirement increases thevariance of opinions but less and less as it grows 119898 coupledwith 119899119863 has a shy S-shaped effect (see Figure 9(d)) on thevariance of opinions meaning that it has a stronger effectwhen 119899119863 is low which suggests a possible trade-off betweenthe number of arguments accepted in the debate and thenumber of debates required before accepting a proposal

In contrast the shifts in opinion are less and less likelyas 119898 grows and this is independent of the variations of theother parameters Again the effect is marginally decreasingand is only truly significant when 120572 = 05 An explanationto these effects may be linked to the design of the systemFirst the variance of opinions is higher when deliberation isasked for because the more deliberation steps there are thehigher the chances are that the central proposal argumentis deemed unacceptable specially when agents are focusedFurthermoremechanically speaking increasing theminimalamount of debates implies that whenever a decision is to betaken at least 119898 times 119905119863 time steps have to take place and if atany moment a proposal argument is considered undecided119905119863 time steps are added to the process So unless the debatedoes not yield 119906119899119889 labels for proposal arguments (highlyunlikely considering that 120590 = grounded semantics) the morenondeliberation steps there are in the decision process thehigher the variance of opinions is Concerning the shiftswhen 120572 = 05 either the system is too stiff to accept anyproposal argument and opinions do not changemuch or theeffects of pair-wise discussion and deliberation cancel out insuch a way that individual shifts are minimal (Figure 9(b))

On the side of labeling-based metrics the more debatesare asked for the more accurate a group is in its judgmentthe effect being smaller as 119898 grows When agents are naivethe effect is quasilinear while when they are focused thestrongest effects of acquiring more deliberation are foundwhen levels of deliberation are already low (Figure 9(f))Thiscan be explained by the fact that the more debates thereare in the decision process the closer one gets to the idealargumentation framework

432 Proportion of the Population in Deliberation Steps (119899119863)Like for 119898 119899119863 has a significant effect on the proportionof extremists and on the shifts and variance of opinions(Figure 9(e)) This may result from the fact that being ableto put more arguments in play at the same debate step mayincrease the odds of revealing cycles around the proposalargument in119860119865120576 Given that we use grounded semantics thearguments in the cycles are labeled 119906119899119889 and in consequence

debates are more often postponed when 119899119863 is higher Post-poning debates in turn increases the number of nondebatessteps in the decision process which increases the varianceof opinions and limits over time the moderating effect ofdeliberation Moreover the effect of this parameter is verydependent on the value of 120572 (Figure 9(e)) For shifts forinstance 120572 = 05 makes the effect of 119899119863 negative while120572 = 0 makes it positive but to a lesser extent Furthermorefor higher requirements of deliberation (119898) adding moreindividuals to the deliberation process has weaker effectson the variance of opinions and on the other metrics It isalso quite interesting to notice that it has the same effect onagents independently of agent behavior during deliberationThis is a surprising result as one would have expected morefocused agents in a deliberation arena to cause a raise inthe proportion of extremists as they play to knock outopposing proposal arguments and thereof reducing the effectof deliberation (lowers variance) on the whole population

Finally similar to119898 addingmore people into the deliber-ation process increases judgment accuracy (see Figure 9(c))and has no clear effect if any on group coherence (seeFigure 9(i))

433 Steps between Deliberation Steps (119905119863) In all configu-rations 119905119863 increases the proportion of extremists (variance)and decreases the shifts of opinions among agents Theshifts and the effect on the variance of opinions are presentfor 120572 = 05 insignificant otherwise (Figure 9(b)) 119905119863 ishighly linked to 119898 by construction When 119898 = 1 thecurve linking the variance of opinions and 119905119863 is convex As119898 increases the curve becomes more and more concavewhich means that 119905119863 has a more important effect on theopinion distribution as collective decisions take longer to beachieved This seems counterintuitive but in reality it reflectsthemultiplicative relationship between deliberation and pair-wise interactions and the semantics in the model If 119898 is lowand 119905119863 is high the effective number of pair-wise interactionsis on average smaller in the deliberation process Hencethe structure of the decision-making process hinders anyincrease in the variance of opinions If119898 is high the oppositeeffect is observed Additionally since the grounded semanticsyields few 119894119899 arguments with respect to other admissibility-based semantics getting closer to the ideal argumentationframeworkmaymake it difficult to obtain the expected effectsof deliberative discussion on agents Lower 119898 constrains thevariance of opinions by making deliberation more influentialon opinions

In terms of judgment accuracy and coherence we observethat 119905119863 does not explain coherence yet it tends to decreasejudgment inaccuracy This may be due to a more variedargument pool in the debate which in turn may be aconsequence of a higher variance of opinions or vice versa

434 Acceptability Voting Quota (120572) By far it is the mostinfluential parameter in our study It changes the directionand the intensity of the effect of all the other proceduralparameters and by construction heavily constrains the roadto accepting a proposal (see Figure 9)

22 Complexity

015

020

025

030

035

040

1 2 3 4 5 6

12345

Values of Ｇ

(a) 119905119863 times 119901119903119900119901119890119909 by119898

0

10

20

30

40

50

1 2 3 4 5 6

005066

Values of

(b) 119905119863 times 119878ℎ by 120572

040

045

050

055

060

1 2 3 4 5

001002003004005

Values of Ｈ＄

(c) 119898 times 119890119888 by 119899119863

015

020

025

030

035

040

1 2 3 4 5

001002003004005

Values of Ｈ＄

(d) 119898 times 119901119903119900119901119890119909 by 119899119863

0

10

20

30

40

50

001 002 003 004 005

00005066

Values of

(e) 119899119863 times 119878ℎ by 120572

040

045

050

055

060

1 2 3 4 5

BehaviorNaiveFocused

(f) 119898 times 119890119888 by 119887119890ℎ119886V119894119900119903

015

020

025

030

035

040

0 05 066Behavior

NaiveFocused

(g) 120572 times 119901119903119900119901119890119909 by 119887119890ℎ119886V119894119900119903

05

06

07

08

09

10

1 2 3 4 5 6

00005066

Values of

(h) 119905119863 times 119894119903 by 120572

08700

08725

08750

08775

08800

001 002 003 004 005

BehaviorNaiveFocused

(i) 119899119863 times 119894119903 by 119887119890ℎ119886V119894119900119903

Figure 9 Curves of mean observations for different parameters and metrics for 36000 runs Bars are 095 confidence intervals Plots withthe same observables (ordinate) except for 119894119903 are on the same scale

Complexity 23

The higher the requirement for accepting the proposal isthe higher the proportion of extremists and the variance ofopinions are and the lower the shift statistic is In few words120572 constrains the world to dyadic discussion or throws it intoa process in which deliberation is much more importantthan dyadic interactions This is why for any 120572 = 05 oneeither gives too much weight to deliberated results whichshadows pair-wise interactions or too little weight in thesense that no deliberated result is ever accepted and thusnever integrated into the opinions of the agents that havevoted for it The reason for the latter is that the latitudeacceptance is too low and thereafter pair-wise interactionsare not enough to unevenly polarize the population in such away that deliberated proposal arguments are voted favorablyby a non-negligeable majority Shifts for 120572 = 0 are stable fordifferent levels of 119905119863 which implies that deliberation keeps incheck all the shifts of opinions related to an increase in thenumber of pair-wise interactions For 120572 = 05 deliberationis less likely to be successful and shifts tend towards the shiftlevels for 120572 = 0 (see Figure 9(b))

Concerning labeling-basedmetrics120572 entirely determinesthe coherence statistic For 120572 = 05 there is no difference incoherence because of how coherence is defined it is maximalwhen 120572 = 0 and strangely maximal when 120572 = 066(see Figure 9(h)) Surprisingly 120572 has no effect on judgmentaccuracy One would have expected a higher 120572 to increasejudgment accuracy since agents would hardly collectivelyaccept a proposal argument discussed during deliberationProposal arguments have given how the argument lattice isgenerated a higher probability of being rejected than of beingaccepted in the ideal argumentation framework

435 A Word on Focused and Naive Agents Focused andnaive specifications for agent behavior are an importantparameter in the decision process since they model howagents choose arguments in the deliberation arena In the sce-narios where agents are focused agents show equal or highervariance of opinions and are therefore more extreme (seeFigure 9(g)) independently of the voting requirements Shiftsof opinion are less likely as well We can see this happeningbecause focused agents knock out proposal arguments moreoften than naive agents provided that 119899119863 is low

One could have expected naive agents to yield morecoherent decisions since they argue sincerely However thisseems not to be the case (see Figure 9(i)) Agents are unable todeliberate in away thatmakes deliberation reflect their votingintentions either because they do not have the necessaryarguments to do so or because such arguments do not exist

In terms of judgment accuracy focused agents do betterthan naive agents in finding the ideal labels for the proposalarguments This is likely because when reconstructing theframework focused agents take into account the deliberatedproposal argument and choose themost pertinent argumentsin their sacks that relate to the state of affairs at the deliber-ation arena In fine debates result in a better approximationofL119886119887120576(119868) even though one could have believed the oppositethat is focused agents are ready to sacrifice a correct collectivelabel of a proposal argument in the goal of defending theirpositions

44 Points of Discussion In this subsection we discuss theresults obtained from the simulations We attempt to makesense of them from a social sciences perspective

441 Opinion Consensus and Dissensus The deliberativeopinion dynamics corresponds to the simulated scenarios inwhich the variance of opinions is at its lowest and judgmentaccuracy at its highest Dyadic opinion dynamics are associ-ated with the scenarios in which the variance of opinions is atits highest judgment accuracy at its lowest and coherence atitsmaximumpossible levelThemixed interactions dynamicsis a combination of these aspects for all the metrics exceptfor coherence at which it is the worsemdashdeliberation leadsto some kind of loose opinion consensus that is reinforcedby pair-wise interactions Likewise the mixed discussionmodel confirms some results exposed in [16] expressingthat opinions and voting intentions in a deliberative contextldquooften changerdquo That being said and more particularly indeliberative opinion dynamics outputs align well with twointeresting results on consensus formation given by aninterpretation of Moscovici and Doisersquos [15] work Sunsteinrsquos[14] account on deliberation and group polarization andsocial impact theory [4] The first states that mild consensusis reached in deliberation as procedure disengages agentsthe second is that within discussion groups majorities tendto become larger and opinions more extrememdashnot on thebasis of competing arguments but of the initial balance ofopinions After several decision processes we find opiniondistributions that reflect these observations even though weare not able to precisely coin the set of parameters that yieldssuch distributions (see Figure 7)

Although differences between mixed and dyadic opiniondynamics in terms of the variance of opinions and shiftsare slim the former does much better in terms of judg-ment accuracy In this sense deliberation does moderateopinions but not as much when non-deliberation steps arerelatively frequent throughout the decision-making processIn terms of Moscovici and Doise theory of consensus [15]this makes sense since pair-wise discussion engages or ego-involve agents in the discussion and pushes to convergenceof opinion at the extremes of the opinion distributionHowever what remains unclear is the reason why proceduralparameters that constrain deliberation do not disengageagents from discussion and thus result in a mild consensusas also expressed by Moscovici and Doise [15] A possibleexplanation of this phenomenon may stem from the factthat deliberation in the model is always accompanied by aseries of ego-involving discussions that shadowor counter themoderating effect of deliberation Not only this deliberationcould also be responsible for crystallizing the opinions ofextremists for it favors one or the other part of the distributionof opinions as explained in [14] and to some extent in [13]Furthermore the mixed discussion model gives an exampleof polarization that results from a minority of dissenterson one side of an issue acceding to views of the majorityrsquosside [4] as opposed to the strong polarization observed indyadic interactions Situations in which the status quo orthe undecidability of the proposal argument lingers cannotpossibly yield situations of moderation In cognitive science

24 Complexity

one can be inclined to believe that the longer a deliberationprocess is the more likely individuals will take an immutablestance on the situation In view of the assumption that longdecision-making processes can be more costly for a group orsociety as a whole agents may feel forced to take a stancejust for the sake of ending the process The stance that isadopted would therefore depend on pair-wise interactionsamong individuals and not on deliberation since deliberationtakes too long to be conclusive

442 For Hypothetical Recommendations in Decision-MakingProcess Elaboration On another tune results from themodelsuggest that procedural parameters have marginally decreas-ing positive or negative effects on the metrics of interest Ina hypothetical policy analysis perspective this may indicatethat after choosing a correct number of individuals andimposing the right amount of debates one can attain rea-sonable levels of judgment accuracy coherence and varianceof opinions As accounted in [18] deliberation increasesknowledge (collective knowledge as well) and therefore itheightens the chances of making correct decisions If onebelieves that social welfare is linked to deliberation andeventually to the metrics studied previously then proceduralparameters can both be used as an instrument to attaindesired levels of social welfare and reinforce the notionof legitimacy of collective decisions It can also provide ajustification for choosing a deliberation regime rather thananother on the basis of how important (or urgent) a topic ofdiscussion is Indeed if we adopt the claim that deliberationis a forerunner of welfare then any state in which deliberativecues are at their maximum (highest number of participantsmany deliberation steps before making a decision and asfrequent as possible) has to be mapped to the highest attain-able social welfare In this case is a situation where judgmentaccuracy is at its highest yet the variance of opinions or theproportion of extremists at its lowest an ideal situation Wecannot say for sure since strong opinion consensus and longdecision-making processes in many situations may result ina loss of welfare rather than in a gain of it

Another interesting result in procedure for deliberationis the very similar estimates we find for the effects of theminimum number (119898) of debates necessary to make adecision on a proposal and the proportion (119899119863) of individualsparticipating in deliberation on judgment accuracy In thescope of social welfare this raises the question of whetheradvocating for longer decision processes by demandingmoredeliberation steps (higher 119898) rather than bigger debates(higher 119899119863) may increase welfare Depending on the objec-tives of the decider of the deliberation procedure he or shemay choose to concentrate on one or the other A recom-mendation we can assert from the mixed social interactionsmodel is that if one prefers low to high variance of opinionsandor high to low coherence the decider is better off ifshe focuses on increasing the size of the debate rather thanorganizing debates very frequently One of the reasons forthis might be that when making bigger debates all agents aretruly considered as equals in the decision-making processIn organizing many debates individuals that are chosen toparticipate in many of these will be more influential and will

somehow bias the set of arguments in the deliberation arenaThemajority rulemodeled through120572 also determines the sizeof the shifts and the distribution of opinions Because a bigger120572 is associated with higher levels of extremism and opinionvariance deliberated cues are harder to accept and thereforeit becomesmore difficult for agents to reach concensus andormoderate their opinions

On the other hand and further into the idea of legitimacythe goal of deliberative processes is the agreement amongall participants This agreement stands for the right or bestdecision from a formal or procedural perspective and it is alsothe best from a substantive point of view In other words thelegitimacy of decisions not only derives fromwell-establishedprocedural requirements (respect of the protocol) but alsobegs for the fulfillment of two essential conditions (1) satisfy-ing the procedural requirements for a correct procedure (for-mal legitimacy) and (2) the rational acceptability of the resultsof this procedure (substantive acceptability) Legitimacy canbe then obtained from the coherence statistic scrutinizedin the model insofar as it measures how well agents acceptthe results of a correct procedure (eg deliberation) So fora proposal to be legitimate one has to choose parametersthat would maximize group coherence in decision-makingprocesses

As deliberation may give rise to consensus in the delib-eration arena it may also sow the seeds of dissensus inthe group Our model clearly illustrates this phenomenonthrough the procedural parameters of deliberation and themetrics observed From another perspective in deliberativedemocracy dissensus in collective decision-making may bedesirable According to Landermore and Pagersquos accounts [36]lack of consensus can be perceived as having agents withalternative ideas that combined with other ideas can providea better approximation of the ideal status of an argumentTheauthors in [36] call this dissensus ldquopositive dissensusrdquo andit makes normative sense when solving complex collectivedecision-making problems Our model captures the ideaof having normative requirements to define a good andlegitimate collective decision requirements of high consen-sual deliberative accuracy (judgment accuracy) of legitimacy(coherence) and of diversity (variance of opinions) are alltaken into account and for such we can conclude that ourmodel is successful in describing a sound process of collectivedecision-making with interpretable outcomes

Of course the present simulation results and analysis donot prove that all opinion dynamics can be accounted forby such simple processes like the one presented here Theydo however shed light on the expressiveness of combiningdifferent paradigms to account formore interpretablemodelsand pave the way to creating original models of the sortThe simulations suggest the desirability of discovering theconsequences of relatively simple laws of communication atdifferent levels (micro meso and macro) to determine whatstill needs to be explained

5 Related Work

We see our model as a contribution to the influence andopinion dynamics field in ABM and a pragmatic application

Complexity 25

of abstract argumentation theory To our knowledge we areunaware of existing literature on agent-based modeling thatexplicitly relates collective choice and the notions of deliber-ation and opinion diffusion through abstract argumentationas we have done it

Most models in the literature of opinion diffusion areinterested in opinions because these have an influence incollective decision and in questions of social order Forinstance in [3] the authors are interested in consensus andin how a group collectively chooses among two alternativesIn other models authors are interested in the emergenceof extremism [8] and on the distribution of opinions whenextremists are introduced in the population [5 33 37] whileother authors coin the notion of opinion polarization as anemergent property of the system [8 12] They show usingmodels of ldquobounded confidencerdquo and opinion diffusion withtrust that three different kinds of steady states (unipolarbipolar and central) are possible depending on whetheragents are sufficiently uncertain about their opinions andsufficiently connected andor a certain proportion of indi-viduals are already extreme Recent articles on informationand opinion dynamics stress the importance of governanceand government intervention in the spread of emotions(opinions) in the frangibility of social consensus systemIn [10] the authors show that government intervention inthe spread of negative emotions can lead to an even fasterspread of negative emotions (single-peeked convergence) andin a faster collapse of the social consensus system Opinionas a function of trust is studied in [12] and relates to howindividuals form their opinions on the basis of how muchthey trust agents in their networks

Another stream of discrete opinion dynamics modelsthat explain the emergence of opinion can be found in theliterature Computer multiagent models of attitude changebased on Latanersquos social impact theory are presented in [4] Inthe theory an agent changes her opinion on the basis of theinformational impact she is subject to which depends on thepersuasiveness (strength) supportiveness and immediacy(group structure) of the environment and information shereceives The simulations predict two emergent groups ofphenomena the shifting of attitudes towards incompletelypolarized equilibria and the formation of coherent clusteringof subgroups with deviant attitudes Similarly but in acontinuous more argumentative fashion Mas and Flache[11] present a model of opinion diffusion using argumentsArguments are considered to be for or against a proposal andare given an agent-dependant relevance in dyadic persuasiveinteractions Arguments also determine the agentsrsquo opinionsand their dissimilarities which posits the rules of pair-wiseencounters in theirmodel as agents are assumed to form theiropinions according to the arguments they own They showthat in an argumentative opinion diffusion model the onlyobservable steady states are bipolarization and consensusand that bipolarization can emerge from interactions amongsimilar agents In other words bipolarization of opinionsis possible with homophily and without negative influenceif there is argumentation They subsequently compare theirresults to experimental data In [38] a similar approach toargumentation and opinion formation is taken but for the fact

that they introduce different types of arguments explicitlyapply social judgment theory [22] and use survey data toestablish model-to-real-world comparisons

Closer to opinion formation and abstract argumentationGabbriellini and Torroni [39] are the first to have origi-nally and soundly merged opinion diffusion and abstractargumentation They define an agentrsquos opinion as a functionof the arguments she holds and the attack relation amongthem They devised a focused peer-to-peer dialogue systemof persuasion (NetArg) inspired from Mercier and Sperberrsquosargumentative theory of reasoning [40] which used onlyabstract argumentation to study opinion polarization andopinion dynamics Moreover they considered networks andthe notions of trust and epistemic vigilance to define adynamics for knowledge revision and trust itself whichwe clearly lack in our model They showed that if a con-servative belief operator in argumentation was applied byagents when they reasoned then their dialogue protocoldid not increase polarization among agents That said theyused the model to study the effect of Groenvetterrsquos weaklink theory in the spreading of arguments and were notparticularly interested in notions like judgment accuracy orcoherence nor in tackling questions related to collectivedecision-making procedures Their system is very expressiveand helps position our work in the litterature our work is atthe frontier of pure argumentative opinion diffusion modelsopinion diffusion with arguments and continuous bounded-confidence models of opinion diffusion a la Deffuant

On another note work on collective cognitive conver-gence [41] and opinion sharing [19] shows that consensustowards a certain ldquocorrectrdquo opinion or cognitive state is alwayspossible yet dependent on noise variability and awarenessof agents In [20] the authors show that learning about anexogenous correct state of the world (represented by bits)under confidence was possible but only if the agents werenot too confident In a homogeneous population they showthat the higher the confidence the worse the learningmdashthevery confident agents do not learn the properties of the truestate of the world and disrupt the learning process of the lessconfident ones Collective cognitive convergence can be seenas a result of deliberation in truth-seeking models

When it comes to abstract argumentation theory we takean approach that wires two types of dialogues that are well-studied in the literature persuasion dialogues [42] and delib-eration dialogues [43] Another interesting line of work is theone on mechanism design [44] or the problem of devisingan argumentation protocol where strategic argumentation isnot a liability for success in debates We tackle mechanismdesign in a different way though Instead of consideringstrategy-proofness we are interested in how differences inprotocol can result in epistemologically ldquobetterrdquo collectivechoices and guarantee that opinion distributions are favorablefor deliberation For a survey on persuasion dialogue see[42]

Work on agent-based argumentation usually assumes thatthe semantic relationship between arguments is fixed Inother words if two individuals were to put two argumentsin the public arena such that one logically attacked the otherthen everyone would agree that such attack exists [45 46]

26 Complexity

Other models which do not make this restrictive assumptioncan also be found in the literature [27 47] and derive fromthe class or family of opponent models [27 48] in which twoopposing sides attempt to win a dialogue Ourmodel is in theintersection of these two but the frame combining opiniondiffusion of the kind and argumentation remains new

The idea of mixing interpersonal influence and verticalcommunication however is not original It is described andimplemented in innovation diffusion models such as [5 49]In both vertical communication is modeled as exogenoustransparent informationmdashagents are aware of the existenceof an innovation thus triggering several processes of choiceand stabilization of opinions Also in [50] an Eulerian modelis implemented to show the effect of media and exogenousinformation on opinion distributions With respect to thispoint the originality of our work is in that the informationemitted as vertical communication is endogenous It is issuedfrom a deliberation model that agents shape on the basisof their opinions arguments and behavior In the spirit of[51] where the authors control for the design of verticalcommunication (to whom the vertical communication isaddressed its timing how it influences agents opinions) wecontrol for the process generating the information for ourown set of variables length of debates majority quotas andthe frequency at which discussions take place

6 Conclusion and Perspectives

The main objective of this article was to build a bridgewhere decision-making argumentation and opinion dif-fusion could come together We proposed a model thatcombined abstract argumentation theory and a boundedconfidence opinion diffusion model and showed to whatextent it could explain variance of opinions extremismcoherence in collective decisions and judgment accuracyon arguments given an ideal state of full information Thesecond objective was to show in what way governance pro-viding agents with arenas of discussion and deliberation wasimportant in addressing the success of collective decision-making processes and the quality of their outcomes

The model revealed that allowing for more deliberationtime though in lower frequency and allowing for widerparticipation in deliberation increased the variance of opin-ions and the proportion of extremists in a group Theseobservations are consistent with the combination of resultsfound in [9] and in [13 14] which stress that deliberationmay polarize groups and may have a meager effect on theshifts of opinions and inconsistent with [16] where it isargued that opinions tend to moderate after deliberationIt happens that deliberation alone did moderate opinionyet when integrated into a more complex system in whichindividuals were allowed to interact with one another andnot all that was deliberated was accepted its influence wasshadowed by other more individualistic dynamics

Undeniably the grounded semantics played an importantrole in the weak effect of deliberation since it models askeptical way of reasoning over arguments For agentsaccepting arguments thus updating their opinions accord-ingly happened not too often Nevertheless asking for more

deliberation did increase judgment accuracy as observed in[18] yet in a marginally decreasing fashion We showed thatvoting within the deliberation protocol not only increasedthe proportion of extremists and the variance of opinions ina group but also determined how coherent deliberation andvoting procedures were with each other Lastly we showedthat agents that are focused judged arguments better hadmore stable opinions and constituted a group with higherproportion of extremists than their naive counterparts

In terms of governance the model says that there is per-haps no trade-off between extremismand judgment accuracyInstead it asserts that the higher the variance of opinions thecloser the group gets to the correct decisions This situationmay be interpreted as Landermore and Pagersquos notion of ldquoposi-tive dissensusrdquo (in complex collective decision-making tasks)[36] as opposed to consensus as a normative requirementfor correct collective decision-making If a decider has toorganize deliberation to legitimize a public policy dependingon what kind of world he or she wants different set ofparameters may be chosen to account for different levels oflegitimacy and correctness For instance one may ask for adecision to be legitimate to respect a certain level of coherenceor of judgment accuracy and to be discussed by a sundryof different-opinionated peoples If so deliberation has tobe made more often more participants have to be includedin debates or in deliberation instances and not too manytime steps should be introduced in between two deliberationinstances So the model may capture normative ideas todefine correct and legitimate collective decisions and tomake recommendations on those grounds Requirements ofhigh consensual deliberative accuracy (judgment accuracy)legitimacy (coherence) and diversity (variance of opinions)are all taken into account and for such a reason the model israther successful in describing a sound process of collectivedecision-making with interpretable outcomes

61 Extensions of the Model and Other Ideas Our model canbe extended in many ways Value-based abstract argumen-tation frameworks (VAFs) for example as an extension ofDungrsquos argumentation framework provide a formal descrip-tion of the process of decision-making in which argumentsare given values and audiences (set of agents) ignore attacksbetween arguments on the basis of their preferences over val-ues Social abstract argumentation frameworks (SAFs) alsoprovide an interesting and seemingly convenient formalismfor our model In SAFs agents vote on the acceptability ofarguments and the resulting labeling is a combination of thearguments that are voted for and labeling-based semantics Inour model we apply the same rule yet only for one argumentand given precise agent-based votingmechanisms and votingrules which are eventually parametrized

To consider our model as a specification of a mixed VAFand SAF model may be of great interest for future workThis formalism can provide solid foundations to describeexplain interpret and eventually extend the results foundhere from an argument-based perspective On those groundsfuture work will focus on creating a hybrid argumentationframeworkmdashwith its semanticsmdashthat takes heed of bothvalues or principles and voting

Complexity 27

Concerning other possible extensions of the model anatural reflex would be to argue on the validity and therobustness of the mix between the bounded confidenceopinion diffusion models and deliberation Consequentiallyadding deliberation to different bounded confidence modelsof opinion diffusion such as the Hegelsmann-Krause model[37] or to models of social impact theory like in [4] canbe of great interest for future work We acknowledge that afiner understanding of argumentation frameworks randomlattices and their respective applications for this modelare necessary Testing for different labeling-based semanticsor even trading labeling-based semantics for scoring-basedsemantics (semantics that defines acceptability on scoresgiven to arguments) such as debate semantics and ranking-based semantics (semantics that yield a preorder) may beinteresting to consider in future work

More into modeling deliberation it is not inadequateto believe that agents may also advance proposals duringdeliberation and replace to some extent the proposal forwhich the deliberation is taking place Endogenizing theprocess that produces the type of argument to be deliberatedon may be interesting to explore either by having agentsstrategically replace proposals at certain moments duringthe debate or having the central authority attempt to chooseproposals such that the probability of having them acceptedis high Agents may also be thought to be autoorganizing(independent of a central authority) and concede on the sizeof the instances of deliberation To consider differences insemantics across agents or even endogenizing the semanticson how urgent or important the proposal that is beingdiscussed is can also be an interesting direction to take

Last but not least many nontrivial modifications of ourmodel are possible and most should include better-thoughtdeliberation protocols It may be interesting to design andobserve protocols in which deliberation only affects agentsthat are actually debating for example Trust network effectsmultidimensionality of opinions new processes of argumentexchange or learning among agents are notions to furtherdevelop in order to make the model more realistic and relaxsome unreliable assumptions In sum the model is to berefined and extended with the objective of either studyingconcrete cases of deliberative polling and opinion dynamicsor implementing more intuitive thought experiments

Appendix

We present below some of the pseudocode we used toimplement the model

Algorithms

(a) Algorithm 1 implements the dyadic influence in themodel It describes how two agents interact andupdate their opinions

(b) Algorithm 2 implements the deliberative influence inthe model which describes how agents update theiropinions in sight of the accepted deliberated proposalarguments

(c) Algorithm 3 is used to obtain the grounded labelingof an argumentation framework

(d) Algorithm 4 provides an overview of a run in thesimulations119872 is the number of proposal argumentsto discuss which we set to 100 in the model

Each object in the system is indexed by a119908ℎ119900 number Hencea set of agents is annotated by a set of natural numbers Weuse this fact in the implementation of the algorithm 119874119887119895119890119888119905119894denotes the object of 119908ℎ119900 number 119894 Agentsrsquo opinions (119900) arecalibrated so that 119900 isin [minus1 1]Data Availability

The program and data used to support the findings of thisstudy are available from the corresponding author uponrequest

Disclosure

A preliminary version of this work was presented anddiscussed at the conference Bridging the Gap between FormalArgumentation and Actual Human Reasoning on October 4-5 2018 at BochumGermanyThe authors thank the audiencefor their valuable comments and suggestions

Conflicts of Interest

The authors declare that they have no conflicts of interestregarding the publication of this article

Acknowledgments

The contribution of Gabriella Pigozzi was supported by theDeutsche Forschungsgemeinschaft (DFG) and the CzechScience Foundation (GACR) as part of the joint project FromShared Evidence to Group Attitudes (RO 45486-1)

Endnotes

1 In other words the superiority of the ldquoright answerrdquo orlabeling in our case will appear as such to all

2 All agents hold the same opinion3 By sensitivity we mean how much an agent is ready to

change her opinion on the sole fact that an informationalcue has been debated on

4 The assumption is strong but this trust among agentsmay be rooted from a genuine individual motivation toreach a democratic consensus

5 We adopted a rule of thumbs that says that if an agentis neutral with respect to the principle then she willvote for any proposal that is presented to her duringdeliberation

6 If the argument is not unique she chooses one atrandom

7 If there are more than one candidate the agent will favorthe argument at minimal distance from 119868

28 Complexity

Require Vector of values for parameters (119881 = (120583 119880 119879))Ensure Successful interactions among agents119896 larr997888 0119873119890 larr997888 0

while 119896 le |119873|2 do ⊳Fill119873119890 with |119873|2 agents119896 larr997888 119896 + 1119873119890 larr997888 119877119886119899119889119900119898(|119873| + 1) ⊳Pseudo-random generator Generates integer between 0 and |119873|119873119903 larr997888 0 120575119894119895 larr997888 0 ⊳Variable to store differences in opinionfor all 119894 isin 119873119890 doRandomly pick a number 119895 isin 119873(119873119890 cup 119873119903)119873119903 larr997888 119873119903 cup 119895 120575119894119895 larr997888 |119900119894 minus 119900119895|procedure dyadic-update(119894 119895) ⊳Two agents 119894 and 119895 discuss and update opinions1199001015840 larr997888 119900119894 (120583119895 119880119895 119879119895) larr997888 (120583119894 119880119894 119879119894) larr997888 (120583 119880 119879) ⊳Save agent irsquos opinion before updating and homogenize agents

if 120575119894119895 lt 119880119894 then119900119894 larr997888 120583119894119900119895 + (1 minus 120583119894)119900119894if 120575119894119895 gt 119879119894 then119900119894 larr997888 (1 + 120583119894)119900119894 minus 120583119894119900119895if 120575119894119895 lt 119880119895 then119900119895 larr997888 1205831198951199001015840 + (1 minus 120583119895)119900119895if 120575119894119895 gt 119879119895 then119900119895 larr997888 (1 + 120583119895)119900119895 minus 1205831198951199001015840

Algorithm 1 Dyadic social influence

Require Vector of values for parameters (119881 = (120574 119901119886 119901119903))Ensure Successful interaction between proposal arguments and agents

for all 119894 isin 119873 doprocedure Probability-Update(V119886 119901119886 119901119903) ⊳Update individual probabilities of change due to deliberated cues119909119894 larr997888 119901|V119886minus119900119894 |2119886 119910119894 larr997888 1199012|V119886minus119900119894 |119903119902 larr997888 0if L119886119887(119868) = 119894119899 then

procedure deliberative-update(119894 119886) ⊳Agent i argument 119886120574119894 larr997888 120574 119902 larr997888 119906 sim U(0 1) ⊳ 119906 is a realization of a uniform random variableif 119902 lt 119909119894 then119900119894 larr997888 120574119894119900119894 + (1 minus 120574119894)V119886break

if 119902 lt 119910119894(1 minus 119909119894) then119900119894 larr997888 (1 + 120574119894)119900119894 minus 120574119894V119886Algorithm 2 Deliberation influence

8 In terms of computational complexity naive agentsmaketheirmoves in119874(|A119894| log(|A119894|)) while focused agents doat least as bad

9 Tables have memory Therefore debates are not neces-sarily trees

10 If there are less than (119899119863 times |119873|)2 agents for any of bothgroups the CA summons all of them to the deliberationtable

11 Or careful consideration of the viewpoints of otherswhich implies that citizens keep an open mind and donot reject arguments outright

12 A ldquohealthyrdquo consensus refers to the case in which manyconsensual decisions are taken but opinion-wise agentsdo not agree with each other Quotes are used on theword healthy because some political science theorists

(eg [36]) believe that in certain situations agreement inopinions for deliberation is epistemologically damaging

13 Simulations on groups of |119873| = 800 and |119873| = 400agents shows that scaling the population has an effect onall themetrics but it is confounding depending onwhichcouples of attitudes structures are considered (119880 119879) Westick to |119873| = 400 as in [9] for comparability andsimulation running time

14 By balanced wemean with as many arguments with V119886 lt0 as with V119886 gt 0

15 Experiments on the system suggest that a bigger(smaller) argument pool 119872 = 1000 (119872 = 200) isassociated with a higher (lower) variance of opinions(and extremism) and higher (lower) judgment accuracyFor the rest of the observations 119872 seem to have noeffect We choose a middle value for 119872 Similarly theargument pool chosen to be balanced or uniformly

Complexity 29

Require Argumentation framework 119860119865 = (AR)Ensure Assignment of labels to arguments from the grounded labeling of 119860119865L(A)

L0(A) larr997888 (0 0 0)119896 larr997888 0do119894119899 L119896+1(A) larr997888 119894119899 L119896(A) cup 119886 | 119886 is not labeled 119894119899 L119896(A) 119886119899119889 forall119887 isin A 119904119905 119887 R 119886 then 119887 isin 119900119906119905 L119896(A)119900119906119905 L119896+1(A) larr997888 119900119906119905 L119896(A) cup 119886 | 119886 is not labeled 119894119899 L119896(A) 119886119899119889 exist119887 isin A 119904119905 119887 R 119886 and 119887 isin 119894119899 L119896+1(A)

while L119896+1(A) = L119896(A) = 0Algorithm 3 Solve argumentation frameworks (with grounded semantics)

Require Vector of values for parameters (119881)Ensure Vector of statistics (119880)

procedure Set-parameters(119881) ⊳Initialize parameters for scenarioprocedure Init-objs(119899119898 119896 ) ⊳Create119898 arguments 119899 agents and 119860119865120576procedure Solve-Argumentation-Framework(119860119865) ⊳Assigns the epistemic labelingL120576 toA119894 larr997888 1119905 larr997888 1while 119894 le 119872 do ⊳ 119872 the number of arguments to discuss119868119894 larr997888 119866119890119899 119886119903119892(1) ⊳Generates proposal argument 119868119860 119894119905 larr997888 (119868119894 0)L119886119887(119868119894) larr997888 119906119899119889 ⊳Initialize label for 119868119895 larr997888 1 ⊳Initialize debate counter 119895while ((L119886119887(119868) = 119906119899119889 or 119898 le 119895) and 119895 le 119898) doif 119905 equiv 0(119898119900119889 119905119863) then

procedure build-argumentation-framework(119899119863 119887119890ℎ119886V119894119900119903) ⊳Build the argument graph agents deliberate(119861119894119905 119878119894119905) larr997888 (119883 119884) sub (⋃119897isin119873119899119863 A119897119894119905 ⋃119897isin119873119899119863 R119897119894119905) ⊳ 119873119899119863 sub 119873 are agents sampled by the CA119860 119894119905 larr997888 119889(119860 119894119905 (119861119894119905 119878119894119905))procedure Solve-Argumentation-Framework(A119889) ⊳FindL(A119889)L119886119887(119868119894) larr997888 L119886119887119863(119868119894)119895 larr997888 119895 + 1

else119896 larr997888 1while 119896 le 119905119863 doprocedure dyadic-social-influence(120583 119879 119880) ⊳Agents discuss one-to-one119896 larr997888 119896 + 1119905 larr997888 119905 + 1

if 120572 gt 0 andL119886119887(119868119894) = 119900119906119905 then ⊳Agents vote for the proposalif (|119897 isin 119873|119900119897 times V119868119894 ge 0| ge 120572 times 119899) then

L119886119887(119868119894) larr997888 119894119899else

L119886119887(119868119894) larr997888 119900119906119905if L119886119887(119868119894) = 119894119899 thenprocedure Deliberation-influence(120574 119901119886 119901119903) ⊳Agents are influenced by the result of the decision process119894 larr997888 119894 + 1119905 larr997888 119905 + 1119880 larr997888 119862119900119898119901119906119905119890 119878119905119886119905119894119904119905119894119888119904(119881)

return 119880 ⊳Get the statisticsAlgorithm 4 Run for one scenario

generated does not seem to play a significant explanatoryrole on any of the metrics

16 Experiments show that themaximumnumber of debatesis not an important parameter even if it is an intuitiveone when considering decision-making processes Inpractice debates end after at most 119898 + 1 deliberation

steps and even then it happens rarely However to avoidunpleasant surprises like infinite debates on a completeargument graph containing the proposal argument weadded the constraint 119898 Its value is precisely 7 becausethe biggest minimum number of debates we studyis 6

30 Complexity

17 To ensure comparability with the deliberative andmixedopinion dynamics models on judgment accuracy andcoherence agents vote on proposal arguments every 119905119863times(119898+1) 119905119863 and119898does not explain variability in judgmentaccuracy nor in coherence

18 Student t-tests for independent samples are robust tothe violation of two hypothesis on the distribution ofthe samples normality and homogeneity of variancesas long as the sample size is big enough (central limittheorem) and if the difference in size of the comparedsamples is smallrespectively

19 Studentrsquos test version forwhen variances are not assumedequal and the difference in size of the compared samplesis big

20 We do not correct for homoskedasticity nor normality oferrors in the estimates

21 Balanced as the same number of runs per scenario orsuch that no significant correlations are found betweenparameters

References

[1] R Axelrod ldquoThe dissemination of culture a model withlocal convergence and global polarizationrdquo Journal of ConflictResolution vol 41 no 2 pp 203ndash226 1997

[2] D M Rousseau Reinforcing the MicroMacro Bridge Organiza-tionalThinking and Pluralistic Vehicles SAGEPublications SageLos Angeles Calif USA 2011

[3] S Galam and SMoscovici ldquoTowards a theory of collective phe-nomena consensus and attitude changes in groupsrdquo EuropeanJournal of Social Psychology vol 21 no 1 pp 49ndash74 1991

[4] A Nowak J Szamrej and B Latane ldquoFrom private attitude topublic opinion a dynamic theory of social impactrdquo Psychologi-cal Review vol 97 no 3 pp 362ndash376 1990

[5] G Deffuant S Huet and F Amblard ldquoAn individual-basedmodel of innovation diffusion mixing social value and individ-ual benefitrdquo American Journal of Sociology vol 110 no 4 pp1041ndash1069 2005

[6] G Deffuant D Neau F Amblard and G Weisbuch ldquoMixingbeliefs among interacting agentsrdquoAdvances in Complex Systems(ACS) vol 3 no 01n04 pp 87ndash98 2000

[7] J Rouchier P Tubaro and C Emery ldquoOpinion transmission inorganizations an agent-based modeling approachrdquo Computa-tional and Mathematical OrganizationTheory vol 20 no 3 pp252ndash277 2014

[8] M Meadows and D Cliff ldquoThe relative disagreement model ofopinion dynamics where do extremists come fromrdquo in Pro-ceedings of the International Workshop on Self-Organizing Sys-tems Lecture Notes in Computer Science pp 66ndash77 Springer2013

[9] W Jager and F Amblard ldquoUniformity bipolarization and plu-riformity captured as generic stylized behavior with an agent-based simulation model of attitude changerdquo Computational ampMathematical Organization Theory vol 10 no 4 pp 295ndash3032005

[10] Y-F Zhang H-Y Duan and Z-L Geng ldquoEvolutionary mech-anism of frangibility in social consensus system based onnegative emotions spreadrdquo Complexity vol 2017 Article ID4037049 8 pages 2017

[11] A Flache andMWMacy ldquoSmall worlds and cultural polariza-tionrdquoThe Journal of Mathematical Sociology vol 35 no 1-3 pp146ndash176 2011

[12] A Tsang and K Larson ldquoOpinion dynamics of skepticalagentsrdquo in Proceedings of the 13th International Conference onAutonomous Agents and Multiagent Systems AAMAS 2014 pp277ndash284 International Foundation forAutonomousAgents andMultiagent Systems France May 2014

[13] K M Hansen Deliberative democracy and opinion formationUniversity Press of Southern Denmark Odense 2004

[14] C R Sunstein ldquoThe law of group polarizationrdquo Journal ofPolitical Philosophy vol 10 no 2 pp 175ndash195 2002 WileyOnline Library

[15] S Moscovici and W Doise Dissensions et Consensus unetheorie generale des decisions collectives Presses Universitairesde France-PUF 1992

[16] J S Fishkin and R C Luskin ldquoExperimenting with a demo-cratic ideal deliberative polling and public opinionrdquo ActaPolitica vol 40 no 3 pp 284ndash298 2005

[17] R C Luskin J S Fishkin and S Iyengar Considered Opinionson US Foreign Policy Face-To-Face versus Online Delibera-tive Polling International Communication Association NewOrleans La USA 2004

[18] J Barabas ldquoHowdeliberation affects policy opinionsrdquoAmericanPolitical Science Review vol 98 no 4 pp 687ndash701 2004

[19] O Pryymak A Rogers and N R Jennings ldquoEfficient opinionsharing in large decentralised teamsrdquo in Proceedings of the 11thInternational Conference on Autonomous Agents andMultiagentSystems vol 1 pp 543ndash550 Spain June 2012

[20] J Rouchier and E Tanimura ldquoWhen overconfident agents slowdown collective learningrdquo Simulation vol 88 no 1 pp 33ndash492012

[21] D Miller ldquoDeliberative democracy and social choicerdquo PoliticalStudies vol 40 S1 pp 54ndash67 1992

[22] M Sherif and C I Hovland Social Judgment Assimilation andContrast Effects in Communication and Attitude Change YaleUniversity Press 1961

[23] P M Dung ldquoOn the acceptability of arguments and its funda-mental role in nonmonotonic reasoning logic programmingand n-person gamesrdquo Artificial Intelligence vol 77 no 2 pp321ndash357 1995

[24] M Caminada and G Pigozzi ldquoOn judgment aggregation inabstract argumentationrdquo Autonomous Agents and Multi-AgentSystems vol 22 no 1 pp 64ndash102 2011

[25] P Baroni M Caminada andM Giacomin ldquoAn introduction toargumentation semanticsrdquoThe Knowledge Engineering Reviewvol 26 no 4 pp 365ndash410 2011

[26] S Chambers ldquoDeliberative democratic theoryrdquo Annual Reviewof Political Science vol 6 no 1 pp 307ndash326 2003

[27] E Bonzon andNMaudet ldquoOn the outcomes of multiparty per-suasionrdquo in Proceedings of the 10th International Conference onAutonomous Agents and Multiagent Systems 2011 AAMAS 2011vol 1 pp 47ndash54 International Foundation for AutonomousAgents and Multiagent Systems Taiwan May 2011

[28] J Ranciere Disagreement Politics and Philosophy University ofMinnesota Press 1999

[29] T Bench-Capon and K Atkinson ldquoAbstract argumentation andvaluesrdquo in Argumentation in Artificial Intelligence pp 45ndash64Springer 2009

[30] M Caminada ldquoOn the issue of reinstatement in argumenta-tionrdquo in Proceedings of the European Workshop on Logics in

Complexity 31

Artificial Intelligence vol 4160 of Lecture Notes in ComputerScience pp 111ndash123 Springer Berlin Heidelberg

[31] GAVreeswijk ldquoAn algorithm to computeminimally groundedand admissible defence sets in argument systemsrdquo in Pro-ceedings of the 2006 conference on Computational Models ofArgument Proceedings of COMMA 2006 vol 144 pp 109ndash120IOS Press Amsterdam The Netherlands 2006

[32] C W Sherif M Kelly H L Rodgers et al ldquoPersonal involve-ment social judgment and actionrdquo Journal of Personality andSocial Psychology vol 27 no 3 pp 311ndash328 1973

[33] G Deffuant F Amblard G Weisbuch and T Faure ldquoHow canextremism prevail A study based on the relative agreementinteraction modelrdquo Journal of Artificial Societies and SocialSimulation vol 5 no 4 p 1 2002

[34] E Awad M W Caminada G Pigozzi M Podlaszewski andI Rahwan ldquoPareto optimality and strategy-proofness in groupargument evaluationrdquo Journal of Logic and Computation vol 27no 8 pp 2581ndash2609 2017

[35] D Grossi and G Pigozzi ldquoJudgment aggregation a primerrdquoSynthesis Lectures on Artificial Intelligence and Machine Learn-ing vol 8 no 2 pp 1ndash151 2014

[36] H Landemore and S E Page ldquoDeliberation and disagreementproblem solving prediction and positive dissensusrdquo PoliticsPhilosophy and Economics vol 14 no 3 pp 229ndash254 2015

[37] R Hegselmann and U Krause ldquoOpinion dynamics andbounded confidence models analysis and simulationrdquo Journalof Artificial Societies and Social Simulation vol 5 no 3 2002

[38] A Stefanelli and R Seidl ldquoModerate and polarized opinionsUsing empirical data for an agent-based simulationrdquo inProceed-ings of the Social Simulation Conference 2014

[39] S Gabbriellini and P Torroni ldquoA new framework for ABMsbased on argumentative reasoningrdquo in Advances in SocialSimulation vol 229 pp 25ndash36 Springer Berlin Germany 2014

[40] H Mercier and D Sperber ldquoWhy do humans reason Argu-ments for an argumentative theoryrdquo Behavioral and BrainSciences vol 34 no 2 pp 57ndash74 2011

[41] H V Parunak T C Belding R Hilscher and S BruecknerldquoModeling and managing collective cognitive convergencerdquoin In Proceedings of the 7th iNternational Joint Conference onAutonomous Agents and Multiagent Systems vol 3 pp 1505ndash1508 International Foundation for Autonomous Agents andMultiagent Systems 2008

[42] H Prakken ldquoFormal systems for persuasion dialoguerdquo TheKnowledge Engineering Review vol 21 no 2 pp 163ndash188 2006

[43] K Atkinson T Bench-Capon and P Mcburney ldquoA dialoguegame protocol for multi-agent argument over proposals foractionrdquoAutonomousAgents andMulti-Agent Systems vol 11 no2 pp 153ndash171 2005

[44] M Thimm and A J Garcıa ldquoClassification and strategicalissues of argumentation games on structured argumentationframeworksrdquo in Proceedings of the 9th International Conferenceon Autonomous Agents and Multiagent Systems vol 1 pp 1247ndash1254 International Foundation for Autonomous Agents andMultiagent Systems Canada 2010

[45] I Rahwan and K Larson ldquoMechanism design for abstractargumentationrdquo in Proceedings of the 7th International JointConference on Autonomous Agents and Multiagent SystemsAAMAS 2008 pp 1031ndash1038 International Foundation forAutonomous Agents and Multiagent Systems Portugal May2008

[46] H Prakken R Riveret A Rotolo and G Sartor ldquoHeuristics inArgumentation A Game-Theoretical Investigationrdquo EuropeanUniversity Institute 2008

[47] N Hadidi Y Dimopoulos and P Moraitis ldquoTactics and con-cessions for argumentation-based negotiationrdquo in Proceedingsof the COMMA vol 245 pp 285ndash296 2012

[48] N Oren and T J Norman ldquoArguing using opponent modelsrdquoin Proceedings of the International Workshop on Argumentationin Multi-Agent Systems pp 160ndash174 Springer 2009

[49] S Delre W Jager T Bijmolt and M Janssen ldquoTargeting andtiming promotional activities An agent-based model for thetakeoff of new productsrdquo Journal of Business Research vol 60no 8 pp 826ndash835 2007

[50] A Mirtabatabaei P Jia and F Bullo ldquoEulerian opinion dynam-ics with bounded confidence and exogenous inputsrdquo SIAMJournal on Applied Dynamical Systems vol 13 no 1 pp 425ndash446 2014

[51] V Barbet N Guiraud V Laperriere and J Rouchier Hagglingon Values Towards Consensus or Trouble 2019

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Complexity 11

weights her opinion to an argument rather than to an opinionThe interpretation of this dynamics is straightforward Ifthe deliberated proposal argument is close to an agentrsquosopinion then it is very likely that the agent shifts its opiniontowards it Please note that the probability 119909119905119894 (119910119905119894 ) that anagent is attracted (repelled) to (from) an accepted deliberatedproposal argument is bounded below (above) by 119901119886 (119901119903)(equivalently 119909119905119894 isin [119901119886 1] 119910119905119894 isin [0 119901119903]) Steps of deliberativeopinion dynamics are associated with debate steps in which aconcluding collective decision is made

253 Mixed Opinion Dynamics Opinion dynamics in thecontext of decision-making processes can be best described asa combination of the two preceding dynamics In this opiniondynamics model agents are engaged in a democratic systemin which they deliberate vote for proposals and occasionallydiscuss with one another on the principle supported by theproposalsThe opinion dynamics for an agent 119894 can be writtenas a sequence of dyadic and deliberative opinion updates

119900119905+1119894 =

119864119902119906119886119905119894119900119899 (5) 119894119891 119905 equiv 0 (119898119900119889119905119863) and 119904119903 (sdot) = 119891119886119897119904119890119864119902119906119886119905119894119900119899 (6) 119894119891 119905 equiv 0 (119898119900119889119905119863) and 119904119903 (sdot) = 119905119903119906119890119900119905119894 119900119905ℎ119890119903119908119894119904119890

(7)

where 119904119903(sdot) is the decision processrsquos stop rule 119905119863 the numberof time steps between each deliberation step Equation (5)describes the dynamics for the dyadic interactions and Equa-tion (6) posits the changes in opinions due to the deliberatedcues Equation (5) applies when agents are not deliberatingand when each agent encounters another agent to exchangeinformation that may lead to local opinion updates Whenthere is deliberation a handful of agents deliberate and ifthey reach the stop condition imposed by the table all agentsvote for the proposal They update (see Equation (6)) theiropinions on the basis of the proposal argumentrsquos support forthe principle and the result of the scrutiny Otherwise thereis no change in the opinions of the agents A vote indicatesthe end of the decision-making process

3 Experiments and Calibration

In this section we introduce the metrics that enable us toobserve the simulations and characterize the calibration ofthe model

31 Observations and Initialization In this section wedescribe the simulations and the protocol used to test andobserve the results of our model We introduce the metricsof interest explain the calibration of the model describesome results obtained from the simulation data used forcalibration and conclude with a brief discussion on theexpected outcomes of the mixed opinion dynamics model

311 Metrics or Statistics of Interest Let 119878 denote the end ofa simulation We are interested in the effect of deliberationand of the modelrsquos procedural parameters on the followingmetrics or statistics

(i) Variance of opinions (119881119886119903(119900)) the variance ofopinions at time 119878 Since opinions live in the unionof the positive and negative unit intervals 119881119886119903(119900) isin[0 1] The higher the variance of the distribution isthe more ldquodiverserdquo the opinions are(ii) Proportion of extremists in the population(119901119903119900119901119890119909) the proportion of agents in the populationwith opinions such that |119900119894| ge 075 A high proportionof extremists makes ldquohealthyrdquo consensus12 difficult toreach and deliberation more or less informative(iii) Shifts of opinions (119878ℎ) [33] statistic that measuresthe aggregated change in individual opinion at time 119878with respect to aggregated individual opinion at thebeginning of the simulation

119878ℎ = 2sum119894isin119873 10038161003816100381610038161199000119894 minus 1199001198781198941003816100381610038161003816max119894isin1198731199000119894 minusmin119894isin1198731199000119894 (8)

119878ℎ is positive and bounded above by 2|119873| since 119900119894 isin[minus1 1] A low shift statistic implies that the processhas a small impact on opinions(iv) Judgment or consensual inaccuracy (119890119888) consen-sual accuracy of a group consists of an ad hoc statisticmeasuring a grouprsquos ability to infer correct labels forproposal arguments from a decision-making processgiven the ideal consensual labeling based on full-informationL120590

120576 (A) We use a Hamming-based dis-tance on labelings [34] to define the statistic

119890119888 = 1|I| sum119868isinI1198861198681003816100381610038161003816L119886119887120576 (119868) = L119886119887 (119868)1003816100381610038161003816 (9)

whereI is the set of all discussed proposal argumentsand 119886119868 = 12 ifL119886119887120576(119868) = 119906119899119889 or ifL119886119887(119868) = 119906119899119889119886119868 = 1 otherwise119890119888 lives in the interval [0 1] An inaccuracy statisticclose to 1 indicates that agents subdued to a particulardecision-making process make many mistakes injudging the labels of the proposal arguments Notethat when there is voting all proposal arguments arelabeled either 119894119899 or 119900119906119905 after deliberation(v) Coherence (119894119903) letL119886119887119863(119868) be the label obtainedfor 119868 from the deliberation process without votingThe coherence statistic measures how well votingresults adjust to results obtained during deliberationonly We use the proportion of arguments that havebeen labeled 119894119899 in the debate and that agents havevoted favorably for

119894119903 = 1003816100381610038161003816119868 isin I | L119886119887119863 (119868) = 119894119899 andL119886119887 (119868) = 11989411989910038161003816100381610038161003816100381610038161003816119868 isin I | L119886119887119863 (119868) = 1198941198991003816100381610038161003816 (10)

The coherence statisticrsquos domain is [0 1] If after 119872debates no deliberated central argument has beenlabeled 119894119899 or voting is not part of the procedure ofcollective decision-making then the statistic equals

12 Complexity

1 or said differently agents are perfectly coherentThis comes for the fact that if the central argumentis labeled 119900119906119905 then it is not even eligible for voting iflabeled 119906119899119889 then the debate will always be coherentwith the preferences of the agents since the result willbe a simple aggregation of their votes If the statisticequals 0 then none of the proposal arguments labeled119894119899 are voted in favorably by the agents It follows that ahigh coherence statistic implies that when agents votefor acceptable proposal arguments the results of thescrutiny reflect the consensual rationality expressedin the deliberation process

312 Parameters of Interest We recall the parameters ofinterest in our study that are linked to the structure of thedecision-making processes

(i) 119898 the minimum number of time steps in which adebate or a scrutiny occurs in a decision-making pro-cess before a final deliberated decision is submitted toa vote

(ii) 119899119863 the number of agents that deliberate as a propor-tion of the population

(iii) 119905119863 the number of time steps between debates inwhich pair-wise interactions among agents mayoccur

(iv) 120572 the proportional majority requirement for theacceptance of a proposal When 120572 = 0 it stands forno voting ldquoany proposal argument that is labeled 119894119899during deliberation is acceptedrdquo

Please recall that in terms of the definition of a decision-making process each parameter controls either the length orthe content of the sequence D119868 (119905119863 119898) or the rules that areapplied during the debate steps (120572 119899119863) (refer to Figure 4)313 Initialization All |119873| = 400 agents start off with anopinion 119900119894 drawn from a uniform distribution U(minus1 1) 13Given 119900119894 every agent 119894 randomly draws a setA119894 (|A119894| = 119896) ofarguments from a balanced14 argument pool A of119872 = 600nonneutral (V119886 = 0) arguments on the basis of 119900119894 if |119900119894| lt 025(the opinion is moderate) then agent 119894 randomly fills halfof her argument sack with arguments such that V119886 lt 0 andthe other half with arguments such that V119886 gt 0 (120601119898119894 = 05)Otherwise she randomly fills 120601119890119894 ge 05 of her sack witharguments such that V119886 and 119900119894 are of the same sign and theremaining (1 minus 120601119890119894 ) with arguments of the opposite sign15

Like for the opinion of agents every argument 119886rsquosadherence to the principle P V119886 is drawn from a uniformdistribution U(minus1 1) The attack relation that gives birth tothe ideal argumentation framework (119860119865120576) is established onthe basis of the V119886rsquos and the argumentsrsquo epistemic reach 119890119903

Let 120579(119886 119887 119903) = (|V119886 + V119887|2)119903 be an auxiliary positivereal-valued function that takes two arguments (119886 119887) and apositive real number (119903 gt 0) as inputThe probability that anyargument 119886 isin A creates a link to any other argument 119887 isin A

is given by the equation 119901 = 119901120579(1198861198872)119890 and thus argumentssupporting opposing views of the principle always attack each

other We fix 119901119890 the epistemic correlation parameter to 015If 119901119890 was greater than 015 then arguments of the same signwould attack each other too often (and focused agents wouldbe rarely incited to advance favorable arguments duringdeliberation) If 119901119890 was lower then the arguments of thesame type would induce an almost empty graph which isunrealistic The number of arguments that any argument 119886can attack is bounded by 119886rsquos epistemic reach that we fix to15 A higher value of epistemic reach makes the argumentlattice too conflicting and nearly bipartite A lower epistemicreachmakes the lattice not sufficiently conflicting in the sensethat too many arguments can attack the proposal argumentrelatively to the few that can defend it If any argument attacksthe proposal argument the chances that another argumentattacks the attacker are low Hence the proposal argumentis almost never accepted and deliberative opinion updatesbecome rare

The arguments in the resulting argumentation frame-work 119860119865120576 are given a permanent labelingL120590

120576 (A) (in shortL120576) using120590 = grounded semanticsWe choose the groundedlabeling-based semantics because it provides a unique admis-sible labeling respects minimal rationality constraints whilesimplifying the model and models a skeptic approach toaccepting arguments (see [23 25 31]) as it maximizes thecardinality of the set 119906119899119889 L120576 To some extent if we considerproposals as committing because they may guide collectiveaction choosing a skeptic semantics seems reasonable insofaras it labels an argument 119894119899 or 119900119906119905 only if it has no reasonto label it 119906119899119889 For instance if a proposal is a public policythat requires large amounts of resources and engages tofuture course of action then it may also determine policycycles and heavily burden a group and its future decisionsIn important situations like these it seems reasonable tolengthen deliberative cues and ask for more demanding andldquogroundedrdquo criteria for policy argument acceptability

On the proposalrsquos side we create an argument 119868 whoseadherence to P is also drawn from aU(minus1 1) and we label itL119886119887(119868) = 119906119899119889 119868 is interpreted as being the main argumentjustifying the discussed proposal P and V119905119868 as its supportfor P at time 119905 119868 cannot attack other arguments but otherarguments can attack it For each argument 119886 isin A a directedarc or attack from 119886 to 119868 is activated with probability 119901119901 =003120579(11988611986802) When 119901119901rsquos lower bound is too high the proposalargument is almost always defeated and thus deliberativeopinion updates do not happen When 119901119901rsquos lower boundis lower than 003 then the opposite occursmdashthe proposalargument is always accepted due to an absence of attackstowards it Finally we set the maximum number of debatesto119898 = 7 for the decision-making processes stop rules1632 Calibration and Simulation Protocol In this subsectionwe discuss the calibration of the models the terminationconditions for runs in the deliberative and mixed model andthe expected outcomes in terms of the observations

321 Calibrating Dyadic Opinion Dynamics Dyadic opiniondynamics correspond to a space of parameters in whichargumentation and deliberation spaces are not taken into

Complexity 13

Table 1 Parameters and parameter types and their domains used to compare dyadic collective and mixed opinion dynamics

Length ofD119868 Decisional in119863119868 Social influence Deliberation influence Cognitive119905119863 119898 119898 119899119863 120572 120583 119879 119880 119901119886 119901119903 120574 119896 119861119890ℎ119886V1198941199001199031 3 6 1 3 6 7 001 002 005 0 05 066 01 14 18 02 06 03 05 01 005 01 02 4 8 16 119873119886119894V119890 119865119900119888119906119904119890119889account Deliberated arguments and informational cues haveno effect on agent opinion but agents still vote for theproposal argument17 We use and calibrate this model forcomparability in terms of all our metrics since they arenot explicitly observed in [9] a reference opinion dynamicsmodel Furthermore for simplicity and comparability againwe suppose that agents are homogeneous in terms of theirattitude structures and opinion weightings We fix 120583119894 =120583 119879119894 = 119879 and 119880119894 = 119880 for all 119894 isin 119873 and for some triplet(120583 119880 119879) isin [0 05] times [0 2] times [119880 2]

Experiments suggest that the strength of the dyadic inter-actions is an explanatory factor of the time of convergence(120583 lt 05) and of the dyamicsrsquo steady states Bigger values of 120583speed up convergence towards a stable set of opinion clustersand affect the size and the relative position of these clustersin the opinion distribution 120583 however does not play a majorrole in the number of opinion clusters observed at the steadystate of the dynamics

We set 120583 to the value given to it in Jagger et alrsquos opiniondynamics model (120583 = 01) [9] For one it is the smallestreal number for which most results found in [9] hold and thedynamics are smooth secondly it seems to be a reasonableldquostrengthrdquo of influence considering that we would not wantpair-wise social interactions to completely shadow or maskan eventual effect of deliberation in the opinion dynamics

For different values of attitude structures 119879 and 119880 weget exactly the same results as in [9] in terms of the numberand density of the opinion clusters The different couplesof values of 119880 and 119879 model the size of the latitude ofnoncommitment (119879 minus 119880) and thus the agentsrsquo propensityto update opinions Whenever 119879 minus 119880 le 02 regularitiesin convergence and opinion clustering appear If 119880 gt 1there is always central convergence if 119879 lt 1 there is alwaysbipolar convergence with clusters of similar sizes For 119879 minus119880 gt 02 relatively low values of 119879 (119879 lt 1) and relativelyhigh values of 119880 always yield extreme bipolar convergencewith a relatively small cluster of moderate agents (bipolar-central convergence) For 119879 sufficiently high (119879 ge 12) and119880 sufficiently low (119880 le 08) in other words not very ego-involved agents there is a bigger spectrum of steady statesFor this reason we studied the metrics of interest for thesevalue differences and kept the couples of (119880 119879) depicted inTable 1 For these couples central bipolar bipolar-centralconvergence (3 clusters) and multicluster convergence (4 or5 clusters) are possible and result frommeaningfully differentattitude structures Judgment accuracy varies significantlyacross the four different attitude structures we retain and soit happens with the variance of opinions These scenariosprovides us with reference results from which to build upon

322 Calibrating Deliberative Social Interactions Delibera-tive social interactions correspond to social situations in

which individuals are not influenced by pair-wise discussionwith peers but are sensitive to collectively deliberated infor-mational cues On these terms parameters such as the sizeof agentsrsquo argument sacks (119896119894) the distribution of argumentsin them (120601119894) the minimum number of debates (119898) andthe proportion of debaters from the population (119899119863) are nolonger mute We assume for simplicity that the formulaexist120601119890exist119896forall119894(119896119894 = 119896 and 120601119890119894 = 120601119890 and 120601119898119894 = 05) is true with 119894 isin 119873for some value 120601119890 isin [05 1] and for 119896 isin N

The chosen domains for these parameters (see Table 1)are justified by the size of the population and the size of theargument pool (119872 = 600) which we set from the start If 119896is too big (119896 ge 32) andor too many agents are allowed inthe debate (119899119863 ge 02) then individuals always disprove thecentral argument and deliberation is never or rarely takeninto account in the update of opinions Similar effects occurwhen 119898 is high yet we only calibrate it with respect to themean running time of a simulation and the values of 119905119863Different distributions of 120601119890119894 seem to significantly or directlyexplain none of the statistics we analyze so we convenientlyset 120601119890119894 = 075 We chose the parameter 119896 to be always divisibleby 4 for it is convenient given the initial conditions regardingthe argument distribution in the agentsrsquo argument sacks

We fix the value space for the parameters 120574 119901119886 119901119903 as inTable 1 to account for different intensities of the effect ofdeliberative voting retroaction on the system For examplethe scenarios where the three parameters are at their lowestvalues correspond to the scenarios with the smallest fixedprior effect of deliberation in the model Increasing any ofthese parameters should mechanically be associated witha world in which agents are more sensitive to collectivedecisions The values for 119901119886 are taken from [13] wherethe author states that by means of deliberation 7 to28 of individuals changed their opinion from agreeingto disagreeing or vice versa on a referendum question forDenmarkrsquos participation in the EuroWe choose two differentvalues for 119901119886 that account for high (03) and very high (05)sensitivity to deliberation in a group 119901119903 on the other handis taken to be small (equal to 01) since we believe that thereexist agents that will always go against the reached consensusTheir numbers are meager

For simplicity we assume that for all 119894 isin 119873 120574119894 = 120574 forsome 120574 isin [0 05] In effect we posit that the heterogeneityof agents in the deliberative context only derives from theirarguments and their opinions We decide to calibrate 120574 inallusion to the strength of the social dynamics in pair-wiseinteractions In doing so we limit ourselves to consider threedifferent scenarios in which deliberation has respectivelyhalf equal and twice as much opinion-shifting power thanone-to-one social influence Finally we include the accept-ability voting quota rule 120572 whose domain is inspired fromclassical majority rules [35] observed empirically (Table 1)

14 Complexity

Table 2 Parameters and their domains used to study the effect of the procedural parameters on the metrics

Procedural parameters Other parameters119905119863 119898 119899119863 120572 119898 120583 119879 119880 119901119886 119901119903 120574 119896 119861119890ℎ119886V1198941199001199031 2 3 4 5 6 1 2 3 4 5 001 002 003 004 005 0 05 066 7 01 16 02 03 005 02 12 119873119886119894V119890 119865119900119888119906119904119890119889

323 Calibrating the Mixed Social Interactions Model Themixed interaction model ascribes to a parameter space inwhich the effect of collective choices on our metrics isnontrivial and where deliberation and voting on proposalsdetermine their acceptability Since the mixed model isequivalent to periodic iterations of pair-wise and collectivesocial interactions the calibration of the parameters in bothpreceding models holds (see Table 1 for the calibration)The first reason for this is comparability the second is thatthe previous calibration takes into account the fact thatboth dynamics are going to be combined We include thefrequency of debates by adding the parameter 119905119863 whichcontrols for the amount of pair-wise interaction amongagents between two distinct deliberation steps

324 Termination Conditions for Runs Simulations stoponce 100 proposals are deliberated on andor voted for inother words after 100 collective decision steps have occuredThe number of proposals discussed seems arbitrary yet it ishigh enough to observe the effects of deliberation on opiniondistributions and on other metrics related to coherence andjudgment accuracy We choose the number of runs as afunction of our research questions which give relevance tothe procedural parameters in the mixed model rather than tothe parameters set to describe the population and its behavior

325 Expected Outcomes of the Simulations We expect moredeliberation in terms of higher 119898 and 119899119863 to increasejudgment accuracy and at the same time to reduce thevariance of opinions In turn a smaller variance of opinionsimplies that the argument pool for deliberation is smaller andtherefore judgment accuracy should be lower Moreovervariance increasing dyadic interactions (or rejection) shouldalso increase judgment accuracy and coherence since bipo-larization and dissensus foster argument diversity in delibera-tionThe coherence statistic should be stronger in simulationsin which central convergence appears quickly and agentsare naive Attitude structures sensitivity to informationalcues and weights given to deliberated cues that makes pathsto bipolar convergence smaller or to central convergencelonger should be associated with higher judgment accuracyShifts of opinion should be more visible in scenarios inwhich extreme agents are pulled away from the extremesHence the sensitivity to deliberated cues (119901119886 119901119903) and 120574should explain the shifts When crossed with high values of119880 and 119879 the shifting power of deliberation should be at itshighest

In the end we do not know how these mechanismswill play out The results of the subsequent experimentsgive us insight on the interplay between the aforementionedeffects

4 Results

Before pointing at any result obtained from the mixedinteractions model we describe and compare the dyadicand deliberative interactions models in the parameter spaceobtained from the calibration in Section 3 (see Table 1)Primarily we require our results to describe two orthogonaltypes of opinion dynamics the pair-wise dyadic and thedeliberative The latter comprises scenarios in which indi-viduals do not influence each other by means of pair-wisediscussion and only update their opinions from delibera-tion the former scenarios where only pair-wise interactionsamong agents determine the dynamics We describe both onthe parameter space given in Table 1 by performing at least20 simulations per scenario Quantitative results from thepair-wise interaction model alone are described during thecalibration and are not treated heremdashdeliberative parametershave no effect on it Moreover we show how these differentmodel specifications yield qualitatively different opiniondistributions and compare them on the basis of the metricsof interest To this end we perform independent two-sampleStudentrsquos t-tests18 or Welch t-tests19 and compute confidenceintervals at a 095 level of confidence Last but not least wecomment ordinary least squares (OLS) regression estimatesto account for the direction and magnitude20 of the effectsof the parameters on the metrics Estimates are declaredsignificant at a 5 level of risk

We perform the same analysis on the mixed interactionsmodel as we compare it to the pair-wise dyadic and deliber-ative interactions models and discuss the marginal effects ofthe parameters on the metrics More precisely we obtain twodifferent types of results for the mixed interactions modelthe first compares the mixed scenarios to their monolithiccounterparts with respect to each metric and with respect tothe regression estimates the second gives a clear idea of themarginal effect of each governance parameter (or parameterof interest) on our observations In the first we allow controland procedural parameters to vary on a parameter spacesimilar to those of the dyadic and deliberative opiniondynamics (see Table 1) In the second we fix all of ourcontrol parameters execute 36000 balanced21 runs andfocus only on the one-way pooled effects of our proceduraland behavioral (all agents are focused or naive) parametersWe generate comprehensive graphs and tables to account forthe obtained results

Please notice the slight change in the size of the parameterspace for the procedural parameters and in the values ofthe nonprocedural ones for the first and second type ofresults (refer to Tables 1 and 2 respectively) Since we wantto have a finer idea of the marginal effects of each of theprocedural parameters on how well a group decides onproposals we make the value jumps for each parameter (1)

Complexity 15

() (U T) = (02 18) (b) ( pa m nD) = (005 05 00 2 02) (＝) ０ＬＧＮＬＭ ＣＨ (＜) (＝) Ｈ＞ tD = 3

(d) (５ ４) = (02 14) (e) ( pa m nD) = (01 03 05 3 02) () ０ＬＧＮＬＭ ＣＨ (＞) () Ｈ＞ tD = 6

Figure 7 From left to right common opinion trajectories and distributions for dyadic deliberative and mixed opinion dynamics within thestructure of different decision-making processes

small enough to detect significant differences in the estimatesand (2) meaningful for interpretation For the fixed one-valued parameters in Table 2 we use Table 1 to set them totheir minimum mean or median values

41 Comments on Dyadic and Deliberative Opinions Dynam-ics In this subsection we present our first results regardingthe deliberative opinion dynamics described in Equation (6)and its differences with the dyadic dynamics (Equation (5))in terms of our observations

411 Qualitative Analysis of Pair-Wise Opinion DynamicsPair-wise opinion dynamics alone produces multiclusterstable convergencewith variable cluster sizes (see Figures 7(a)and 7(d)) Insight on the complexity of these dynamics wasdiscussed in Section 321

In Figures 7(a) and 7(d) agents discuss randomly withone another in pairs Clusters in the extremes form quicklysince either agents that are close to the extremes attractone another or are convinced to stay close to the extremesby interacting with other agents they disagree with Otheragents with near-extreme positions may be simply attractedto one of the several moderate foci of agents in the opiniondistribution On the other hand moderate agents are eitherattracted to the closest opinion focus or pushed towards theextremes of the distribution Over time agents in the centralfocus either attract other agents into it or ignore the opinions

of the extreme agents that interact with them The result ofthese dynamics yields amulticlustered convergencewhere theopinion foci are at least at distance 119880 from one another andonly the extreme foci (clusters of agents with |119900119894| ge 075) areat a distance greater than or equal to 119879 from one anotherThefoci at the bounds of the opinion distribution are very stableand in analogy to the (12119889)-rule in assimilation bounded-confidence models [33] the number of clusters that form isroughly equal to 119880minus1412 Qualitative Analysis of Deliberative Opinion DynamicsQualitatively deliberative opinion dynamics yield a looseunipolar convergence of opinions near the center (centralconvergence see Figure 7(b)) near the center of either the leftor right portions of the distribution (group polarization seeFigure 7(e)) or to a sparse bipolar opinion distribution withclusters at the center and at the extremes (see Figure 7(c)) ifmixed with pair-wise interactions This is probably a reasonto expect the variance of opinions and the proportion ofextremists (almost perfectly correlated 120588 = 089 119901 lt 0001)to be low for these scenarios The side towards which theopinion cluster skews depends essentially on the valence ofthe first argument that is collectively accepted

In Figures 7(e) and 7(b) agents that update their opin-ions do so at the same time and only when deliberationis successful In these scenarios every time a decision iscollectively accepted agents update their opinions towards

16 Complexity

Table 3 Ordinary least squares estimates for pair-wise interactions scenarios for most of the observed metrics

Dependent variable119881119886119903(119900) 119878ℎ 119890119888 119894119903(1) (2) (3) (4)Constant 0610lowastlowastlowast 4429lowastlowastlowast 0472lowastlowastlowast 0000

(0002) (0206) (0003) (0000)

119880 = 06 (119880 = 02) minus0012lowastlowastlowast 2981lowastlowastlowast 0061lowastlowastlowast 0000(0002) (0260) (0004) (0000)

119879 = 18 (ref 119879 = 14) minus0264lowastlowastlowast minus3261lowastlowastlowast 0004 0000(0002) (0260) (0004) (0000)

120572 = 066 (ref 120572 = 05) 0002 minus0059 minus0100lowastlowastlowast 0000(0002) (0184) (0003) (0000)

(119880 119879) = (06 18) (ref (119880 119879) = (02 14)) minus0324lowastlowastlowast minus3728lowastlowastlowast minus0033lowastlowastlowast 0000(0003) (0368) (0006) (0000)

Observations 1440 1440 1440 1440R2 0983 0389 0529Adjusted R2 0983 0388 0527Residual Std Error (df = 1435) 0032 3492 0053 0000F Statistic (df = 4 1435) 21 272740lowastlowastlowast 228686lowastlowastlowast 402326lowastlowastlowastNote lowastplt01 lowastlowastplt005 lowastlowastlowastplt001

the argumentrsquos defended position Hence if there is votingand if agents happen to obtain 119894119899 proposal arguments theyinexorably cluster towards one or the other side of the opiniondistribution resulting in opinion convergence remeniscent ofgroup polarization (see Figure 7(e)) Furthermore as agentscluster in the same side of the opinion spectrum the argu-ments that may be advanced in deliberation are fewer andmore skewed towards that one side of the opinion spectrumAgents become more easily persuaded in deliberation andonly by arguments supporting one side of the spectrumConvergence of opinions towards one loose opinion focusbecomes faster and more certain If there is no vote (seeFigure 7(b)) the dynamics are similar but for the fact thatit is the uniform randomness of the proposal argument thatguarantees a central convergence of opinions

413 Quantitative Analysis of Pair-Wise Opinion DynamicsThe three parameters that allow for quantitative analysis are119880 and119879 the agentsrsquo attitude structure and themajority quotavoting rule 120572 Table 3 supports the claim that going from 120572 =05 to 120572 = 066 does not affect the variance nor the shifts ofopinions However judgment accuracy is significantly higherwhen accepting proposals becomes more difficult This israther surprising but also intuitive since decisions that aretaken using more restrictive methods of scrutiny should bemore accurate

Concerning the attitude structures the interaction termbetween 119880 and 119879 shows that 119880 has a small effect on thevariance of opinions given that 119879 is low This shows thateven when the assimilation threshold is high a sufficientlysmall rejection threshold can make its one-way effect vanishHowever high levels of both 119880 and 119879 reduce the variance ofopinions dramatically In contrast when it comes to overallchanges in opinion the magnitude of 119880rsquos one-way marginal

effect increases the shifts of opinions whereas 119879 lowers itThis can be interpreted as having the rejection dynamics pushmoderate-to-extreme agents to positions that are close tothe extremes and very moderate agents to the center of thedistribution Not too many quantitative changes (on average)can therefore be observedThe fact that higher119880 is associatedwith faster convergence of opinions seems to imply lowerjudgment accuracy At some point voting for the same type ofproposals for too longmay result inmany collectivemistakesThis is an interesting conclusion to consider in the analysis ofthe mixed model Finally the interaction term shows that 119879has no strong impact on the metric

414 Quantitative Analysis of Deliberative OpinionDynamicsSeven parameters can be considered for quantitative anal-ysis in this dynamics the parameters of interest recalledin Section 312 the size of the argument sacks (119896) agentbehavior during deliberation the strength of the deliberationdynamics (120574) and the measure sensitivity to deliberatedresults (119901119886) The most ldquounexpectedrdquo result obtained fromthe deliberation dynamics is that it shows how deliberationcontributes to the proportion of extremists and the varianceof opinions It happens that having more agents discussproposals and having more of these discussions increasesthe variance of opinions and the proportion of extremistswhile lowering judgment inaccuracy and coherence For theformer the fact that far too many arguments are advancedduring deliberation plummets the chances that the centralargument is accepted hence deliberation does not moveopinions too often For the latter we observe that only ldquobigrdquodifferences (going from the lowest value of a parameter to thehighest) in protocol actually prove to have a preponderantimpact on the metrics These observations hint at a possibletrade-off between judgment accuracy and the variance of

Complexity 17

Table 4 Ordinary least squares estimates for the deliberative interactions scenarios for most of the observed metrics

Dependent variable119881119886119903(119900) 119878ℎ 119890119888 119894119903(1) (2) (3) (4)Constant 0032lowastlowastlowast 13092lowastlowastlowast 0370lowastlowastlowast 1007lowastlowastlowast

(0001) (0406) (0001) (0001)

119898 = 3 (ref119898 = 1) 0012lowastlowastlowast minus2062lowastlowastlowast minus0012lowastlowastlowast minus0001(0001) (0276) (0001) (0001)

119898 = 6 (ref119898 = 1) 0023lowastlowastlowast minus3285lowastlowastlowast minus0018lowastlowastlowast minus0004lowastlowastlowast(0001) (0276) (0001) (0001)

119899119863 = 002 (ref 119899119863 = 001) 0007lowastlowastlowast minus0965lowastlowastlowast minus0006lowastlowastlowast minus0002lowastlowast(0001) (0276) (0001) (0001)

119899119863 = 005 (ref 119899119863 = 001) 0020lowastlowastlowast minus3316lowastlowastlowast minus0015lowastlowastlowast minus0004lowastlowastlowast(0001) (0276) (0001) (0001)

120572 = 05 (ref 120572 = 0) 0146lowastlowastlowast 68932lowastlowastlowast 0068lowastlowastlowast minus0507lowastlowastlowast(0001) (0276) (0001) (0001)

120572 = 066 (ref 120572 = 0) 0291lowastlowastlowast minus13842lowastlowastlowast 00004 minus0000(0001) (0276) (0001) (0001)

119896 = 8 (ref 119896 = 4) 0004lowastlowastlowast minus0815lowastlowastlowast minus0002lowastlowastlowast minus0001(0001) (0276) (0001) (0001)

119896 = 16 (ref 119896 = 4) 0008lowastlowastlowast minus1118lowastlowastlowast minus0004lowastlowastlowast minus0003lowastlowastlowast(0001) (0276) (0001) (0001)

119861119890ℎ119886V119894119900119903 = ldquoFocusedrdquo (ref ldquoNaiverdquo) 0015lowastlowastlowast minus1980lowastlowastlowast minus0012lowastlowastlowast minus0004lowastlowastlowast(00005) (0225) (00004) (0001)

120574 = 01 (ref 120574 = 005) minus0020lowastlowastlowast 5762lowastlowastlowast minus00003 minus0001(0001) (0276) (0001) (0001)

120574 = 02 (ref 120574 = 005) minus0025lowastlowastlowast 8648lowastlowastlowast minus00003 minus0001(0001) (0276) (0001) (0001)

119901119886 = 05 (ref 119901119886 = 03) minus0013lowastlowastlowast 1581lowastlowastlowast 00004 00003(00005) (0225) (00004) (0001)

Observations 19440 19440 19440 19440R2 0931 0844 0562 0958Adjusted R2 0931 0844 0562 0958Residual Std Error (df = 19427) 0033 15701 0030 0050F Statistic (df = 12 19427) 21 844790lowastlowastlowast 8 733563lowastlowastlowast 2 079585lowastlowastlowast 36 761250lowastlowastlowastNote lowastplt01 lowastlowastplt005 lowastlowastlowastplt001

opinions that may be of interest in the mixed discussionmodel

Group coherence shifts of opinions and the varianceof opinions are highly affected by the voting thresholdthe coherence metric is reduced by half when allowing forclassic majority voting shifts explode and the variance ofopinions increases dramatically This is not surprising sincethe majority quota rule works as a buffer to the success ofdecision-making processes and in consequence it hampersthe deliberative opinion updates For the shifts of opinions itis clear that the two-third majority voting rule results in littleopinion change and that the simple majority rule inducesgroup polarization As expected there are less extremists inscenarios where there is no voting and the shifts of opinionare more likely to happen when agents are more sensitiveto deliberation When we include voting in the system

the grouprsquos judgment accuracy decreases due to the correctdeliberated decisions that are vetoed by a salient opinion-agent majority Strangely the effect only holds for the simplemajority decision rule Lastly the proportion of individualsthat participate in the debate and the minimal amount ofdebates have similar marginal effects on all the metrics (seeTable 4)

415 Comparing Deliberative and Pair-Wise Opinion Dynam-ics in a Nuthshell Qualitatively both models yield similaroutcomes but for the fact that deliberation never producesopinion polarization with both types of extremists (seeFigure 7) Furthermore clusters in the deliberative interac-tions model are less homogeneous in size and multiclusterconvergence occurs only when deliberation is rarely suc-cessful which happens sporadically given the calibration of

18 Complexity

Table 5 Mean difference 095 confidence intervals for metrics by model comparison

Model Metric119881119886119903(119900) 119901119903119900119901119890119909 119878ℎ 119890119888 119894119903Dyadic vs Deliberative [0191 0216] [0170 0197] [-2948 -2828] [0074 0082] [0165 0172]Mixed vs Deliberative [0142 0147] [0150 0155] [-2085 -1976] [0003 0002] [-0038 -0030]Mixed vs Dyadic [-0072 -0046] [-0045 -0018] [8332 8859] [-0079 -0071] [-0205 -0201]the model On the other hand pair-wise opinion dynamicsproduces multicluster convergence most of the time andcentral convergence only in the infrequent cases where thelatitude of acceptance is large

The comparison of these two scenarios in terms ofthe metrics is straightforward Whilst dyadic interactionsallow for more variability in opinions through pair-wiseinteractions and more mistakes in judging proposals delib-erative updates simply act as an antagonist force On aver-age the deliberative interaction model produces opiniondistributions with smaller variance and yields steady stateswith higher judgment accuracy a higher shift statistic andnaturally lower group coherence since only proposals that aredeliberated 119894119899 are eligible for voting (see Table 5)

42 Articulating Dyadic and Deliberative Social Interactionsin Opinion Dynamics So far we have considered scenarioswhere argumentation had no place and where individualsmade decisions according to opinions that were purelyconstructed from pair-wise discussions We also observedcases in which agents formed their opinions only by inte-grating deliberated proposals into their opinion updates Thescenarios we present in this section combine the two pre-ceding scenarios to account for both the effects of pair-wisediscussion and the effects of deliberation on opinions judg-ment accuracy and group coherence From the precedingobservations we conclude the following deliberative opiniondynamics reduces the variance and the proportion of extrem-ists because it shifts opinions greatly and polarizes groupsby eliminating at least one type of extremist the one whoseopinions are initially opposed to the deliberated resultsor it brings opinions close to neutrality (see Figure 7(b))This may not be desired in a group since the diversity andstability of opinions are necessary for deliberation to makesense and lead to the collective acceptance of controversialproposals Dyadic opinion dynamics on the other handtend to increase the variance of opinions and the numberof extremists while having a lesser aggregated effect on theshifts of opinion This is an immediate result of the pair-wiseinteraction dynamics inasmuch as it polarizes individualsvery quickly

To sum up in the mixed interactions model deliberationhinders the effect of voting on opinions by making someproposals that would normally be submitted for voting noteligible for voting It may also undo opinion changes dueto pair-wise interactions The mixed model can be seen asthe deliberation opinion dynamics model in which pair-wiseinteractions occur in between collective discussion and mayaffect which arguments are advanced during deliberationEquivalently it can be seen as the dyadic opinion dynamics

model in which deliberation accounts for endogenousldquoshocksrdquo on the opinion distribution

421 Qualitative Analysis of the Mixed Dynamics In Figures7(c) and 7(f) agents update their opinions through delibera-tion and pair-wise interactions Opinion trajectories visualsshow a combination of both previously described opiniontrajectories Over time as agents discuss with one anotheropinion clusters form as in the pair-wise or dyadic opiniondynamics Deliberation may disrupt the formation of theseclusters and three different situations may occur

(i) Deliberation disrupts the formation of the opinionclusters but not sufficiently to jeopardize pair-wiseopinion cluster formation (eg for high values 119905119863)Two phenomena may be responsible for it (1) thevariability of the proposal argumentrsquos support for theprinciple P and (2) too few (successive) deliberationsteps that result in opinion updates Once the opinionclusters are formed successful deliberation foreshad-ows the merging of the previously formed clustersmay bring extremists closer to moderate agents andthanks to the assimilation component of the pair-wisedynamics allow bigger and more moderate opinionclusters to form (refer to Figure 7(c))

(ii) Deliberation significantly disrupts the pair-wisedynamic formation of clusters This may happenwhen the requirements for accepting proposals areweak in the decision-making process (eg low 120572) Inthis case one has either central convergence sinceall agents follow the deliberated results in the samemanner or 3-cluster convergence with two extremeopinion foci In the latter situation the extremegroups form quickly (like in the pair-wise model) anddeliberation cannot pull extreme agents away fromthe extreme foci they are in The main reason for thisis that immediately after an extreme agent attempts toleave her foci through deliberation updates she getspulled or pushed back into it by means of a dyadicinteraction with another extreme agent

(iii) Deliberation does not disrupt the formation of theopinions cluster This is the rare case scenario forvery demanding decision-making processes in termsof inclusion (eg 119899119863) and argument acceptability (eg120572) Opinions converge as in the pair-wise opiniondynamics model (refer to Figures 7(a) and 7(d))

Last when there is voting (120572 = 0) the position ofthe moderate opinion cluster is determined by the first

Complexity 19

minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

Var(o)

sh

ec

ir

Var(o)

propex

sh

ec

ir

1

098

minus02

minus004

001

1

minus014

minus001

minus003

1

026

minus028

1

minus057 1

ＪＬＩＪ

Ｒ

Figure 8 Significant correlation estimates for themetrics of interestin the mixed opinion dyamics scenarios

accepted proposal argument andor the opinionmajority thatis formed from previous dyadic interactions (see Figure 7(f))

422 Relating the Metrics of Interest in Mixed OpinionDynamics The first thing to notice is that for all of thesescenarios all the metrics are significantly correlated (seeFigure 8) The variance of opinions and the proportion ofextremists are almost perfectly correlated To some extentthis means that a high variance of opinions known thatthe mean of opinions is statistically null equates to havingmany extremists in the population be towards one or theother side of the opinion distribution (eg extreme bipolar-ization) Similarly we find a negative and strong correlationbetween the judgment inaccuracymeasure and the coherencecoefficient (see Figure 8)mdashwhen judgment inaccuracy is high(low) coherence to deliberation is low (high) Cases likethese are telling since they may be related to sequences ofcollective decisions in which agents do not make too manymistakes when judging proposal arguments and still makedeliberated decisions that are on average in line with theirprinciples

The shift statistic interestingly is negatively correlatedwith the variance of opinions with judgment accuracy andwith the coherence coefficient In a way this means thathaving more moderate agents conduces on average to moremistakes in judging arguments and to less coherent collectivedecisions An explanation for this phenomenon could be thatas agents reach consensus in opinion important (possiblyextreme) arguments for correctly judging argument propos-als are not advanced in the deliberation arenawhich results inmore mistakes and in possibly accepting proposal argumentsthat after voting are not meant to be accepted Indeed the

negative correlation between the variance of opinions andjudgment inaccuracy points towards this direction since onaverage the more shifts there are the smaller the variance ofopinions is (see Figure 8)

423 Comparing Mixed Dynamics with Other MonolithicDynamics of Interactions Qualitatively opinion trajectoriesin the mixed model are more chaotic than in the pair-wise and deliberative interaction models (see Figure 7) Asanalyzed previously deliberation dynamics disrupt the pair-wise dynamics and create situations in which central andmulticlustered convergence (with two clusters at the extremesof the distribution) are possible at the expense of more(pair-wise dynamics) or less (deliberative dynamics) clustersIt tolerates the survival of very small isolated group ofagents that the previous models rarely tolerate in a hundredcollective decision-making steps (see Figure 7(c))

One of the most interesting observations one can makeof this family of models regards how groups change (in termsof our metrics) when dyadic social interactions are addedto deliberative interactions and vice-versa When comparingthe mixed interactions with only deliberative ones shiftsagentsrsquo aggregated change in opinions are on average lesspronounced The variance of opinions and the proportionof extremists are significantly higher for the former than forthe latter group coherence in contrast is lower with respectto the scenarios where opinions are formed only throughdeliberation Judgment accuracy follows that same trendA reason for this might be that mixed dynamics are fasterto yield semistable clusters andor opinion convergence Inconsequence the potential argument pool for deliberativeexchange shrinks faster and deliberation becomes less effec-tive over time which results in more collective mistakes Theoutcomeon coherence is predictable inasmuch as deliberativeinteractions produce group polarization If a group agrees inopinion debates are sterile because themembers of the groupwill most likely refrain from attacking the proposal argu-ments they all support and will work together to disparagethe proposal arguments they do not support (see Table 5)

In the same manner we report that the proportion ofextremists and the variance of opinions are barely lower inthe pair-wise interaction model when including deliberationin the decision-making process Shifts of opinions are alsobarely lower when there is no deliberation This can indicatethat deliberation does not undo the changes in opiniondue to pair-wise interactions nor does it amplifies them Incontrast when we consider the labeling-based metrics themixed interaction model ensures lower judgment inaccuracybecause of deliberation and lower group coherence (seeTable 5)

In sum the mixed discussion model is a compromisebetween the pair-wise interaction and deliberative opiniondynamic models in that it offers some guarantees in termsof judgment accuracy and coherence while allowing forldquoreasonablerdquo variance in the opinion pool It establishes somekind of trade-off between labeling-based metrics and thevariance of opinions that may be of interest for decision-making process design

20 Complexity

Table 6 Summary of sign or relative marginal effects for all varying parameters on dyadic deliberative and mixed interactions models inand for all experiments

Metrics Parameters119905119863 119898 119899119863 120572 119879 119880 119901119886 120574 119896 119861119890ℎ119886V119894119900119903119881119886119903(119900) XX X X O X X X X X X119901119903119900119901119890119909 XX X X O X X X X X X119878ℎ XX X X O X X X X X119890119888 XX X X XO X XOO XOO X X119894119903 XX X X O XO O X OOX XOO X XNota gray symbols correspond to dyadic dynamics light gray to deliberative dynamics and bold black to mixed dynamics Given a scenario the symbol 119883stands for constant parameter and 119874 for insignificant marginal effect and as parameters grow for positive (negative) marginal effect for the parameterFrom left to right at the parameter level the leftmost arrow corresponds to the scenario with the strongest (in absolute value) marginal effect of the parameter(with respect to to its factors) Example Let 119901 be a parameter and 119874 stands for ldquo119901 has a significant positive effect in dyadic discussion scenario negativein the deliberative one and insignificant in mixed discussion The positive effect in dyadic discussion is greater than the absolute value of the negative one inthe purely deliberative scenariordquo

424 Comments on the Differences between the Three Modelsfor OLS Regression Estimates The significance of most of theregression parameters estimated for the mixed interactionsscenarios are similar to those estimated for the pure dyadicanddeliberative scenariosNevertheless themagnitude of theeffects is different insomuch that the dynamics themselvesare different So instead of commenting the linear regressionestimates of the mixed model we compare when possiblethe standardized significant coefficients of the regressors ofthe mixed model with those of the pair-wise and deliberativeinteractionmodels (see Table 6 for a summary of the results)

In mixed interactions for instance the influence of119898 onthe opinion distribution is much more preponderant than inthe deliberative dynamics scenarios Going from 119898 = 1 to119898 = 3 increases the standard variation units of the variance ofopinions by twice as much as the one in the deliberative sce-nariosThis in particular is due to the design of the decision-making process If 119898 is higher then the number of socialinfluence steps (119905119863) in the decision process is also higher(compared to 119905119863 = 0 in the deliberative case) If we decideto consider these steps as contributing to the variance ofopinions then the effect of119898 on the variance of opinions hasto be stronger in the mixed interactions model In contrastthe number of agents participating in the debate has a weakereffect on the variance of opinions in the mixed interactionsmodel than in the deliberative dynamics specially for 119899119863 =005 (more than twice as much) The difference may comefrom the fact that in mixed discussions the part of theargument pool used in deliberation is more restrictive thanin the deliberative case We can argue that since pair-wisedynamics may get individuals closer to one another aftersome deliberation the same happens to the arguments thatindividuals will most likely use in deliberation Taking thisinto account means that deliberation will at many instancesbe unfruitful and have no effect on opinions thus failing tomoderate extreme opinions andor polarize the group In thesame direction a high number of participants in deliberationcan make the deliberation dynamics stiff In this case thedistribution of opinion is close to the uniform distributionwhich has a relatively high-to-moderate variancewith respectto the reference results

The strength of the deliberative dynamics has a positivestronger effect on the variance of opinions in themixedmodelthan in the deliberative model when 120574 = 02 and a negativeweaker one when 120574 = 01 A sound interpretation of thisresult is that when deliberation is successful and its effect onopinions does not get agents to move sufficiently away fromtheir current positions agents are pulled or pushed back totheir previously held opinions the variance of opinions doesnot changemuchOn the other hand if the effect of successfuldeliberation is strong opinions can change sufficiently tomake agents much more prone to assimilation into moderateopinion foci the variance of opinions falls

The deliberative opinion dynamics model is much moresensitive to 120572 than the mixed discussion model Differ-ences in marginal effects can double The probability ofattraction to deliberated results (119901119886) for instance has astrongmoderating effect (negative) on agentsrsquo opinions in thedeliberative opinion dynamics but a meager one (two-fold)in the mixed and dyadic dynamic models This is becauseopinion updates in the deliberative model depend only onthat parameter whereas in the mixed model more numeroustypes of interactions make opinion updates possible

In terms of judgment accuracy all parameters except thesize of the argument sacks (119896) have a significant slightlystronger effect on the deliberative dynamics than in themixedmodel In effect the difference may be attributed to theupshot of pair-wise interactions on the distribution of opin-ions and on the arguments that would be advanced duringdeliberation The result respecting parameter 119896 however iscounterintuitive Since argument sacks are static scenarioswhere shifts are higher should make 119896 more important fordeliberation to succeed and it is precisely in the scenarios ofpure deliberative dynamics that we find the biggest opinionshifts This may point to the hypothesis that in deliberativeinteractions agents change opinions substantially but do notoften change their adherence to the principle (go from anegative opinion to a positive one or vice versa) when thereis no group polarization

43 Sensitivity Analysis for the Procedural Parameters ofInterest Weperform a sensitivity analysis of the observationson the parameters of interest in themixed interactionsmodel

Complexity 21

We extend the parameter domains of the pure dyadic anddeliberative scenarios as we make it explicit in Table 2

431 Minimum Number of Debates (119898) We observe that theminimum number of debates has a significant well-observedeffect on all of our metrics excluding group coherence Forthe variance of opinions we can observe that the moredebates there are the bigger the value of the metric isalthough the higher 119899119863 and 119905119863 are the weaker the overalleffect of119898 becomes (Figures 9(d) and 9(a)) Furthermore themarginal effect of increasing minimal debates is decreasingan additional deliberation step requirement increases thevariance of opinions but less and less as it grows 119898 coupledwith 119899119863 has a shy S-shaped effect (see Figure 9(d)) on thevariance of opinions meaning that it has a stronger effectwhen 119899119863 is low which suggests a possible trade-off betweenthe number of arguments accepted in the debate and thenumber of debates required before accepting a proposal

In contrast the shifts in opinion are less and less likelyas 119898 grows and this is independent of the variations of theother parameters Again the effect is marginally decreasingand is only truly significant when 120572 = 05 An explanationto these effects may be linked to the design of the systemFirst the variance of opinions is higher when deliberation isasked for because the more deliberation steps there are thehigher the chances are that the central proposal argumentis deemed unacceptable specially when agents are focusedFurthermoremechanically speaking increasing theminimalamount of debates implies that whenever a decision is to betaken at least 119898 times 119905119863 time steps have to take place and if atany moment a proposal argument is considered undecided119905119863 time steps are added to the process So unless the debatedoes not yield 119906119899119889 labels for proposal arguments (highlyunlikely considering that 120590 = grounded semantics) the morenondeliberation steps there are in the decision process thehigher the variance of opinions is Concerning the shiftswhen 120572 = 05 either the system is too stiff to accept anyproposal argument and opinions do not changemuch or theeffects of pair-wise discussion and deliberation cancel out insuch a way that individual shifts are minimal (Figure 9(b))

On the side of labeling-based metrics the more debatesare asked for the more accurate a group is in its judgmentthe effect being smaller as 119898 grows When agents are naivethe effect is quasilinear while when they are focused thestrongest effects of acquiring more deliberation are foundwhen levels of deliberation are already low (Figure 9(f))Thiscan be explained by the fact that the more debates thereare in the decision process the closer one gets to the idealargumentation framework

432 Proportion of the Population in Deliberation Steps (119899119863)Like for 119898 119899119863 has a significant effect on the proportionof extremists and on the shifts and variance of opinions(Figure 9(e)) This may result from the fact that being ableto put more arguments in play at the same debate step mayincrease the odds of revealing cycles around the proposalargument in119860119865120576 Given that we use grounded semantics thearguments in the cycles are labeled 119906119899119889 and in consequence

debates are more often postponed when 119899119863 is higher Post-poning debates in turn increases the number of nondebatessteps in the decision process which increases the varianceof opinions and limits over time the moderating effect ofdeliberation Moreover the effect of this parameter is verydependent on the value of 120572 (Figure 9(e)) For shifts forinstance 120572 = 05 makes the effect of 119899119863 negative while120572 = 0 makes it positive but to a lesser extent Furthermorefor higher requirements of deliberation (119898) adding moreindividuals to the deliberation process has weaker effectson the variance of opinions and on the other metrics It isalso quite interesting to notice that it has the same effect onagents independently of agent behavior during deliberationThis is a surprising result as one would have expected morefocused agents in a deliberation arena to cause a raise inthe proportion of extremists as they play to knock outopposing proposal arguments and thereof reducing the effectof deliberation (lowers variance) on the whole population

Finally similar to119898 addingmore people into the deliber-ation process increases judgment accuracy (see Figure 9(c))and has no clear effect if any on group coherence (seeFigure 9(i))

433 Steps between Deliberation Steps (119905119863) In all configu-rations 119905119863 increases the proportion of extremists (variance)and decreases the shifts of opinions among agents Theshifts and the effect on the variance of opinions are presentfor 120572 = 05 insignificant otherwise (Figure 9(b)) 119905119863 ishighly linked to 119898 by construction When 119898 = 1 thecurve linking the variance of opinions and 119905119863 is convex As119898 increases the curve becomes more and more concavewhich means that 119905119863 has a more important effect on theopinion distribution as collective decisions take longer to beachieved This seems counterintuitive but in reality it reflectsthemultiplicative relationship between deliberation and pair-wise interactions and the semantics in the model If 119898 is lowand 119905119863 is high the effective number of pair-wise interactionsis on average smaller in the deliberation process Hencethe structure of the decision-making process hinders anyincrease in the variance of opinions If119898 is high the oppositeeffect is observed Additionally since the grounded semanticsyields few 119894119899 arguments with respect to other admissibility-based semantics getting closer to the ideal argumentationframeworkmaymake it difficult to obtain the expected effectsof deliberative discussion on agents Lower 119898 constrains thevariance of opinions by making deliberation more influentialon opinions

In terms of judgment accuracy and coherence we observethat 119905119863 does not explain coherence yet it tends to decreasejudgment inaccuracy This may be due to a more variedargument pool in the debate which in turn may be aconsequence of a higher variance of opinions or vice versa

434 Acceptability Voting Quota (120572) By far it is the mostinfluential parameter in our study It changes the directionand the intensity of the effect of all the other proceduralparameters and by construction heavily constrains the roadto accepting a proposal (see Figure 9)

22 Complexity

015

020

025

030

035

040

1 2 3 4 5 6

12345

Values of Ｇ

(a) 119905119863 times 119901119903119900119901119890119909 by119898

0

10

20

30

40

50

1 2 3 4 5 6

005066

Values of

(b) 119905119863 times 119878ℎ by 120572

040

045

050

055

060

1 2 3 4 5

001002003004005

Values of Ｈ＄

(c) 119898 times 119890119888 by 119899119863

015

020

025

030

035

040

1 2 3 4 5

001002003004005

Values of Ｈ＄

(d) 119898 times 119901119903119900119901119890119909 by 119899119863

0

10

20

30

40

50

001 002 003 004 005

00005066

Values of

(e) 119899119863 times 119878ℎ by 120572

040

045

050

055

060

1 2 3 4 5

BehaviorNaiveFocused

(f) 119898 times 119890119888 by 119887119890ℎ119886V119894119900119903

015

020

025

030

035

040

0 05 066Behavior

NaiveFocused

(g) 120572 times 119901119903119900119901119890119909 by 119887119890ℎ119886V119894119900119903

05

06

07

08

09

10

1 2 3 4 5 6

00005066

Values of

(h) 119905119863 times 119894119903 by 120572

08700

08725

08750

08775

08800

001 002 003 004 005

BehaviorNaiveFocused

(i) 119899119863 times 119894119903 by 119887119890ℎ119886V119894119900119903

Figure 9 Curves of mean observations for different parameters and metrics for 36000 runs Bars are 095 confidence intervals Plots withthe same observables (ordinate) except for 119894119903 are on the same scale

Complexity 23

The higher the requirement for accepting the proposal isthe higher the proportion of extremists and the variance ofopinions are and the lower the shift statistic is In few words120572 constrains the world to dyadic discussion or throws it intoa process in which deliberation is much more importantthan dyadic interactions This is why for any 120572 = 05 oneeither gives too much weight to deliberated results whichshadows pair-wise interactions or too little weight in thesense that no deliberated result is ever accepted and thusnever integrated into the opinions of the agents that havevoted for it The reason for the latter is that the latitudeacceptance is too low and thereafter pair-wise interactionsare not enough to unevenly polarize the population in such away that deliberated proposal arguments are voted favorablyby a non-negligeable majority Shifts for 120572 = 0 are stable fordifferent levels of 119905119863 which implies that deliberation keeps incheck all the shifts of opinions related to an increase in thenumber of pair-wise interactions For 120572 = 05 deliberationis less likely to be successful and shifts tend towards the shiftlevels for 120572 = 0 (see Figure 9(b))

Concerning labeling-basedmetrics120572 entirely determinesthe coherence statistic For 120572 = 05 there is no difference incoherence because of how coherence is defined it is maximalwhen 120572 = 0 and strangely maximal when 120572 = 066(see Figure 9(h)) Surprisingly 120572 has no effect on judgmentaccuracy One would have expected a higher 120572 to increasejudgment accuracy since agents would hardly collectivelyaccept a proposal argument discussed during deliberationProposal arguments have given how the argument lattice isgenerated a higher probability of being rejected than of beingaccepted in the ideal argumentation framework

435 A Word on Focused and Naive Agents Focused andnaive specifications for agent behavior are an importantparameter in the decision process since they model howagents choose arguments in the deliberation arena In the sce-narios where agents are focused agents show equal or highervariance of opinions and are therefore more extreme (seeFigure 9(g)) independently of the voting requirements Shiftsof opinion are less likely as well We can see this happeningbecause focused agents knock out proposal arguments moreoften than naive agents provided that 119899119863 is low

One could have expected naive agents to yield morecoherent decisions since they argue sincerely However thisseems not to be the case (see Figure 9(i)) Agents are unable todeliberate in away thatmakes deliberation reflect their votingintentions either because they do not have the necessaryarguments to do so or because such arguments do not exist

In terms of judgment accuracy focused agents do betterthan naive agents in finding the ideal labels for the proposalarguments This is likely because when reconstructing theframework focused agents take into account the deliberatedproposal argument and choose themost pertinent argumentsin their sacks that relate to the state of affairs at the deliber-ation arena In fine debates result in a better approximationofL119886119887120576(119868) even though one could have believed the oppositethat is focused agents are ready to sacrifice a correct collectivelabel of a proposal argument in the goal of defending theirpositions

44 Points of Discussion In this subsection we discuss theresults obtained from the simulations We attempt to makesense of them from a social sciences perspective

441 Opinion Consensus and Dissensus The deliberativeopinion dynamics corresponds to the simulated scenarios inwhich the variance of opinions is at its lowest and judgmentaccuracy at its highest Dyadic opinion dynamics are associ-ated with the scenarios in which the variance of opinions is atits highest judgment accuracy at its lowest and coherence atitsmaximumpossible levelThemixed interactions dynamicsis a combination of these aspects for all the metrics exceptfor coherence at which it is the worsemdashdeliberation leadsto some kind of loose opinion consensus that is reinforcedby pair-wise interactions Likewise the mixed discussionmodel confirms some results exposed in [16] expressingthat opinions and voting intentions in a deliberative contextldquooften changerdquo That being said and more particularly indeliberative opinion dynamics outputs align well with twointeresting results on consensus formation given by aninterpretation of Moscovici and Doisersquos [15] work Sunsteinrsquos[14] account on deliberation and group polarization andsocial impact theory [4] The first states that mild consensusis reached in deliberation as procedure disengages agentsthe second is that within discussion groups majorities tendto become larger and opinions more extrememdashnot on thebasis of competing arguments but of the initial balance ofopinions After several decision processes we find opiniondistributions that reflect these observations even though weare not able to precisely coin the set of parameters that yieldssuch distributions (see Figure 7)

Although differences between mixed and dyadic opiniondynamics in terms of the variance of opinions and shiftsare slim the former does much better in terms of judg-ment accuracy In this sense deliberation does moderateopinions but not as much when non-deliberation steps arerelatively frequent throughout the decision-making processIn terms of Moscovici and Doise theory of consensus [15]this makes sense since pair-wise discussion engages or ego-involve agents in the discussion and pushes to convergenceof opinion at the extremes of the opinion distributionHowever what remains unclear is the reason why proceduralparameters that constrain deliberation do not disengageagents from discussion and thus result in a mild consensusas also expressed by Moscovici and Doise [15] A possibleexplanation of this phenomenon may stem from the factthat deliberation in the model is always accompanied by aseries of ego-involving discussions that shadowor counter themoderating effect of deliberation Not only this deliberationcould also be responsible for crystallizing the opinions ofextremists for it favors one or the other part of the distributionof opinions as explained in [14] and to some extent in [13]Furthermore the mixed discussion model gives an exampleof polarization that results from a minority of dissenterson one side of an issue acceding to views of the majorityrsquosside [4] as opposed to the strong polarization observed indyadic interactions Situations in which the status quo orthe undecidability of the proposal argument lingers cannotpossibly yield situations of moderation In cognitive science

24 Complexity

one can be inclined to believe that the longer a deliberationprocess is the more likely individuals will take an immutablestance on the situation In view of the assumption that longdecision-making processes can be more costly for a group orsociety as a whole agents may feel forced to take a stancejust for the sake of ending the process The stance that isadopted would therefore depend on pair-wise interactionsamong individuals and not on deliberation since deliberationtakes too long to be conclusive

442 For Hypothetical Recommendations in Decision-MakingProcess Elaboration On another tune results from themodelsuggest that procedural parameters have marginally decreas-ing positive or negative effects on the metrics of interest Ina hypothetical policy analysis perspective this may indicatethat after choosing a correct number of individuals andimposing the right amount of debates one can attain rea-sonable levels of judgment accuracy coherence and varianceof opinions As accounted in [18] deliberation increasesknowledge (collective knowledge as well) and therefore itheightens the chances of making correct decisions If onebelieves that social welfare is linked to deliberation andeventually to the metrics studied previously then proceduralparameters can both be used as an instrument to attaindesired levels of social welfare and reinforce the notionof legitimacy of collective decisions It can also provide ajustification for choosing a deliberation regime rather thananother on the basis of how important (or urgent) a topic ofdiscussion is Indeed if we adopt the claim that deliberationis a forerunner of welfare then any state in which deliberativecues are at their maximum (highest number of participantsmany deliberation steps before making a decision and asfrequent as possible) has to be mapped to the highest attain-able social welfare In this case is a situation where judgmentaccuracy is at its highest yet the variance of opinions or theproportion of extremists at its lowest an ideal situation Wecannot say for sure since strong opinion consensus and longdecision-making processes in many situations may result ina loss of welfare rather than in a gain of it

Another interesting result in procedure for deliberationis the very similar estimates we find for the effects of theminimum number (119898) of debates necessary to make adecision on a proposal and the proportion (119899119863) of individualsparticipating in deliberation on judgment accuracy In thescope of social welfare this raises the question of whetheradvocating for longer decision processes by demandingmoredeliberation steps (higher 119898) rather than bigger debates(higher 119899119863) may increase welfare Depending on the objec-tives of the decider of the deliberation procedure he or shemay choose to concentrate on one or the other A recom-mendation we can assert from the mixed social interactionsmodel is that if one prefers low to high variance of opinionsandor high to low coherence the decider is better off ifshe focuses on increasing the size of the debate rather thanorganizing debates very frequently One of the reasons forthis might be that when making bigger debates all agents aretruly considered as equals in the decision-making processIn organizing many debates individuals that are chosen toparticipate in many of these will be more influential and will

somehow bias the set of arguments in the deliberation arenaThemajority rulemodeled through120572 also determines the sizeof the shifts and the distribution of opinions Because a bigger120572 is associated with higher levels of extremism and opinionvariance deliberated cues are harder to accept and thereforeit becomesmore difficult for agents to reach concensus andormoderate their opinions

On the other hand and further into the idea of legitimacythe goal of deliberative processes is the agreement amongall participants This agreement stands for the right or bestdecision from a formal or procedural perspective and it is alsothe best from a substantive point of view In other words thelegitimacy of decisions not only derives fromwell-establishedprocedural requirements (respect of the protocol) but alsobegs for the fulfillment of two essential conditions (1) satisfy-ing the procedural requirements for a correct procedure (for-mal legitimacy) and (2) the rational acceptability of the resultsof this procedure (substantive acceptability) Legitimacy canbe then obtained from the coherence statistic scrutinizedin the model insofar as it measures how well agents acceptthe results of a correct procedure (eg deliberation) So fora proposal to be legitimate one has to choose parametersthat would maximize group coherence in decision-makingprocesses

As deliberation may give rise to consensus in the delib-eration arena it may also sow the seeds of dissensus inthe group Our model clearly illustrates this phenomenonthrough the procedural parameters of deliberation and themetrics observed From another perspective in deliberativedemocracy dissensus in collective decision-making may bedesirable According to Landermore and Pagersquos accounts [36]lack of consensus can be perceived as having agents withalternative ideas that combined with other ideas can providea better approximation of the ideal status of an argumentTheauthors in [36] call this dissensus ldquopositive dissensusrdquo andit makes normative sense when solving complex collectivedecision-making problems Our model captures the ideaof having normative requirements to define a good andlegitimate collective decision requirements of high consen-sual deliberative accuracy (judgment accuracy) of legitimacy(coherence) and of diversity (variance of opinions) are alltaken into account and for such we can conclude that ourmodel is successful in describing a sound process of collectivedecision-making with interpretable outcomes

Of course the present simulation results and analysis donot prove that all opinion dynamics can be accounted forby such simple processes like the one presented here Theydo however shed light on the expressiveness of combiningdifferent paradigms to account formore interpretablemodelsand pave the way to creating original models of the sortThe simulations suggest the desirability of discovering theconsequences of relatively simple laws of communication atdifferent levels (micro meso and macro) to determine whatstill needs to be explained

5 Related Work

We see our model as a contribution to the influence andopinion dynamics field in ABM and a pragmatic application

Complexity 25

of abstract argumentation theory To our knowledge we areunaware of existing literature on agent-based modeling thatexplicitly relates collective choice and the notions of deliber-ation and opinion diffusion through abstract argumentationas we have done it

Most models in the literature of opinion diffusion areinterested in opinions because these have an influence incollective decision and in questions of social order Forinstance in [3] the authors are interested in consensus andin how a group collectively chooses among two alternativesIn other models authors are interested in the emergenceof extremism [8] and on the distribution of opinions whenextremists are introduced in the population [5 33 37] whileother authors coin the notion of opinion polarization as anemergent property of the system [8 12] They show usingmodels of ldquobounded confidencerdquo and opinion diffusion withtrust that three different kinds of steady states (unipolarbipolar and central) are possible depending on whetheragents are sufficiently uncertain about their opinions andsufficiently connected andor a certain proportion of indi-viduals are already extreme Recent articles on informationand opinion dynamics stress the importance of governanceand government intervention in the spread of emotions(opinions) in the frangibility of social consensus systemIn [10] the authors show that government intervention inthe spread of negative emotions can lead to an even fasterspread of negative emotions (single-peeked convergence) andin a faster collapse of the social consensus system Opinionas a function of trust is studied in [12] and relates to howindividuals form their opinions on the basis of how muchthey trust agents in their networks

Another stream of discrete opinion dynamics modelsthat explain the emergence of opinion can be found in theliterature Computer multiagent models of attitude changebased on Latanersquos social impact theory are presented in [4] Inthe theory an agent changes her opinion on the basis of theinformational impact she is subject to which depends on thepersuasiveness (strength) supportiveness and immediacy(group structure) of the environment and information shereceives The simulations predict two emergent groups ofphenomena the shifting of attitudes towards incompletelypolarized equilibria and the formation of coherent clusteringof subgroups with deviant attitudes Similarly but in acontinuous more argumentative fashion Mas and Flache[11] present a model of opinion diffusion using argumentsArguments are considered to be for or against a proposal andare given an agent-dependant relevance in dyadic persuasiveinteractions Arguments also determine the agentsrsquo opinionsand their dissimilarities which posits the rules of pair-wiseencounters in theirmodel as agents are assumed to form theiropinions according to the arguments they own They showthat in an argumentative opinion diffusion model the onlyobservable steady states are bipolarization and consensusand that bipolarization can emerge from interactions amongsimilar agents In other words bipolarization of opinionsis possible with homophily and without negative influenceif there is argumentation They subsequently compare theirresults to experimental data In [38] a similar approach toargumentation and opinion formation is taken but for the fact

that they introduce different types of arguments explicitlyapply social judgment theory [22] and use survey data toestablish model-to-real-world comparisons

Closer to opinion formation and abstract argumentationGabbriellini and Torroni [39] are the first to have origi-nally and soundly merged opinion diffusion and abstractargumentation They define an agentrsquos opinion as a functionof the arguments she holds and the attack relation amongthem They devised a focused peer-to-peer dialogue systemof persuasion (NetArg) inspired from Mercier and Sperberrsquosargumentative theory of reasoning [40] which used onlyabstract argumentation to study opinion polarization andopinion dynamics Moreover they considered networks andthe notions of trust and epistemic vigilance to define adynamics for knowledge revision and trust itself whichwe clearly lack in our model They showed that if a con-servative belief operator in argumentation was applied byagents when they reasoned then their dialogue protocoldid not increase polarization among agents That said theyused the model to study the effect of Groenvetterrsquos weaklink theory in the spreading of arguments and were notparticularly interested in notions like judgment accuracy orcoherence nor in tackling questions related to collectivedecision-making procedures Their system is very expressiveand helps position our work in the litterature our work is atthe frontier of pure argumentative opinion diffusion modelsopinion diffusion with arguments and continuous bounded-confidence models of opinion diffusion a la Deffuant

On another note work on collective cognitive conver-gence [41] and opinion sharing [19] shows that consensustowards a certain ldquocorrectrdquo opinion or cognitive state is alwayspossible yet dependent on noise variability and awarenessof agents In [20] the authors show that learning about anexogenous correct state of the world (represented by bits)under confidence was possible but only if the agents werenot too confident In a homogeneous population they showthat the higher the confidence the worse the learningmdashthevery confident agents do not learn the properties of the truestate of the world and disrupt the learning process of the lessconfident ones Collective cognitive convergence can be seenas a result of deliberation in truth-seeking models

When it comes to abstract argumentation theory we takean approach that wires two types of dialogues that are well-studied in the literature persuasion dialogues [42] and delib-eration dialogues [43] Another interesting line of work is theone on mechanism design [44] or the problem of devisingan argumentation protocol where strategic argumentation isnot a liability for success in debates We tackle mechanismdesign in a different way though Instead of consideringstrategy-proofness we are interested in how differences inprotocol can result in epistemologically ldquobetterrdquo collectivechoices and guarantee that opinion distributions are favorablefor deliberation For a survey on persuasion dialogue see[42]

Work on agent-based argumentation usually assumes thatthe semantic relationship between arguments is fixed Inother words if two individuals were to put two argumentsin the public arena such that one logically attacked the otherthen everyone would agree that such attack exists [45 46]

26 Complexity

Other models which do not make this restrictive assumptioncan also be found in the literature [27 47] and derive fromthe class or family of opponent models [27 48] in which twoopposing sides attempt to win a dialogue Ourmodel is in theintersection of these two but the frame combining opiniondiffusion of the kind and argumentation remains new

The idea of mixing interpersonal influence and verticalcommunication however is not original It is described andimplemented in innovation diffusion models such as [5 49]In both vertical communication is modeled as exogenoustransparent informationmdashagents are aware of the existenceof an innovation thus triggering several processes of choiceand stabilization of opinions Also in [50] an Eulerian modelis implemented to show the effect of media and exogenousinformation on opinion distributions With respect to thispoint the originality of our work is in that the informationemitted as vertical communication is endogenous It is issuedfrom a deliberation model that agents shape on the basisof their opinions arguments and behavior In the spirit of[51] where the authors control for the design of verticalcommunication (to whom the vertical communication isaddressed its timing how it influences agents opinions) wecontrol for the process generating the information for ourown set of variables length of debates majority quotas andthe frequency at which discussions take place

6 Conclusion and Perspectives

The main objective of this article was to build a bridgewhere decision-making argumentation and opinion dif-fusion could come together We proposed a model thatcombined abstract argumentation theory and a boundedconfidence opinion diffusion model and showed to whatextent it could explain variance of opinions extremismcoherence in collective decisions and judgment accuracyon arguments given an ideal state of full information Thesecond objective was to show in what way governance pro-viding agents with arenas of discussion and deliberation wasimportant in addressing the success of collective decision-making processes and the quality of their outcomes

The model revealed that allowing for more deliberationtime though in lower frequency and allowing for widerparticipation in deliberation increased the variance of opin-ions and the proportion of extremists in a group Theseobservations are consistent with the combination of resultsfound in [9] and in [13 14] which stress that deliberationmay polarize groups and may have a meager effect on theshifts of opinions and inconsistent with [16] where it isargued that opinions tend to moderate after deliberationIt happens that deliberation alone did moderate opinionyet when integrated into a more complex system in whichindividuals were allowed to interact with one another andnot all that was deliberated was accepted its influence wasshadowed by other more individualistic dynamics

Undeniably the grounded semantics played an importantrole in the weak effect of deliberation since it models askeptical way of reasoning over arguments For agentsaccepting arguments thus updating their opinions accord-ingly happened not too often Nevertheless asking for more

deliberation did increase judgment accuracy as observed in[18] yet in a marginally decreasing fashion We showed thatvoting within the deliberation protocol not only increasedthe proportion of extremists and the variance of opinions ina group but also determined how coherent deliberation andvoting procedures were with each other Lastly we showedthat agents that are focused judged arguments better hadmore stable opinions and constituted a group with higherproportion of extremists than their naive counterparts

In terms of governance the model says that there is per-haps no trade-off between extremismand judgment accuracyInstead it asserts that the higher the variance of opinions thecloser the group gets to the correct decisions This situationmay be interpreted as Landermore and Pagersquos notion of ldquoposi-tive dissensusrdquo (in complex collective decision-making tasks)[36] as opposed to consensus as a normative requirementfor correct collective decision-making If a decider has toorganize deliberation to legitimize a public policy dependingon what kind of world he or she wants different set ofparameters may be chosen to account for different levels oflegitimacy and correctness For instance one may ask for adecision to be legitimate to respect a certain level of coherenceor of judgment accuracy and to be discussed by a sundryof different-opinionated peoples If so deliberation has tobe made more often more participants have to be includedin debates or in deliberation instances and not too manytime steps should be introduced in between two deliberationinstances So the model may capture normative ideas todefine correct and legitimate collective decisions and tomake recommendations on those grounds Requirements ofhigh consensual deliberative accuracy (judgment accuracy)legitimacy (coherence) and diversity (variance of opinions)are all taken into account and for such a reason the model israther successful in describing a sound process of collectivedecision-making with interpretable outcomes

61 Extensions of the Model and Other Ideas Our model canbe extended in many ways Value-based abstract argumen-tation frameworks (VAFs) for example as an extension ofDungrsquos argumentation framework provide a formal descrip-tion of the process of decision-making in which argumentsare given values and audiences (set of agents) ignore attacksbetween arguments on the basis of their preferences over val-ues Social abstract argumentation frameworks (SAFs) alsoprovide an interesting and seemingly convenient formalismfor our model In SAFs agents vote on the acceptability ofarguments and the resulting labeling is a combination of thearguments that are voted for and labeling-based semantics Inour model we apply the same rule yet only for one argumentand given precise agent-based votingmechanisms and votingrules which are eventually parametrized

To consider our model as a specification of a mixed VAFand SAF model may be of great interest for future workThis formalism can provide solid foundations to describeexplain interpret and eventually extend the results foundhere from an argument-based perspective On those groundsfuture work will focus on creating a hybrid argumentationframeworkmdashwith its semanticsmdashthat takes heed of bothvalues or principles and voting

Complexity 27

Concerning other possible extensions of the model anatural reflex would be to argue on the validity and therobustness of the mix between the bounded confidenceopinion diffusion models and deliberation Consequentiallyadding deliberation to different bounded confidence modelsof opinion diffusion such as the Hegelsmann-Krause model[37] or to models of social impact theory like in [4] canbe of great interest for future work We acknowledge that afiner understanding of argumentation frameworks randomlattices and their respective applications for this modelare necessary Testing for different labeling-based semanticsor even trading labeling-based semantics for scoring-basedsemantics (semantics that defines acceptability on scoresgiven to arguments) such as debate semantics and ranking-based semantics (semantics that yield a preorder) may beinteresting to consider in future work

More into modeling deliberation it is not inadequateto believe that agents may also advance proposals duringdeliberation and replace to some extent the proposal forwhich the deliberation is taking place Endogenizing theprocess that produces the type of argument to be deliberatedon may be interesting to explore either by having agentsstrategically replace proposals at certain moments duringthe debate or having the central authority attempt to chooseproposals such that the probability of having them acceptedis high Agents may also be thought to be autoorganizing(independent of a central authority) and concede on the sizeof the instances of deliberation To consider differences insemantics across agents or even endogenizing the semanticson how urgent or important the proposal that is beingdiscussed is can also be an interesting direction to take

Last but not least many nontrivial modifications of ourmodel are possible and most should include better-thoughtdeliberation protocols It may be interesting to design andobserve protocols in which deliberation only affects agentsthat are actually debating for example Trust network effectsmultidimensionality of opinions new processes of argumentexchange or learning among agents are notions to furtherdevelop in order to make the model more realistic and relaxsome unreliable assumptions In sum the model is to berefined and extended with the objective of either studyingconcrete cases of deliberative polling and opinion dynamicsor implementing more intuitive thought experiments

Appendix

We present below some of the pseudocode we used toimplement the model

Algorithms

(a) Algorithm 1 implements the dyadic influence in themodel It describes how two agents interact andupdate their opinions

(b) Algorithm 2 implements the deliberative influence inthe model which describes how agents update theiropinions in sight of the accepted deliberated proposalarguments

(c) Algorithm 3 is used to obtain the grounded labelingof an argumentation framework

(d) Algorithm 4 provides an overview of a run in thesimulations119872 is the number of proposal argumentsto discuss which we set to 100 in the model

Each object in the system is indexed by a119908ℎ119900 number Hencea set of agents is annotated by a set of natural numbers Weuse this fact in the implementation of the algorithm 119874119887119895119890119888119905119894denotes the object of 119908ℎ119900 number 119894 Agentsrsquo opinions (119900) arecalibrated so that 119900 isin [minus1 1]Data Availability

The program and data used to support the findings of thisstudy are available from the corresponding author uponrequest

Disclosure

A preliminary version of this work was presented anddiscussed at the conference Bridging the Gap between FormalArgumentation and Actual Human Reasoning on October 4-5 2018 at BochumGermanyThe authors thank the audiencefor their valuable comments and suggestions

Conflicts of Interest

The authors declare that they have no conflicts of interestregarding the publication of this article

Acknowledgments

The contribution of Gabriella Pigozzi was supported by theDeutsche Forschungsgemeinschaft (DFG) and the CzechScience Foundation (GACR) as part of the joint project FromShared Evidence to Group Attitudes (RO 45486-1)

Endnotes

1 In other words the superiority of the ldquoright answerrdquo orlabeling in our case will appear as such to all

2 All agents hold the same opinion3 By sensitivity we mean how much an agent is ready to

change her opinion on the sole fact that an informationalcue has been debated on

4 The assumption is strong but this trust among agentsmay be rooted from a genuine individual motivation toreach a democratic consensus

5 We adopted a rule of thumbs that says that if an agentis neutral with respect to the principle then she willvote for any proposal that is presented to her duringdeliberation

6 If the argument is not unique she chooses one atrandom

7 If there are more than one candidate the agent will favorthe argument at minimal distance from 119868

28 Complexity

Require Vector of values for parameters (119881 = (120583 119880 119879))Ensure Successful interactions among agents119896 larr997888 0119873119890 larr997888 0

while 119896 le |119873|2 do ⊳Fill119873119890 with |119873|2 agents119896 larr997888 119896 + 1119873119890 larr997888 119877119886119899119889119900119898(|119873| + 1) ⊳Pseudo-random generator Generates integer between 0 and |119873|119873119903 larr997888 0 120575119894119895 larr997888 0 ⊳Variable to store differences in opinionfor all 119894 isin 119873119890 doRandomly pick a number 119895 isin 119873(119873119890 cup 119873119903)119873119903 larr997888 119873119903 cup 119895 120575119894119895 larr997888 |119900119894 minus 119900119895|procedure dyadic-update(119894 119895) ⊳Two agents 119894 and 119895 discuss and update opinions1199001015840 larr997888 119900119894 (120583119895 119880119895 119879119895) larr997888 (120583119894 119880119894 119879119894) larr997888 (120583 119880 119879) ⊳Save agent irsquos opinion before updating and homogenize agents

if 120575119894119895 lt 119880119894 then119900119894 larr997888 120583119894119900119895 + (1 minus 120583119894)119900119894if 120575119894119895 gt 119879119894 then119900119894 larr997888 (1 + 120583119894)119900119894 minus 120583119894119900119895if 120575119894119895 lt 119880119895 then119900119895 larr997888 1205831198951199001015840 + (1 minus 120583119895)119900119895if 120575119894119895 gt 119879119895 then119900119895 larr997888 (1 + 120583119895)119900119895 minus 1205831198951199001015840

Algorithm 1 Dyadic social influence

Require Vector of values for parameters (119881 = (120574 119901119886 119901119903))Ensure Successful interaction between proposal arguments and agents

for all 119894 isin 119873 doprocedure Probability-Update(V119886 119901119886 119901119903) ⊳Update individual probabilities of change due to deliberated cues119909119894 larr997888 119901|V119886minus119900119894 |2119886 119910119894 larr997888 1199012|V119886minus119900119894 |119903119902 larr997888 0if L119886119887(119868) = 119894119899 then

procedure deliberative-update(119894 119886) ⊳Agent i argument 119886120574119894 larr997888 120574 119902 larr997888 119906 sim U(0 1) ⊳ 119906 is a realization of a uniform random variableif 119902 lt 119909119894 then119900119894 larr997888 120574119894119900119894 + (1 minus 120574119894)V119886break

if 119902 lt 119910119894(1 minus 119909119894) then119900119894 larr997888 (1 + 120574119894)119900119894 minus 120574119894V119886Algorithm 2 Deliberation influence

8 In terms of computational complexity naive agentsmaketheirmoves in119874(|A119894| log(|A119894|)) while focused agents doat least as bad

9 Tables have memory Therefore debates are not neces-sarily trees

10 If there are less than (119899119863 times |119873|)2 agents for any of bothgroups the CA summons all of them to the deliberationtable

11 Or careful consideration of the viewpoints of otherswhich implies that citizens keep an open mind and donot reject arguments outright

12 A ldquohealthyrdquo consensus refers to the case in which manyconsensual decisions are taken but opinion-wise agentsdo not agree with each other Quotes are used on theword healthy because some political science theorists

(eg [36]) believe that in certain situations agreement inopinions for deliberation is epistemologically damaging

13 Simulations on groups of |119873| = 800 and |119873| = 400agents shows that scaling the population has an effect onall themetrics but it is confounding depending onwhichcouples of attitudes structures are considered (119880 119879) Westick to |119873| = 400 as in [9] for comparability andsimulation running time

14 By balanced wemean with as many arguments with V119886 lt0 as with V119886 gt 0

15 Experiments on the system suggest that a bigger(smaller) argument pool 119872 = 1000 (119872 = 200) isassociated with a higher (lower) variance of opinions(and extremism) and higher (lower) judgment accuracyFor the rest of the observations 119872 seem to have noeffect We choose a middle value for 119872 Similarly theargument pool chosen to be balanced or uniformly

Complexity 29

Require Argumentation framework 119860119865 = (AR)Ensure Assignment of labels to arguments from the grounded labeling of 119860119865L(A)

L0(A) larr997888 (0 0 0)119896 larr997888 0do119894119899 L119896+1(A) larr997888 119894119899 L119896(A) cup 119886 | 119886 is not labeled 119894119899 L119896(A) 119886119899119889 forall119887 isin A 119904119905 119887 R 119886 then 119887 isin 119900119906119905 L119896(A)119900119906119905 L119896+1(A) larr997888 119900119906119905 L119896(A) cup 119886 | 119886 is not labeled 119894119899 L119896(A) 119886119899119889 exist119887 isin A 119904119905 119887 R 119886 and 119887 isin 119894119899 L119896+1(A)

while L119896+1(A) = L119896(A) = 0Algorithm 3 Solve argumentation frameworks (with grounded semantics)

Require Vector of values for parameters (119881)Ensure Vector of statistics (119880)

procedure Set-parameters(119881) ⊳Initialize parameters for scenarioprocedure Init-objs(119899119898 119896 ) ⊳Create119898 arguments 119899 agents and 119860119865120576procedure Solve-Argumentation-Framework(119860119865) ⊳Assigns the epistemic labelingL120576 toA119894 larr997888 1119905 larr997888 1while 119894 le 119872 do ⊳ 119872 the number of arguments to discuss119868119894 larr997888 119866119890119899 119886119903119892(1) ⊳Generates proposal argument 119868119860 119894119905 larr997888 (119868119894 0)L119886119887(119868119894) larr997888 119906119899119889 ⊳Initialize label for 119868119895 larr997888 1 ⊳Initialize debate counter 119895while ((L119886119887(119868) = 119906119899119889 or 119898 le 119895) and 119895 le 119898) doif 119905 equiv 0(119898119900119889 119905119863) then

procedure build-argumentation-framework(119899119863 119887119890ℎ119886V119894119900119903) ⊳Build the argument graph agents deliberate(119861119894119905 119878119894119905) larr997888 (119883 119884) sub (⋃119897isin119873119899119863 A119897119894119905 ⋃119897isin119873119899119863 R119897119894119905) ⊳ 119873119899119863 sub 119873 are agents sampled by the CA119860 119894119905 larr997888 119889(119860 119894119905 (119861119894119905 119878119894119905))procedure Solve-Argumentation-Framework(A119889) ⊳FindL(A119889)L119886119887(119868119894) larr997888 L119886119887119863(119868119894)119895 larr997888 119895 + 1

else119896 larr997888 1while 119896 le 119905119863 doprocedure dyadic-social-influence(120583 119879 119880) ⊳Agents discuss one-to-one119896 larr997888 119896 + 1119905 larr997888 119905 + 1

if 120572 gt 0 andL119886119887(119868119894) = 119900119906119905 then ⊳Agents vote for the proposalif (|119897 isin 119873|119900119897 times V119868119894 ge 0| ge 120572 times 119899) then

L119886119887(119868119894) larr997888 119894119899else

L119886119887(119868119894) larr997888 119900119906119905if L119886119887(119868119894) = 119894119899 thenprocedure Deliberation-influence(120574 119901119886 119901119903) ⊳Agents are influenced by the result of the decision process119894 larr997888 119894 + 1119905 larr997888 119905 + 1119880 larr997888 119862119900119898119901119906119905119890 119878119905119886119905119894119904119905119894119888119904(119881)

return 119880 ⊳Get the statisticsAlgorithm 4 Run for one scenario

generated does not seem to play a significant explanatoryrole on any of the metrics

16 Experiments show that themaximumnumber of debatesis not an important parameter even if it is an intuitiveone when considering decision-making processes Inpractice debates end after at most 119898 + 1 deliberation

steps and even then it happens rarely However to avoidunpleasant surprises like infinite debates on a completeargument graph containing the proposal argument weadded the constraint 119898 Its value is precisely 7 becausethe biggest minimum number of debates we studyis 6

30 Complexity

17 To ensure comparability with the deliberative andmixedopinion dynamics models on judgment accuracy andcoherence agents vote on proposal arguments every 119905119863times(119898+1) 119905119863 and119898does not explain variability in judgmentaccuracy nor in coherence

18 Student t-tests for independent samples are robust tothe violation of two hypothesis on the distribution ofthe samples normality and homogeneity of variancesas long as the sample size is big enough (central limittheorem) and if the difference in size of the comparedsamples is smallrespectively

19 Studentrsquos test version forwhen variances are not assumedequal and the difference in size of the compared samplesis big

20 We do not correct for homoskedasticity nor normality oferrors in the estimates

21 Balanced as the same number of runs per scenario orsuch that no significant correlations are found betweenparameters

References

[1] R Axelrod ldquoThe dissemination of culture a model withlocal convergence and global polarizationrdquo Journal of ConflictResolution vol 41 no 2 pp 203ndash226 1997

[2] D M Rousseau Reinforcing the MicroMacro Bridge Organiza-tionalThinking and Pluralistic Vehicles SAGEPublications SageLos Angeles Calif USA 2011

[3] S Galam and SMoscovici ldquoTowards a theory of collective phe-nomena consensus and attitude changes in groupsrdquo EuropeanJournal of Social Psychology vol 21 no 1 pp 49ndash74 1991

[4] A Nowak J Szamrej and B Latane ldquoFrom private attitude topublic opinion a dynamic theory of social impactrdquo Psychologi-cal Review vol 97 no 3 pp 362ndash376 1990

[5] G Deffuant S Huet and F Amblard ldquoAn individual-basedmodel of innovation diffusion mixing social value and individ-ual benefitrdquo American Journal of Sociology vol 110 no 4 pp1041ndash1069 2005

[6] G Deffuant D Neau F Amblard and G Weisbuch ldquoMixingbeliefs among interacting agentsrdquoAdvances in Complex Systems(ACS) vol 3 no 01n04 pp 87ndash98 2000

[7] J Rouchier P Tubaro and C Emery ldquoOpinion transmission inorganizations an agent-based modeling approachrdquo Computa-tional and Mathematical OrganizationTheory vol 20 no 3 pp252ndash277 2014

[8] M Meadows and D Cliff ldquoThe relative disagreement model ofopinion dynamics where do extremists come fromrdquo in Pro-ceedings of the International Workshop on Self-Organizing Sys-tems Lecture Notes in Computer Science pp 66ndash77 Springer2013

[9] W Jager and F Amblard ldquoUniformity bipolarization and plu-riformity captured as generic stylized behavior with an agent-based simulation model of attitude changerdquo Computational ampMathematical Organization Theory vol 10 no 4 pp 295ndash3032005

[10] Y-F Zhang H-Y Duan and Z-L Geng ldquoEvolutionary mech-anism of frangibility in social consensus system based onnegative emotions spreadrdquo Complexity vol 2017 Article ID4037049 8 pages 2017

[11] A Flache andMWMacy ldquoSmall worlds and cultural polariza-tionrdquoThe Journal of Mathematical Sociology vol 35 no 1-3 pp146ndash176 2011

[12] A Tsang and K Larson ldquoOpinion dynamics of skepticalagentsrdquo in Proceedings of the 13th International Conference onAutonomous Agents and Multiagent Systems AAMAS 2014 pp277ndash284 International Foundation forAutonomousAgents andMultiagent Systems France May 2014

[13] K M Hansen Deliberative democracy and opinion formationUniversity Press of Southern Denmark Odense 2004

[14] C R Sunstein ldquoThe law of group polarizationrdquo Journal ofPolitical Philosophy vol 10 no 2 pp 175ndash195 2002 WileyOnline Library

[15] S Moscovici and W Doise Dissensions et Consensus unetheorie generale des decisions collectives Presses Universitairesde France-PUF 1992

[16] J S Fishkin and R C Luskin ldquoExperimenting with a demo-cratic ideal deliberative polling and public opinionrdquo ActaPolitica vol 40 no 3 pp 284ndash298 2005

[17] R C Luskin J S Fishkin and S Iyengar Considered Opinionson US Foreign Policy Face-To-Face versus Online Delibera-tive Polling International Communication Association NewOrleans La USA 2004

[18] J Barabas ldquoHowdeliberation affects policy opinionsrdquoAmericanPolitical Science Review vol 98 no 4 pp 687ndash701 2004

[19] O Pryymak A Rogers and N R Jennings ldquoEfficient opinionsharing in large decentralised teamsrdquo in Proceedings of the 11thInternational Conference on Autonomous Agents andMultiagentSystems vol 1 pp 543ndash550 Spain June 2012

[20] J Rouchier and E Tanimura ldquoWhen overconfident agents slowdown collective learningrdquo Simulation vol 88 no 1 pp 33ndash492012

[21] D Miller ldquoDeliberative democracy and social choicerdquo PoliticalStudies vol 40 S1 pp 54ndash67 1992

[22] M Sherif and C I Hovland Social Judgment Assimilation andContrast Effects in Communication and Attitude Change YaleUniversity Press 1961

[23] P M Dung ldquoOn the acceptability of arguments and its funda-mental role in nonmonotonic reasoning logic programmingand n-person gamesrdquo Artificial Intelligence vol 77 no 2 pp321ndash357 1995

[24] M Caminada and G Pigozzi ldquoOn judgment aggregation inabstract argumentationrdquo Autonomous Agents and Multi-AgentSystems vol 22 no 1 pp 64ndash102 2011

[25] P Baroni M Caminada andM Giacomin ldquoAn introduction toargumentation semanticsrdquoThe Knowledge Engineering Reviewvol 26 no 4 pp 365ndash410 2011

[26] S Chambers ldquoDeliberative democratic theoryrdquo Annual Reviewof Political Science vol 6 no 1 pp 307ndash326 2003

[27] E Bonzon andNMaudet ldquoOn the outcomes of multiparty per-suasionrdquo in Proceedings of the 10th International Conference onAutonomous Agents and Multiagent Systems 2011 AAMAS 2011vol 1 pp 47ndash54 International Foundation for AutonomousAgents and Multiagent Systems Taiwan May 2011

[28] J Ranciere Disagreement Politics and Philosophy University ofMinnesota Press 1999

[29] T Bench-Capon and K Atkinson ldquoAbstract argumentation andvaluesrdquo in Argumentation in Artificial Intelligence pp 45ndash64Springer 2009

[30] M Caminada ldquoOn the issue of reinstatement in argumenta-tionrdquo in Proceedings of the European Workshop on Logics in

Complexity 31

Artificial Intelligence vol 4160 of Lecture Notes in ComputerScience pp 111ndash123 Springer Berlin Heidelberg

[31] GAVreeswijk ldquoAn algorithm to computeminimally groundedand admissible defence sets in argument systemsrdquo in Pro-ceedings of the 2006 conference on Computational Models ofArgument Proceedings of COMMA 2006 vol 144 pp 109ndash120IOS Press Amsterdam The Netherlands 2006

[32] C W Sherif M Kelly H L Rodgers et al ldquoPersonal involve-ment social judgment and actionrdquo Journal of Personality andSocial Psychology vol 27 no 3 pp 311ndash328 1973

[33] G Deffuant F Amblard G Weisbuch and T Faure ldquoHow canextremism prevail A study based on the relative agreementinteraction modelrdquo Journal of Artificial Societies and SocialSimulation vol 5 no 4 p 1 2002

[34] E Awad M W Caminada G Pigozzi M Podlaszewski andI Rahwan ldquoPareto optimality and strategy-proofness in groupargument evaluationrdquo Journal of Logic and Computation vol 27no 8 pp 2581ndash2609 2017

[35] D Grossi and G Pigozzi ldquoJudgment aggregation a primerrdquoSynthesis Lectures on Artificial Intelligence and Machine Learn-ing vol 8 no 2 pp 1ndash151 2014

[36] H Landemore and S E Page ldquoDeliberation and disagreementproblem solving prediction and positive dissensusrdquo PoliticsPhilosophy and Economics vol 14 no 3 pp 229ndash254 2015

[37] R Hegselmann and U Krause ldquoOpinion dynamics andbounded confidence models analysis and simulationrdquo Journalof Artificial Societies and Social Simulation vol 5 no 3 2002

[38] A Stefanelli and R Seidl ldquoModerate and polarized opinionsUsing empirical data for an agent-based simulationrdquo inProceed-ings of the Social Simulation Conference 2014

[39] S Gabbriellini and P Torroni ldquoA new framework for ABMsbased on argumentative reasoningrdquo in Advances in SocialSimulation vol 229 pp 25ndash36 Springer Berlin Germany 2014

[40] H Mercier and D Sperber ldquoWhy do humans reason Argu-ments for an argumentative theoryrdquo Behavioral and BrainSciences vol 34 no 2 pp 57ndash74 2011

[41] H V Parunak T C Belding R Hilscher and S BruecknerldquoModeling and managing collective cognitive convergencerdquoin In Proceedings of the 7th iNternational Joint Conference onAutonomous Agents and Multiagent Systems vol 3 pp 1505ndash1508 International Foundation for Autonomous Agents andMultiagent Systems 2008

[42] H Prakken ldquoFormal systems for persuasion dialoguerdquo TheKnowledge Engineering Review vol 21 no 2 pp 163ndash188 2006

[43] K Atkinson T Bench-Capon and P Mcburney ldquoA dialoguegame protocol for multi-agent argument over proposals foractionrdquoAutonomousAgents andMulti-Agent Systems vol 11 no2 pp 153ndash171 2005

[44] M Thimm and A J Garcıa ldquoClassification and strategicalissues of argumentation games on structured argumentationframeworksrdquo in Proceedings of the 9th International Conferenceon Autonomous Agents and Multiagent Systems vol 1 pp 1247ndash1254 International Foundation for Autonomous Agents andMultiagent Systems Canada 2010

[45] I Rahwan and K Larson ldquoMechanism design for abstractargumentationrdquo in Proceedings of the 7th International JointConference on Autonomous Agents and Multiagent SystemsAAMAS 2008 pp 1031ndash1038 International Foundation forAutonomous Agents and Multiagent Systems Portugal May2008

[46] H Prakken R Riveret A Rotolo and G Sartor ldquoHeuristics inArgumentation A Game-Theoretical Investigationrdquo EuropeanUniversity Institute 2008

[47] N Hadidi Y Dimopoulos and P Moraitis ldquoTactics and con-cessions for argumentation-based negotiationrdquo in Proceedingsof the COMMA vol 245 pp 285ndash296 2012

[48] N Oren and T J Norman ldquoArguing using opponent modelsrdquoin Proceedings of the International Workshop on Argumentationin Multi-Agent Systems pp 160ndash174 Springer 2009

[49] S Delre W Jager T Bijmolt and M Janssen ldquoTargeting andtiming promotional activities An agent-based model for thetakeoff of new productsrdquo Journal of Business Research vol 60no 8 pp 826ndash835 2007

[50] A Mirtabatabaei P Jia and F Bullo ldquoEulerian opinion dynam-ics with bounded confidence and exogenous inputsrdquo SIAMJournal on Applied Dynamical Systems vol 13 no 1 pp 425ndash446 2014

[51] V Barbet N Guiraud V Laperriere and J Rouchier Hagglingon Values Towards Consensus or Trouble 2019

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Submit your manuscripts atwwwhindawicom

12 Complexity

1 or said differently agents are perfectly coherentThis comes for the fact that if the central argumentis labeled 119900119906119905 then it is not even eligible for voting iflabeled 119906119899119889 then the debate will always be coherentwith the preferences of the agents since the result willbe a simple aggregation of their votes If the statisticequals 0 then none of the proposal arguments labeled119894119899 are voted in favorably by the agents It follows that ahigh coherence statistic implies that when agents votefor acceptable proposal arguments the results of thescrutiny reflect the consensual rationality expressedin the deliberation process

312 Parameters of Interest We recall the parameters ofinterest in our study that are linked to the structure of thedecision-making processes

(i) 119898 the minimum number of time steps in which adebate or a scrutiny occurs in a decision-making pro-cess before a final deliberated decision is submitted toa vote

(ii) 119899119863 the number of agents that deliberate as a propor-tion of the population

(iii) 119905119863 the number of time steps between debates inwhich pair-wise interactions among agents mayoccur

(iv) 120572 the proportional majority requirement for theacceptance of a proposal When 120572 = 0 it stands forno voting ldquoany proposal argument that is labeled 119894119899during deliberation is acceptedrdquo

Please recall that in terms of the definition of a decision-making process each parameter controls either the length orthe content of the sequence D119868 (119905119863 119898) or the rules that areapplied during the debate steps (120572 119899119863) (refer to Figure 4)313 Initialization All |119873| = 400 agents start off with anopinion 119900119894 drawn from a uniform distribution U(minus1 1) 13Given 119900119894 every agent 119894 randomly draws a setA119894 (|A119894| = 119896) ofarguments from a balanced14 argument pool A of119872 = 600nonneutral (V119886 = 0) arguments on the basis of 119900119894 if |119900119894| lt 025(the opinion is moderate) then agent 119894 randomly fills halfof her argument sack with arguments such that V119886 lt 0 andthe other half with arguments such that V119886 gt 0 (120601119898119894 = 05)Otherwise she randomly fills 120601119890119894 ge 05 of her sack witharguments such that V119886 and 119900119894 are of the same sign and theremaining (1 minus 120601119890119894 ) with arguments of the opposite sign15

Like for the opinion of agents every argument 119886rsquosadherence to the principle P V119886 is drawn from a uniformdistribution U(minus1 1) The attack relation that gives birth tothe ideal argumentation framework (119860119865120576) is established onthe basis of the V119886rsquos and the argumentsrsquo epistemic reach 119890119903

Let 120579(119886 119887 119903) = (|V119886 + V119887|2)119903 be an auxiliary positivereal-valued function that takes two arguments (119886 119887) and apositive real number (119903 gt 0) as inputThe probability that anyargument 119886 isin A creates a link to any other argument 119887 isin A

is given by the equation 119901 = 119901120579(1198861198872)119890 and thus argumentssupporting opposing views of the principle always attack each

other We fix 119901119890 the epistemic correlation parameter to 015If 119901119890 was greater than 015 then arguments of the same signwould attack each other too often (and focused agents wouldbe rarely incited to advance favorable arguments duringdeliberation) If 119901119890 was lower then the arguments of thesame type would induce an almost empty graph which isunrealistic The number of arguments that any argument 119886can attack is bounded by 119886rsquos epistemic reach that we fix to15 A higher value of epistemic reach makes the argumentlattice too conflicting and nearly bipartite A lower epistemicreachmakes the lattice not sufficiently conflicting in the sensethat too many arguments can attack the proposal argumentrelatively to the few that can defend it If any argument attacksthe proposal argument the chances that another argumentattacks the attacker are low Hence the proposal argumentis almost never accepted and deliberative opinion updatesbecome rare

The arguments in the resulting argumentation frame-work 119860119865120576 are given a permanent labelingL120590

120576 (A) (in shortL120576) using120590 = grounded semanticsWe choose the groundedlabeling-based semantics because it provides a unique admis-sible labeling respects minimal rationality constraints whilesimplifying the model and models a skeptic approach toaccepting arguments (see [23 25 31]) as it maximizes thecardinality of the set 119906119899119889 L120576 To some extent if we considerproposals as committing because they may guide collectiveaction choosing a skeptic semantics seems reasonable insofaras it labels an argument 119894119899 or 119900119906119905 only if it has no reasonto label it 119906119899119889 For instance if a proposal is a public policythat requires large amounts of resources and engages tofuture course of action then it may also determine policycycles and heavily burden a group and its future decisionsIn important situations like these it seems reasonable tolengthen deliberative cues and ask for more demanding andldquogroundedrdquo criteria for policy argument acceptability

On the proposalrsquos side we create an argument 119868 whoseadherence to P is also drawn from aU(minus1 1) and we label itL119886119887(119868) = 119906119899119889 119868 is interpreted as being the main argumentjustifying the discussed proposal P and V119905119868 as its supportfor P at time 119905 119868 cannot attack other arguments but otherarguments can attack it For each argument 119886 isin A a directedarc or attack from 119886 to 119868 is activated with probability 119901119901 =003120579(11988611986802) When 119901119901rsquos lower bound is too high the proposalargument is almost always defeated and thus deliberativeopinion updates do not happen When 119901119901rsquos lower boundis lower than 003 then the opposite occursmdashthe proposalargument is always accepted due to an absence of attackstowards it Finally we set the maximum number of debatesto119898 = 7 for the decision-making processes stop rules1632 Calibration and Simulation Protocol In this subsectionwe discuss the calibration of the models the terminationconditions for runs in the deliberative and mixed model andthe expected outcomes in terms of the observations

321 Calibrating Dyadic Opinion Dynamics Dyadic opiniondynamics correspond to a space of parameters in whichargumentation and deliberation spaces are not taken into

Complexity 13

Table 1 Parameters and parameter types and their domains used to compare dyadic collective and mixed opinion dynamics

Length ofD119868 Decisional in119863119868 Social influence Deliberation influence Cognitive119905119863 119898 119898 119899119863 120572 120583 119879 119880 119901119886 119901119903 120574 119896 119861119890ℎ119886V1198941199001199031 3 6 1 3 6 7 001 002 005 0 05 066 01 14 18 02 06 03 05 01 005 01 02 4 8 16 119873119886119894V119890 119865119900119888119906119904119890119889account Deliberated arguments and informational cues haveno effect on agent opinion but agents still vote for theproposal argument17 We use and calibrate this model forcomparability in terms of all our metrics since they arenot explicitly observed in [9] a reference opinion dynamicsmodel Furthermore for simplicity and comparability againwe suppose that agents are homogeneous in terms of theirattitude structures and opinion weightings We fix 120583119894 =120583 119879119894 = 119879 and 119880119894 = 119880 for all 119894 isin 119873 and for some triplet(120583 119880 119879) isin [0 05] times [0 2] times [119880 2]

Experiments suggest that the strength of the dyadic inter-actions is an explanatory factor of the time of convergence(120583 lt 05) and of the dyamicsrsquo steady states Bigger values of 120583speed up convergence towards a stable set of opinion clustersand affect the size and the relative position of these clustersin the opinion distribution 120583 however does not play a majorrole in the number of opinion clusters observed at the steadystate of the dynamics

We set 120583 to the value given to it in Jagger et alrsquos opiniondynamics model (120583 = 01) [9] For one it is the smallestreal number for which most results found in [9] hold and thedynamics are smooth secondly it seems to be a reasonableldquostrengthrdquo of influence considering that we would not wantpair-wise social interactions to completely shadow or maskan eventual effect of deliberation in the opinion dynamics

For different values of attitude structures 119879 and 119880 weget exactly the same results as in [9] in terms of the numberand density of the opinion clusters The different couplesof values of 119880 and 119879 model the size of the latitude ofnoncommitment (119879 minus 119880) and thus the agentsrsquo propensityto update opinions Whenever 119879 minus 119880 le 02 regularitiesin convergence and opinion clustering appear If 119880 gt 1there is always central convergence if 119879 lt 1 there is alwaysbipolar convergence with clusters of similar sizes For 119879 minus119880 gt 02 relatively low values of 119879 (119879 lt 1) and relativelyhigh values of 119880 always yield extreme bipolar convergencewith a relatively small cluster of moderate agents (bipolar-central convergence) For 119879 sufficiently high (119879 ge 12) and119880 sufficiently low (119880 le 08) in other words not very ego-involved agents there is a bigger spectrum of steady statesFor this reason we studied the metrics of interest for thesevalue differences and kept the couples of (119880 119879) depicted inTable 1 For these couples central bipolar bipolar-centralconvergence (3 clusters) and multicluster convergence (4 or5 clusters) are possible and result frommeaningfully differentattitude structures Judgment accuracy varies significantlyacross the four different attitude structures we retain and soit happens with the variance of opinions These scenariosprovides us with reference results from which to build upon

322 Calibrating Deliberative Social Interactions Delibera-tive social interactions correspond to social situations in

which individuals are not influenced by pair-wise discussionwith peers but are sensitive to collectively deliberated infor-mational cues On these terms parameters such as the sizeof agentsrsquo argument sacks (119896119894) the distribution of argumentsin them (120601119894) the minimum number of debates (119898) andthe proportion of debaters from the population (119899119863) are nolonger mute We assume for simplicity that the formulaexist120601119890exist119896forall119894(119896119894 = 119896 and 120601119890119894 = 120601119890 and 120601119898119894 = 05) is true with 119894 isin 119873for some value 120601119890 isin [05 1] and for 119896 isin N

The chosen domains for these parameters (see Table 1)are justified by the size of the population and the size of theargument pool (119872 = 600) which we set from the start If 119896is too big (119896 ge 32) andor too many agents are allowed inthe debate (119899119863 ge 02) then individuals always disprove thecentral argument and deliberation is never or rarely takeninto account in the update of opinions Similar effects occurwhen 119898 is high yet we only calibrate it with respect to themean running time of a simulation and the values of 119905119863Different distributions of 120601119890119894 seem to significantly or directlyexplain none of the statistics we analyze so we convenientlyset 120601119890119894 = 075 We chose the parameter 119896 to be always divisibleby 4 for it is convenient given the initial conditions regardingthe argument distribution in the agentsrsquo argument sacks

We fix the value space for the parameters 120574 119901119886 119901119903 as inTable 1 to account for different intensities of the effect ofdeliberative voting retroaction on the system For examplethe scenarios where the three parameters are at their lowestvalues correspond to the scenarios with the smallest fixedprior effect of deliberation in the model Increasing any ofthese parameters should mechanically be associated witha world in which agents are more sensitive to collectivedecisions The values for 119901119886 are taken from [13] wherethe author states that by means of deliberation 7 to28 of individuals changed their opinion from agreeingto disagreeing or vice versa on a referendum question forDenmarkrsquos participation in the EuroWe choose two differentvalues for 119901119886 that account for high (03) and very high (05)sensitivity to deliberation in a group 119901119903 on the other handis taken to be small (equal to 01) since we believe that thereexist agents that will always go against the reached consensusTheir numbers are meager

For simplicity we assume that for all 119894 isin 119873 120574119894 = 120574 forsome 120574 isin [0 05] In effect we posit that the heterogeneityof agents in the deliberative context only derives from theirarguments and their opinions We decide to calibrate 120574 inallusion to the strength of the social dynamics in pair-wiseinteractions In doing so we limit ourselves to consider threedifferent scenarios in which deliberation has respectivelyhalf equal and twice as much opinion-shifting power thanone-to-one social influence Finally we include the accept-ability voting quota rule 120572 whose domain is inspired fromclassical majority rules [35] observed empirically (Table 1)

14 Complexity

Table 2 Parameters and their domains used to study the effect of the procedural parameters on the metrics

Procedural parameters Other parameters119905119863 119898 119899119863 120572 119898 120583 119879 119880 119901119886 119901119903 120574 119896 119861119890ℎ119886V1198941199001199031 2 3 4 5 6 1 2 3 4 5 001 002 003 004 005 0 05 066 7 01 16 02 03 005 02 12 119873119886119894V119890 119865119900119888119906119904119890119889

323 Calibrating the Mixed Social Interactions Model Themixed interaction model ascribes to a parameter space inwhich the effect of collective choices on our metrics isnontrivial and where deliberation and voting on proposalsdetermine their acceptability Since the mixed model isequivalent to periodic iterations of pair-wise and collectivesocial interactions the calibration of the parameters in bothpreceding models holds (see Table 1 for the calibration)The first reason for this is comparability the second is thatthe previous calibration takes into account the fact thatboth dynamics are going to be combined We include thefrequency of debates by adding the parameter 119905119863 whichcontrols for the amount of pair-wise interaction amongagents between two distinct deliberation steps

324 Termination Conditions for Runs Simulations stoponce 100 proposals are deliberated on andor voted for inother words after 100 collective decision steps have occuredThe number of proposals discussed seems arbitrary yet it ishigh enough to observe the effects of deliberation on opiniondistributions and on other metrics related to coherence andjudgment accuracy We choose the number of runs as afunction of our research questions which give relevance tothe procedural parameters in the mixed model rather than tothe parameters set to describe the population and its behavior

325 Expected Outcomes of the Simulations We expect moredeliberation in terms of higher 119898 and 119899119863 to increasejudgment accuracy and at the same time to reduce thevariance of opinions In turn a smaller variance of opinionsimplies that the argument pool for deliberation is smaller andtherefore judgment accuracy should be lower Moreovervariance increasing dyadic interactions (or rejection) shouldalso increase judgment accuracy and coherence since bipo-larization and dissensus foster argument diversity in delibera-tionThe coherence statistic should be stronger in simulationsin which central convergence appears quickly and agentsare naive Attitude structures sensitivity to informationalcues and weights given to deliberated cues that makes pathsto bipolar convergence smaller or to central convergencelonger should be associated with higher judgment accuracyShifts of opinion should be more visible in scenarios inwhich extreme agents are pulled away from the extremesHence the sensitivity to deliberated cues (119901119886 119901119903) and 120574should explain the shifts When crossed with high values of119880 and 119879 the shifting power of deliberation should be at itshighest

In the end we do not know how these mechanismswill play out The results of the subsequent experimentsgive us insight on the interplay between the aforementionedeffects

4 Results

Before pointing at any result obtained from the mixedinteractions model we describe and compare the dyadicand deliberative interactions models in the parameter spaceobtained from the calibration in Section 3 (see Table 1)Primarily we require our results to describe two orthogonaltypes of opinion dynamics the pair-wise dyadic and thedeliberative The latter comprises scenarios in which indi-viduals do not influence each other by means of pair-wisediscussion and only update their opinions from delibera-tion the former scenarios where only pair-wise interactionsamong agents determine the dynamics We describe both onthe parameter space given in Table 1 by performing at least20 simulations per scenario Quantitative results from thepair-wise interaction model alone are described during thecalibration and are not treated heremdashdeliberative parametershave no effect on it Moreover we show how these differentmodel specifications yield qualitatively different opiniondistributions and compare them on the basis of the metricsof interest To this end we perform independent two-sampleStudentrsquos t-tests18 or Welch t-tests19 and compute confidenceintervals at a 095 level of confidence Last but not least wecomment ordinary least squares (OLS) regression estimatesto account for the direction and magnitude20 of the effectsof the parameters on the metrics Estimates are declaredsignificant at a 5 level of risk

We perform the same analysis on the mixed interactionsmodel as we compare it to the pair-wise dyadic and deliber-ative interactions models and discuss the marginal effects ofthe parameters on the metrics More precisely we obtain twodifferent types of results for the mixed interactions modelthe first compares the mixed scenarios to their monolithiccounterparts with respect to each metric and with respect tothe regression estimates the second gives a clear idea of themarginal effect of each governance parameter (or parameterof interest) on our observations In the first we allow controland procedural parameters to vary on a parameter spacesimilar to those of the dyadic and deliberative opiniondynamics (see Table 1) In the second we fix all of ourcontrol parameters execute 36000 balanced21 runs andfocus only on the one-way pooled effects of our proceduraland behavioral (all agents are focused or naive) parametersWe generate comprehensive graphs and tables to account forthe obtained results

Please notice the slight change in the size of the parameterspace for the procedural parameters and in the values ofthe nonprocedural ones for the first and second type ofresults (refer to Tables 1 and 2 respectively) Since we wantto have a finer idea of the marginal effects of each of theprocedural parameters on how well a group decides onproposals we make the value jumps for each parameter (1)

Complexity 15

() (U T) = (02 18) (b) ( pa m nD) = (005 05 00 2 02) (＝) ０ＬＧＮＬＭ ＣＨ (＜) (＝) Ｈ＞ tD = 3

(d) (５ ４) = (02 14) (e) ( pa m nD) = (01 03 05 3 02) () ０ＬＧＮＬＭ ＣＨ (＞) () Ｈ＞ tD = 6

Figure 7 From left to right common opinion trajectories and distributions for dyadic deliberative and mixed opinion dynamics within thestructure of different decision-making processes

small enough to detect significant differences in the estimatesand (2) meaningful for interpretation For the fixed one-valued parameters in Table 2 we use Table 1 to set them totheir minimum mean or median values

41 Comments on Dyadic and Deliberative Opinions Dynam-ics In this subsection we present our first results regardingthe deliberative opinion dynamics described in Equation (6)and its differences with the dyadic dynamics (Equation (5))in terms of our observations

411 Qualitative Analysis of Pair-Wise Opinion DynamicsPair-wise opinion dynamics alone produces multiclusterstable convergencewith variable cluster sizes (see Figures 7(a)and 7(d)) Insight on the complexity of these dynamics wasdiscussed in Section 321

In Figures 7(a) and 7(d) agents discuss randomly withone another in pairs Clusters in the extremes form quicklysince either agents that are close to the extremes attractone another or are convinced to stay close to the extremesby interacting with other agents they disagree with Otheragents with near-extreme positions may be simply attractedto one of the several moderate foci of agents in the opiniondistribution On the other hand moderate agents are eitherattracted to the closest opinion focus or pushed towards theextremes of the distribution Over time agents in the centralfocus either attract other agents into it or ignore the opinions

of the extreme agents that interact with them The result ofthese dynamics yields amulticlustered convergencewhere theopinion foci are at least at distance 119880 from one another andonly the extreme foci (clusters of agents with |119900119894| ge 075) areat a distance greater than or equal to 119879 from one anotherThefoci at the bounds of the opinion distribution are very stableand in analogy to the (12119889)-rule in assimilation bounded-confidence models [33] the number of clusters that form isroughly equal to 119880minus1412 Qualitative Analysis of Deliberative Opinion DynamicsQualitatively deliberative opinion dynamics yield a looseunipolar convergence of opinions near the center (centralconvergence see Figure 7(b)) near the center of either the leftor right portions of the distribution (group polarization seeFigure 7(e)) or to a sparse bipolar opinion distribution withclusters at the center and at the extremes (see Figure 7(c)) ifmixed with pair-wise interactions This is probably a reasonto expect the variance of opinions and the proportion ofextremists (almost perfectly correlated 120588 = 089 119901 lt 0001)to be low for these scenarios The side towards which theopinion cluster skews depends essentially on the valence ofthe first argument that is collectively accepted

In Figures 7(e) and 7(b) agents that update their opin-ions do so at the same time and only when deliberationis successful In these scenarios every time a decision iscollectively accepted agents update their opinions towards

16 Complexity

Table 3 Ordinary least squares estimates for pair-wise interactions scenarios for most of the observed metrics

Dependent variable119881119886119903(119900) 119878ℎ 119890119888 119894119903(1) (2) (3) (4)Constant 0610lowastlowastlowast 4429lowastlowastlowast 0472lowastlowastlowast 0000

(0002) (0206) (0003) (0000)

119880 = 06 (119880 = 02) minus0012lowastlowastlowast 2981lowastlowastlowast 0061lowastlowastlowast 0000(0002) (0260) (0004) (0000)

119879 = 18 (ref 119879 = 14) minus0264lowastlowastlowast minus3261lowastlowastlowast 0004 0000(0002) (0260) (0004) (0000)

120572 = 066 (ref 120572 = 05) 0002 minus0059 minus0100lowastlowastlowast 0000(0002) (0184) (0003) (0000)

(119880 119879) = (06 18) (ref (119880 119879) = (02 14)) minus0324lowastlowastlowast minus3728lowastlowastlowast minus0033lowastlowastlowast 0000(0003) (0368) (0006) (0000)

Observations 1440 1440 1440 1440R2 0983 0389 0529Adjusted R2 0983 0388 0527Residual Std Error (df = 1435) 0032 3492 0053 0000F Statistic (df = 4 1435) 21 272740lowastlowastlowast 228686lowastlowastlowast 402326lowastlowastlowastNote lowastplt01 lowastlowastplt005 lowastlowastlowastplt001

the argumentrsquos defended position Hence if there is votingand if agents happen to obtain 119894119899 proposal arguments theyinexorably cluster towards one or the other side of the opiniondistribution resulting in opinion convergence remeniscent ofgroup polarization (see Figure 7(e)) Furthermore as agentscluster in the same side of the opinion spectrum the argu-ments that may be advanced in deliberation are fewer andmore skewed towards that one side of the opinion spectrumAgents become more easily persuaded in deliberation andonly by arguments supporting one side of the spectrumConvergence of opinions towards one loose opinion focusbecomes faster and more certain If there is no vote (seeFigure 7(b)) the dynamics are similar but for the fact thatit is the uniform randomness of the proposal argument thatguarantees a central convergence of opinions

413 Quantitative Analysis of Pair-Wise Opinion DynamicsThe three parameters that allow for quantitative analysis are119880 and119879 the agentsrsquo attitude structure and themajority quotavoting rule 120572 Table 3 supports the claim that going from 120572 =05 to 120572 = 066 does not affect the variance nor the shifts ofopinions However judgment accuracy is significantly higherwhen accepting proposals becomes more difficult This israther surprising but also intuitive since decisions that aretaken using more restrictive methods of scrutiny should bemore accurate

Concerning the attitude structures the interaction termbetween 119880 and 119879 shows that 119880 has a small effect on thevariance of opinions given that 119879 is low This shows thateven when the assimilation threshold is high a sufficientlysmall rejection threshold can make its one-way effect vanishHowever high levels of both 119880 and 119879 reduce the variance ofopinions dramatically In contrast when it comes to overallchanges in opinion the magnitude of 119880rsquos one-way marginal

effect increases the shifts of opinions whereas 119879 lowers itThis can be interpreted as having the rejection dynamics pushmoderate-to-extreme agents to positions that are close tothe extremes and very moderate agents to the center of thedistribution Not too many quantitative changes (on average)can therefore be observedThe fact that higher119880 is associatedwith faster convergence of opinions seems to imply lowerjudgment accuracy At some point voting for the same type ofproposals for too longmay result inmany collectivemistakesThis is an interesting conclusion to consider in the analysis ofthe mixed model Finally the interaction term shows that 119879has no strong impact on the metric

414 Quantitative Analysis of Deliberative OpinionDynamicsSeven parameters can be considered for quantitative anal-ysis in this dynamics the parameters of interest recalledin Section 312 the size of the argument sacks (119896) agentbehavior during deliberation the strength of the deliberationdynamics (120574) and the measure sensitivity to deliberatedresults (119901119886) The most ldquounexpectedrdquo result obtained fromthe deliberation dynamics is that it shows how deliberationcontributes to the proportion of extremists and the varianceof opinions It happens that having more agents discussproposals and having more of these discussions increasesthe variance of opinions and the proportion of extremistswhile lowering judgment inaccuracy and coherence For theformer the fact that far too many arguments are advancedduring deliberation plummets the chances that the centralargument is accepted hence deliberation does not moveopinions too often For the latter we observe that only ldquobigrdquodifferences (going from the lowest value of a parameter to thehighest) in protocol actually prove to have a preponderantimpact on the metrics These observations hint at a possibletrade-off between judgment accuracy and the variance of

Complexity 17

Table 4 Ordinary least squares estimates for the deliberative interactions scenarios for most of the observed metrics

Dependent variable119881119886119903(119900) 119878ℎ 119890119888 119894119903(1) (2) (3) (4)Constant 0032lowastlowastlowast 13092lowastlowastlowast 0370lowastlowastlowast 1007lowastlowastlowast

(0001) (0406) (0001) (0001)

119898 = 3 (ref119898 = 1) 0012lowastlowastlowast minus2062lowastlowastlowast minus0012lowastlowastlowast minus0001(0001) (0276) (0001) (0001)

119898 = 6 (ref119898 = 1) 0023lowastlowastlowast minus3285lowastlowastlowast minus0018lowastlowastlowast minus0004lowastlowastlowast(0001) (0276) (0001) (0001)

119899119863 = 002 (ref 119899119863 = 001) 0007lowastlowastlowast minus0965lowastlowastlowast minus0006lowastlowastlowast minus0002lowastlowast(0001) (0276) (0001) (0001)

119899119863 = 005 (ref 119899119863 = 001) 0020lowastlowastlowast minus3316lowastlowastlowast minus0015lowastlowastlowast minus0004lowastlowastlowast(0001) (0276) (0001) (0001)

120572 = 05 (ref 120572 = 0) 0146lowastlowastlowast 68932lowastlowastlowast 0068lowastlowastlowast minus0507lowastlowastlowast(0001) (0276) (0001) (0001)

120572 = 066 (ref 120572 = 0) 0291lowastlowastlowast minus13842lowastlowastlowast 00004 minus0000(0001) (0276) (0001) (0001)

119896 = 8 (ref 119896 = 4) 0004lowastlowastlowast minus0815lowastlowastlowast minus0002lowastlowastlowast minus0001(0001) (0276) (0001) (0001)

119896 = 16 (ref 119896 = 4) 0008lowastlowastlowast minus1118lowastlowastlowast minus0004lowastlowastlowast minus0003lowastlowastlowast(0001) (0276) (0001) (0001)

119861119890ℎ119886V119894119900119903 = ldquoFocusedrdquo (ref ldquoNaiverdquo) 0015lowastlowastlowast minus1980lowastlowastlowast minus0012lowastlowastlowast minus0004lowastlowastlowast(00005) (0225) (00004) (0001)

120574 = 01 (ref 120574 = 005) minus0020lowastlowastlowast 5762lowastlowastlowast minus00003 minus0001(0001) (0276) (0001) (0001)

120574 = 02 (ref 120574 = 005) minus0025lowastlowastlowast 8648lowastlowastlowast minus00003 minus0001(0001) (0276) (0001) (0001)

119901119886 = 05 (ref 119901119886 = 03) minus0013lowastlowastlowast 1581lowastlowastlowast 00004 00003(00005) (0225) (00004) (0001)

Observations 19440 19440 19440 19440R2 0931 0844 0562 0958Adjusted R2 0931 0844 0562 0958Residual Std Error (df = 19427) 0033 15701 0030 0050F Statistic (df = 12 19427) 21 844790lowastlowastlowast 8 733563lowastlowastlowast 2 079585lowastlowastlowast 36 761250lowastlowastlowastNote lowastplt01 lowastlowastplt005 lowastlowastlowastplt001

opinions that may be of interest in the mixed discussionmodel

Group coherence shifts of opinions and the varianceof opinions are highly affected by the voting thresholdthe coherence metric is reduced by half when allowing forclassic majority voting shifts explode and the variance ofopinions increases dramatically This is not surprising sincethe majority quota rule works as a buffer to the success ofdecision-making processes and in consequence it hampersthe deliberative opinion updates For the shifts of opinions itis clear that the two-third majority voting rule results in littleopinion change and that the simple majority rule inducesgroup polarization As expected there are less extremists inscenarios where there is no voting and the shifts of opinionare more likely to happen when agents are more sensitiveto deliberation When we include voting in the system

the grouprsquos judgment accuracy decreases due to the correctdeliberated decisions that are vetoed by a salient opinion-agent majority Strangely the effect only holds for the simplemajority decision rule Lastly the proportion of individualsthat participate in the debate and the minimal amount ofdebates have similar marginal effects on all the metrics (seeTable 4)

415 Comparing Deliberative and Pair-Wise Opinion Dynam-ics in a Nuthshell Qualitatively both models yield similaroutcomes but for the fact that deliberation never producesopinion polarization with both types of extremists (seeFigure 7) Furthermore clusters in the deliberative interac-tions model are less homogeneous in size and multiclusterconvergence occurs only when deliberation is rarely suc-cessful which happens sporadically given the calibration of

18 Complexity

Table 5 Mean difference 095 confidence intervals for metrics by model comparison

Model Metric119881119886119903(119900) 119901119903119900119901119890119909 119878ℎ 119890119888 119894119903Dyadic vs Deliberative [0191 0216] [0170 0197] [-2948 -2828] [0074 0082] [0165 0172]Mixed vs Deliberative [0142 0147] [0150 0155] [-2085 -1976] [0003 0002] [-0038 -0030]Mixed vs Dyadic [-0072 -0046] [-0045 -0018] [8332 8859] [-0079 -0071] [-0205 -0201]the model On the other hand pair-wise opinion dynamicsproduces multicluster convergence most of the time andcentral convergence only in the infrequent cases where thelatitude of acceptance is large

The comparison of these two scenarios in terms ofthe metrics is straightforward Whilst dyadic interactionsallow for more variability in opinions through pair-wiseinteractions and more mistakes in judging proposals delib-erative updates simply act as an antagonist force On aver-age the deliberative interaction model produces opiniondistributions with smaller variance and yields steady stateswith higher judgment accuracy a higher shift statistic andnaturally lower group coherence since only proposals that aredeliberated 119894119899 are eligible for voting (see Table 5)

42 Articulating Dyadic and Deliberative Social Interactionsin Opinion Dynamics So far we have considered scenarioswhere argumentation had no place and where individualsmade decisions according to opinions that were purelyconstructed from pair-wise discussions We also observedcases in which agents formed their opinions only by inte-grating deliberated proposals into their opinion updates Thescenarios we present in this section combine the two pre-ceding scenarios to account for both the effects of pair-wisediscussion and the effects of deliberation on opinions judg-ment accuracy and group coherence From the precedingobservations we conclude the following deliberative opiniondynamics reduces the variance and the proportion of extrem-ists because it shifts opinions greatly and polarizes groupsby eliminating at least one type of extremist the one whoseopinions are initially opposed to the deliberated resultsor it brings opinions close to neutrality (see Figure 7(b))This may not be desired in a group since the diversity andstability of opinions are necessary for deliberation to makesense and lead to the collective acceptance of controversialproposals Dyadic opinion dynamics on the other handtend to increase the variance of opinions and the numberof extremists while having a lesser aggregated effect on theshifts of opinion This is an immediate result of the pair-wiseinteraction dynamics inasmuch as it polarizes individualsvery quickly

To sum up in the mixed interactions model deliberationhinders the effect of voting on opinions by making someproposals that would normally be submitted for voting noteligible for voting It may also undo opinion changes dueto pair-wise interactions The mixed model can be seen asthe deliberation opinion dynamics model in which pair-wiseinteractions occur in between collective discussion and mayaffect which arguments are advanced during deliberationEquivalently it can be seen as the dyadic opinion dynamics

model in which deliberation accounts for endogenousldquoshocksrdquo on the opinion distribution

421 Qualitative Analysis of the Mixed Dynamics In Figures7(c) and 7(f) agents update their opinions through delibera-tion and pair-wise interactions Opinion trajectories visualsshow a combination of both previously described opiniontrajectories Over time as agents discuss with one anotheropinion clusters form as in the pair-wise or dyadic opiniondynamics Deliberation may disrupt the formation of theseclusters and three different situations may occur

(i) Deliberation disrupts the formation of the opinionclusters but not sufficiently to jeopardize pair-wiseopinion cluster formation (eg for high values 119905119863)Two phenomena may be responsible for it (1) thevariability of the proposal argumentrsquos support for theprinciple P and (2) too few (successive) deliberationsteps that result in opinion updates Once the opinionclusters are formed successful deliberation foreshad-ows the merging of the previously formed clustersmay bring extremists closer to moderate agents andthanks to the assimilation component of the pair-wisedynamics allow bigger and more moderate opinionclusters to form (refer to Figure 7(c))

(ii) Deliberation significantly disrupts the pair-wisedynamic formation of clusters This may happenwhen the requirements for accepting proposals areweak in the decision-making process (eg low 120572) Inthis case one has either central convergence sinceall agents follow the deliberated results in the samemanner or 3-cluster convergence with two extremeopinion foci In the latter situation the extremegroups form quickly (like in the pair-wise model) anddeliberation cannot pull extreme agents away fromthe extreme foci they are in The main reason for thisis that immediately after an extreme agent attempts toleave her foci through deliberation updates she getspulled or pushed back into it by means of a dyadicinteraction with another extreme agent

(iii) Deliberation does not disrupt the formation of theopinions cluster This is the rare case scenario forvery demanding decision-making processes in termsof inclusion (eg 119899119863) and argument acceptability (eg120572) Opinions converge as in the pair-wise opiniondynamics model (refer to Figures 7(a) and 7(d))

Last when there is voting (120572 = 0) the position ofthe moderate opinion cluster is determined by the first

Complexity 19

minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

Var(o)

sh

ec

ir

Var(o)

propex

sh

ec

ir

1

098

minus02

minus004

001

1

minus014

minus001

minus003

1

026

minus028

1

minus057 1

ＪＬＩＪ

Ｒ

Figure 8 Significant correlation estimates for themetrics of interestin the mixed opinion dyamics scenarios

accepted proposal argument andor the opinionmajority thatis formed from previous dyadic interactions (see Figure 7(f))

422 Relating the Metrics of Interest in Mixed OpinionDynamics The first thing to notice is that for all of thesescenarios all the metrics are significantly correlated (seeFigure 8) The variance of opinions and the proportion ofextremists are almost perfectly correlated To some extentthis means that a high variance of opinions known thatthe mean of opinions is statistically null equates to havingmany extremists in the population be towards one or theother side of the opinion distribution (eg extreme bipolar-ization) Similarly we find a negative and strong correlationbetween the judgment inaccuracymeasure and the coherencecoefficient (see Figure 8)mdashwhen judgment inaccuracy is high(low) coherence to deliberation is low (high) Cases likethese are telling since they may be related to sequences ofcollective decisions in which agents do not make too manymistakes when judging proposal arguments and still makedeliberated decisions that are on average in line with theirprinciples

The shift statistic interestingly is negatively correlatedwith the variance of opinions with judgment accuracy andwith the coherence coefficient In a way this means thathaving more moderate agents conduces on average to moremistakes in judging arguments and to less coherent collectivedecisions An explanation for this phenomenon could be thatas agents reach consensus in opinion important (possiblyextreme) arguments for correctly judging argument propos-als are not advanced in the deliberation arenawhich results inmore mistakes and in possibly accepting proposal argumentsthat after voting are not meant to be accepted Indeed the

negative correlation between the variance of opinions andjudgment inaccuracy points towards this direction since onaverage the more shifts there are the smaller the variance ofopinions is (see Figure 8)

423 Comparing Mixed Dynamics with Other MonolithicDynamics of Interactions Qualitatively opinion trajectoriesin the mixed model are more chaotic than in the pair-wise and deliberative interaction models (see Figure 7) Asanalyzed previously deliberation dynamics disrupt the pair-wise dynamics and create situations in which central andmulticlustered convergence (with two clusters at the extremesof the distribution) are possible at the expense of more(pair-wise dynamics) or less (deliberative dynamics) clustersIt tolerates the survival of very small isolated group ofagents that the previous models rarely tolerate in a hundredcollective decision-making steps (see Figure 7(c))

One of the most interesting observations one can makeof this family of models regards how groups change (in termsof our metrics) when dyadic social interactions are addedto deliberative interactions and vice-versa When comparingthe mixed interactions with only deliberative ones shiftsagentsrsquo aggregated change in opinions are on average lesspronounced The variance of opinions and the proportionof extremists are significantly higher for the former than forthe latter group coherence in contrast is lower with respectto the scenarios where opinions are formed only throughdeliberation Judgment accuracy follows that same trendA reason for this might be that mixed dynamics are fasterto yield semistable clusters andor opinion convergence Inconsequence the potential argument pool for deliberativeexchange shrinks faster and deliberation becomes less effec-tive over time which results in more collective mistakes Theoutcomeon coherence is predictable inasmuch as deliberativeinteractions produce group polarization If a group agrees inopinion debates are sterile because themembers of the groupwill most likely refrain from attacking the proposal argu-ments they all support and will work together to disparagethe proposal arguments they do not support (see Table 5)

In the same manner we report that the proportion ofextremists and the variance of opinions are barely lower inthe pair-wise interaction model when including deliberationin the decision-making process Shifts of opinions are alsobarely lower when there is no deliberation This can indicatethat deliberation does not undo the changes in opiniondue to pair-wise interactions nor does it amplifies them Incontrast when we consider the labeling-based metrics themixed interaction model ensures lower judgment inaccuracybecause of deliberation and lower group coherence (seeTable 5)

In sum the mixed discussion model is a compromisebetween the pair-wise interaction and deliberative opiniondynamic models in that it offers some guarantees in termsof judgment accuracy and coherence while allowing forldquoreasonablerdquo variance in the opinion pool It establishes somekind of trade-off between labeling-based metrics and thevariance of opinions that may be of interest for decision-making process design

20 Complexity

Table 6 Summary of sign or relative marginal effects for all varying parameters on dyadic deliberative and mixed interactions models inand for all experiments

Metrics Parameters119905119863 119898 119899119863 120572 119879 119880 119901119886 120574 119896 119861119890ℎ119886V119894119900119903119881119886119903(119900) XX X X O X X X X X X119901119903119900119901119890119909 XX X X O X X X X X X119878ℎ XX X X O X X X X X119890119888 XX X X XO X XOO XOO X X119894119903 XX X X O XO O X OOX XOO X XNota gray symbols correspond to dyadic dynamics light gray to deliberative dynamics and bold black to mixed dynamics Given a scenario the symbol 119883stands for constant parameter and 119874 for insignificant marginal effect and as parameters grow for positive (negative) marginal effect for the parameterFrom left to right at the parameter level the leftmost arrow corresponds to the scenario with the strongest (in absolute value) marginal effect of the parameter(with respect to to its factors) Example Let 119901 be a parameter and 119874 stands for ldquo119901 has a significant positive effect in dyadic discussion scenario negativein the deliberative one and insignificant in mixed discussion The positive effect in dyadic discussion is greater than the absolute value of the negative one inthe purely deliberative scenariordquo

424 Comments on the Differences between the Three Modelsfor OLS Regression Estimates The significance of most of theregression parameters estimated for the mixed interactionsscenarios are similar to those estimated for the pure dyadicanddeliberative scenariosNevertheless themagnitude of theeffects is different insomuch that the dynamics themselvesare different So instead of commenting the linear regressionestimates of the mixed model we compare when possiblethe standardized significant coefficients of the regressors ofthe mixed model with those of the pair-wise and deliberativeinteractionmodels (see Table 6 for a summary of the results)

In mixed interactions for instance the influence of119898 onthe opinion distribution is much more preponderant than inthe deliberative dynamics scenarios Going from 119898 = 1 to119898 = 3 increases the standard variation units of the variance ofopinions by twice as much as the one in the deliberative sce-nariosThis in particular is due to the design of the decision-making process If 119898 is higher then the number of socialinfluence steps (119905119863) in the decision process is also higher(compared to 119905119863 = 0 in the deliberative case) If we decideto consider these steps as contributing to the variance ofopinions then the effect of119898 on the variance of opinions hasto be stronger in the mixed interactions model In contrastthe number of agents participating in the debate has a weakereffect on the variance of opinions in the mixed interactionsmodel than in the deliberative dynamics specially for 119899119863 =005 (more than twice as much) The difference may comefrom the fact that in mixed discussions the part of theargument pool used in deliberation is more restrictive thanin the deliberative case We can argue that since pair-wisedynamics may get individuals closer to one another aftersome deliberation the same happens to the arguments thatindividuals will most likely use in deliberation Taking thisinto account means that deliberation will at many instancesbe unfruitful and have no effect on opinions thus failing tomoderate extreme opinions andor polarize the group In thesame direction a high number of participants in deliberationcan make the deliberation dynamics stiff In this case thedistribution of opinion is close to the uniform distributionwhich has a relatively high-to-moderate variancewith respectto the reference results

The strength of the deliberative dynamics has a positivestronger effect on the variance of opinions in themixedmodelthan in the deliberative model when 120574 = 02 and a negativeweaker one when 120574 = 01 A sound interpretation of thisresult is that when deliberation is successful and its effect onopinions does not get agents to move sufficiently away fromtheir current positions agents are pulled or pushed back totheir previously held opinions the variance of opinions doesnot changemuchOn the other hand if the effect of successfuldeliberation is strong opinions can change sufficiently tomake agents much more prone to assimilation into moderateopinion foci the variance of opinions falls

The deliberative opinion dynamics model is much moresensitive to 120572 than the mixed discussion model Differ-ences in marginal effects can double The probability ofattraction to deliberated results (119901119886) for instance has astrongmoderating effect (negative) on agentsrsquo opinions in thedeliberative opinion dynamics but a meager one (two-fold)in the mixed and dyadic dynamic models This is becauseopinion updates in the deliberative model depend only onthat parameter whereas in the mixed model more numeroustypes of interactions make opinion updates possible

In terms of judgment accuracy all parameters except thesize of the argument sacks (119896) have a significant slightlystronger effect on the deliberative dynamics than in themixedmodel In effect the difference may be attributed to theupshot of pair-wise interactions on the distribution of opin-ions and on the arguments that would be advanced duringdeliberation The result respecting parameter 119896 however iscounterintuitive Since argument sacks are static scenarioswhere shifts are higher should make 119896 more important fordeliberation to succeed and it is precisely in the scenarios ofpure deliberative dynamics that we find the biggest opinionshifts This may point to the hypothesis that in deliberativeinteractions agents change opinions substantially but do notoften change their adherence to the principle (go from anegative opinion to a positive one or vice versa) when thereis no group polarization

43 Sensitivity Analysis for the Procedural Parameters ofInterest Weperform a sensitivity analysis of the observationson the parameters of interest in themixed interactionsmodel

Complexity 21

We extend the parameter domains of the pure dyadic anddeliberative scenarios as we make it explicit in Table 2

431 Minimum Number of Debates (119898) We observe that theminimum number of debates has a significant well-observedeffect on all of our metrics excluding group coherence Forthe variance of opinions we can observe that the moredebates there are the bigger the value of the metric isalthough the higher 119899119863 and 119905119863 are the weaker the overalleffect of119898 becomes (Figures 9(d) and 9(a)) Furthermore themarginal effect of increasing minimal debates is decreasingan additional deliberation step requirement increases thevariance of opinions but less and less as it grows 119898 coupledwith 119899119863 has a shy S-shaped effect (see Figure 9(d)) on thevariance of opinions meaning that it has a stronger effectwhen 119899119863 is low which suggests a possible trade-off betweenthe number of arguments accepted in the debate and thenumber of debates required before accepting a proposal

In contrast the shifts in opinion are less and less likelyas 119898 grows and this is independent of the variations of theother parameters Again the effect is marginally decreasingand is only truly significant when 120572 = 05 An explanationto these effects may be linked to the design of the systemFirst the variance of opinions is higher when deliberation isasked for because the more deliberation steps there are thehigher the chances are that the central proposal argumentis deemed unacceptable specially when agents are focusedFurthermoremechanically speaking increasing theminimalamount of debates implies that whenever a decision is to betaken at least 119898 times 119905119863 time steps have to take place and if atany moment a proposal argument is considered undecided119905119863 time steps are added to the process So unless the debatedoes not yield 119906119899119889 labels for proposal arguments (highlyunlikely considering that 120590 = grounded semantics) the morenondeliberation steps there are in the decision process thehigher the variance of opinions is Concerning the shiftswhen 120572 = 05 either the system is too stiff to accept anyproposal argument and opinions do not changemuch or theeffects of pair-wise discussion and deliberation cancel out insuch a way that individual shifts are minimal (Figure 9(b))

On the side of labeling-based metrics the more debatesare asked for the more accurate a group is in its judgmentthe effect being smaller as 119898 grows When agents are naivethe effect is quasilinear while when they are focused thestrongest effects of acquiring more deliberation are foundwhen levels of deliberation are already low (Figure 9(f))Thiscan be explained by the fact that the more debates thereare in the decision process the closer one gets to the idealargumentation framework

432 Proportion of the Population in Deliberation Steps (119899119863)Like for 119898 119899119863 has a significant effect on the proportionof extremists and on the shifts and variance of opinions(Figure 9(e)) This may result from the fact that being ableto put more arguments in play at the same debate step mayincrease the odds of revealing cycles around the proposalargument in119860119865120576 Given that we use grounded semantics thearguments in the cycles are labeled 119906119899119889 and in consequence

debates are more often postponed when 119899119863 is higher Post-poning debates in turn increases the number of nondebatessteps in the decision process which increases the varianceof opinions and limits over time the moderating effect ofdeliberation Moreover the effect of this parameter is verydependent on the value of 120572 (Figure 9(e)) For shifts forinstance 120572 = 05 makes the effect of 119899119863 negative while120572 = 0 makes it positive but to a lesser extent Furthermorefor higher requirements of deliberation (119898) adding moreindividuals to the deliberation process has weaker effectson the variance of opinions and on the other metrics It isalso quite interesting to notice that it has the same effect onagents independently of agent behavior during deliberationThis is a surprising result as one would have expected morefocused agents in a deliberation arena to cause a raise inthe proportion of extremists as they play to knock outopposing proposal arguments and thereof reducing the effectof deliberation (lowers variance) on the whole population

Finally similar to119898 addingmore people into the deliber-ation process increases judgment accuracy (see Figure 9(c))and has no clear effect if any on group coherence (seeFigure 9(i))

433 Steps between Deliberation Steps (119905119863) In all configu-rations 119905119863 increases the proportion of extremists (variance)and decreases the shifts of opinions among agents Theshifts and the effect on the variance of opinions are presentfor 120572 = 05 insignificant otherwise (Figure 9(b)) 119905119863 ishighly linked to 119898 by construction When 119898 = 1 thecurve linking the variance of opinions and 119905119863 is convex As119898 increases the curve becomes more and more concavewhich means that 119905119863 has a more important effect on theopinion distribution as collective decisions take longer to beachieved This seems counterintuitive but in reality it reflectsthemultiplicative relationship between deliberation and pair-wise interactions and the semantics in the model If 119898 is lowand 119905119863 is high the effective number of pair-wise interactionsis on average smaller in the deliberation process Hencethe structure of the decision-making process hinders anyincrease in the variance of opinions If119898 is high the oppositeeffect is observed Additionally since the grounded semanticsyields few 119894119899 arguments with respect to other admissibility-based semantics getting closer to the ideal argumentationframeworkmaymake it difficult to obtain the expected effectsof deliberative discussion on agents Lower 119898 constrains thevariance of opinions by making deliberation more influentialon opinions

In terms of judgment accuracy and coherence we observethat 119905119863 does not explain coherence yet it tends to decreasejudgment inaccuracy This may be due to a more variedargument pool in the debate which in turn may be aconsequence of a higher variance of opinions or vice versa

434 Acceptability Voting Quota (120572) By far it is the mostinfluential parameter in our study It changes the directionand the intensity of the effect of all the other proceduralparameters and by construction heavily constrains the roadto accepting a proposal (see Figure 9)

22 Complexity

015

020

025

030

035

040

1 2 3 4 5 6

12345

Values of Ｇ

(a) 119905119863 times 119901119903119900119901119890119909 by119898

0

10

20

30

40

50

1 2 3 4 5 6

005066

Values of

(b) 119905119863 times 119878ℎ by 120572

040

045

050

055

060

1 2 3 4 5

001002003004005

Values of Ｈ＄

(c) 119898 times 119890119888 by 119899119863

015

020

025

030

035

040

1 2 3 4 5

001002003004005

Values of Ｈ＄

(d) 119898 times 119901119903119900119901119890119909 by 119899119863

0

10

20

30

40

50

001 002 003 004 005

00005066

Values of

(e) 119899119863 times 119878ℎ by 120572

040

045

050

055

060

1 2 3 4 5

BehaviorNaiveFocused

(f) 119898 times 119890119888 by 119887119890ℎ119886V119894119900119903

015

020

025

030

035

040

0 05 066Behavior

NaiveFocused

(g) 120572 times 119901119903119900119901119890119909 by 119887119890ℎ119886V119894119900119903

05

06

07

08

09

10

1 2 3 4 5 6

00005066

Values of

(h) 119905119863 times 119894119903 by 120572

08700

08725

08750

08775

08800

001 002 003 004 005

BehaviorNaiveFocused

(i) 119899119863 times 119894119903 by 119887119890ℎ119886V119894119900119903

Figure 9 Curves of mean observations for different parameters and metrics for 36000 runs Bars are 095 confidence intervals Plots withthe same observables (ordinate) except for 119894119903 are on the same scale

Complexity 23

The higher the requirement for accepting the proposal isthe higher the proportion of extremists and the variance ofopinions are and the lower the shift statistic is In few words120572 constrains the world to dyadic discussion or throws it intoa process in which deliberation is much more importantthan dyadic interactions This is why for any 120572 = 05 oneeither gives too much weight to deliberated results whichshadows pair-wise interactions or too little weight in thesense that no deliberated result is ever accepted and thusnever integrated into the opinions of the agents that havevoted for it The reason for the latter is that the latitudeacceptance is too low and thereafter pair-wise interactionsare not enough to unevenly polarize the population in such away that deliberated proposal arguments are voted favorablyby a non-negligeable majority Shifts for 120572 = 0 are stable fordifferent levels of 119905119863 which implies that deliberation keeps incheck all the shifts of opinions related to an increase in thenumber of pair-wise interactions For 120572 = 05 deliberationis less likely to be successful and shifts tend towards the shiftlevels for 120572 = 0 (see Figure 9(b))

Concerning labeling-basedmetrics120572 entirely determinesthe coherence statistic For 120572 = 05 there is no difference incoherence because of how coherence is defined it is maximalwhen 120572 = 0 and strangely maximal when 120572 = 066(see Figure 9(h)) Surprisingly 120572 has no effect on judgmentaccuracy One would have expected a higher 120572 to increasejudgment accuracy since agents would hardly collectivelyaccept a proposal argument discussed during deliberationProposal arguments have given how the argument lattice isgenerated a higher probability of being rejected than of beingaccepted in the ideal argumentation framework

435 A Word on Focused and Naive Agents Focused andnaive specifications for agent behavior are an importantparameter in the decision process since they model howagents choose arguments in the deliberation arena In the sce-narios where agents are focused agents show equal or highervariance of opinions and are therefore more extreme (seeFigure 9(g)) independently of the voting requirements Shiftsof opinion are less likely as well We can see this happeningbecause focused agents knock out proposal arguments moreoften than naive agents provided that 119899119863 is low

One could have expected naive agents to yield morecoherent decisions since they argue sincerely However thisseems not to be the case (see Figure 9(i)) Agents are unable todeliberate in away thatmakes deliberation reflect their votingintentions either because they do not have the necessaryarguments to do so or because such arguments do not exist

In terms of judgment accuracy focused agents do betterthan naive agents in finding the ideal labels for the proposalarguments This is likely because when reconstructing theframework focused agents take into account the deliberatedproposal argument and choose themost pertinent argumentsin their sacks that relate to the state of affairs at the deliber-ation arena In fine debates result in a better approximationofL119886119887120576(119868) even though one could have believed the oppositethat is focused agents are ready to sacrifice a correct collectivelabel of a proposal argument in the goal of defending theirpositions

44 Points of Discussion In this subsection we discuss theresults obtained from the simulations We attempt to makesense of them from a social sciences perspective

441 Opinion Consensus and Dissensus The deliberativeopinion dynamics corresponds to the simulated scenarios inwhich the variance of opinions is at its lowest and judgmentaccuracy at its highest Dyadic opinion dynamics are associ-ated with the scenarios in which the variance of opinions is atits highest judgment accuracy at its lowest and coherence atitsmaximumpossible levelThemixed interactions dynamicsis a combination of these aspects for all the metrics exceptfor coherence at which it is the worsemdashdeliberation leadsto some kind of loose opinion consensus that is reinforcedby pair-wise interactions Likewise the mixed discussionmodel confirms some results exposed in [16] expressingthat opinions and voting intentions in a deliberative contextldquooften changerdquo That being said and more particularly indeliberative opinion dynamics outputs align well with twointeresting results on consensus formation given by aninterpretation of Moscovici and Doisersquos [15] work Sunsteinrsquos[14] account on deliberation and group polarization andsocial impact theory [4] The first states that mild consensusis reached in deliberation as procedure disengages agentsthe second is that within discussion groups majorities tendto become larger and opinions more extrememdashnot on thebasis of competing arguments but of the initial balance ofopinions After several decision processes we find opiniondistributions that reflect these observations even though weare not able to precisely coin the set of parameters that yieldssuch distributions (see Figure 7)

Although differences between mixed and dyadic opiniondynamics in terms of the variance of opinions and shiftsare slim the former does much better in terms of judg-ment accuracy In this sense deliberation does moderateopinions but not as much when non-deliberation steps arerelatively frequent throughout the decision-making processIn terms of Moscovici and Doise theory of consensus [15]this makes sense since pair-wise discussion engages or ego-involve agents in the discussion and pushes to convergenceof opinion at the extremes of the opinion distributionHowever what remains unclear is the reason why proceduralparameters that constrain deliberation do not disengageagents from discussion and thus result in a mild consensusas also expressed by Moscovici and Doise [15] A possibleexplanation of this phenomenon may stem from the factthat deliberation in the model is always accompanied by aseries of ego-involving discussions that shadowor counter themoderating effect of deliberation Not only this deliberationcould also be responsible for crystallizing the opinions ofextremists for it favors one or the other part of the distributionof opinions as explained in [14] and to some extent in [13]Furthermore the mixed discussion model gives an exampleof polarization that results from a minority of dissenterson one side of an issue acceding to views of the majorityrsquosside [4] as opposed to the strong polarization observed indyadic interactions Situations in which the status quo orthe undecidability of the proposal argument lingers cannotpossibly yield situations of moderation In cognitive science

24 Complexity

one can be inclined to believe that the longer a deliberationprocess is the more likely individuals will take an immutablestance on the situation In view of the assumption that longdecision-making processes can be more costly for a group orsociety as a whole agents may feel forced to take a stancejust for the sake of ending the process The stance that isadopted would therefore depend on pair-wise interactionsamong individuals and not on deliberation since deliberationtakes too long to be conclusive

442 For Hypothetical Recommendations in Decision-MakingProcess Elaboration On another tune results from themodelsuggest that procedural parameters have marginally decreas-ing positive or negative effects on the metrics of interest Ina hypothetical policy analysis perspective this may indicatethat after choosing a correct number of individuals andimposing the right amount of debates one can attain rea-sonable levels of judgment accuracy coherence and varianceof opinions As accounted in [18] deliberation increasesknowledge (collective knowledge as well) and therefore itheightens the chances of making correct decisions If onebelieves that social welfare is linked to deliberation andeventually to the metrics studied previously then proceduralparameters can both be used as an instrument to attaindesired levels of social welfare and reinforce the notionof legitimacy of collective decisions It can also provide ajustification for choosing a deliberation regime rather thananother on the basis of how important (or urgent) a topic ofdiscussion is Indeed if we adopt the claim that deliberationis a forerunner of welfare then any state in which deliberativecues are at their maximum (highest number of participantsmany deliberation steps before making a decision and asfrequent as possible) has to be mapped to the highest attain-able social welfare In this case is a situation where judgmentaccuracy is at its highest yet the variance of opinions or theproportion of extremists at its lowest an ideal situation Wecannot say for sure since strong opinion consensus and longdecision-making processes in many situations may result ina loss of welfare rather than in a gain of it

Another interesting result in procedure for deliberationis the very similar estimates we find for the effects of theminimum number (119898) of debates necessary to make adecision on a proposal and the proportion (119899119863) of individualsparticipating in deliberation on judgment accuracy In thescope of social welfare this raises the question of whetheradvocating for longer decision processes by demandingmoredeliberation steps (higher 119898) rather than bigger debates(higher 119899119863) may increase welfare Depending on the objec-tives of the decider of the deliberation procedure he or shemay choose to concentrate on one or the other A recom-mendation we can assert from the mixed social interactionsmodel is that if one prefers low to high variance of opinionsandor high to low coherence the decider is better off ifshe focuses on increasing the size of the debate rather thanorganizing debates very frequently One of the reasons forthis might be that when making bigger debates all agents aretruly considered as equals in the decision-making processIn organizing many debates individuals that are chosen toparticipate in many of these will be more influential and will

somehow bias the set of arguments in the deliberation arenaThemajority rulemodeled through120572 also determines the sizeof the shifts and the distribution of opinions Because a bigger120572 is associated with higher levels of extremism and opinionvariance deliberated cues are harder to accept and thereforeit becomesmore difficult for agents to reach concensus andormoderate their opinions

On the other hand and further into the idea of legitimacythe goal of deliberative processes is the agreement amongall participants This agreement stands for the right or bestdecision from a formal or procedural perspective and it is alsothe best from a substantive point of view In other words thelegitimacy of decisions not only derives fromwell-establishedprocedural requirements (respect of the protocol) but alsobegs for the fulfillment of two essential conditions (1) satisfy-ing the procedural requirements for a correct procedure (for-mal legitimacy) and (2) the rational acceptability of the resultsof this procedure (substantive acceptability) Legitimacy canbe then obtained from the coherence statistic scrutinizedin the model insofar as it measures how well agents acceptthe results of a correct procedure (eg deliberation) So fora proposal to be legitimate one has to choose parametersthat would maximize group coherence in decision-makingprocesses

As deliberation may give rise to consensus in the delib-eration arena it may also sow the seeds of dissensus inthe group Our model clearly illustrates this phenomenonthrough the procedural parameters of deliberation and themetrics observed From another perspective in deliberativedemocracy dissensus in collective decision-making may bedesirable According to Landermore and Pagersquos accounts [36]lack of consensus can be perceived as having agents withalternative ideas that combined with other ideas can providea better approximation of the ideal status of an argumentTheauthors in [36] call this dissensus ldquopositive dissensusrdquo andit makes normative sense when solving complex collectivedecision-making problems Our model captures the ideaof having normative requirements to define a good andlegitimate collective decision requirements of high consen-sual deliberative accuracy (judgment accuracy) of legitimacy(coherence) and of diversity (variance of opinions) are alltaken into account and for such we can conclude that ourmodel is successful in describing a sound process of collectivedecision-making with interpretable outcomes

Of course the present simulation results and analysis donot prove that all opinion dynamics can be accounted forby such simple processes like the one presented here Theydo however shed light on the expressiveness of combiningdifferent paradigms to account formore interpretablemodelsand pave the way to creating original models of the sortThe simulations suggest the desirability of discovering theconsequences of relatively simple laws of communication atdifferent levels (micro meso and macro) to determine whatstill needs to be explained

5 Related Work

We see our model as a contribution to the influence andopinion dynamics field in ABM and a pragmatic application

Complexity 25

of abstract argumentation theory To our knowledge we areunaware of existing literature on agent-based modeling thatexplicitly relates collective choice and the notions of deliber-ation and opinion diffusion through abstract argumentationas we have done it

Most models in the literature of opinion diffusion areinterested in opinions because these have an influence incollective decision and in questions of social order Forinstance in [3] the authors are interested in consensus andin how a group collectively chooses among two alternativesIn other models authors are interested in the emergenceof extremism [8] and on the distribution of opinions whenextremists are introduced in the population [5 33 37] whileother authors coin the notion of opinion polarization as anemergent property of the system [8 12] They show usingmodels of ldquobounded confidencerdquo and opinion diffusion withtrust that three different kinds of steady states (unipolarbipolar and central) are possible depending on whetheragents are sufficiently uncertain about their opinions andsufficiently connected andor a certain proportion of indi-viduals are already extreme Recent articles on informationand opinion dynamics stress the importance of governanceand government intervention in the spread of emotions(opinions) in the frangibility of social consensus systemIn [10] the authors show that government intervention inthe spread of negative emotions can lead to an even fasterspread of negative emotions (single-peeked convergence) andin a faster collapse of the social consensus system Opinionas a function of trust is studied in [12] and relates to howindividuals form their opinions on the basis of how muchthey trust agents in their networks

Another stream of discrete opinion dynamics modelsthat explain the emergence of opinion can be found in theliterature Computer multiagent models of attitude changebased on Latanersquos social impact theory are presented in [4] Inthe theory an agent changes her opinion on the basis of theinformational impact she is subject to which depends on thepersuasiveness (strength) supportiveness and immediacy(group structure) of the environment and information shereceives The simulations predict two emergent groups ofphenomena the shifting of attitudes towards incompletelypolarized equilibria and the formation of coherent clusteringof subgroups with deviant attitudes Similarly but in acontinuous more argumentative fashion Mas and Flache[11] present a model of opinion diffusion using argumentsArguments are considered to be for or against a proposal andare given an agent-dependant relevance in dyadic persuasiveinteractions Arguments also determine the agentsrsquo opinionsand their dissimilarities which posits the rules of pair-wiseencounters in theirmodel as agents are assumed to form theiropinions according to the arguments they own They showthat in an argumentative opinion diffusion model the onlyobservable steady states are bipolarization and consensusand that bipolarization can emerge from interactions amongsimilar agents In other words bipolarization of opinionsis possible with homophily and without negative influenceif there is argumentation They subsequently compare theirresults to experimental data In [38] a similar approach toargumentation and opinion formation is taken but for the fact

that they introduce different types of arguments explicitlyapply social judgment theory [22] and use survey data toestablish model-to-real-world comparisons

Closer to opinion formation and abstract argumentationGabbriellini and Torroni [39] are the first to have origi-nally and soundly merged opinion diffusion and abstractargumentation They define an agentrsquos opinion as a functionof the arguments she holds and the attack relation amongthem They devised a focused peer-to-peer dialogue systemof persuasion (NetArg) inspired from Mercier and Sperberrsquosargumentative theory of reasoning [40] which used onlyabstract argumentation to study opinion polarization andopinion dynamics Moreover they considered networks andthe notions of trust and epistemic vigilance to define adynamics for knowledge revision and trust itself whichwe clearly lack in our model They showed that if a con-servative belief operator in argumentation was applied byagents when they reasoned then their dialogue protocoldid not increase polarization among agents That said theyused the model to study the effect of Groenvetterrsquos weaklink theory in the spreading of arguments and were notparticularly interested in notions like judgment accuracy orcoherence nor in tackling questions related to collectivedecision-making procedures Their system is very expressiveand helps position our work in the litterature our work is atthe frontier of pure argumentative opinion diffusion modelsopinion diffusion with arguments and continuous bounded-confidence models of opinion diffusion a la Deffuant

On another note work on collective cognitive conver-gence [41] and opinion sharing [19] shows that consensustowards a certain ldquocorrectrdquo opinion or cognitive state is alwayspossible yet dependent on noise variability and awarenessof agents In [20] the authors show that learning about anexogenous correct state of the world (represented by bits)under confidence was possible but only if the agents werenot too confident In a homogeneous population they showthat the higher the confidence the worse the learningmdashthevery confident agents do not learn the properties of the truestate of the world and disrupt the learning process of the lessconfident ones Collective cognitive convergence can be seenas a result of deliberation in truth-seeking models

When it comes to abstract argumentation theory we takean approach that wires two types of dialogues that are well-studied in the literature persuasion dialogues [42] and delib-eration dialogues [43] Another interesting line of work is theone on mechanism design [44] or the problem of devisingan argumentation protocol where strategic argumentation isnot a liability for success in debates We tackle mechanismdesign in a different way though Instead of consideringstrategy-proofness we are interested in how differences inprotocol can result in epistemologically ldquobetterrdquo collectivechoices and guarantee that opinion distributions are favorablefor deliberation For a survey on persuasion dialogue see[42]

Work on agent-based argumentation usually assumes thatthe semantic relationship between arguments is fixed Inother words if two individuals were to put two argumentsin the public arena such that one logically attacked the otherthen everyone would agree that such attack exists [45 46]

26 Complexity

Other models which do not make this restrictive assumptioncan also be found in the literature [27 47] and derive fromthe class or family of opponent models [27 48] in which twoopposing sides attempt to win a dialogue Ourmodel is in theintersection of these two but the frame combining opiniondiffusion of the kind and argumentation remains new

The idea of mixing interpersonal influence and verticalcommunication however is not original It is described andimplemented in innovation diffusion models such as [5 49]In both vertical communication is modeled as exogenoustransparent informationmdashagents are aware of the existenceof an innovation thus triggering several processes of choiceand stabilization of opinions Also in [50] an Eulerian modelis implemented to show the effect of media and exogenousinformation on opinion distributions With respect to thispoint the originality of our work is in that the informationemitted as vertical communication is endogenous It is issuedfrom a deliberation model that agents shape on the basisof their opinions arguments and behavior In the spirit of[51] where the authors control for the design of verticalcommunication (to whom the vertical communication isaddressed its timing how it influences agents opinions) wecontrol for the process generating the information for ourown set of variables length of debates majority quotas andthe frequency at which discussions take place

6 Conclusion and Perspectives

The main objective of this article was to build a bridgewhere decision-making argumentation and opinion dif-fusion could come together We proposed a model thatcombined abstract argumentation theory and a boundedconfidence opinion diffusion model and showed to whatextent it could explain variance of opinions extremismcoherence in collective decisions and judgment accuracyon arguments given an ideal state of full information Thesecond objective was to show in what way governance pro-viding agents with arenas of discussion and deliberation wasimportant in addressing the success of collective decision-making processes and the quality of their outcomes

The model revealed that allowing for more deliberationtime though in lower frequency and allowing for widerparticipation in deliberation increased the variance of opin-ions and the proportion of extremists in a group Theseobservations are consistent with the combination of resultsfound in [9] and in [13 14] which stress that deliberationmay polarize groups and may have a meager effect on theshifts of opinions and inconsistent with [16] where it isargued that opinions tend to moderate after deliberationIt happens that deliberation alone did moderate opinionyet when integrated into a more complex system in whichindividuals were allowed to interact with one another andnot all that was deliberated was accepted its influence wasshadowed by other more individualistic dynamics

Undeniably the grounded semantics played an importantrole in the weak effect of deliberation since it models askeptical way of reasoning over arguments For agentsaccepting arguments thus updating their opinions accord-ingly happened not too often Nevertheless asking for more

deliberation did increase judgment accuracy as observed in[18] yet in a marginally decreasing fashion We showed thatvoting within the deliberation protocol not only increasedthe proportion of extremists and the variance of opinions ina group but also determined how coherent deliberation andvoting procedures were with each other Lastly we showedthat agents that are focused judged arguments better hadmore stable opinions and constituted a group with higherproportion of extremists than their naive counterparts

In terms of governance the model says that there is per-haps no trade-off between extremismand judgment accuracyInstead it asserts that the higher the variance of opinions thecloser the group gets to the correct decisions This situationmay be interpreted as Landermore and Pagersquos notion of ldquoposi-tive dissensusrdquo (in complex collective decision-making tasks)[36] as opposed to consensus as a normative requirementfor correct collective decision-making If a decider has toorganize deliberation to legitimize a public policy dependingon what kind of world he or she wants different set ofparameters may be chosen to account for different levels oflegitimacy and correctness For instance one may ask for adecision to be legitimate to respect a certain level of coherenceor of judgment accuracy and to be discussed by a sundryof different-opinionated peoples If so deliberation has tobe made more often more participants have to be includedin debates or in deliberation instances and not too manytime steps should be introduced in between two deliberationinstances So the model may capture normative ideas todefine correct and legitimate collective decisions and tomake recommendations on those grounds Requirements ofhigh consensual deliberative accuracy (judgment accuracy)legitimacy (coherence) and diversity (variance of opinions)are all taken into account and for such a reason the model israther successful in describing a sound process of collectivedecision-making with interpretable outcomes

61 Extensions of the Model and Other Ideas Our model canbe extended in many ways Value-based abstract argumen-tation frameworks (VAFs) for example as an extension ofDungrsquos argumentation framework provide a formal descrip-tion of the process of decision-making in which argumentsare given values and audiences (set of agents) ignore attacksbetween arguments on the basis of their preferences over val-ues Social abstract argumentation frameworks (SAFs) alsoprovide an interesting and seemingly convenient formalismfor our model In SAFs agents vote on the acceptability ofarguments and the resulting labeling is a combination of thearguments that are voted for and labeling-based semantics Inour model we apply the same rule yet only for one argumentand given precise agent-based votingmechanisms and votingrules which are eventually parametrized

To consider our model as a specification of a mixed VAFand SAF model may be of great interest for future workThis formalism can provide solid foundations to describeexplain interpret and eventually extend the results foundhere from an argument-based perspective On those groundsfuture work will focus on creating a hybrid argumentationframeworkmdashwith its semanticsmdashthat takes heed of bothvalues or principles and voting

Complexity 27

Concerning other possible extensions of the model anatural reflex would be to argue on the validity and therobustness of the mix between the bounded confidenceopinion diffusion models and deliberation Consequentiallyadding deliberation to different bounded confidence modelsof opinion diffusion such as the Hegelsmann-Krause model[37] or to models of social impact theory like in [4] canbe of great interest for future work We acknowledge that afiner understanding of argumentation frameworks randomlattices and their respective applications for this modelare necessary Testing for different labeling-based semanticsor even trading labeling-based semantics for scoring-basedsemantics (semantics that defines acceptability on scoresgiven to arguments) such as debate semantics and ranking-based semantics (semantics that yield a preorder) may beinteresting to consider in future work

More into modeling deliberation it is not inadequateto believe that agents may also advance proposals duringdeliberation and replace to some extent the proposal forwhich the deliberation is taking place Endogenizing theprocess that produces the type of argument to be deliberatedon may be interesting to explore either by having agentsstrategically replace proposals at certain moments duringthe debate or having the central authority attempt to chooseproposals such that the probability of having them acceptedis high Agents may also be thought to be autoorganizing(independent of a central authority) and concede on the sizeof the instances of deliberation To consider differences insemantics across agents or even endogenizing the semanticson how urgent or important the proposal that is beingdiscussed is can also be an interesting direction to take

Last but not least many nontrivial modifications of ourmodel are possible and most should include better-thoughtdeliberation protocols It may be interesting to design andobserve protocols in which deliberation only affects agentsthat are actually debating for example Trust network effectsmultidimensionality of opinions new processes of argumentexchange or learning among agents are notions to furtherdevelop in order to make the model more realistic and relaxsome unreliable assumptions In sum the model is to berefined and extended with the objective of either studyingconcrete cases of deliberative polling and opinion dynamicsor implementing more intuitive thought experiments

Appendix

We present below some of the pseudocode we used toimplement the model

Algorithms

(a) Algorithm 1 implements the dyadic influence in themodel It describes how two agents interact andupdate their opinions

(b) Algorithm 2 implements the deliberative influence inthe model which describes how agents update theiropinions in sight of the accepted deliberated proposalarguments

(c) Algorithm 3 is used to obtain the grounded labelingof an argumentation framework

(d) Algorithm 4 provides an overview of a run in thesimulations119872 is the number of proposal argumentsto discuss which we set to 100 in the model

Each object in the system is indexed by a119908ℎ119900 number Hencea set of agents is annotated by a set of natural numbers Weuse this fact in the implementation of the algorithm 119874119887119895119890119888119905119894denotes the object of 119908ℎ119900 number 119894 Agentsrsquo opinions (119900) arecalibrated so that 119900 isin [minus1 1]Data Availability

The program and data used to support the findings of thisstudy are available from the corresponding author uponrequest

Disclosure

A preliminary version of this work was presented anddiscussed at the conference Bridging the Gap between FormalArgumentation and Actual Human Reasoning on October 4-5 2018 at BochumGermanyThe authors thank the audiencefor their valuable comments and suggestions

Conflicts of Interest

The authors declare that they have no conflicts of interestregarding the publication of this article

Acknowledgments

The contribution of Gabriella Pigozzi was supported by theDeutsche Forschungsgemeinschaft (DFG) and the CzechScience Foundation (GACR) as part of the joint project FromShared Evidence to Group Attitudes (RO 45486-1)

Endnotes

1 In other words the superiority of the ldquoright answerrdquo orlabeling in our case will appear as such to all

2 All agents hold the same opinion3 By sensitivity we mean how much an agent is ready to

change her opinion on the sole fact that an informationalcue has been debated on

4 The assumption is strong but this trust among agentsmay be rooted from a genuine individual motivation toreach a democratic consensus

5 We adopted a rule of thumbs that says that if an agentis neutral with respect to the principle then she willvote for any proposal that is presented to her duringdeliberation

6 If the argument is not unique she chooses one atrandom

7 If there are more than one candidate the agent will favorthe argument at minimal distance from 119868

28 Complexity

Require Vector of values for parameters (119881 = (120583 119880 119879))Ensure Successful interactions among agents119896 larr997888 0119873119890 larr997888 0

while 119896 le |119873|2 do ⊳Fill119873119890 with |119873|2 agents119896 larr997888 119896 + 1119873119890 larr997888 119877119886119899119889119900119898(|119873| + 1) ⊳Pseudo-random generator Generates integer between 0 and |119873|119873119903 larr997888 0 120575119894119895 larr997888 0 ⊳Variable to store differences in opinionfor all 119894 isin 119873119890 doRandomly pick a number 119895 isin 119873(119873119890 cup 119873119903)119873119903 larr997888 119873119903 cup 119895 120575119894119895 larr997888 |119900119894 minus 119900119895|procedure dyadic-update(119894 119895) ⊳Two agents 119894 and 119895 discuss and update opinions1199001015840 larr997888 119900119894 (120583119895 119880119895 119879119895) larr997888 (120583119894 119880119894 119879119894) larr997888 (120583 119880 119879) ⊳Save agent irsquos opinion before updating and homogenize agents

if 120575119894119895 lt 119880119894 then119900119894 larr997888 120583119894119900119895 + (1 minus 120583119894)119900119894if 120575119894119895 gt 119879119894 then119900119894 larr997888 (1 + 120583119894)119900119894 minus 120583119894119900119895if 120575119894119895 lt 119880119895 then119900119895 larr997888 1205831198951199001015840 + (1 minus 120583119895)119900119895if 120575119894119895 gt 119879119895 then119900119895 larr997888 (1 + 120583119895)119900119895 minus 1205831198951199001015840

Algorithm 1 Dyadic social influence

Require Vector of values for parameters (119881 = (120574 119901119886 119901119903))Ensure Successful interaction between proposal arguments and agents

for all 119894 isin 119873 doprocedure Probability-Update(V119886 119901119886 119901119903) ⊳Update individual probabilities of change due to deliberated cues119909119894 larr997888 119901|V119886minus119900119894 |2119886 119910119894 larr997888 1199012|V119886minus119900119894 |119903119902 larr997888 0if L119886119887(119868) = 119894119899 then

procedure deliberative-update(119894 119886) ⊳Agent i argument 119886120574119894 larr997888 120574 119902 larr997888 119906 sim U(0 1) ⊳ 119906 is a realization of a uniform random variableif 119902 lt 119909119894 then119900119894 larr997888 120574119894119900119894 + (1 minus 120574119894)V119886break

if 119902 lt 119910119894(1 minus 119909119894) then119900119894 larr997888 (1 + 120574119894)119900119894 minus 120574119894V119886Algorithm 2 Deliberation influence

8 In terms of computational complexity naive agentsmaketheirmoves in119874(|A119894| log(|A119894|)) while focused agents doat least as bad

9 Tables have memory Therefore debates are not neces-sarily trees

10 If there are less than (119899119863 times |119873|)2 agents for any of bothgroups the CA summons all of them to the deliberationtable

11 Or careful consideration of the viewpoints of otherswhich implies that citizens keep an open mind and donot reject arguments outright

12 A ldquohealthyrdquo consensus refers to the case in which manyconsensual decisions are taken but opinion-wise agentsdo not agree with each other Quotes are used on theword healthy because some political science theorists

(eg [36]) believe that in certain situations agreement inopinions for deliberation is epistemologically damaging

13 Simulations on groups of |119873| = 800 and |119873| = 400agents shows that scaling the population has an effect onall themetrics but it is confounding depending onwhichcouples of attitudes structures are considered (119880 119879) Westick to |119873| = 400 as in [9] for comparability andsimulation running time

14 By balanced wemean with as many arguments with V119886 lt0 as with V119886 gt 0

15 Experiments on the system suggest that a bigger(smaller) argument pool 119872 = 1000 (119872 = 200) isassociated with a higher (lower) variance of opinions(and extremism) and higher (lower) judgment accuracyFor the rest of the observations 119872 seem to have noeffect We choose a middle value for 119872 Similarly theargument pool chosen to be balanced or uniformly

Complexity 29

Require Argumentation framework 119860119865 = (AR)Ensure Assignment of labels to arguments from the grounded labeling of 119860119865L(A)

L0(A) larr997888 (0 0 0)119896 larr997888 0do119894119899 L119896+1(A) larr997888 119894119899 L119896(A) cup 119886 | 119886 is not labeled 119894119899 L119896(A) 119886119899119889 forall119887 isin A 119904119905 119887 R 119886 then 119887 isin 119900119906119905 L119896(A)119900119906119905 L119896+1(A) larr997888 119900119906119905 L119896(A) cup 119886 | 119886 is not labeled 119894119899 L119896(A) 119886119899119889 exist119887 isin A 119904119905 119887 R 119886 and 119887 isin 119894119899 L119896+1(A)

while L119896+1(A) = L119896(A) = 0Algorithm 3 Solve argumentation frameworks (with grounded semantics)

Require Vector of values for parameters (119881)Ensure Vector of statistics (119880)

procedure Set-parameters(119881) ⊳Initialize parameters for scenarioprocedure Init-objs(119899119898 119896 ) ⊳Create119898 arguments 119899 agents and 119860119865120576procedure Solve-Argumentation-Framework(119860119865) ⊳Assigns the epistemic labelingL120576 toA119894 larr997888 1119905 larr997888 1while 119894 le 119872 do ⊳ 119872 the number of arguments to discuss119868119894 larr997888 119866119890119899 119886119903119892(1) ⊳Generates proposal argument 119868119860 119894119905 larr997888 (119868119894 0)L119886119887(119868119894) larr997888 119906119899119889 ⊳Initialize label for 119868119895 larr997888 1 ⊳Initialize debate counter 119895while ((L119886119887(119868) = 119906119899119889 or 119898 le 119895) and 119895 le 119898) doif 119905 equiv 0(119898119900119889 119905119863) then

procedure build-argumentation-framework(119899119863 119887119890ℎ119886V119894119900119903) ⊳Build the argument graph agents deliberate(119861119894119905 119878119894119905) larr997888 (119883 119884) sub (⋃119897isin119873119899119863 A119897119894119905 ⋃119897isin119873119899119863 R119897119894119905) ⊳ 119873119899119863 sub 119873 are agents sampled by the CA119860 119894119905 larr997888 119889(119860 119894119905 (119861119894119905 119878119894119905))procedure Solve-Argumentation-Framework(A119889) ⊳FindL(A119889)L119886119887(119868119894) larr997888 L119886119887119863(119868119894)119895 larr997888 119895 + 1

else119896 larr997888 1while 119896 le 119905119863 doprocedure dyadic-social-influence(120583 119879 119880) ⊳Agents discuss one-to-one119896 larr997888 119896 + 1119905 larr997888 119905 + 1

if 120572 gt 0 andL119886119887(119868119894) = 119900119906119905 then ⊳Agents vote for the proposalif (|119897 isin 119873|119900119897 times V119868119894 ge 0| ge 120572 times 119899) then

L119886119887(119868119894) larr997888 119894119899else

L119886119887(119868119894) larr997888 119900119906119905if L119886119887(119868119894) = 119894119899 thenprocedure Deliberation-influence(120574 119901119886 119901119903) ⊳Agents are influenced by the result of the decision process119894 larr997888 119894 + 1119905 larr997888 119905 + 1119880 larr997888 119862119900119898119901119906119905119890 119878119905119886119905119894119904119905119894119888119904(119881)

return 119880 ⊳Get the statisticsAlgorithm 4 Run for one scenario

generated does not seem to play a significant explanatoryrole on any of the metrics

16 Experiments show that themaximumnumber of debatesis not an important parameter even if it is an intuitiveone when considering decision-making processes Inpractice debates end after at most 119898 + 1 deliberation

steps and even then it happens rarely However to avoidunpleasant surprises like infinite debates on a completeargument graph containing the proposal argument weadded the constraint 119898 Its value is precisely 7 becausethe biggest minimum number of debates we studyis 6

30 Complexity

17 To ensure comparability with the deliberative andmixedopinion dynamics models on judgment accuracy andcoherence agents vote on proposal arguments every 119905119863times(119898+1) 119905119863 and119898does not explain variability in judgmentaccuracy nor in coherence

18 Student t-tests for independent samples are robust tothe violation of two hypothesis on the distribution ofthe samples normality and homogeneity of variancesas long as the sample size is big enough (central limittheorem) and if the difference in size of the comparedsamples is smallrespectively

19 Studentrsquos test version forwhen variances are not assumedequal and the difference in size of the compared samplesis big

20 We do not correct for homoskedasticity nor normality oferrors in the estimates

21 Balanced as the same number of runs per scenario orsuch that no significant correlations are found betweenparameters

References

[1] R Axelrod ldquoThe dissemination of culture a model withlocal convergence and global polarizationrdquo Journal of ConflictResolution vol 41 no 2 pp 203ndash226 1997

[2] D M Rousseau Reinforcing the MicroMacro Bridge Organiza-tionalThinking and Pluralistic Vehicles SAGEPublications SageLos Angeles Calif USA 2011

[3] S Galam and SMoscovici ldquoTowards a theory of collective phe-nomena consensus and attitude changes in groupsrdquo EuropeanJournal of Social Psychology vol 21 no 1 pp 49ndash74 1991

[4] A Nowak J Szamrej and B Latane ldquoFrom private attitude topublic opinion a dynamic theory of social impactrdquo Psychologi-cal Review vol 97 no 3 pp 362ndash376 1990

[5] G Deffuant S Huet and F Amblard ldquoAn individual-basedmodel of innovation diffusion mixing social value and individ-ual benefitrdquo American Journal of Sociology vol 110 no 4 pp1041ndash1069 2005

[6] G Deffuant D Neau F Amblard and G Weisbuch ldquoMixingbeliefs among interacting agentsrdquoAdvances in Complex Systems(ACS) vol 3 no 01n04 pp 87ndash98 2000

[7] J Rouchier P Tubaro and C Emery ldquoOpinion transmission inorganizations an agent-based modeling approachrdquo Computa-tional and Mathematical OrganizationTheory vol 20 no 3 pp252ndash277 2014

[8] M Meadows and D Cliff ldquoThe relative disagreement model ofopinion dynamics where do extremists come fromrdquo in Pro-ceedings of the International Workshop on Self-Organizing Sys-tems Lecture Notes in Computer Science pp 66ndash77 Springer2013

[9] W Jager and F Amblard ldquoUniformity bipolarization and plu-riformity captured as generic stylized behavior with an agent-based simulation model of attitude changerdquo Computational ampMathematical Organization Theory vol 10 no 4 pp 295ndash3032005

[10] Y-F Zhang H-Y Duan and Z-L Geng ldquoEvolutionary mech-anism of frangibility in social consensus system based onnegative emotions spreadrdquo Complexity vol 2017 Article ID4037049 8 pages 2017

[11] A Flache andMWMacy ldquoSmall worlds and cultural polariza-tionrdquoThe Journal of Mathematical Sociology vol 35 no 1-3 pp146ndash176 2011

[12] A Tsang and K Larson ldquoOpinion dynamics of skepticalagentsrdquo in Proceedings of the 13th International Conference onAutonomous Agents and Multiagent Systems AAMAS 2014 pp277ndash284 International Foundation forAutonomousAgents andMultiagent Systems France May 2014

[13] K M Hansen Deliberative democracy and opinion formationUniversity Press of Southern Denmark Odense 2004

[14] C R Sunstein ldquoThe law of group polarizationrdquo Journal ofPolitical Philosophy vol 10 no 2 pp 175ndash195 2002 WileyOnline Library

[15] S Moscovici and W Doise Dissensions et Consensus unetheorie generale des decisions collectives Presses Universitairesde France-PUF 1992

[16] J S Fishkin and R C Luskin ldquoExperimenting with a demo-cratic ideal deliberative polling and public opinionrdquo ActaPolitica vol 40 no 3 pp 284ndash298 2005

[17] R C Luskin J S Fishkin and S Iyengar Considered Opinionson US Foreign Policy Face-To-Face versus Online Delibera-tive Polling International Communication Association NewOrleans La USA 2004

[18] J Barabas ldquoHowdeliberation affects policy opinionsrdquoAmericanPolitical Science Review vol 98 no 4 pp 687ndash701 2004

[19] O Pryymak A Rogers and N R Jennings ldquoEfficient opinionsharing in large decentralised teamsrdquo in Proceedings of the 11thInternational Conference on Autonomous Agents andMultiagentSystems vol 1 pp 543ndash550 Spain June 2012

[20] J Rouchier and E Tanimura ldquoWhen overconfident agents slowdown collective learningrdquo Simulation vol 88 no 1 pp 33ndash492012

[21] D Miller ldquoDeliberative democracy and social choicerdquo PoliticalStudies vol 40 S1 pp 54ndash67 1992

[22] M Sherif and C I Hovland Social Judgment Assimilation andContrast Effects in Communication and Attitude Change YaleUniversity Press 1961

[23] P M Dung ldquoOn the acceptability of arguments and its funda-mental role in nonmonotonic reasoning logic programmingand n-person gamesrdquo Artificial Intelligence vol 77 no 2 pp321ndash357 1995

[24] M Caminada and G Pigozzi ldquoOn judgment aggregation inabstract argumentationrdquo Autonomous Agents and Multi-AgentSystems vol 22 no 1 pp 64ndash102 2011

[25] P Baroni M Caminada andM Giacomin ldquoAn introduction toargumentation semanticsrdquoThe Knowledge Engineering Reviewvol 26 no 4 pp 365ndash410 2011

[26] S Chambers ldquoDeliberative democratic theoryrdquo Annual Reviewof Political Science vol 6 no 1 pp 307ndash326 2003

[27] E Bonzon andNMaudet ldquoOn the outcomes of multiparty per-suasionrdquo in Proceedings of the 10th International Conference onAutonomous Agents and Multiagent Systems 2011 AAMAS 2011vol 1 pp 47ndash54 International Foundation for AutonomousAgents and Multiagent Systems Taiwan May 2011

[28] J Ranciere Disagreement Politics and Philosophy University ofMinnesota Press 1999

[29] T Bench-Capon and K Atkinson ldquoAbstract argumentation andvaluesrdquo in Argumentation in Artificial Intelligence pp 45ndash64Springer 2009

[30] M Caminada ldquoOn the issue of reinstatement in argumenta-tionrdquo in Proceedings of the European Workshop on Logics in

Complexity 31

Artificial Intelligence vol 4160 of Lecture Notes in ComputerScience pp 111ndash123 Springer Berlin Heidelberg

[31] GAVreeswijk ldquoAn algorithm to computeminimally groundedand admissible defence sets in argument systemsrdquo in Pro-ceedings of the 2006 conference on Computational Models ofArgument Proceedings of COMMA 2006 vol 144 pp 109ndash120IOS Press Amsterdam The Netherlands 2006

[32] C W Sherif M Kelly H L Rodgers et al ldquoPersonal involve-ment social judgment and actionrdquo Journal of Personality andSocial Psychology vol 27 no 3 pp 311ndash328 1973

[33] G Deffuant F Amblard G Weisbuch and T Faure ldquoHow canextremism prevail A study based on the relative agreementinteraction modelrdquo Journal of Artificial Societies and SocialSimulation vol 5 no 4 p 1 2002

[34] E Awad M W Caminada G Pigozzi M Podlaszewski andI Rahwan ldquoPareto optimality and strategy-proofness in groupargument evaluationrdquo Journal of Logic and Computation vol 27no 8 pp 2581ndash2609 2017

[35] D Grossi and G Pigozzi ldquoJudgment aggregation a primerrdquoSynthesis Lectures on Artificial Intelligence and Machine Learn-ing vol 8 no 2 pp 1ndash151 2014

[36] H Landemore and S E Page ldquoDeliberation and disagreementproblem solving prediction and positive dissensusrdquo PoliticsPhilosophy and Economics vol 14 no 3 pp 229ndash254 2015

[37] R Hegselmann and U Krause ldquoOpinion dynamics andbounded confidence models analysis and simulationrdquo Journalof Artificial Societies and Social Simulation vol 5 no 3 2002

[38] A Stefanelli and R Seidl ldquoModerate and polarized opinionsUsing empirical data for an agent-based simulationrdquo inProceed-ings of the Social Simulation Conference 2014

[39] S Gabbriellini and P Torroni ldquoA new framework for ABMsbased on argumentative reasoningrdquo in Advances in SocialSimulation vol 229 pp 25ndash36 Springer Berlin Germany 2014

[40] H Mercier and D Sperber ldquoWhy do humans reason Argu-ments for an argumentative theoryrdquo Behavioral and BrainSciences vol 34 no 2 pp 57ndash74 2011

[41] H V Parunak T C Belding R Hilscher and S BruecknerldquoModeling and managing collective cognitive convergencerdquoin In Proceedings of the 7th iNternational Joint Conference onAutonomous Agents and Multiagent Systems vol 3 pp 1505ndash1508 International Foundation for Autonomous Agents andMultiagent Systems 2008

[42] H Prakken ldquoFormal systems for persuasion dialoguerdquo TheKnowledge Engineering Review vol 21 no 2 pp 163ndash188 2006

[43] K Atkinson T Bench-Capon and P Mcburney ldquoA dialoguegame protocol for multi-agent argument over proposals foractionrdquoAutonomousAgents andMulti-Agent Systems vol 11 no2 pp 153ndash171 2005

[44] M Thimm and A J Garcıa ldquoClassification and strategicalissues of argumentation games on structured argumentationframeworksrdquo in Proceedings of the 9th International Conferenceon Autonomous Agents and Multiagent Systems vol 1 pp 1247ndash1254 International Foundation for Autonomous Agents andMultiagent Systems Canada 2010

[45] I Rahwan and K Larson ldquoMechanism design for abstractargumentationrdquo in Proceedings of the 7th International JointConference on Autonomous Agents and Multiagent SystemsAAMAS 2008 pp 1031ndash1038 International Foundation forAutonomous Agents and Multiagent Systems Portugal May2008

[46] H Prakken R Riveret A Rotolo and G Sartor ldquoHeuristics inArgumentation A Game-Theoretical Investigationrdquo EuropeanUniversity Institute 2008

[47] N Hadidi Y Dimopoulos and P Moraitis ldquoTactics and con-cessions for argumentation-based negotiationrdquo in Proceedingsof the COMMA vol 245 pp 285ndash296 2012

[48] N Oren and T J Norman ldquoArguing using opponent modelsrdquoin Proceedings of the International Workshop on Argumentationin Multi-Agent Systems pp 160ndash174 Springer 2009

[49] S Delre W Jager T Bijmolt and M Janssen ldquoTargeting andtiming promotional activities An agent-based model for thetakeoff of new productsrdquo Journal of Business Research vol 60no 8 pp 826ndash835 2007

[50] A Mirtabatabaei P Jia and F Bullo ldquoEulerian opinion dynam-ics with bounded confidence and exogenous inputsrdquo SIAMJournal on Applied Dynamical Systems vol 13 no 1 pp 425ndash446 2014

[51] V Barbet N Guiraud V Laperriere and J Rouchier Hagglingon Values Towards Consensus or Trouble 2019

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Complexity 13

Table 1 Parameters and parameter types and their domains used to compare dyadic collective and mixed opinion dynamics

Length ofD119868 Decisional in119863119868 Social influence Deliberation influence Cognitive119905119863 119898 119898 119899119863 120572 120583 119879 119880 119901119886 119901119903 120574 119896 119861119890ℎ119886V1198941199001199031 3 6 1 3 6 7 001 002 005 0 05 066 01 14 18 02 06 03 05 01 005 01 02 4 8 16 119873119886119894V119890 119865119900119888119906119904119890119889account Deliberated arguments and informational cues haveno effect on agent opinion but agents still vote for theproposal argument17 We use and calibrate this model forcomparability in terms of all our metrics since they arenot explicitly observed in [9] a reference opinion dynamicsmodel Furthermore for simplicity and comparability againwe suppose that agents are homogeneous in terms of theirattitude structures and opinion weightings We fix 120583119894 =120583 119879119894 = 119879 and 119880119894 = 119880 for all 119894 isin 119873 and for some triplet(120583 119880 119879) isin [0 05] times [0 2] times [119880 2]

Experiments suggest that the strength of the dyadic inter-actions is an explanatory factor of the time of convergence(120583 lt 05) and of the dyamicsrsquo steady states Bigger values of 120583speed up convergence towards a stable set of opinion clustersand affect the size and the relative position of these clustersin the opinion distribution 120583 however does not play a majorrole in the number of opinion clusters observed at the steadystate of the dynamics

We set 120583 to the value given to it in Jagger et alrsquos opiniondynamics model (120583 = 01) [9] For one it is the smallestreal number for which most results found in [9] hold and thedynamics are smooth secondly it seems to be a reasonableldquostrengthrdquo of influence considering that we would not wantpair-wise social interactions to completely shadow or maskan eventual effect of deliberation in the opinion dynamics

For different values of attitude structures 119879 and 119880 weget exactly the same results as in [9] in terms of the numberand density of the opinion clusters The different couplesof values of 119880 and 119879 model the size of the latitude ofnoncommitment (119879 minus 119880) and thus the agentsrsquo propensityto update opinions Whenever 119879 minus 119880 le 02 regularitiesin convergence and opinion clustering appear If 119880 gt 1there is always central convergence if 119879 lt 1 there is alwaysbipolar convergence with clusters of similar sizes For 119879 minus119880 gt 02 relatively low values of 119879 (119879 lt 1) and relativelyhigh values of 119880 always yield extreme bipolar convergencewith a relatively small cluster of moderate agents (bipolar-central convergence) For 119879 sufficiently high (119879 ge 12) and119880 sufficiently low (119880 le 08) in other words not very ego-involved agents there is a bigger spectrum of steady statesFor this reason we studied the metrics of interest for thesevalue differences and kept the couples of (119880 119879) depicted inTable 1 For these couples central bipolar bipolar-centralconvergence (3 clusters) and multicluster convergence (4 or5 clusters) are possible and result frommeaningfully differentattitude structures Judgment accuracy varies significantlyacross the four different attitude structures we retain and soit happens with the variance of opinions These scenariosprovides us with reference results from which to build upon

322 Calibrating Deliberative Social Interactions Delibera-tive social interactions correspond to social situations in

which individuals are not influenced by pair-wise discussionwith peers but are sensitive to collectively deliberated infor-mational cues On these terms parameters such as the sizeof agentsrsquo argument sacks (119896119894) the distribution of argumentsin them (120601119894) the minimum number of debates (119898) andthe proportion of debaters from the population (119899119863) are nolonger mute We assume for simplicity that the formulaexist120601119890exist119896forall119894(119896119894 = 119896 and 120601119890119894 = 120601119890 and 120601119898119894 = 05) is true with 119894 isin 119873for some value 120601119890 isin [05 1] and for 119896 isin N

The chosen domains for these parameters (see Table 1)are justified by the size of the population and the size of theargument pool (119872 = 600) which we set from the start If 119896is too big (119896 ge 32) andor too many agents are allowed inthe debate (119899119863 ge 02) then individuals always disprove thecentral argument and deliberation is never or rarely takeninto account in the update of opinions Similar effects occurwhen 119898 is high yet we only calibrate it with respect to themean running time of a simulation and the values of 119905119863Different distributions of 120601119890119894 seem to significantly or directlyexplain none of the statistics we analyze so we convenientlyset 120601119890119894 = 075 We chose the parameter 119896 to be always divisibleby 4 for it is convenient given the initial conditions regardingthe argument distribution in the agentsrsquo argument sacks

We fix the value space for the parameters 120574 119901119886 119901119903 as inTable 1 to account for different intensities of the effect ofdeliberative voting retroaction on the system For examplethe scenarios where the three parameters are at their lowestvalues correspond to the scenarios with the smallest fixedprior effect of deliberation in the model Increasing any ofthese parameters should mechanically be associated witha world in which agents are more sensitive to collectivedecisions The values for 119901119886 are taken from [13] wherethe author states that by means of deliberation 7 to28 of individuals changed their opinion from agreeingto disagreeing or vice versa on a referendum question forDenmarkrsquos participation in the EuroWe choose two differentvalues for 119901119886 that account for high (03) and very high (05)sensitivity to deliberation in a group 119901119903 on the other handis taken to be small (equal to 01) since we believe that thereexist agents that will always go against the reached consensusTheir numbers are meager

For simplicity we assume that for all 119894 isin 119873 120574119894 = 120574 forsome 120574 isin [0 05] In effect we posit that the heterogeneityof agents in the deliberative context only derives from theirarguments and their opinions We decide to calibrate 120574 inallusion to the strength of the social dynamics in pair-wiseinteractions In doing so we limit ourselves to consider threedifferent scenarios in which deliberation has respectivelyhalf equal and twice as much opinion-shifting power thanone-to-one social influence Finally we include the accept-ability voting quota rule 120572 whose domain is inspired fromclassical majority rules [35] observed empirically (Table 1)

14 Complexity

Table 2 Parameters and their domains used to study the effect of the procedural parameters on the metrics

Procedural parameters Other parameters119905119863 119898 119899119863 120572 119898 120583 119879 119880 119901119886 119901119903 120574 119896 119861119890ℎ119886V1198941199001199031 2 3 4 5 6 1 2 3 4 5 001 002 003 004 005 0 05 066 7 01 16 02 03 005 02 12 119873119886119894V119890 119865119900119888119906119904119890119889

323 Calibrating the Mixed Social Interactions Model Themixed interaction model ascribes to a parameter space inwhich the effect of collective choices on our metrics isnontrivial and where deliberation and voting on proposalsdetermine their acceptability Since the mixed model isequivalent to periodic iterations of pair-wise and collectivesocial interactions the calibration of the parameters in bothpreceding models holds (see Table 1 for the calibration)The first reason for this is comparability the second is thatthe previous calibration takes into account the fact thatboth dynamics are going to be combined We include thefrequency of debates by adding the parameter 119905119863 whichcontrols for the amount of pair-wise interaction amongagents between two distinct deliberation steps

324 Termination Conditions for Runs Simulations stoponce 100 proposals are deliberated on andor voted for inother words after 100 collective decision steps have occuredThe number of proposals discussed seems arbitrary yet it ishigh enough to observe the effects of deliberation on opiniondistributions and on other metrics related to coherence andjudgment accuracy We choose the number of runs as afunction of our research questions which give relevance tothe procedural parameters in the mixed model rather than tothe parameters set to describe the population and its behavior

325 Expected Outcomes of the Simulations We expect moredeliberation in terms of higher 119898 and 119899119863 to increasejudgment accuracy and at the same time to reduce thevariance of opinions In turn a smaller variance of opinionsimplies that the argument pool for deliberation is smaller andtherefore judgment accuracy should be lower Moreovervariance increasing dyadic interactions (or rejection) shouldalso increase judgment accuracy and coherence since bipo-larization and dissensus foster argument diversity in delibera-tionThe coherence statistic should be stronger in simulationsin which central convergence appears quickly and agentsare naive Attitude structures sensitivity to informationalcues and weights given to deliberated cues that makes pathsto bipolar convergence smaller or to central convergencelonger should be associated with higher judgment accuracyShifts of opinion should be more visible in scenarios inwhich extreme agents are pulled away from the extremesHence the sensitivity to deliberated cues (119901119886 119901119903) and 120574should explain the shifts When crossed with high values of119880 and 119879 the shifting power of deliberation should be at itshighest

In the end we do not know how these mechanismswill play out The results of the subsequent experimentsgive us insight on the interplay between the aforementionedeffects

4 Results

Before pointing at any result obtained from the mixedinteractions model we describe and compare the dyadicand deliberative interactions models in the parameter spaceobtained from the calibration in Section 3 (see Table 1)Primarily we require our results to describe two orthogonaltypes of opinion dynamics the pair-wise dyadic and thedeliberative The latter comprises scenarios in which indi-viduals do not influence each other by means of pair-wisediscussion and only update their opinions from delibera-tion the former scenarios where only pair-wise interactionsamong agents determine the dynamics We describe both onthe parameter space given in Table 1 by performing at least20 simulations per scenario Quantitative results from thepair-wise interaction model alone are described during thecalibration and are not treated heremdashdeliberative parametershave no effect on it Moreover we show how these differentmodel specifications yield qualitatively different opiniondistributions and compare them on the basis of the metricsof interest To this end we perform independent two-sampleStudentrsquos t-tests18 or Welch t-tests19 and compute confidenceintervals at a 095 level of confidence Last but not least wecomment ordinary least squares (OLS) regression estimatesto account for the direction and magnitude20 of the effectsof the parameters on the metrics Estimates are declaredsignificant at a 5 level of risk

We perform the same analysis on the mixed interactionsmodel as we compare it to the pair-wise dyadic and deliber-ative interactions models and discuss the marginal effects ofthe parameters on the metrics More precisely we obtain twodifferent types of results for the mixed interactions modelthe first compares the mixed scenarios to their monolithiccounterparts with respect to each metric and with respect tothe regression estimates the second gives a clear idea of themarginal effect of each governance parameter (or parameterof interest) on our observations In the first we allow controland procedural parameters to vary on a parameter spacesimilar to those of the dyadic and deliberative opiniondynamics (see Table 1) In the second we fix all of ourcontrol parameters execute 36000 balanced21 runs andfocus only on the one-way pooled effects of our proceduraland behavioral (all agents are focused or naive) parametersWe generate comprehensive graphs and tables to account forthe obtained results

Please notice the slight change in the size of the parameterspace for the procedural parameters and in the values ofthe nonprocedural ones for the first and second type ofresults (refer to Tables 1 and 2 respectively) Since we wantto have a finer idea of the marginal effects of each of theprocedural parameters on how well a group decides onproposals we make the value jumps for each parameter (1)

Complexity 15

() (U T) = (02 18) (b) ( pa m nD) = (005 05 00 2 02) (＝) ０ＬＧＮＬＭ ＣＨ (＜) (＝) Ｈ＞ tD = 3

(d) (５ ４) = (02 14) (e) ( pa m nD) = (01 03 05 3 02) () ０ＬＧＮＬＭ ＣＨ (＞) () Ｈ＞ tD = 6

Figure 7 From left to right common opinion trajectories and distributions for dyadic deliberative and mixed opinion dynamics within thestructure of different decision-making processes

small enough to detect significant differences in the estimatesand (2) meaningful for interpretation For the fixed one-valued parameters in Table 2 we use Table 1 to set them totheir minimum mean or median values

41 Comments on Dyadic and Deliberative Opinions Dynam-ics In this subsection we present our first results regardingthe deliberative opinion dynamics described in Equation (6)and its differences with the dyadic dynamics (Equation (5))in terms of our observations

411 Qualitative Analysis of Pair-Wise Opinion DynamicsPair-wise opinion dynamics alone produces multiclusterstable convergencewith variable cluster sizes (see Figures 7(a)and 7(d)) Insight on the complexity of these dynamics wasdiscussed in Section 321

In Figures 7(a) and 7(d) agents discuss randomly withone another in pairs Clusters in the extremes form quicklysince either agents that are close to the extremes attractone another or are convinced to stay close to the extremesby interacting with other agents they disagree with Otheragents with near-extreme positions may be simply attractedto one of the several moderate foci of agents in the opiniondistribution On the other hand moderate agents are eitherattracted to the closest opinion focus or pushed towards theextremes of the distribution Over time agents in the centralfocus either attract other agents into it or ignore the opinions

of the extreme agents that interact with them The result ofthese dynamics yields amulticlustered convergencewhere theopinion foci are at least at distance 119880 from one another andonly the extreme foci (clusters of agents with |119900119894| ge 075) areat a distance greater than or equal to 119879 from one anotherThefoci at the bounds of the opinion distribution are very stableand in analogy to the (12119889)-rule in assimilation bounded-confidence models [33] the number of clusters that form isroughly equal to 119880minus1412 Qualitative Analysis of Deliberative Opinion DynamicsQualitatively deliberative opinion dynamics yield a looseunipolar convergence of opinions near the center (centralconvergence see Figure 7(b)) near the center of either the leftor right portions of the distribution (group polarization seeFigure 7(e)) or to a sparse bipolar opinion distribution withclusters at the center and at the extremes (see Figure 7(c)) ifmixed with pair-wise interactions This is probably a reasonto expect the variance of opinions and the proportion ofextremists (almost perfectly correlated 120588 = 089 119901 lt 0001)to be low for these scenarios The side towards which theopinion cluster skews depends essentially on the valence ofthe first argument that is collectively accepted

In Figures 7(e) and 7(b) agents that update their opin-ions do so at the same time and only when deliberationis successful In these scenarios every time a decision iscollectively accepted agents update their opinions towards

16 Complexity

Table 3 Ordinary least squares estimates for pair-wise interactions scenarios for most of the observed metrics

Dependent variable119881119886119903(119900) 119878ℎ 119890119888 119894119903(1) (2) (3) (4)Constant 0610lowastlowastlowast 4429lowastlowastlowast 0472lowastlowastlowast 0000

(0002) (0206) (0003) (0000)

119880 = 06 (119880 = 02) minus0012lowastlowastlowast 2981lowastlowastlowast 0061lowastlowastlowast 0000(0002) (0260) (0004) (0000)

119879 = 18 (ref 119879 = 14) minus0264lowastlowastlowast minus3261lowastlowastlowast 0004 0000(0002) (0260) (0004) (0000)

120572 = 066 (ref 120572 = 05) 0002 minus0059 minus0100lowastlowastlowast 0000(0002) (0184) (0003) (0000)

(119880 119879) = (06 18) (ref (119880 119879) = (02 14)) minus0324lowastlowastlowast minus3728lowastlowastlowast minus0033lowastlowastlowast 0000(0003) (0368) (0006) (0000)

Observations 1440 1440 1440 1440R2 0983 0389 0529Adjusted R2 0983 0388 0527Residual Std Error (df = 1435) 0032 3492 0053 0000F Statistic (df = 4 1435) 21 272740lowastlowastlowast 228686lowastlowastlowast 402326lowastlowastlowastNote lowastplt01 lowastlowastplt005 lowastlowastlowastplt001

the argumentrsquos defended position Hence if there is votingand if agents happen to obtain 119894119899 proposal arguments theyinexorably cluster towards one or the other side of the opiniondistribution resulting in opinion convergence remeniscent ofgroup polarization (see Figure 7(e)) Furthermore as agentscluster in the same side of the opinion spectrum the argu-ments that may be advanced in deliberation are fewer andmore skewed towards that one side of the opinion spectrumAgents become more easily persuaded in deliberation andonly by arguments supporting one side of the spectrumConvergence of opinions towards one loose opinion focusbecomes faster and more certain If there is no vote (seeFigure 7(b)) the dynamics are similar but for the fact thatit is the uniform randomness of the proposal argument thatguarantees a central convergence of opinions

413 Quantitative Analysis of Pair-Wise Opinion DynamicsThe three parameters that allow for quantitative analysis are119880 and119879 the agentsrsquo attitude structure and themajority quotavoting rule 120572 Table 3 supports the claim that going from 120572 =05 to 120572 = 066 does not affect the variance nor the shifts ofopinions However judgment accuracy is significantly higherwhen accepting proposals becomes more difficult This israther surprising but also intuitive since decisions that aretaken using more restrictive methods of scrutiny should bemore accurate

Concerning the attitude structures the interaction termbetween 119880 and 119879 shows that 119880 has a small effect on thevariance of opinions given that 119879 is low This shows thateven when the assimilation threshold is high a sufficientlysmall rejection threshold can make its one-way effect vanishHowever high levels of both 119880 and 119879 reduce the variance ofopinions dramatically In contrast when it comes to overallchanges in opinion the magnitude of 119880rsquos one-way marginal

effect increases the shifts of opinions whereas 119879 lowers itThis can be interpreted as having the rejection dynamics pushmoderate-to-extreme agents to positions that are close tothe extremes and very moderate agents to the center of thedistribution Not too many quantitative changes (on average)can therefore be observedThe fact that higher119880 is associatedwith faster convergence of opinions seems to imply lowerjudgment accuracy At some point voting for the same type ofproposals for too longmay result inmany collectivemistakesThis is an interesting conclusion to consider in the analysis ofthe mixed model Finally the interaction term shows that 119879has no strong impact on the metric

414 Quantitative Analysis of Deliberative OpinionDynamicsSeven parameters can be considered for quantitative anal-ysis in this dynamics the parameters of interest recalledin Section 312 the size of the argument sacks (119896) agentbehavior during deliberation the strength of the deliberationdynamics (120574) and the measure sensitivity to deliberatedresults (119901119886) The most ldquounexpectedrdquo result obtained fromthe deliberation dynamics is that it shows how deliberationcontributes to the proportion of extremists and the varianceof opinions It happens that having more agents discussproposals and having more of these discussions increasesthe variance of opinions and the proportion of extremistswhile lowering judgment inaccuracy and coherence For theformer the fact that far too many arguments are advancedduring deliberation plummets the chances that the centralargument is accepted hence deliberation does not moveopinions too often For the latter we observe that only ldquobigrdquodifferences (going from the lowest value of a parameter to thehighest) in protocol actually prove to have a preponderantimpact on the metrics These observations hint at a possibletrade-off between judgment accuracy and the variance of

Complexity 17

Table 4 Ordinary least squares estimates for the deliberative interactions scenarios for most of the observed metrics

Dependent variable119881119886119903(119900) 119878ℎ 119890119888 119894119903(1) (2) (3) (4)Constant 0032lowastlowastlowast 13092lowastlowastlowast 0370lowastlowastlowast 1007lowastlowastlowast

(0001) (0406) (0001) (0001)

119898 = 3 (ref119898 = 1) 0012lowastlowastlowast minus2062lowastlowastlowast minus0012lowastlowastlowast minus0001(0001) (0276) (0001) (0001)

119898 = 6 (ref119898 = 1) 0023lowastlowastlowast minus3285lowastlowastlowast minus0018lowastlowastlowast minus0004lowastlowastlowast(0001) (0276) (0001) (0001)

119899119863 = 002 (ref 119899119863 = 001) 0007lowastlowastlowast minus0965lowastlowastlowast minus0006lowastlowastlowast minus0002lowastlowast(0001) (0276) (0001) (0001)

119899119863 = 005 (ref 119899119863 = 001) 0020lowastlowastlowast minus3316lowastlowastlowast minus0015lowastlowastlowast minus0004lowastlowastlowast(0001) (0276) (0001) (0001)

120572 = 05 (ref 120572 = 0) 0146lowastlowastlowast 68932lowastlowastlowast 0068lowastlowastlowast minus0507lowastlowastlowast(0001) (0276) (0001) (0001)

120572 = 066 (ref 120572 = 0) 0291lowastlowastlowast minus13842lowastlowastlowast 00004 minus0000(0001) (0276) (0001) (0001)

119896 = 8 (ref 119896 = 4) 0004lowastlowastlowast minus0815lowastlowastlowast minus0002lowastlowastlowast minus0001(0001) (0276) (0001) (0001)

119896 = 16 (ref 119896 = 4) 0008lowastlowastlowast minus1118lowastlowastlowast minus0004lowastlowastlowast minus0003lowastlowastlowast(0001) (0276) (0001) (0001)

119861119890ℎ119886V119894119900119903 = ldquoFocusedrdquo (ref ldquoNaiverdquo) 0015lowastlowastlowast minus1980lowastlowastlowast minus0012lowastlowastlowast minus0004lowastlowastlowast(00005) (0225) (00004) (0001)

120574 = 01 (ref 120574 = 005) minus0020lowastlowastlowast 5762lowastlowastlowast minus00003 minus0001(0001) (0276) (0001) (0001)

120574 = 02 (ref 120574 = 005) minus0025lowastlowastlowast 8648lowastlowastlowast minus00003 minus0001(0001) (0276) (0001) (0001)

119901119886 = 05 (ref 119901119886 = 03) minus0013lowastlowastlowast 1581lowastlowastlowast 00004 00003(00005) (0225) (00004) (0001)

Observations 19440 19440 19440 19440R2 0931 0844 0562 0958Adjusted R2 0931 0844 0562 0958Residual Std Error (df = 19427) 0033 15701 0030 0050F Statistic (df = 12 19427) 21 844790lowastlowastlowast 8 733563lowastlowastlowast 2 079585lowastlowastlowast 36 761250lowastlowastlowastNote lowastplt01 lowastlowastplt005 lowastlowastlowastplt001

opinions that may be of interest in the mixed discussionmodel

Group coherence shifts of opinions and the varianceof opinions are highly affected by the voting thresholdthe coherence metric is reduced by half when allowing forclassic majority voting shifts explode and the variance ofopinions increases dramatically This is not surprising sincethe majority quota rule works as a buffer to the success ofdecision-making processes and in consequence it hampersthe deliberative opinion updates For the shifts of opinions itis clear that the two-third majority voting rule results in littleopinion change and that the simple majority rule inducesgroup polarization As expected there are less extremists inscenarios where there is no voting and the shifts of opinionare more likely to happen when agents are more sensitiveto deliberation When we include voting in the system

the grouprsquos judgment accuracy decreases due to the correctdeliberated decisions that are vetoed by a salient opinion-agent majority Strangely the effect only holds for the simplemajority decision rule Lastly the proportion of individualsthat participate in the debate and the minimal amount ofdebates have similar marginal effects on all the metrics (seeTable 4)

415 Comparing Deliberative and Pair-Wise Opinion Dynam-ics in a Nuthshell Qualitatively both models yield similaroutcomes but for the fact that deliberation never producesopinion polarization with both types of extremists (seeFigure 7) Furthermore clusters in the deliberative interac-tions model are less homogeneous in size and multiclusterconvergence occurs only when deliberation is rarely suc-cessful which happens sporadically given the calibration of

18 Complexity

Table 5 Mean difference 095 confidence intervals for metrics by model comparison

Model Metric119881119886119903(119900) 119901119903119900119901119890119909 119878ℎ 119890119888 119894119903Dyadic vs Deliberative [0191 0216] [0170 0197] [-2948 -2828] [0074 0082] [0165 0172]Mixed vs Deliberative [0142 0147] [0150 0155] [-2085 -1976] [0003 0002] [-0038 -0030]Mixed vs Dyadic [-0072 -0046] [-0045 -0018] [8332 8859] [-0079 -0071] [-0205 -0201]the model On the other hand pair-wise opinion dynamicsproduces multicluster convergence most of the time andcentral convergence only in the infrequent cases where thelatitude of acceptance is large

The comparison of these two scenarios in terms ofthe metrics is straightforward Whilst dyadic interactionsallow for more variability in opinions through pair-wiseinteractions and more mistakes in judging proposals delib-erative updates simply act as an antagonist force On aver-age the deliberative interaction model produces opiniondistributions with smaller variance and yields steady stateswith higher judgment accuracy a higher shift statistic andnaturally lower group coherence since only proposals that aredeliberated 119894119899 are eligible for voting (see Table 5)

42 Articulating Dyadic and Deliberative Social Interactionsin Opinion Dynamics So far we have considered scenarioswhere argumentation had no place and where individualsmade decisions according to opinions that were purelyconstructed from pair-wise discussions We also observedcases in which agents formed their opinions only by inte-grating deliberated proposals into their opinion updates Thescenarios we present in this section combine the two pre-ceding scenarios to account for both the effects of pair-wisediscussion and the effects of deliberation on opinions judg-ment accuracy and group coherence From the precedingobservations we conclude the following deliberative opiniondynamics reduces the variance and the proportion of extrem-ists because it shifts opinions greatly and polarizes groupsby eliminating at least one type of extremist the one whoseopinions are initially opposed to the deliberated resultsor it brings opinions close to neutrality (see Figure 7(b))This may not be desired in a group since the diversity andstability of opinions are necessary for deliberation to makesense and lead to the collective acceptance of controversialproposals Dyadic opinion dynamics on the other handtend to increase the variance of opinions and the numberof extremists while having a lesser aggregated effect on theshifts of opinion This is an immediate result of the pair-wiseinteraction dynamics inasmuch as it polarizes individualsvery quickly

To sum up in the mixed interactions model deliberationhinders the effect of voting on opinions by making someproposals that would normally be submitted for voting noteligible for voting It may also undo opinion changes dueto pair-wise interactions The mixed model can be seen asthe deliberation opinion dynamics model in which pair-wiseinteractions occur in between collective discussion and mayaffect which arguments are advanced during deliberationEquivalently it can be seen as the dyadic opinion dynamics

model in which deliberation accounts for endogenousldquoshocksrdquo on the opinion distribution

421 Qualitative Analysis of the Mixed Dynamics In Figures7(c) and 7(f) agents update their opinions through delibera-tion and pair-wise interactions Opinion trajectories visualsshow a combination of both previously described opiniontrajectories Over time as agents discuss with one anotheropinion clusters form as in the pair-wise or dyadic opiniondynamics Deliberation may disrupt the formation of theseclusters and three different situations may occur

(i) Deliberation disrupts the formation of the opinionclusters but not sufficiently to jeopardize pair-wiseopinion cluster formation (eg for high values 119905119863)Two phenomena may be responsible for it (1) thevariability of the proposal argumentrsquos support for theprinciple P and (2) too few (successive) deliberationsteps that result in opinion updates Once the opinionclusters are formed successful deliberation foreshad-ows the merging of the previously formed clustersmay bring extremists closer to moderate agents andthanks to the assimilation component of the pair-wisedynamics allow bigger and more moderate opinionclusters to form (refer to Figure 7(c))

(ii) Deliberation significantly disrupts the pair-wisedynamic formation of clusters This may happenwhen the requirements for accepting proposals areweak in the decision-making process (eg low 120572) Inthis case one has either central convergence sinceall agents follow the deliberated results in the samemanner or 3-cluster convergence with two extremeopinion foci In the latter situation the extremegroups form quickly (like in the pair-wise model) anddeliberation cannot pull extreme agents away fromthe extreme foci they are in The main reason for thisis that immediately after an extreme agent attempts toleave her foci through deliberation updates she getspulled or pushed back into it by means of a dyadicinteraction with another extreme agent

(iii) Deliberation does not disrupt the formation of theopinions cluster This is the rare case scenario forvery demanding decision-making processes in termsof inclusion (eg 119899119863) and argument acceptability (eg120572) Opinions converge as in the pair-wise opiniondynamics model (refer to Figures 7(a) and 7(d))

Last when there is voting (120572 = 0) the position ofthe moderate opinion cluster is determined by the first

Complexity 19

minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

Var(o)

sh

ec

ir

Var(o)

propex

sh

ec

ir

1

098

minus02

minus004

001

1

minus014

minus001

minus003

1

026

minus028

1

minus057 1

ＪＬＩＪ

Ｒ

Figure 8 Significant correlation estimates for themetrics of interestin the mixed opinion dyamics scenarios

accepted proposal argument andor the opinionmajority thatis formed from previous dyadic interactions (see Figure 7(f))

422 Relating the Metrics of Interest in Mixed OpinionDynamics The first thing to notice is that for all of thesescenarios all the metrics are significantly correlated (seeFigure 8) The variance of opinions and the proportion ofextremists are almost perfectly correlated To some extentthis means that a high variance of opinions known thatthe mean of opinions is statistically null equates to havingmany extremists in the population be towards one or theother side of the opinion distribution (eg extreme bipolar-ization) Similarly we find a negative and strong correlationbetween the judgment inaccuracymeasure and the coherencecoefficient (see Figure 8)mdashwhen judgment inaccuracy is high(low) coherence to deliberation is low (high) Cases likethese are telling since they may be related to sequences ofcollective decisions in which agents do not make too manymistakes when judging proposal arguments and still makedeliberated decisions that are on average in line with theirprinciples

The shift statistic interestingly is negatively correlatedwith the variance of opinions with judgment accuracy andwith the coherence coefficient In a way this means thathaving more moderate agents conduces on average to moremistakes in judging arguments and to less coherent collectivedecisions An explanation for this phenomenon could be thatas agents reach consensus in opinion important (possiblyextreme) arguments for correctly judging argument propos-als are not advanced in the deliberation arenawhich results inmore mistakes and in possibly accepting proposal argumentsthat after voting are not meant to be accepted Indeed the

negative correlation between the variance of opinions andjudgment inaccuracy points towards this direction since onaverage the more shifts there are the smaller the variance ofopinions is (see Figure 8)

423 Comparing Mixed Dynamics with Other MonolithicDynamics of Interactions Qualitatively opinion trajectoriesin the mixed model are more chaotic than in the pair-wise and deliberative interaction models (see Figure 7) Asanalyzed previously deliberation dynamics disrupt the pair-wise dynamics and create situations in which central andmulticlustered convergence (with two clusters at the extremesof the distribution) are possible at the expense of more(pair-wise dynamics) or less (deliberative dynamics) clustersIt tolerates the survival of very small isolated group ofagents that the previous models rarely tolerate in a hundredcollective decision-making steps (see Figure 7(c))

One of the most interesting observations one can makeof this family of models regards how groups change (in termsof our metrics) when dyadic social interactions are addedto deliberative interactions and vice-versa When comparingthe mixed interactions with only deliberative ones shiftsagentsrsquo aggregated change in opinions are on average lesspronounced The variance of opinions and the proportionof extremists are significantly higher for the former than forthe latter group coherence in contrast is lower with respectto the scenarios where opinions are formed only throughdeliberation Judgment accuracy follows that same trendA reason for this might be that mixed dynamics are fasterto yield semistable clusters andor opinion convergence Inconsequence the potential argument pool for deliberativeexchange shrinks faster and deliberation becomes less effec-tive over time which results in more collective mistakes Theoutcomeon coherence is predictable inasmuch as deliberativeinteractions produce group polarization If a group agrees inopinion debates are sterile because themembers of the groupwill most likely refrain from attacking the proposal argu-ments they all support and will work together to disparagethe proposal arguments they do not support (see Table 5)

In the same manner we report that the proportion ofextremists and the variance of opinions are barely lower inthe pair-wise interaction model when including deliberationin the decision-making process Shifts of opinions are alsobarely lower when there is no deliberation This can indicatethat deliberation does not undo the changes in opiniondue to pair-wise interactions nor does it amplifies them Incontrast when we consider the labeling-based metrics themixed interaction model ensures lower judgment inaccuracybecause of deliberation and lower group coherence (seeTable 5)

In sum the mixed discussion model is a compromisebetween the pair-wise interaction and deliberative opiniondynamic models in that it offers some guarantees in termsof judgment accuracy and coherence while allowing forldquoreasonablerdquo variance in the opinion pool It establishes somekind of trade-off between labeling-based metrics and thevariance of opinions that may be of interest for decision-making process design

20 Complexity

Table 6 Summary of sign or relative marginal effects for all varying parameters on dyadic deliberative and mixed interactions models inand for all experiments

Metrics Parameters119905119863 119898 119899119863 120572 119879 119880 119901119886 120574 119896 119861119890ℎ119886V119894119900119903119881119886119903(119900) XX X X O X X X X X X119901119903119900119901119890119909 XX X X O X X X X X X119878ℎ XX X X O X X X X X119890119888 XX X X XO X XOO XOO X X119894119903 XX X X O XO O X OOX XOO X XNota gray symbols correspond to dyadic dynamics light gray to deliberative dynamics and bold black to mixed dynamics Given a scenario the symbol 119883stands for constant parameter and 119874 for insignificant marginal effect and as parameters grow for positive (negative) marginal effect for the parameterFrom left to right at the parameter level the leftmost arrow corresponds to the scenario with the strongest (in absolute value) marginal effect of the parameter(with respect to to its factors) Example Let 119901 be a parameter and 119874 stands for ldquo119901 has a significant positive effect in dyadic discussion scenario negativein the deliberative one and insignificant in mixed discussion The positive effect in dyadic discussion is greater than the absolute value of the negative one inthe purely deliberative scenariordquo

424 Comments on the Differences between the Three Modelsfor OLS Regression Estimates The significance of most of theregression parameters estimated for the mixed interactionsscenarios are similar to those estimated for the pure dyadicanddeliberative scenariosNevertheless themagnitude of theeffects is different insomuch that the dynamics themselvesare different So instead of commenting the linear regressionestimates of the mixed model we compare when possiblethe standardized significant coefficients of the regressors ofthe mixed model with those of the pair-wise and deliberativeinteractionmodels (see Table 6 for a summary of the results)

In mixed interactions for instance the influence of119898 onthe opinion distribution is much more preponderant than inthe deliberative dynamics scenarios Going from 119898 = 1 to119898 = 3 increases the standard variation units of the variance ofopinions by twice as much as the one in the deliberative sce-nariosThis in particular is due to the design of the decision-making process If 119898 is higher then the number of socialinfluence steps (119905119863) in the decision process is also higher(compared to 119905119863 = 0 in the deliberative case) If we decideto consider these steps as contributing to the variance ofopinions then the effect of119898 on the variance of opinions hasto be stronger in the mixed interactions model In contrastthe number of agents participating in the debate has a weakereffect on the variance of opinions in the mixed interactionsmodel than in the deliberative dynamics specially for 119899119863 =005 (more than twice as much) The difference may comefrom the fact that in mixed discussions the part of theargument pool used in deliberation is more restrictive thanin the deliberative case We can argue that since pair-wisedynamics may get individuals closer to one another aftersome deliberation the same happens to the arguments thatindividuals will most likely use in deliberation Taking thisinto account means that deliberation will at many instancesbe unfruitful and have no effect on opinions thus failing tomoderate extreme opinions andor polarize the group In thesame direction a high number of participants in deliberationcan make the deliberation dynamics stiff In this case thedistribution of opinion is close to the uniform distributionwhich has a relatively high-to-moderate variancewith respectto the reference results

The strength of the deliberative dynamics has a positivestronger effect on the variance of opinions in themixedmodelthan in the deliberative model when 120574 = 02 and a negativeweaker one when 120574 = 01 A sound interpretation of thisresult is that when deliberation is successful and its effect onopinions does not get agents to move sufficiently away fromtheir current positions agents are pulled or pushed back totheir previously held opinions the variance of opinions doesnot changemuchOn the other hand if the effect of successfuldeliberation is strong opinions can change sufficiently tomake agents much more prone to assimilation into moderateopinion foci the variance of opinions falls

The deliberative opinion dynamics model is much moresensitive to 120572 than the mixed discussion model Differ-ences in marginal effects can double The probability ofattraction to deliberated results (119901119886) for instance has astrongmoderating effect (negative) on agentsrsquo opinions in thedeliberative opinion dynamics but a meager one (two-fold)in the mixed and dyadic dynamic models This is becauseopinion updates in the deliberative model depend only onthat parameter whereas in the mixed model more numeroustypes of interactions make opinion updates possible

In terms of judgment accuracy all parameters except thesize of the argument sacks (119896) have a significant slightlystronger effect on the deliberative dynamics than in themixedmodel In effect the difference may be attributed to theupshot of pair-wise interactions on the distribution of opin-ions and on the arguments that would be advanced duringdeliberation The result respecting parameter 119896 however iscounterintuitive Since argument sacks are static scenarioswhere shifts are higher should make 119896 more important fordeliberation to succeed and it is precisely in the scenarios ofpure deliberative dynamics that we find the biggest opinionshifts This may point to the hypothesis that in deliberativeinteractions agents change opinions substantially but do notoften change their adherence to the principle (go from anegative opinion to a positive one or vice versa) when thereis no group polarization

43 Sensitivity Analysis for the Procedural Parameters ofInterest Weperform a sensitivity analysis of the observationson the parameters of interest in themixed interactionsmodel

Complexity 21

We extend the parameter domains of the pure dyadic anddeliberative scenarios as we make it explicit in Table 2

431 Minimum Number of Debates (119898) We observe that theminimum number of debates has a significant well-observedeffect on all of our metrics excluding group coherence Forthe variance of opinions we can observe that the moredebates there are the bigger the value of the metric isalthough the higher 119899119863 and 119905119863 are the weaker the overalleffect of119898 becomes (Figures 9(d) and 9(a)) Furthermore themarginal effect of increasing minimal debates is decreasingan additional deliberation step requirement increases thevariance of opinions but less and less as it grows 119898 coupledwith 119899119863 has a shy S-shaped effect (see Figure 9(d)) on thevariance of opinions meaning that it has a stronger effectwhen 119899119863 is low which suggests a possible trade-off betweenthe number of arguments accepted in the debate and thenumber of debates required before accepting a proposal

In contrast the shifts in opinion are less and less likelyas 119898 grows and this is independent of the variations of theother parameters Again the effect is marginally decreasingand is only truly significant when 120572 = 05 An explanationto these effects may be linked to the design of the systemFirst the variance of opinions is higher when deliberation isasked for because the more deliberation steps there are thehigher the chances are that the central proposal argumentis deemed unacceptable specially when agents are focusedFurthermoremechanically speaking increasing theminimalamount of debates implies that whenever a decision is to betaken at least 119898 times 119905119863 time steps have to take place and if atany moment a proposal argument is considered undecided119905119863 time steps are added to the process So unless the debatedoes not yield 119906119899119889 labels for proposal arguments (highlyunlikely considering that 120590 = grounded semantics) the morenondeliberation steps there are in the decision process thehigher the variance of opinions is Concerning the shiftswhen 120572 = 05 either the system is too stiff to accept anyproposal argument and opinions do not changemuch or theeffects of pair-wise discussion and deliberation cancel out insuch a way that individual shifts are minimal (Figure 9(b))

On the side of labeling-based metrics the more debatesare asked for the more accurate a group is in its judgmentthe effect being smaller as 119898 grows When agents are naivethe effect is quasilinear while when they are focused thestrongest effects of acquiring more deliberation are foundwhen levels of deliberation are already low (Figure 9(f))Thiscan be explained by the fact that the more debates thereare in the decision process the closer one gets to the idealargumentation framework

432 Proportion of the Population in Deliberation Steps (119899119863)Like for 119898 119899119863 has a significant effect on the proportionof extremists and on the shifts and variance of opinions(Figure 9(e)) This may result from the fact that being ableto put more arguments in play at the same debate step mayincrease the odds of revealing cycles around the proposalargument in119860119865120576 Given that we use grounded semantics thearguments in the cycles are labeled 119906119899119889 and in consequence

debates are more often postponed when 119899119863 is higher Post-poning debates in turn increases the number of nondebatessteps in the decision process which increases the varianceof opinions and limits over time the moderating effect ofdeliberation Moreover the effect of this parameter is verydependent on the value of 120572 (Figure 9(e)) For shifts forinstance 120572 = 05 makes the effect of 119899119863 negative while120572 = 0 makes it positive but to a lesser extent Furthermorefor higher requirements of deliberation (119898) adding moreindividuals to the deliberation process has weaker effectson the variance of opinions and on the other metrics It isalso quite interesting to notice that it has the same effect onagents independently of agent behavior during deliberationThis is a surprising result as one would have expected morefocused agents in a deliberation arena to cause a raise inthe proportion of extremists as they play to knock outopposing proposal arguments and thereof reducing the effectof deliberation (lowers variance) on the whole population

Finally similar to119898 addingmore people into the deliber-ation process increases judgment accuracy (see Figure 9(c))and has no clear effect if any on group coherence (seeFigure 9(i))

433 Steps between Deliberation Steps (119905119863) In all configu-rations 119905119863 increases the proportion of extremists (variance)and decreases the shifts of opinions among agents Theshifts and the effect on the variance of opinions are presentfor 120572 = 05 insignificant otherwise (Figure 9(b)) 119905119863 ishighly linked to 119898 by construction When 119898 = 1 thecurve linking the variance of opinions and 119905119863 is convex As119898 increases the curve becomes more and more concavewhich means that 119905119863 has a more important effect on theopinion distribution as collective decisions take longer to beachieved This seems counterintuitive but in reality it reflectsthemultiplicative relationship between deliberation and pair-wise interactions and the semantics in the model If 119898 is lowand 119905119863 is high the effective number of pair-wise interactionsis on average smaller in the deliberation process Hencethe structure of the decision-making process hinders anyincrease in the variance of opinions If119898 is high the oppositeeffect is observed Additionally since the grounded semanticsyields few 119894119899 arguments with respect to other admissibility-based semantics getting closer to the ideal argumentationframeworkmaymake it difficult to obtain the expected effectsof deliberative discussion on agents Lower 119898 constrains thevariance of opinions by making deliberation more influentialon opinions

In terms of judgment accuracy and coherence we observethat 119905119863 does not explain coherence yet it tends to decreasejudgment inaccuracy This may be due to a more variedargument pool in the debate which in turn may be aconsequence of a higher variance of opinions or vice versa

434 Acceptability Voting Quota (120572) By far it is the mostinfluential parameter in our study It changes the directionand the intensity of the effect of all the other proceduralparameters and by construction heavily constrains the roadto accepting a proposal (see Figure 9)

22 Complexity

015

020

025

030

035

040

1 2 3 4 5 6

12345

Values of Ｇ

(a) 119905119863 times 119901119903119900119901119890119909 by119898

0

10

20

30

40

50

1 2 3 4 5 6

005066

Values of

(b) 119905119863 times 119878ℎ by 120572

040

045

050

055

060

1 2 3 4 5

001002003004005

Values of Ｈ＄

(c) 119898 times 119890119888 by 119899119863

015

020

025

030

035

040

1 2 3 4 5

001002003004005

Values of Ｈ＄

(d) 119898 times 119901119903119900119901119890119909 by 119899119863

0

10

20

30

40

50

001 002 003 004 005

00005066

Values of

(e) 119899119863 times 119878ℎ by 120572

040

045

050

055

060

1 2 3 4 5

BehaviorNaiveFocused

(f) 119898 times 119890119888 by 119887119890ℎ119886V119894119900119903

015

020

025

030

035

040

0 05 066Behavior

NaiveFocused

(g) 120572 times 119901119903119900119901119890119909 by 119887119890ℎ119886V119894119900119903

05

06

07

08

09

10

1 2 3 4 5 6

00005066

Values of

(h) 119905119863 times 119894119903 by 120572

08700

08725

08750

08775

08800

001 002 003 004 005

BehaviorNaiveFocused

(i) 119899119863 times 119894119903 by 119887119890ℎ119886V119894119900119903

Figure 9 Curves of mean observations for different parameters and metrics for 36000 runs Bars are 095 confidence intervals Plots withthe same observables (ordinate) except for 119894119903 are on the same scale

Complexity 23

The higher the requirement for accepting the proposal isthe higher the proportion of extremists and the variance ofopinions are and the lower the shift statistic is In few words120572 constrains the world to dyadic discussion or throws it intoa process in which deliberation is much more importantthan dyadic interactions This is why for any 120572 = 05 oneeither gives too much weight to deliberated results whichshadows pair-wise interactions or too little weight in thesense that no deliberated result is ever accepted and thusnever integrated into the opinions of the agents that havevoted for it The reason for the latter is that the latitudeacceptance is too low and thereafter pair-wise interactionsare not enough to unevenly polarize the population in such away that deliberated proposal arguments are voted favorablyby a non-negligeable majority Shifts for 120572 = 0 are stable fordifferent levels of 119905119863 which implies that deliberation keeps incheck all the shifts of opinions related to an increase in thenumber of pair-wise interactions For 120572 = 05 deliberationis less likely to be successful and shifts tend towards the shiftlevels for 120572 = 0 (see Figure 9(b))

Concerning labeling-basedmetrics120572 entirely determinesthe coherence statistic For 120572 = 05 there is no difference incoherence because of how coherence is defined it is maximalwhen 120572 = 0 and strangely maximal when 120572 = 066(see Figure 9(h)) Surprisingly 120572 has no effect on judgmentaccuracy One would have expected a higher 120572 to increasejudgment accuracy since agents would hardly collectivelyaccept a proposal argument discussed during deliberationProposal arguments have given how the argument lattice isgenerated a higher probability of being rejected than of beingaccepted in the ideal argumentation framework

435 A Word on Focused and Naive Agents Focused andnaive specifications for agent behavior are an importantparameter in the decision process since they model howagents choose arguments in the deliberation arena In the sce-narios where agents are focused agents show equal or highervariance of opinions and are therefore more extreme (seeFigure 9(g)) independently of the voting requirements Shiftsof opinion are less likely as well We can see this happeningbecause focused agents knock out proposal arguments moreoften than naive agents provided that 119899119863 is low

One could have expected naive agents to yield morecoherent decisions since they argue sincerely However thisseems not to be the case (see Figure 9(i)) Agents are unable todeliberate in away thatmakes deliberation reflect their votingintentions either because they do not have the necessaryarguments to do so or because such arguments do not exist

In terms of judgment accuracy focused agents do betterthan naive agents in finding the ideal labels for the proposalarguments This is likely because when reconstructing theframework focused agents take into account the deliberatedproposal argument and choose themost pertinent argumentsin their sacks that relate to the state of affairs at the deliber-ation arena In fine debates result in a better approximationofL119886119887120576(119868) even though one could have believed the oppositethat is focused agents are ready to sacrifice a correct collectivelabel of a proposal argument in the goal of defending theirpositions

44 Points of Discussion In this subsection we discuss theresults obtained from the simulations We attempt to makesense of them from a social sciences perspective

441 Opinion Consensus and Dissensus The deliberativeopinion dynamics corresponds to the simulated scenarios inwhich the variance of opinions is at its lowest and judgmentaccuracy at its highest Dyadic opinion dynamics are associ-ated with the scenarios in which the variance of opinions is atits highest judgment accuracy at its lowest and coherence atitsmaximumpossible levelThemixed interactions dynamicsis a combination of these aspects for all the metrics exceptfor coherence at which it is the worsemdashdeliberation leadsto some kind of loose opinion consensus that is reinforcedby pair-wise interactions Likewise the mixed discussionmodel confirms some results exposed in [16] expressingthat opinions and voting intentions in a deliberative contextldquooften changerdquo That being said and more particularly indeliberative opinion dynamics outputs align well with twointeresting results on consensus formation given by aninterpretation of Moscovici and Doisersquos [15] work Sunsteinrsquos[14] account on deliberation and group polarization andsocial impact theory [4] The first states that mild consensusis reached in deliberation as procedure disengages agentsthe second is that within discussion groups majorities tendto become larger and opinions more extrememdashnot on thebasis of competing arguments but of the initial balance ofopinions After several decision processes we find opiniondistributions that reflect these observations even though weare not able to precisely coin the set of parameters that yieldssuch distributions (see Figure 7)

Although differences between mixed and dyadic opiniondynamics in terms of the variance of opinions and shiftsare slim the former does much better in terms of judg-ment accuracy In this sense deliberation does moderateopinions but not as much when non-deliberation steps arerelatively frequent throughout the decision-making processIn terms of Moscovici and Doise theory of consensus [15]this makes sense since pair-wise discussion engages or ego-involve agents in the discussion and pushes to convergenceof opinion at the extremes of the opinion distributionHowever what remains unclear is the reason why proceduralparameters that constrain deliberation do not disengageagents from discussion and thus result in a mild consensusas also expressed by Moscovici and Doise [15] A possibleexplanation of this phenomenon may stem from the factthat deliberation in the model is always accompanied by aseries of ego-involving discussions that shadowor counter themoderating effect of deliberation Not only this deliberationcould also be responsible for crystallizing the opinions ofextremists for it favors one or the other part of the distributionof opinions as explained in [14] and to some extent in [13]Furthermore the mixed discussion model gives an exampleof polarization that results from a minority of dissenterson one side of an issue acceding to views of the majorityrsquosside [4] as opposed to the strong polarization observed indyadic interactions Situations in which the status quo orthe undecidability of the proposal argument lingers cannotpossibly yield situations of moderation In cognitive science

24 Complexity

one can be inclined to believe that the longer a deliberationprocess is the more likely individuals will take an immutablestance on the situation In view of the assumption that longdecision-making processes can be more costly for a group orsociety as a whole agents may feel forced to take a stancejust for the sake of ending the process The stance that isadopted would therefore depend on pair-wise interactionsamong individuals and not on deliberation since deliberationtakes too long to be conclusive

442 For Hypothetical Recommendations in Decision-MakingProcess Elaboration On another tune results from themodelsuggest that procedural parameters have marginally decreas-ing positive or negative effects on the metrics of interest Ina hypothetical policy analysis perspective this may indicatethat after choosing a correct number of individuals andimposing the right amount of debates one can attain rea-sonable levels of judgment accuracy coherence and varianceof opinions As accounted in [18] deliberation increasesknowledge (collective knowledge as well) and therefore itheightens the chances of making correct decisions If onebelieves that social welfare is linked to deliberation andeventually to the metrics studied previously then proceduralparameters can both be used as an instrument to attaindesired levels of social welfare and reinforce the notionof legitimacy of collective decisions It can also provide ajustification for choosing a deliberation regime rather thananother on the basis of how important (or urgent) a topic ofdiscussion is Indeed if we adopt the claim that deliberationis a forerunner of welfare then any state in which deliberativecues are at their maximum (highest number of participantsmany deliberation steps before making a decision and asfrequent as possible) has to be mapped to the highest attain-able social welfare In this case is a situation where judgmentaccuracy is at its highest yet the variance of opinions or theproportion of extremists at its lowest an ideal situation Wecannot say for sure since strong opinion consensus and longdecision-making processes in many situations may result ina loss of welfare rather than in a gain of it

Another interesting result in procedure for deliberationis the very similar estimates we find for the effects of theminimum number (119898) of debates necessary to make adecision on a proposal and the proportion (119899119863) of individualsparticipating in deliberation on judgment accuracy In thescope of social welfare this raises the question of whetheradvocating for longer decision processes by demandingmoredeliberation steps (higher 119898) rather than bigger debates(higher 119899119863) may increase welfare Depending on the objec-tives of the decider of the deliberation procedure he or shemay choose to concentrate on one or the other A recom-mendation we can assert from the mixed social interactionsmodel is that if one prefers low to high variance of opinionsandor high to low coherence the decider is better off ifshe focuses on increasing the size of the debate rather thanorganizing debates very frequently One of the reasons forthis might be that when making bigger debates all agents aretruly considered as equals in the decision-making processIn organizing many debates individuals that are chosen toparticipate in many of these will be more influential and will

somehow bias the set of arguments in the deliberation arenaThemajority rulemodeled through120572 also determines the sizeof the shifts and the distribution of opinions Because a bigger120572 is associated with higher levels of extremism and opinionvariance deliberated cues are harder to accept and thereforeit becomesmore difficult for agents to reach concensus andormoderate their opinions

On the other hand and further into the idea of legitimacythe goal of deliberative processes is the agreement amongall participants This agreement stands for the right or bestdecision from a formal or procedural perspective and it is alsothe best from a substantive point of view In other words thelegitimacy of decisions not only derives fromwell-establishedprocedural requirements (respect of the protocol) but alsobegs for the fulfillment of two essential conditions (1) satisfy-ing the procedural requirements for a correct procedure (for-mal legitimacy) and (2) the rational acceptability of the resultsof this procedure (substantive acceptability) Legitimacy canbe then obtained from the coherence statistic scrutinizedin the model insofar as it measures how well agents acceptthe results of a correct procedure (eg deliberation) So fora proposal to be legitimate one has to choose parametersthat would maximize group coherence in decision-makingprocesses

As deliberation may give rise to consensus in the delib-eration arena it may also sow the seeds of dissensus inthe group Our model clearly illustrates this phenomenonthrough the procedural parameters of deliberation and themetrics observed From another perspective in deliberativedemocracy dissensus in collective decision-making may bedesirable According to Landermore and Pagersquos accounts [36]lack of consensus can be perceived as having agents withalternative ideas that combined with other ideas can providea better approximation of the ideal status of an argumentTheauthors in [36] call this dissensus ldquopositive dissensusrdquo andit makes normative sense when solving complex collectivedecision-making problems Our model captures the ideaof having normative requirements to define a good andlegitimate collective decision requirements of high consen-sual deliberative accuracy (judgment accuracy) of legitimacy(coherence) and of diversity (variance of opinions) are alltaken into account and for such we can conclude that ourmodel is successful in describing a sound process of collectivedecision-making with interpretable outcomes

Of course the present simulation results and analysis donot prove that all opinion dynamics can be accounted forby such simple processes like the one presented here Theydo however shed light on the expressiveness of combiningdifferent paradigms to account formore interpretablemodelsand pave the way to creating original models of the sortThe simulations suggest the desirability of discovering theconsequences of relatively simple laws of communication atdifferent levels (micro meso and macro) to determine whatstill needs to be explained

5 Related Work

We see our model as a contribution to the influence andopinion dynamics field in ABM and a pragmatic application

Complexity 25

of abstract argumentation theory To our knowledge we areunaware of existing literature on agent-based modeling thatexplicitly relates collective choice and the notions of deliber-ation and opinion diffusion through abstract argumentationas we have done it

Most models in the literature of opinion diffusion areinterested in opinions because these have an influence incollective decision and in questions of social order Forinstance in [3] the authors are interested in consensus andin how a group collectively chooses among two alternativesIn other models authors are interested in the emergenceof extremism [8] and on the distribution of opinions whenextremists are introduced in the population [5 33 37] whileother authors coin the notion of opinion polarization as anemergent property of the system [8 12] They show usingmodels of ldquobounded confidencerdquo and opinion diffusion withtrust that three different kinds of steady states (unipolarbipolar and central) are possible depending on whetheragents are sufficiently uncertain about their opinions andsufficiently connected andor a certain proportion of indi-viduals are already extreme Recent articles on informationand opinion dynamics stress the importance of governanceand government intervention in the spread of emotions(opinions) in the frangibility of social consensus systemIn [10] the authors show that government intervention inthe spread of negative emotions can lead to an even fasterspread of negative emotions (single-peeked convergence) andin a faster collapse of the social consensus system Opinionas a function of trust is studied in [12] and relates to howindividuals form their opinions on the basis of how muchthey trust agents in their networks

Another stream of discrete opinion dynamics modelsthat explain the emergence of opinion can be found in theliterature Computer multiagent models of attitude changebased on Latanersquos social impact theory are presented in [4] Inthe theory an agent changes her opinion on the basis of theinformational impact she is subject to which depends on thepersuasiveness (strength) supportiveness and immediacy(group structure) of the environment and information shereceives The simulations predict two emergent groups ofphenomena the shifting of attitudes towards incompletelypolarized equilibria and the formation of coherent clusteringof subgroups with deviant attitudes Similarly but in acontinuous more argumentative fashion Mas and Flache[11] present a model of opinion diffusion using argumentsArguments are considered to be for or against a proposal andare given an agent-dependant relevance in dyadic persuasiveinteractions Arguments also determine the agentsrsquo opinionsand their dissimilarities which posits the rules of pair-wiseencounters in theirmodel as agents are assumed to form theiropinions according to the arguments they own They showthat in an argumentative opinion diffusion model the onlyobservable steady states are bipolarization and consensusand that bipolarization can emerge from interactions amongsimilar agents In other words bipolarization of opinionsis possible with homophily and without negative influenceif there is argumentation They subsequently compare theirresults to experimental data In [38] a similar approach toargumentation and opinion formation is taken but for the fact

that they introduce different types of arguments explicitlyapply social judgment theory [22] and use survey data toestablish model-to-real-world comparisons

Closer to opinion formation and abstract argumentationGabbriellini and Torroni [39] are the first to have origi-nally and soundly merged opinion diffusion and abstractargumentation They define an agentrsquos opinion as a functionof the arguments she holds and the attack relation amongthem They devised a focused peer-to-peer dialogue systemof persuasion (NetArg) inspired from Mercier and Sperberrsquosargumentative theory of reasoning [40] which used onlyabstract argumentation to study opinion polarization andopinion dynamics Moreover they considered networks andthe notions of trust and epistemic vigilance to define adynamics for knowledge revision and trust itself whichwe clearly lack in our model They showed that if a con-servative belief operator in argumentation was applied byagents when they reasoned then their dialogue protocoldid not increase polarization among agents That said theyused the model to study the effect of Groenvetterrsquos weaklink theory in the spreading of arguments and were notparticularly interested in notions like judgment accuracy orcoherence nor in tackling questions related to collectivedecision-making procedures Their system is very expressiveand helps position our work in the litterature our work is atthe frontier of pure argumentative opinion diffusion modelsopinion diffusion with arguments and continuous bounded-confidence models of opinion diffusion a la Deffuant

On another note work on collective cognitive conver-gence [41] and opinion sharing [19] shows that consensustowards a certain ldquocorrectrdquo opinion or cognitive state is alwayspossible yet dependent on noise variability and awarenessof agents In [20] the authors show that learning about anexogenous correct state of the world (represented by bits)under confidence was possible but only if the agents werenot too confident In a homogeneous population they showthat the higher the confidence the worse the learningmdashthevery confident agents do not learn the properties of the truestate of the world and disrupt the learning process of the lessconfident ones Collective cognitive convergence can be seenas a result of deliberation in truth-seeking models

When it comes to abstract argumentation theory we takean approach that wires two types of dialogues that are well-studied in the literature persuasion dialogues [42] and delib-eration dialogues [43] Another interesting line of work is theone on mechanism design [44] or the problem of devisingan argumentation protocol where strategic argumentation isnot a liability for success in debates We tackle mechanismdesign in a different way though Instead of consideringstrategy-proofness we are interested in how differences inprotocol can result in epistemologically ldquobetterrdquo collectivechoices and guarantee that opinion distributions are favorablefor deliberation For a survey on persuasion dialogue see[42]

Work on agent-based argumentation usually assumes thatthe semantic relationship between arguments is fixed Inother words if two individuals were to put two argumentsin the public arena such that one logically attacked the otherthen everyone would agree that such attack exists [45 46]

26 Complexity

Other models which do not make this restrictive assumptioncan also be found in the literature [27 47] and derive fromthe class or family of opponent models [27 48] in which twoopposing sides attempt to win a dialogue Ourmodel is in theintersection of these two but the frame combining opiniondiffusion of the kind and argumentation remains new

The idea of mixing interpersonal influence and verticalcommunication however is not original It is described andimplemented in innovation diffusion models such as [5 49]In both vertical communication is modeled as exogenoustransparent informationmdashagents are aware of the existenceof an innovation thus triggering several processes of choiceand stabilization of opinions Also in [50] an Eulerian modelis implemented to show the effect of media and exogenousinformation on opinion distributions With respect to thispoint the originality of our work is in that the informationemitted as vertical communication is endogenous It is issuedfrom a deliberation model that agents shape on the basisof their opinions arguments and behavior In the spirit of[51] where the authors control for the design of verticalcommunication (to whom the vertical communication isaddressed its timing how it influences agents opinions) wecontrol for the process generating the information for ourown set of variables length of debates majority quotas andthe frequency at which discussions take place

6 Conclusion and Perspectives

The main objective of this article was to build a bridgewhere decision-making argumentation and opinion dif-fusion could come together We proposed a model thatcombined abstract argumentation theory and a boundedconfidence opinion diffusion model and showed to whatextent it could explain variance of opinions extremismcoherence in collective decisions and judgment accuracyon arguments given an ideal state of full information Thesecond objective was to show in what way governance pro-viding agents with arenas of discussion and deliberation wasimportant in addressing the success of collective decision-making processes and the quality of their outcomes

The model revealed that allowing for more deliberationtime though in lower frequency and allowing for widerparticipation in deliberation increased the variance of opin-ions and the proportion of extremists in a group Theseobservations are consistent with the combination of resultsfound in [9] and in [13 14] which stress that deliberationmay polarize groups and may have a meager effect on theshifts of opinions and inconsistent with [16] where it isargued that opinions tend to moderate after deliberationIt happens that deliberation alone did moderate opinionyet when integrated into a more complex system in whichindividuals were allowed to interact with one another andnot all that was deliberated was accepted its influence wasshadowed by other more individualistic dynamics

Undeniably the grounded semantics played an importantrole in the weak effect of deliberation since it models askeptical way of reasoning over arguments For agentsaccepting arguments thus updating their opinions accord-ingly happened not too often Nevertheless asking for more

deliberation did increase judgment accuracy as observed in[18] yet in a marginally decreasing fashion We showed thatvoting within the deliberation protocol not only increasedthe proportion of extremists and the variance of opinions ina group but also determined how coherent deliberation andvoting procedures were with each other Lastly we showedthat agents that are focused judged arguments better hadmore stable opinions and constituted a group with higherproportion of extremists than their naive counterparts

In terms of governance the model says that there is per-haps no trade-off between extremismand judgment accuracyInstead it asserts that the higher the variance of opinions thecloser the group gets to the correct decisions This situationmay be interpreted as Landermore and Pagersquos notion of ldquoposi-tive dissensusrdquo (in complex collective decision-making tasks)[36] as opposed to consensus as a normative requirementfor correct collective decision-making If a decider has toorganize deliberation to legitimize a public policy dependingon what kind of world he or she wants different set ofparameters may be chosen to account for different levels oflegitimacy and correctness For instance one may ask for adecision to be legitimate to respect a certain level of coherenceor of judgment accuracy and to be discussed by a sundryof different-opinionated peoples If so deliberation has tobe made more often more participants have to be includedin debates or in deliberation instances and not too manytime steps should be introduced in between two deliberationinstances So the model may capture normative ideas todefine correct and legitimate collective decisions and tomake recommendations on those grounds Requirements ofhigh consensual deliberative accuracy (judgment accuracy)legitimacy (coherence) and diversity (variance of opinions)are all taken into account and for such a reason the model israther successful in describing a sound process of collectivedecision-making with interpretable outcomes

61 Extensions of the Model and Other Ideas Our model canbe extended in many ways Value-based abstract argumen-tation frameworks (VAFs) for example as an extension ofDungrsquos argumentation framework provide a formal descrip-tion of the process of decision-making in which argumentsare given values and audiences (set of agents) ignore attacksbetween arguments on the basis of their preferences over val-ues Social abstract argumentation frameworks (SAFs) alsoprovide an interesting and seemingly convenient formalismfor our model In SAFs agents vote on the acceptability ofarguments and the resulting labeling is a combination of thearguments that are voted for and labeling-based semantics Inour model we apply the same rule yet only for one argumentand given precise agent-based votingmechanisms and votingrules which are eventually parametrized

To consider our model as a specification of a mixed VAFand SAF model may be of great interest for future workThis formalism can provide solid foundations to describeexplain interpret and eventually extend the results foundhere from an argument-based perspective On those groundsfuture work will focus on creating a hybrid argumentationframeworkmdashwith its semanticsmdashthat takes heed of bothvalues or principles and voting

Complexity 27

Concerning other possible extensions of the model anatural reflex would be to argue on the validity and therobustness of the mix between the bounded confidenceopinion diffusion models and deliberation Consequentiallyadding deliberation to different bounded confidence modelsof opinion diffusion such as the Hegelsmann-Krause model[37] or to models of social impact theory like in [4] canbe of great interest for future work We acknowledge that afiner understanding of argumentation frameworks randomlattices and their respective applications for this modelare necessary Testing for different labeling-based semanticsor even trading labeling-based semantics for scoring-basedsemantics (semantics that defines acceptability on scoresgiven to arguments) such as debate semantics and ranking-based semantics (semantics that yield a preorder) may beinteresting to consider in future work

More into modeling deliberation it is not inadequateto believe that agents may also advance proposals duringdeliberation and replace to some extent the proposal forwhich the deliberation is taking place Endogenizing theprocess that produces the type of argument to be deliberatedon may be interesting to explore either by having agentsstrategically replace proposals at certain moments duringthe debate or having the central authority attempt to chooseproposals such that the probability of having them acceptedis high Agents may also be thought to be autoorganizing(independent of a central authority) and concede on the sizeof the instances of deliberation To consider differences insemantics across agents or even endogenizing the semanticson how urgent or important the proposal that is beingdiscussed is can also be an interesting direction to take

Last but not least many nontrivial modifications of ourmodel are possible and most should include better-thoughtdeliberation protocols It may be interesting to design andobserve protocols in which deliberation only affects agentsthat are actually debating for example Trust network effectsmultidimensionality of opinions new processes of argumentexchange or learning among agents are notions to furtherdevelop in order to make the model more realistic and relaxsome unreliable assumptions In sum the model is to berefined and extended with the objective of either studyingconcrete cases of deliberative polling and opinion dynamicsor implementing more intuitive thought experiments

Appendix

We present below some of the pseudocode we used toimplement the model

Algorithms

(a) Algorithm 1 implements the dyadic influence in themodel It describes how two agents interact andupdate their opinions

(b) Algorithm 2 implements the deliberative influence inthe model which describes how agents update theiropinions in sight of the accepted deliberated proposalarguments

(c) Algorithm 3 is used to obtain the grounded labelingof an argumentation framework

(d) Algorithm 4 provides an overview of a run in thesimulations119872 is the number of proposal argumentsto discuss which we set to 100 in the model

Each object in the system is indexed by a119908ℎ119900 number Hencea set of agents is annotated by a set of natural numbers Weuse this fact in the implementation of the algorithm 119874119887119895119890119888119905119894denotes the object of 119908ℎ119900 number 119894 Agentsrsquo opinions (119900) arecalibrated so that 119900 isin [minus1 1]Data Availability

The program and data used to support the findings of thisstudy are available from the corresponding author uponrequest

Disclosure

A preliminary version of this work was presented anddiscussed at the conference Bridging the Gap between FormalArgumentation and Actual Human Reasoning on October 4-5 2018 at BochumGermanyThe authors thank the audiencefor their valuable comments and suggestions

Conflicts of Interest

The authors declare that they have no conflicts of interestregarding the publication of this article

Acknowledgments

The contribution of Gabriella Pigozzi was supported by theDeutsche Forschungsgemeinschaft (DFG) and the CzechScience Foundation (GACR) as part of the joint project FromShared Evidence to Group Attitudes (RO 45486-1)

Endnotes

1 In other words the superiority of the ldquoright answerrdquo orlabeling in our case will appear as such to all

2 All agents hold the same opinion3 By sensitivity we mean how much an agent is ready to

change her opinion on the sole fact that an informationalcue has been debated on

4 The assumption is strong but this trust among agentsmay be rooted from a genuine individual motivation toreach a democratic consensus

5 We adopted a rule of thumbs that says that if an agentis neutral with respect to the principle then she willvote for any proposal that is presented to her duringdeliberation

6 If the argument is not unique she chooses one atrandom

7 If there are more than one candidate the agent will favorthe argument at minimal distance from 119868

28 Complexity

Require Vector of values for parameters (119881 = (120583 119880 119879))Ensure Successful interactions among agents119896 larr997888 0119873119890 larr997888 0

while 119896 le |119873|2 do ⊳Fill119873119890 with |119873|2 agents119896 larr997888 119896 + 1119873119890 larr997888 119877119886119899119889119900119898(|119873| + 1) ⊳Pseudo-random generator Generates integer between 0 and |119873|119873119903 larr997888 0 120575119894119895 larr997888 0 ⊳Variable to store differences in opinionfor all 119894 isin 119873119890 doRandomly pick a number 119895 isin 119873(119873119890 cup 119873119903)119873119903 larr997888 119873119903 cup 119895 120575119894119895 larr997888 |119900119894 minus 119900119895|procedure dyadic-update(119894 119895) ⊳Two agents 119894 and 119895 discuss and update opinions1199001015840 larr997888 119900119894 (120583119895 119880119895 119879119895) larr997888 (120583119894 119880119894 119879119894) larr997888 (120583 119880 119879) ⊳Save agent irsquos opinion before updating and homogenize agents

if 120575119894119895 lt 119880119894 then119900119894 larr997888 120583119894119900119895 + (1 minus 120583119894)119900119894if 120575119894119895 gt 119879119894 then119900119894 larr997888 (1 + 120583119894)119900119894 minus 120583119894119900119895if 120575119894119895 lt 119880119895 then119900119895 larr997888 1205831198951199001015840 + (1 minus 120583119895)119900119895if 120575119894119895 gt 119879119895 then119900119895 larr997888 (1 + 120583119895)119900119895 minus 1205831198951199001015840

Algorithm 1 Dyadic social influence

Require Vector of values for parameters (119881 = (120574 119901119886 119901119903))Ensure Successful interaction between proposal arguments and agents

for all 119894 isin 119873 doprocedure Probability-Update(V119886 119901119886 119901119903) ⊳Update individual probabilities of change due to deliberated cues119909119894 larr997888 119901|V119886minus119900119894 |2119886 119910119894 larr997888 1199012|V119886minus119900119894 |119903119902 larr997888 0if L119886119887(119868) = 119894119899 then

procedure deliberative-update(119894 119886) ⊳Agent i argument 119886120574119894 larr997888 120574 119902 larr997888 119906 sim U(0 1) ⊳ 119906 is a realization of a uniform random variableif 119902 lt 119909119894 then119900119894 larr997888 120574119894119900119894 + (1 minus 120574119894)V119886break

if 119902 lt 119910119894(1 minus 119909119894) then119900119894 larr997888 (1 + 120574119894)119900119894 minus 120574119894V119886Algorithm 2 Deliberation influence

8 In terms of computational complexity naive agentsmaketheirmoves in119874(|A119894| log(|A119894|)) while focused agents doat least as bad

9 Tables have memory Therefore debates are not neces-sarily trees

10 If there are less than (119899119863 times |119873|)2 agents for any of bothgroups the CA summons all of them to the deliberationtable

11 Or careful consideration of the viewpoints of otherswhich implies that citizens keep an open mind and donot reject arguments outright

12 A ldquohealthyrdquo consensus refers to the case in which manyconsensual decisions are taken but opinion-wise agentsdo not agree with each other Quotes are used on theword healthy because some political science theorists

(eg [36]) believe that in certain situations agreement inopinions for deliberation is epistemologically damaging

13 Simulations on groups of |119873| = 800 and |119873| = 400agents shows that scaling the population has an effect onall themetrics but it is confounding depending onwhichcouples of attitudes structures are considered (119880 119879) Westick to |119873| = 400 as in [9] for comparability andsimulation running time

14 By balanced wemean with as many arguments with V119886 lt0 as with V119886 gt 0

15 Experiments on the system suggest that a bigger(smaller) argument pool 119872 = 1000 (119872 = 200) isassociated with a higher (lower) variance of opinions(and extremism) and higher (lower) judgment accuracyFor the rest of the observations 119872 seem to have noeffect We choose a middle value for 119872 Similarly theargument pool chosen to be balanced or uniformly

Complexity 29

Require Argumentation framework 119860119865 = (AR)Ensure Assignment of labels to arguments from the grounded labeling of 119860119865L(A)

L0(A) larr997888 (0 0 0)119896 larr997888 0do119894119899 L119896+1(A) larr997888 119894119899 L119896(A) cup 119886 | 119886 is not labeled 119894119899 L119896(A) 119886119899119889 forall119887 isin A 119904119905 119887 R 119886 then 119887 isin 119900119906119905 L119896(A)119900119906119905 L119896+1(A) larr997888 119900119906119905 L119896(A) cup 119886 | 119886 is not labeled 119894119899 L119896(A) 119886119899119889 exist119887 isin A 119904119905 119887 R 119886 and 119887 isin 119894119899 L119896+1(A)

while L119896+1(A) = L119896(A) = 0Algorithm 3 Solve argumentation frameworks (with grounded semantics)

Require Vector of values for parameters (119881)Ensure Vector of statistics (119880)

procedure Set-parameters(119881) ⊳Initialize parameters for scenarioprocedure Init-objs(119899119898 119896 ) ⊳Create119898 arguments 119899 agents and 119860119865120576procedure Solve-Argumentation-Framework(119860119865) ⊳Assigns the epistemic labelingL120576 toA119894 larr997888 1119905 larr997888 1while 119894 le 119872 do ⊳ 119872 the number of arguments to discuss119868119894 larr997888 119866119890119899 119886119903119892(1) ⊳Generates proposal argument 119868119860 119894119905 larr997888 (119868119894 0)L119886119887(119868119894) larr997888 119906119899119889 ⊳Initialize label for 119868119895 larr997888 1 ⊳Initialize debate counter 119895while ((L119886119887(119868) = 119906119899119889 or 119898 le 119895) and 119895 le 119898) doif 119905 equiv 0(119898119900119889 119905119863) then

procedure build-argumentation-framework(119899119863 119887119890ℎ119886V119894119900119903) ⊳Build the argument graph agents deliberate(119861119894119905 119878119894119905) larr997888 (119883 119884) sub (⋃119897isin119873119899119863 A119897119894119905 ⋃119897isin119873119899119863 R119897119894119905) ⊳ 119873119899119863 sub 119873 are agents sampled by the CA119860 119894119905 larr997888 119889(119860 119894119905 (119861119894119905 119878119894119905))procedure Solve-Argumentation-Framework(A119889) ⊳FindL(A119889)L119886119887(119868119894) larr997888 L119886119887119863(119868119894)119895 larr997888 119895 + 1

else119896 larr997888 1while 119896 le 119905119863 doprocedure dyadic-social-influence(120583 119879 119880) ⊳Agents discuss one-to-one119896 larr997888 119896 + 1119905 larr997888 119905 + 1

if 120572 gt 0 andL119886119887(119868119894) = 119900119906119905 then ⊳Agents vote for the proposalif (|119897 isin 119873|119900119897 times V119868119894 ge 0| ge 120572 times 119899) then

L119886119887(119868119894) larr997888 119894119899else

L119886119887(119868119894) larr997888 119900119906119905if L119886119887(119868119894) = 119894119899 thenprocedure Deliberation-influence(120574 119901119886 119901119903) ⊳Agents are influenced by the result of the decision process119894 larr997888 119894 + 1119905 larr997888 119905 + 1119880 larr997888 119862119900119898119901119906119905119890 119878119905119886119905119894119904119905119894119888119904(119881)

return 119880 ⊳Get the statisticsAlgorithm 4 Run for one scenario

generated does not seem to play a significant explanatoryrole on any of the metrics

16 Experiments show that themaximumnumber of debatesis not an important parameter even if it is an intuitiveone when considering decision-making processes Inpractice debates end after at most 119898 + 1 deliberation

steps and even then it happens rarely However to avoidunpleasant surprises like infinite debates on a completeargument graph containing the proposal argument weadded the constraint 119898 Its value is precisely 7 becausethe biggest minimum number of debates we studyis 6

30 Complexity

17 To ensure comparability with the deliberative andmixedopinion dynamics models on judgment accuracy andcoherence agents vote on proposal arguments every 119905119863times(119898+1) 119905119863 and119898does not explain variability in judgmentaccuracy nor in coherence

18 Student t-tests for independent samples are robust tothe violation of two hypothesis on the distribution ofthe samples normality and homogeneity of variancesas long as the sample size is big enough (central limittheorem) and if the difference in size of the comparedsamples is smallrespectively

19 Studentrsquos test version forwhen variances are not assumedequal and the difference in size of the compared samplesis big

20 We do not correct for homoskedasticity nor normality oferrors in the estimates

21 Balanced as the same number of runs per scenario orsuch that no significant correlations are found betweenparameters

References

[1] R Axelrod ldquoThe dissemination of culture a model withlocal convergence and global polarizationrdquo Journal of ConflictResolution vol 41 no 2 pp 203ndash226 1997

[2] D M Rousseau Reinforcing the MicroMacro Bridge Organiza-tionalThinking and Pluralistic Vehicles SAGEPublications SageLos Angeles Calif USA 2011

[3] S Galam and SMoscovici ldquoTowards a theory of collective phe-nomena consensus and attitude changes in groupsrdquo EuropeanJournal of Social Psychology vol 21 no 1 pp 49ndash74 1991

[4] A Nowak J Szamrej and B Latane ldquoFrom private attitude topublic opinion a dynamic theory of social impactrdquo Psychologi-cal Review vol 97 no 3 pp 362ndash376 1990

[5] G Deffuant S Huet and F Amblard ldquoAn individual-basedmodel of innovation diffusion mixing social value and individ-ual benefitrdquo American Journal of Sociology vol 110 no 4 pp1041ndash1069 2005

[6] G Deffuant D Neau F Amblard and G Weisbuch ldquoMixingbeliefs among interacting agentsrdquoAdvances in Complex Systems(ACS) vol 3 no 01n04 pp 87ndash98 2000

[7] J Rouchier P Tubaro and C Emery ldquoOpinion transmission inorganizations an agent-based modeling approachrdquo Computa-tional and Mathematical OrganizationTheory vol 20 no 3 pp252ndash277 2014

[8] M Meadows and D Cliff ldquoThe relative disagreement model ofopinion dynamics where do extremists come fromrdquo in Pro-ceedings of the International Workshop on Self-Organizing Sys-tems Lecture Notes in Computer Science pp 66ndash77 Springer2013

[9] W Jager and F Amblard ldquoUniformity bipolarization and plu-riformity captured as generic stylized behavior with an agent-based simulation model of attitude changerdquo Computational ampMathematical Organization Theory vol 10 no 4 pp 295ndash3032005

[10] Y-F Zhang H-Y Duan and Z-L Geng ldquoEvolutionary mech-anism of frangibility in social consensus system based onnegative emotions spreadrdquo Complexity vol 2017 Article ID4037049 8 pages 2017

[11] A Flache andMWMacy ldquoSmall worlds and cultural polariza-tionrdquoThe Journal of Mathematical Sociology vol 35 no 1-3 pp146ndash176 2011

[12] A Tsang and K Larson ldquoOpinion dynamics of skepticalagentsrdquo in Proceedings of the 13th International Conference onAutonomous Agents and Multiagent Systems AAMAS 2014 pp277ndash284 International Foundation forAutonomousAgents andMultiagent Systems France May 2014

[13] K M Hansen Deliberative democracy and opinion formationUniversity Press of Southern Denmark Odense 2004

[14] C R Sunstein ldquoThe law of group polarizationrdquo Journal ofPolitical Philosophy vol 10 no 2 pp 175ndash195 2002 WileyOnline Library

[15] S Moscovici and W Doise Dissensions et Consensus unetheorie generale des decisions collectives Presses Universitairesde France-PUF 1992

[16] J S Fishkin and R C Luskin ldquoExperimenting with a demo-cratic ideal deliberative polling and public opinionrdquo ActaPolitica vol 40 no 3 pp 284ndash298 2005

[17] R C Luskin J S Fishkin and S Iyengar Considered Opinionson US Foreign Policy Face-To-Face versus Online Delibera-tive Polling International Communication Association NewOrleans La USA 2004

[18] J Barabas ldquoHowdeliberation affects policy opinionsrdquoAmericanPolitical Science Review vol 98 no 4 pp 687ndash701 2004

[19] O Pryymak A Rogers and N R Jennings ldquoEfficient opinionsharing in large decentralised teamsrdquo in Proceedings of the 11thInternational Conference on Autonomous Agents andMultiagentSystems vol 1 pp 543ndash550 Spain June 2012

[20] J Rouchier and E Tanimura ldquoWhen overconfident agents slowdown collective learningrdquo Simulation vol 88 no 1 pp 33ndash492012

[21] D Miller ldquoDeliberative democracy and social choicerdquo PoliticalStudies vol 40 S1 pp 54ndash67 1992

[22] M Sherif and C I Hovland Social Judgment Assimilation andContrast Effects in Communication and Attitude Change YaleUniversity Press 1961

[23] P M Dung ldquoOn the acceptability of arguments and its funda-mental role in nonmonotonic reasoning logic programmingand n-person gamesrdquo Artificial Intelligence vol 77 no 2 pp321ndash357 1995

[24] M Caminada and G Pigozzi ldquoOn judgment aggregation inabstract argumentationrdquo Autonomous Agents and Multi-AgentSystems vol 22 no 1 pp 64ndash102 2011

[25] P Baroni M Caminada andM Giacomin ldquoAn introduction toargumentation semanticsrdquoThe Knowledge Engineering Reviewvol 26 no 4 pp 365ndash410 2011

[26] S Chambers ldquoDeliberative democratic theoryrdquo Annual Reviewof Political Science vol 6 no 1 pp 307ndash326 2003

[27] E Bonzon andNMaudet ldquoOn the outcomes of multiparty per-suasionrdquo in Proceedings of the 10th International Conference onAutonomous Agents and Multiagent Systems 2011 AAMAS 2011vol 1 pp 47ndash54 International Foundation for AutonomousAgents and Multiagent Systems Taiwan May 2011

[28] J Ranciere Disagreement Politics and Philosophy University ofMinnesota Press 1999

[29] T Bench-Capon and K Atkinson ldquoAbstract argumentation andvaluesrdquo in Argumentation in Artificial Intelligence pp 45ndash64Springer 2009

[30] M Caminada ldquoOn the issue of reinstatement in argumenta-tionrdquo in Proceedings of the European Workshop on Logics in

Complexity 31

Artificial Intelligence vol 4160 of Lecture Notes in ComputerScience pp 111ndash123 Springer Berlin Heidelberg

[31] GAVreeswijk ldquoAn algorithm to computeminimally groundedand admissible defence sets in argument systemsrdquo in Pro-ceedings of the 2006 conference on Computational Models ofArgument Proceedings of COMMA 2006 vol 144 pp 109ndash120IOS Press Amsterdam The Netherlands 2006

[32] C W Sherif M Kelly H L Rodgers et al ldquoPersonal involve-ment social judgment and actionrdquo Journal of Personality andSocial Psychology vol 27 no 3 pp 311ndash328 1973

[33] G Deffuant F Amblard G Weisbuch and T Faure ldquoHow canextremism prevail A study based on the relative agreementinteraction modelrdquo Journal of Artificial Societies and SocialSimulation vol 5 no 4 p 1 2002

[34] E Awad M W Caminada G Pigozzi M Podlaszewski andI Rahwan ldquoPareto optimality and strategy-proofness in groupargument evaluationrdquo Journal of Logic and Computation vol 27no 8 pp 2581ndash2609 2017

[35] D Grossi and G Pigozzi ldquoJudgment aggregation a primerrdquoSynthesis Lectures on Artificial Intelligence and Machine Learn-ing vol 8 no 2 pp 1ndash151 2014

[36] H Landemore and S E Page ldquoDeliberation and disagreementproblem solving prediction and positive dissensusrdquo PoliticsPhilosophy and Economics vol 14 no 3 pp 229ndash254 2015

[37] R Hegselmann and U Krause ldquoOpinion dynamics andbounded confidence models analysis and simulationrdquo Journalof Artificial Societies and Social Simulation vol 5 no 3 2002

[38] A Stefanelli and R Seidl ldquoModerate and polarized opinionsUsing empirical data for an agent-based simulationrdquo inProceed-ings of the Social Simulation Conference 2014

[39] S Gabbriellini and P Torroni ldquoA new framework for ABMsbased on argumentative reasoningrdquo in Advances in SocialSimulation vol 229 pp 25ndash36 Springer Berlin Germany 2014

[40] H Mercier and D Sperber ldquoWhy do humans reason Argu-ments for an argumentative theoryrdquo Behavioral and BrainSciences vol 34 no 2 pp 57ndash74 2011

[41] H V Parunak T C Belding R Hilscher and S BruecknerldquoModeling and managing collective cognitive convergencerdquoin In Proceedings of the 7th iNternational Joint Conference onAutonomous Agents and Multiagent Systems vol 3 pp 1505ndash1508 International Foundation for Autonomous Agents andMultiagent Systems 2008

[42] H Prakken ldquoFormal systems for persuasion dialoguerdquo TheKnowledge Engineering Review vol 21 no 2 pp 163ndash188 2006

[43] K Atkinson T Bench-Capon and P Mcburney ldquoA dialoguegame protocol for multi-agent argument over proposals foractionrdquoAutonomousAgents andMulti-Agent Systems vol 11 no2 pp 153ndash171 2005

[44] M Thimm and A J Garcıa ldquoClassification and strategicalissues of argumentation games on structured argumentationframeworksrdquo in Proceedings of the 9th International Conferenceon Autonomous Agents and Multiagent Systems vol 1 pp 1247ndash1254 International Foundation for Autonomous Agents andMultiagent Systems Canada 2010

[45] I Rahwan and K Larson ldquoMechanism design for abstractargumentationrdquo in Proceedings of the 7th International JointConference on Autonomous Agents and Multiagent SystemsAAMAS 2008 pp 1031ndash1038 International Foundation forAutonomous Agents and Multiagent Systems Portugal May2008

[46] H Prakken R Riveret A Rotolo and G Sartor ldquoHeuristics inArgumentation A Game-Theoretical Investigationrdquo EuropeanUniversity Institute 2008

[47] N Hadidi Y Dimopoulos and P Moraitis ldquoTactics and con-cessions for argumentation-based negotiationrdquo in Proceedingsof the COMMA vol 245 pp 285ndash296 2012

[48] N Oren and T J Norman ldquoArguing using opponent modelsrdquoin Proceedings of the International Workshop on Argumentationin Multi-Agent Systems pp 160ndash174 Springer 2009

[49] S Delre W Jager T Bijmolt and M Janssen ldquoTargeting andtiming promotional activities An agent-based model for thetakeoff of new productsrdquo Journal of Business Research vol 60no 8 pp 826ndash835 2007

[50] A Mirtabatabaei P Jia and F Bullo ldquoEulerian opinion dynam-ics with bounded confidence and exogenous inputsrdquo SIAMJournal on Applied Dynamical Systems vol 13 no 1 pp 425ndash446 2014

[51] V Barbet N Guiraud V Laperriere and J Rouchier Hagglingon Values Towards Consensus or Trouble 2019

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Submit your manuscripts atwwwhindawicom

14 Complexity

Table 2 Parameters and their domains used to study the effect of the procedural parameters on the metrics

Procedural parameters Other parameters119905119863 119898 119899119863 120572 119898 120583 119879 119880 119901119886 119901119903 120574 119896 119861119890ℎ119886V1198941199001199031 2 3 4 5 6 1 2 3 4 5 001 002 003 004 005 0 05 066 7 01 16 02 03 005 02 12 119873119886119894V119890 119865119900119888119906119904119890119889

323 Calibrating the Mixed Social Interactions Model Themixed interaction model ascribes to a parameter space inwhich the effect of collective choices on our metrics isnontrivial and where deliberation and voting on proposalsdetermine their acceptability Since the mixed model isequivalent to periodic iterations of pair-wise and collectivesocial interactions the calibration of the parameters in bothpreceding models holds (see Table 1 for the calibration)The first reason for this is comparability the second is thatthe previous calibration takes into account the fact thatboth dynamics are going to be combined We include thefrequency of debates by adding the parameter 119905119863 whichcontrols for the amount of pair-wise interaction amongagents between two distinct deliberation steps

324 Termination Conditions for Runs Simulations stoponce 100 proposals are deliberated on andor voted for inother words after 100 collective decision steps have occuredThe number of proposals discussed seems arbitrary yet it ishigh enough to observe the effects of deliberation on opiniondistributions and on other metrics related to coherence andjudgment accuracy We choose the number of runs as afunction of our research questions which give relevance tothe procedural parameters in the mixed model rather than tothe parameters set to describe the population and its behavior

325 Expected Outcomes of the Simulations We expect moredeliberation in terms of higher 119898 and 119899119863 to increasejudgment accuracy and at the same time to reduce thevariance of opinions In turn a smaller variance of opinionsimplies that the argument pool for deliberation is smaller andtherefore judgment accuracy should be lower Moreovervariance increasing dyadic interactions (or rejection) shouldalso increase judgment accuracy and coherence since bipo-larization and dissensus foster argument diversity in delibera-tionThe coherence statistic should be stronger in simulationsin which central convergence appears quickly and agentsare naive Attitude structures sensitivity to informationalcues and weights given to deliberated cues that makes pathsto bipolar convergence smaller or to central convergencelonger should be associated with higher judgment accuracyShifts of opinion should be more visible in scenarios inwhich extreme agents are pulled away from the extremesHence the sensitivity to deliberated cues (119901119886 119901119903) and 120574should explain the shifts When crossed with high values of119880 and 119879 the shifting power of deliberation should be at itshighest

In the end we do not know how these mechanismswill play out The results of the subsequent experimentsgive us insight on the interplay between the aforementionedeffects

4 Results

Before pointing at any result obtained from the mixedinteractions model we describe and compare the dyadicand deliberative interactions models in the parameter spaceobtained from the calibration in Section 3 (see Table 1)Primarily we require our results to describe two orthogonaltypes of opinion dynamics the pair-wise dyadic and thedeliberative The latter comprises scenarios in which indi-viduals do not influence each other by means of pair-wisediscussion and only update their opinions from delibera-tion the former scenarios where only pair-wise interactionsamong agents determine the dynamics We describe both onthe parameter space given in Table 1 by performing at least20 simulations per scenario Quantitative results from thepair-wise interaction model alone are described during thecalibration and are not treated heremdashdeliberative parametershave no effect on it Moreover we show how these differentmodel specifications yield qualitatively different opiniondistributions and compare them on the basis of the metricsof interest To this end we perform independent two-sampleStudentrsquos t-tests18 or Welch t-tests19 and compute confidenceintervals at a 095 level of confidence Last but not least wecomment ordinary least squares (OLS) regression estimatesto account for the direction and magnitude20 of the effectsof the parameters on the metrics Estimates are declaredsignificant at a 5 level of risk

We perform the same analysis on the mixed interactionsmodel as we compare it to the pair-wise dyadic and deliber-ative interactions models and discuss the marginal effects ofthe parameters on the metrics More precisely we obtain twodifferent types of results for the mixed interactions modelthe first compares the mixed scenarios to their monolithiccounterparts with respect to each metric and with respect tothe regression estimates the second gives a clear idea of themarginal effect of each governance parameter (or parameterof interest) on our observations In the first we allow controland procedural parameters to vary on a parameter spacesimilar to those of the dyadic and deliberative opiniondynamics (see Table 1) In the second we fix all of ourcontrol parameters execute 36000 balanced21 runs andfocus only on the one-way pooled effects of our proceduraland behavioral (all agents are focused or naive) parametersWe generate comprehensive graphs and tables to account forthe obtained results

Please notice the slight change in the size of the parameterspace for the procedural parameters and in the values ofthe nonprocedural ones for the first and second type ofresults (refer to Tables 1 and 2 respectively) Since we wantto have a finer idea of the marginal effects of each of theprocedural parameters on how well a group decides onproposals we make the value jumps for each parameter (1)

Complexity 15

() (U T) = (02 18) (b) ( pa m nD) = (005 05 00 2 02) (＝) ０ＬＧＮＬＭ ＣＨ (＜) (＝) Ｈ＞ tD = 3

(d) (５ ４) = (02 14) (e) ( pa m nD) = (01 03 05 3 02) () ０ＬＧＮＬＭ ＣＨ (＞) () Ｈ＞ tD = 6

Figure 7 From left to right common opinion trajectories and distributions for dyadic deliberative and mixed opinion dynamics within thestructure of different decision-making processes

small enough to detect significant differences in the estimatesand (2) meaningful for interpretation For the fixed one-valued parameters in Table 2 we use Table 1 to set them totheir minimum mean or median values

41 Comments on Dyadic and Deliberative Opinions Dynam-ics In this subsection we present our first results regardingthe deliberative opinion dynamics described in Equation (6)and its differences with the dyadic dynamics (Equation (5))in terms of our observations

411 Qualitative Analysis of Pair-Wise Opinion DynamicsPair-wise opinion dynamics alone produces multiclusterstable convergencewith variable cluster sizes (see Figures 7(a)and 7(d)) Insight on the complexity of these dynamics wasdiscussed in Section 321

In Figures 7(a) and 7(d) agents discuss randomly withone another in pairs Clusters in the extremes form quicklysince either agents that are close to the extremes attractone another or are convinced to stay close to the extremesby interacting with other agents they disagree with Otheragents with near-extreme positions may be simply attractedto one of the several moderate foci of agents in the opiniondistribution On the other hand moderate agents are eitherattracted to the closest opinion focus or pushed towards theextremes of the distribution Over time agents in the centralfocus either attract other agents into it or ignore the opinions

of the extreme agents that interact with them The result ofthese dynamics yields amulticlustered convergencewhere theopinion foci are at least at distance 119880 from one another andonly the extreme foci (clusters of agents with |119900119894| ge 075) areat a distance greater than or equal to 119879 from one anotherThefoci at the bounds of the opinion distribution are very stableand in analogy to the (12119889)-rule in assimilation bounded-confidence models [33] the number of clusters that form isroughly equal to 119880minus1412 Qualitative Analysis of Deliberative Opinion DynamicsQualitatively deliberative opinion dynamics yield a looseunipolar convergence of opinions near the center (centralconvergence see Figure 7(b)) near the center of either the leftor right portions of the distribution (group polarization seeFigure 7(e)) or to a sparse bipolar opinion distribution withclusters at the center and at the extremes (see Figure 7(c)) ifmixed with pair-wise interactions This is probably a reasonto expect the variance of opinions and the proportion ofextremists (almost perfectly correlated 120588 = 089 119901 lt 0001)to be low for these scenarios The side towards which theopinion cluster skews depends essentially on the valence ofthe first argument that is collectively accepted

In Figures 7(e) and 7(b) agents that update their opin-ions do so at the same time and only when deliberationis successful In these scenarios every time a decision iscollectively accepted agents update their opinions towards

16 Complexity

Table 3 Ordinary least squares estimates for pair-wise interactions scenarios for most of the observed metrics

Dependent variable119881119886119903(119900) 119878ℎ 119890119888 119894119903(1) (2) (3) (4)Constant 0610lowastlowastlowast 4429lowastlowastlowast 0472lowastlowastlowast 0000

(0002) (0206) (0003) (0000)

119880 = 06 (119880 = 02) minus0012lowastlowastlowast 2981lowastlowastlowast 0061lowastlowastlowast 0000(0002) (0260) (0004) (0000)

119879 = 18 (ref 119879 = 14) minus0264lowastlowastlowast minus3261lowastlowastlowast 0004 0000(0002) (0260) (0004) (0000)

120572 = 066 (ref 120572 = 05) 0002 minus0059 minus0100lowastlowastlowast 0000(0002) (0184) (0003) (0000)

(119880 119879) = (06 18) (ref (119880 119879) = (02 14)) minus0324lowastlowastlowast minus3728lowastlowastlowast minus0033lowastlowastlowast 0000(0003) (0368) (0006) (0000)

Observations 1440 1440 1440 1440R2 0983 0389 0529Adjusted R2 0983 0388 0527Residual Std Error (df = 1435) 0032 3492 0053 0000F Statistic (df = 4 1435) 21 272740lowastlowastlowast 228686lowastlowastlowast 402326lowastlowastlowastNote lowastplt01 lowastlowastplt005 lowastlowastlowastplt001

the argumentrsquos defended position Hence if there is votingand if agents happen to obtain 119894119899 proposal arguments theyinexorably cluster towards one or the other side of the opiniondistribution resulting in opinion convergence remeniscent ofgroup polarization (see Figure 7(e)) Furthermore as agentscluster in the same side of the opinion spectrum the argu-ments that may be advanced in deliberation are fewer andmore skewed towards that one side of the opinion spectrumAgents become more easily persuaded in deliberation andonly by arguments supporting one side of the spectrumConvergence of opinions towards one loose opinion focusbecomes faster and more certain If there is no vote (seeFigure 7(b)) the dynamics are similar but for the fact thatit is the uniform randomness of the proposal argument thatguarantees a central convergence of opinions

413 Quantitative Analysis of Pair-Wise Opinion DynamicsThe three parameters that allow for quantitative analysis are119880 and119879 the agentsrsquo attitude structure and themajority quotavoting rule 120572 Table 3 supports the claim that going from 120572 =05 to 120572 = 066 does not affect the variance nor the shifts ofopinions However judgment accuracy is significantly higherwhen accepting proposals becomes more difficult This israther surprising but also intuitive since decisions that aretaken using more restrictive methods of scrutiny should bemore accurate

Concerning the attitude structures the interaction termbetween 119880 and 119879 shows that 119880 has a small effect on thevariance of opinions given that 119879 is low This shows thateven when the assimilation threshold is high a sufficientlysmall rejection threshold can make its one-way effect vanishHowever high levels of both 119880 and 119879 reduce the variance ofopinions dramatically In contrast when it comes to overallchanges in opinion the magnitude of 119880rsquos one-way marginal

effect increases the shifts of opinions whereas 119879 lowers itThis can be interpreted as having the rejection dynamics pushmoderate-to-extreme agents to positions that are close tothe extremes and very moderate agents to the center of thedistribution Not too many quantitative changes (on average)can therefore be observedThe fact that higher119880 is associatedwith faster convergence of opinions seems to imply lowerjudgment accuracy At some point voting for the same type ofproposals for too longmay result inmany collectivemistakesThis is an interesting conclusion to consider in the analysis ofthe mixed model Finally the interaction term shows that 119879has no strong impact on the metric

414 Quantitative Analysis of Deliberative OpinionDynamicsSeven parameters can be considered for quantitative anal-ysis in this dynamics the parameters of interest recalledin Section 312 the size of the argument sacks (119896) agentbehavior during deliberation the strength of the deliberationdynamics (120574) and the measure sensitivity to deliberatedresults (119901119886) The most ldquounexpectedrdquo result obtained fromthe deliberation dynamics is that it shows how deliberationcontributes to the proportion of extremists and the varianceof opinions It happens that having more agents discussproposals and having more of these discussions increasesthe variance of opinions and the proportion of extremistswhile lowering judgment inaccuracy and coherence For theformer the fact that far too many arguments are advancedduring deliberation plummets the chances that the centralargument is accepted hence deliberation does not moveopinions too often For the latter we observe that only ldquobigrdquodifferences (going from the lowest value of a parameter to thehighest) in protocol actually prove to have a preponderantimpact on the metrics These observations hint at a possibletrade-off between judgment accuracy and the variance of

Complexity 17

Table 4 Ordinary least squares estimates for the deliberative interactions scenarios for most of the observed metrics

Dependent variable119881119886119903(119900) 119878ℎ 119890119888 119894119903(1) (2) (3) (4)Constant 0032lowastlowastlowast 13092lowastlowastlowast 0370lowastlowastlowast 1007lowastlowastlowast

(0001) (0406) (0001) (0001)

119898 = 3 (ref119898 = 1) 0012lowastlowastlowast minus2062lowastlowastlowast minus0012lowastlowastlowast minus0001(0001) (0276) (0001) (0001)

119898 = 6 (ref119898 = 1) 0023lowastlowastlowast minus3285lowastlowastlowast minus0018lowastlowastlowast minus0004lowastlowastlowast(0001) (0276) (0001) (0001)

119899119863 = 002 (ref 119899119863 = 001) 0007lowastlowastlowast minus0965lowastlowastlowast minus0006lowastlowastlowast minus0002lowastlowast(0001) (0276) (0001) (0001)

119899119863 = 005 (ref 119899119863 = 001) 0020lowastlowastlowast minus3316lowastlowastlowast minus0015lowastlowastlowast minus0004lowastlowastlowast(0001) (0276) (0001) (0001)

120572 = 05 (ref 120572 = 0) 0146lowastlowastlowast 68932lowastlowastlowast 0068lowastlowastlowast minus0507lowastlowastlowast(0001) (0276) (0001) (0001)

120572 = 066 (ref 120572 = 0) 0291lowastlowastlowast minus13842lowastlowastlowast 00004 minus0000(0001) (0276) (0001) (0001)

119896 = 8 (ref 119896 = 4) 0004lowastlowastlowast minus0815lowastlowastlowast minus0002lowastlowastlowast minus0001(0001) (0276) (0001) (0001)

119896 = 16 (ref 119896 = 4) 0008lowastlowastlowast minus1118lowastlowastlowast minus0004lowastlowastlowast minus0003lowastlowastlowast(0001) (0276) (0001) (0001)

119861119890ℎ119886V119894119900119903 = ldquoFocusedrdquo (ref ldquoNaiverdquo) 0015lowastlowastlowast minus1980lowastlowastlowast minus0012lowastlowastlowast minus0004lowastlowastlowast(00005) (0225) (00004) (0001)

120574 = 01 (ref 120574 = 005) minus0020lowastlowastlowast 5762lowastlowastlowast minus00003 minus0001(0001) (0276) (0001) (0001)

120574 = 02 (ref 120574 = 005) minus0025lowastlowastlowast 8648lowastlowastlowast minus00003 minus0001(0001) (0276) (0001) (0001)

119901119886 = 05 (ref 119901119886 = 03) minus0013lowastlowastlowast 1581lowastlowastlowast 00004 00003(00005) (0225) (00004) (0001)

Observations 19440 19440 19440 19440R2 0931 0844 0562 0958Adjusted R2 0931 0844 0562 0958Residual Std Error (df = 19427) 0033 15701 0030 0050F Statistic (df = 12 19427) 21 844790lowastlowastlowast 8 733563lowastlowastlowast 2 079585lowastlowastlowast 36 761250lowastlowastlowastNote lowastplt01 lowastlowastplt005 lowastlowastlowastplt001

opinions that may be of interest in the mixed discussionmodel

Group coherence shifts of opinions and the varianceof opinions are highly affected by the voting thresholdthe coherence metric is reduced by half when allowing forclassic majority voting shifts explode and the variance ofopinions increases dramatically This is not surprising sincethe majority quota rule works as a buffer to the success ofdecision-making processes and in consequence it hampersthe deliberative opinion updates For the shifts of opinions itis clear that the two-third majority voting rule results in littleopinion change and that the simple majority rule inducesgroup polarization As expected there are less extremists inscenarios where there is no voting and the shifts of opinionare more likely to happen when agents are more sensitiveto deliberation When we include voting in the system

the grouprsquos judgment accuracy decreases due to the correctdeliberated decisions that are vetoed by a salient opinion-agent majority Strangely the effect only holds for the simplemajority decision rule Lastly the proportion of individualsthat participate in the debate and the minimal amount ofdebates have similar marginal effects on all the metrics (seeTable 4)

415 Comparing Deliberative and Pair-Wise Opinion Dynam-ics in a Nuthshell Qualitatively both models yield similaroutcomes but for the fact that deliberation never producesopinion polarization with both types of extremists (seeFigure 7) Furthermore clusters in the deliberative interac-tions model are less homogeneous in size and multiclusterconvergence occurs only when deliberation is rarely suc-cessful which happens sporadically given the calibration of

18 Complexity

Table 5 Mean difference 095 confidence intervals for metrics by model comparison

Model Metric119881119886119903(119900) 119901119903119900119901119890119909 119878ℎ 119890119888 119894119903Dyadic vs Deliberative [0191 0216] [0170 0197] [-2948 -2828] [0074 0082] [0165 0172]Mixed vs Deliberative [0142 0147] [0150 0155] [-2085 -1976] [0003 0002] [-0038 -0030]Mixed vs Dyadic [-0072 -0046] [-0045 -0018] [8332 8859] [-0079 -0071] [-0205 -0201]the model On the other hand pair-wise opinion dynamicsproduces multicluster convergence most of the time andcentral convergence only in the infrequent cases where thelatitude of acceptance is large

The comparison of these two scenarios in terms ofthe metrics is straightforward Whilst dyadic interactionsallow for more variability in opinions through pair-wiseinteractions and more mistakes in judging proposals delib-erative updates simply act as an antagonist force On aver-age the deliberative interaction model produces opiniondistributions with smaller variance and yields steady stateswith higher judgment accuracy a higher shift statistic andnaturally lower group coherence since only proposals that aredeliberated 119894119899 are eligible for voting (see Table 5)

42 Articulating Dyadic and Deliberative Social Interactionsin Opinion Dynamics So far we have considered scenarioswhere argumentation had no place and where individualsmade decisions according to opinions that were purelyconstructed from pair-wise discussions We also observedcases in which agents formed their opinions only by inte-grating deliberated proposals into their opinion updates Thescenarios we present in this section combine the two pre-ceding scenarios to account for both the effects of pair-wisediscussion and the effects of deliberation on opinions judg-ment accuracy and group coherence From the precedingobservations we conclude the following deliberative opiniondynamics reduces the variance and the proportion of extrem-ists because it shifts opinions greatly and polarizes groupsby eliminating at least one type of extremist the one whoseopinions are initially opposed to the deliberated resultsor it brings opinions close to neutrality (see Figure 7(b))This may not be desired in a group since the diversity andstability of opinions are necessary for deliberation to makesense and lead to the collective acceptance of controversialproposals Dyadic opinion dynamics on the other handtend to increase the variance of opinions and the numberof extremists while having a lesser aggregated effect on theshifts of opinion This is an immediate result of the pair-wiseinteraction dynamics inasmuch as it polarizes individualsvery quickly

To sum up in the mixed interactions model deliberationhinders the effect of voting on opinions by making someproposals that would normally be submitted for voting noteligible for voting It may also undo opinion changes dueto pair-wise interactions The mixed model can be seen asthe deliberation opinion dynamics model in which pair-wiseinteractions occur in between collective discussion and mayaffect which arguments are advanced during deliberationEquivalently it can be seen as the dyadic opinion dynamics

model in which deliberation accounts for endogenousldquoshocksrdquo on the opinion distribution

421 Qualitative Analysis of the Mixed Dynamics In Figures7(c) and 7(f) agents update their opinions through delibera-tion and pair-wise interactions Opinion trajectories visualsshow a combination of both previously described opiniontrajectories Over time as agents discuss with one anotheropinion clusters form as in the pair-wise or dyadic opiniondynamics Deliberation may disrupt the formation of theseclusters and three different situations may occur

(i) Deliberation disrupts the formation of the opinionclusters but not sufficiently to jeopardize pair-wiseopinion cluster formation (eg for high values 119905119863)Two phenomena may be responsible for it (1) thevariability of the proposal argumentrsquos support for theprinciple P and (2) too few (successive) deliberationsteps that result in opinion updates Once the opinionclusters are formed successful deliberation foreshad-ows the merging of the previously formed clustersmay bring extremists closer to moderate agents andthanks to the assimilation component of the pair-wisedynamics allow bigger and more moderate opinionclusters to form (refer to Figure 7(c))

(ii) Deliberation significantly disrupts the pair-wisedynamic formation of clusters This may happenwhen the requirements for accepting proposals areweak in the decision-making process (eg low 120572) Inthis case one has either central convergence sinceall agents follow the deliberated results in the samemanner or 3-cluster convergence with two extremeopinion foci In the latter situation the extremegroups form quickly (like in the pair-wise model) anddeliberation cannot pull extreme agents away fromthe extreme foci they are in The main reason for thisis that immediately after an extreme agent attempts toleave her foci through deliberation updates she getspulled or pushed back into it by means of a dyadicinteraction with another extreme agent

(iii) Deliberation does not disrupt the formation of theopinions cluster This is the rare case scenario forvery demanding decision-making processes in termsof inclusion (eg 119899119863) and argument acceptability (eg120572) Opinions converge as in the pair-wise opiniondynamics model (refer to Figures 7(a) and 7(d))

Last when there is voting (120572 = 0) the position ofthe moderate opinion cluster is determined by the first

Complexity 19

minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

Var(o)

sh

ec

ir

Var(o)

propex

sh

ec

ir

1

098

minus02

minus004

001

1

minus014

minus001

minus003

1

026

minus028

1

minus057 1

ＪＬＩＪ

Ｒ

Figure 8 Significant correlation estimates for themetrics of interestin the mixed opinion dyamics scenarios

accepted proposal argument andor the opinionmajority thatis formed from previous dyadic interactions (see Figure 7(f))

422 Relating the Metrics of Interest in Mixed OpinionDynamics The first thing to notice is that for all of thesescenarios all the metrics are significantly correlated (seeFigure 8) The variance of opinions and the proportion ofextremists are almost perfectly correlated To some extentthis means that a high variance of opinions known thatthe mean of opinions is statistically null equates to havingmany extremists in the population be towards one or theother side of the opinion distribution (eg extreme bipolar-ization) Similarly we find a negative and strong correlationbetween the judgment inaccuracymeasure and the coherencecoefficient (see Figure 8)mdashwhen judgment inaccuracy is high(low) coherence to deliberation is low (high) Cases likethese are telling since they may be related to sequences ofcollective decisions in which agents do not make too manymistakes when judging proposal arguments and still makedeliberated decisions that are on average in line with theirprinciples

The shift statistic interestingly is negatively correlatedwith the variance of opinions with judgment accuracy andwith the coherence coefficient In a way this means thathaving more moderate agents conduces on average to moremistakes in judging arguments and to less coherent collectivedecisions An explanation for this phenomenon could be thatas agents reach consensus in opinion important (possiblyextreme) arguments for correctly judging argument propos-als are not advanced in the deliberation arenawhich results inmore mistakes and in possibly accepting proposal argumentsthat after voting are not meant to be accepted Indeed the

negative correlation between the variance of opinions andjudgment inaccuracy points towards this direction since onaverage the more shifts there are the smaller the variance ofopinions is (see Figure 8)

423 Comparing Mixed Dynamics with Other MonolithicDynamics of Interactions Qualitatively opinion trajectoriesin the mixed model are more chaotic than in the pair-wise and deliberative interaction models (see Figure 7) Asanalyzed previously deliberation dynamics disrupt the pair-wise dynamics and create situations in which central andmulticlustered convergence (with two clusters at the extremesof the distribution) are possible at the expense of more(pair-wise dynamics) or less (deliberative dynamics) clustersIt tolerates the survival of very small isolated group ofagents that the previous models rarely tolerate in a hundredcollective decision-making steps (see Figure 7(c))

One of the most interesting observations one can makeof this family of models regards how groups change (in termsof our metrics) when dyadic social interactions are addedto deliberative interactions and vice-versa When comparingthe mixed interactions with only deliberative ones shiftsagentsrsquo aggregated change in opinions are on average lesspronounced The variance of opinions and the proportionof extremists are significantly higher for the former than forthe latter group coherence in contrast is lower with respectto the scenarios where opinions are formed only throughdeliberation Judgment accuracy follows that same trendA reason for this might be that mixed dynamics are fasterto yield semistable clusters andor opinion convergence Inconsequence the potential argument pool for deliberativeexchange shrinks faster and deliberation becomes less effec-tive over time which results in more collective mistakes Theoutcomeon coherence is predictable inasmuch as deliberativeinteractions produce group polarization If a group agrees inopinion debates are sterile because themembers of the groupwill most likely refrain from attacking the proposal argu-ments they all support and will work together to disparagethe proposal arguments they do not support (see Table 5)

In the same manner we report that the proportion ofextremists and the variance of opinions are barely lower inthe pair-wise interaction model when including deliberationin the decision-making process Shifts of opinions are alsobarely lower when there is no deliberation This can indicatethat deliberation does not undo the changes in opiniondue to pair-wise interactions nor does it amplifies them Incontrast when we consider the labeling-based metrics themixed interaction model ensures lower judgment inaccuracybecause of deliberation and lower group coherence (seeTable 5)

In sum the mixed discussion model is a compromisebetween the pair-wise interaction and deliberative opiniondynamic models in that it offers some guarantees in termsof judgment accuracy and coherence while allowing forldquoreasonablerdquo variance in the opinion pool It establishes somekind of trade-off between labeling-based metrics and thevariance of opinions that may be of interest for decision-making process design

20 Complexity

Table 6 Summary of sign or relative marginal effects for all varying parameters on dyadic deliberative and mixed interactions models inand for all experiments

Metrics Parameters119905119863 119898 119899119863 120572 119879 119880 119901119886 120574 119896 119861119890ℎ119886V119894119900119903119881119886119903(119900) XX X X O X X X X X X119901119903119900119901119890119909 XX X X O X X X X X X119878ℎ XX X X O X X X X X119890119888 XX X X XO X XOO XOO X X119894119903 XX X X O XO O X OOX XOO X XNota gray symbols correspond to dyadic dynamics light gray to deliberative dynamics and bold black to mixed dynamics Given a scenario the symbol 119883stands for constant parameter and 119874 for insignificant marginal effect and as parameters grow for positive (negative) marginal effect for the parameterFrom left to right at the parameter level the leftmost arrow corresponds to the scenario with the strongest (in absolute value) marginal effect of the parameter(with respect to to its factors) Example Let 119901 be a parameter and 119874 stands for ldquo119901 has a significant positive effect in dyadic discussion scenario negativein the deliberative one and insignificant in mixed discussion The positive effect in dyadic discussion is greater than the absolute value of the negative one inthe purely deliberative scenariordquo

424 Comments on the Differences between the Three Modelsfor OLS Regression Estimates The significance of most of theregression parameters estimated for the mixed interactionsscenarios are similar to those estimated for the pure dyadicanddeliberative scenariosNevertheless themagnitude of theeffects is different insomuch that the dynamics themselvesare different So instead of commenting the linear regressionestimates of the mixed model we compare when possiblethe standardized significant coefficients of the regressors ofthe mixed model with those of the pair-wise and deliberativeinteractionmodels (see Table 6 for a summary of the results)

In mixed interactions for instance the influence of119898 onthe opinion distribution is much more preponderant than inthe deliberative dynamics scenarios Going from 119898 = 1 to119898 = 3 increases the standard variation units of the variance ofopinions by twice as much as the one in the deliberative sce-nariosThis in particular is due to the design of the decision-making process If 119898 is higher then the number of socialinfluence steps (119905119863) in the decision process is also higher(compared to 119905119863 = 0 in the deliberative case) If we decideto consider these steps as contributing to the variance ofopinions then the effect of119898 on the variance of opinions hasto be stronger in the mixed interactions model In contrastthe number of agents participating in the debate has a weakereffect on the variance of opinions in the mixed interactionsmodel than in the deliberative dynamics specially for 119899119863 =005 (more than twice as much) The difference may comefrom the fact that in mixed discussions the part of theargument pool used in deliberation is more restrictive thanin the deliberative case We can argue that since pair-wisedynamics may get individuals closer to one another aftersome deliberation the same happens to the arguments thatindividuals will most likely use in deliberation Taking thisinto account means that deliberation will at many instancesbe unfruitful and have no effect on opinions thus failing tomoderate extreme opinions andor polarize the group In thesame direction a high number of participants in deliberationcan make the deliberation dynamics stiff In this case thedistribution of opinion is close to the uniform distributionwhich has a relatively high-to-moderate variancewith respectto the reference results

The strength of the deliberative dynamics has a positivestronger effect on the variance of opinions in themixedmodelthan in the deliberative model when 120574 = 02 and a negativeweaker one when 120574 = 01 A sound interpretation of thisresult is that when deliberation is successful and its effect onopinions does not get agents to move sufficiently away fromtheir current positions agents are pulled or pushed back totheir previously held opinions the variance of opinions doesnot changemuchOn the other hand if the effect of successfuldeliberation is strong opinions can change sufficiently tomake agents much more prone to assimilation into moderateopinion foci the variance of opinions falls

The deliberative opinion dynamics model is much moresensitive to 120572 than the mixed discussion model Differ-ences in marginal effects can double The probability ofattraction to deliberated results (119901119886) for instance has astrongmoderating effect (negative) on agentsrsquo opinions in thedeliberative opinion dynamics but a meager one (two-fold)in the mixed and dyadic dynamic models This is becauseopinion updates in the deliberative model depend only onthat parameter whereas in the mixed model more numeroustypes of interactions make opinion updates possible

In terms of judgment accuracy all parameters except thesize of the argument sacks (119896) have a significant slightlystronger effect on the deliberative dynamics than in themixedmodel In effect the difference may be attributed to theupshot of pair-wise interactions on the distribution of opin-ions and on the arguments that would be advanced duringdeliberation The result respecting parameter 119896 however iscounterintuitive Since argument sacks are static scenarioswhere shifts are higher should make 119896 more important fordeliberation to succeed and it is precisely in the scenarios ofpure deliberative dynamics that we find the biggest opinionshifts This may point to the hypothesis that in deliberativeinteractions agents change opinions substantially but do notoften change their adherence to the principle (go from anegative opinion to a positive one or vice versa) when thereis no group polarization

43 Sensitivity Analysis for the Procedural Parameters ofInterest Weperform a sensitivity analysis of the observationson the parameters of interest in themixed interactionsmodel

Complexity 21

We extend the parameter domains of the pure dyadic anddeliberative scenarios as we make it explicit in Table 2

431 Minimum Number of Debates (119898) We observe that theminimum number of debates has a significant well-observedeffect on all of our metrics excluding group coherence Forthe variance of opinions we can observe that the moredebates there are the bigger the value of the metric isalthough the higher 119899119863 and 119905119863 are the weaker the overalleffect of119898 becomes (Figures 9(d) and 9(a)) Furthermore themarginal effect of increasing minimal debates is decreasingan additional deliberation step requirement increases thevariance of opinions but less and less as it grows 119898 coupledwith 119899119863 has a shy S-shaped effect (see Figure 9(d)) on thevariance of opinions meaning that it has a stronger effectwhen 119899119863 is low which suggests a possible trade-off betweenthe number of arguments accepted in the debate and thenumber of debates required before accepting a proposal

In contrast the shifts in opinion are less and less likelyas 119898 grows and this is independent of the variations of theother parameters Again the effect is marginally decreasingand is only truly significant when 120572 = 05 An explanationto these effects may be linked to the design of the systemFirst the variance of opinions is higher when deliberation isasked for because the more deliberation steps there are thehigher the chances are that the central proposal argumentis deemed unacceptable specially when agents are focusedFurthermoremechanically speaking increasing theminimalamount of debates implies that whenever a decision is to betaken at least 119898 times 119905119863 time steps have to take place and if atany moment a proposal argument is considered undecided119905119863 time steps are added to the process So unless the debatedoes not yield 119906119899119889 labels for proposal arguments (highlyunlikely considering that 120590 = grounded semantics) the morenondeliberation steps there are in the decision process thehigher the variance of opinions is Concerning the shiftswhen 120572 = 05 either the system is too stiff to accept anyproposal argument and opinions do not changemuch or theeffects of pair-wise discussion and deliberation cancel out insuch a way that individual shifts are minimal (Figure 9(b))

On the side of labeling-based metrics the more debatesare asked for the more accurate a group is in its judgmentthe effect being smaller as 119898 grows When agents are naivethe effect is quasilinear while when they are focused thestrongest effects of acquiring more deliberation are foundwhen levels of deliberation are already low (Figure 9(f))Thiscan be explained by the fact that the more debates thereare in the decision process the closer one gets to the idealargumentation framework

432 Proportion of the Population in Deliberation Steps (119899119863)Like for 119898 119899119863 has a significant effect on the proportionof extremists and on the shifts and variance of opinions(Figure 9(e)) This may result from the fact that being ableto put more arguments in play at the same debate step mayincrease the odds of revealing cycles around the proposalargument in119860119865120576 Given that we use grounded semantics thearguments in the cycles are labeled 119906119899119889 and in consequence

debates are more often postponed when 119899119863 is higher Post-poning debates in turn increases the number of nondebatessteps in the decision process which increases the varianceof opinions and limits over time the moderating effect ofdeliberation Moreover the effect of this parameter is verydependent on the value of 120572 (Figure 9(e)) For shifts forinstance 120572 = 05 makes the effect of 119899119863 negative while120572 = 0 makes it positive but to a lesser extent Furthermorefor higher requirements of deliberation (119898) adding moreindividuals to the deliberation process has weaker effectson the variance of opinions and on the other metrics It isalso quite interesting to notice that it has the same effect onagents independently of agent behavior during deliberationThis is a surprising result as one would have expected morefocused agents in a deliberation arena to cause a raise inthe proportion of extremists as they play to knock outopposing proposal arguments and thereof reducing the effectof deliberation (lowers variance) on the whole population

Finally similar to119898 addingmore people into the deliber-ation process increases judgment accuracy (see Figure 9(c))and has no clear effect if any on group coherence (seeFigure 9(i))

433 Steps between Deliberation Steps (119905119863) In all configu-rations 119905119863 increases the proportion of extremists (variance)and decreases the shifts of opinions among agents Theshifts and the effect on the variance of opinions are presentfor 120572 = 05 insignificant otherwise (Figure 9(b)) 119905119863 ishighly linked to 119898 by construction When 119898 = 1 thecurve linking the variance of opinions and 119905119863 is convex As119898 increases the curve becomes more and more concavewhich means that 119905119863 has a more important effect on theopinion distribution as collective decisions take longer to beachieved This seems counterintuitive but in reality it reflectsthemultiplicative relationship between deliberation and pair-wise interactions and the semantics in the model If 119898 is lowand 119905119863 is high the effective number of pair-wise interactionsis on average smaller in the deliberation process Hencethe structure of the decision-making process hinders anyincrease in the variance of opinions If119898 is high the oppositeeffect is observed Additionally since the grounded semanticsyields few 119894119899 arguments with respect to other admissibility-based semantics getting closer to the ideal argumentationframeworkmaymake it difficult to obtain the expected effectsof deliberative discussion on agents Lower 119898 constrains thevariance of opinions by making deliberation more influentialon opinions

In terms of judgment accuracy and coherence we observethat 119905119863 does not explain coherence yet it tends to decreasejudgment inaccuracy This may be due to a more variedargument pool in the debate which in turn may be aconsequence of a higher variance of opinions or vice versa

434 Acceptability Voting Quota (120572) By far it is the mostinfluential parameter in our study It changes the directionand the intensity of the effect of all the other proceduralparameters and by construction heavily constrains the roadto accepting a proposal (see Figure 9)

22 Complexity

015

020

025

030

035

040

1 2 3 4 5 6

12345

Values of Ｇ

(a) 119905119863 times 119901119903119900119901119890119909 by119898

0

10

20

30

40

50

1 2 3 4 5 6

005066

Values of

(b) 119905119863 times 119878ℎ by 120572

040

045

050

055

060

1 2 3 4 5

001002003004005

Values of Ｈ＄

(c) 119898 times 119890119888 by 119899119863

015

020

025

030

035

040

1 2 3 4 5

001002003004005

Values of Ｈ＄

(d) 119898 times 119901119903119900119901119890119909 by 119899119863

0

10

20

30

40

50

001 002 003 004 005

00005066

Values of

(e) 119899119863 times 119878ℎ by 120572

040

045

050

055

060

1 2 3 4 5

BehaviorNaiveFocused

(f) 119898 times 119890119888 by 119887119890ℎ119886V119894119900119903

015

020

025

030

035

040

0 05 066Behavior

NaiveFocused

(g) 120572 times 119901119903119900119901119890119909 by 119887119890ℎ119886V119894119900119903

05

06

07

08

09

10

1 2 3 4 5 6

00005066

Values of

(h) 119905119863 times 119894119903 by 120572

08700

08725

08750

08775

08800

001 002 003 004 005

BehaviorNaiveFocused

(i) 119899119863 times 119894119903 by 119887119890ℎ119886V119894119900119903

Figure 9 Curves of mean observations for different parameters and metrics for 36000 runs Bars are 095 confidence intervals Plots withthe same observables (ordinate) except for 119894119903 are on the same scale

Complexity 23

The higher the requirement for accepting the proposal isthe higher the proportion of extremists and the variance ofopinions are and the lower the shift statistic is In few words120572 constrains the world to dyadic discussion or throws it intoa process in which deliberation is much more importantthan dyadic interactions This is why for any 120572 = 05 oneeither gives too much weight to deliberated results whichshadows pair-wise interactions or too little weight in thesense that no deliberated result is ever accepted and thusnever integrated into the opinions of the agents that havevoted for it The reason for the latter is that the latitudeacceptance is too low and thereafter pair-wise interactionsare not enough to unevenly polarize the population in such away that deliberated proposal arguments are voted favorablyby a non-negligeable majority Shifts for 120572 = 0 are stable fordifferent levels of 119905119863 which implies that deliberation keeps incheck all the shifts of opinions related to an increase in thenumber of pair-wise interactions For 120572 = 05 deliberationis less likely to be successful and shifts tend towards the shiftlevels for 120572 = 0 (see Figure 9(b))

Concerning labeling-basedmetrics120572 entirely determinesthe coherence statistic For 120572 = 05 there is no difference incoherence because of how coherence is defined it is maximalwhen 120572 = 0 and strangely maximal when 120572 = 066(see Figure 9(h)) Surprisingly 120572 has no effect on judgmentaccuracy One would have expected a higher 120572 to increasejudgment accuracy since agents would hardly collectivelyaccept a proposal argument discussed during deliberationProposal arguments have given how the argument lattice isgenerated a higher probability of being rejected than of beingaccepted in the ideal argumentation framework

435 A Word on Focused and Naive Agents Focused andnaive specifications for agent behavior are an importantparameter in the decision process since they model howagents choose arguments in the deliberation arena In the sce-narios where agents are focused agents show equal or highervariance of opinions and are therefore more extreme (seeFigure 9(g)) independently of the voting requirements Shiftsof opinion are less likely as well We can see this happeningbecause focused agents knock out proposal arguments moreoften than naive agents provided that 119899119863 is low

One could have expected naive agents to yield morecoherent decisions since they argue sincerely However thisseems not to be the case (see Figure 9(i)) Agents are unable todeliberate in away thatmakes deliberation reflect their votingintentions either because they do not have the necessaryarguments to do so or because such arguments do not exist

In terms of judgment accuracy focused agents do betterthan naive agents in finding the ideal labels for the proposalarguments This is likely because when reconstructing theframework focused agents take into account the deliberatedproposal argument and choose themost pertinent argumentsin their sacks that relate to the state of affairs at the deliber-ation arena In fine debates result in a better approximationofL119886119887120576(119868) even though one could have believed the oppositethat is focused agents are ready to sacrifice a correct collectivelabel of a proposal argument in the goal of defending theirpositions

44 Points of Discussion In this subsection we discuss theresults obtained from the simulations We attempt to makesense of them from a social sciences perspective

441 Opinion Consensus and Dissensus The deliberativeopinion dynamics corresponds to the simulated scenarios inwhich the variance of opinions is at its lowest and judgmentaccuracy at its highest Dyadic opinion dynamics are associ-ated with the scenarios in which the variance of opinions is atits highest judgment accuracy at its lowest and coherence atitsmaximumpossible levelThemixed interactions dynamicsis a combination of these aspects for all the metrics exceptfor coherence at which it is the worsemdashdeliberation leadsto some kind of loose opinion consensus that is reinforcedby pair-wise interactions Likewise the mixed discussionmodel confirms some results exposed in [16] expressingthat opinions and voting intentions in a deliberative contextldquooften changerdquo That being said and more particularly indeliberative opinion dynamics outputs align well with twointeresting results on consensus formation given by aninterpretation of Moscovici and Doisersquos [15] work Sunsteinrsquos[14] account on deliberation and group polarization andsocial impact theory [4] The first states that mild consensusis reached in deliberation as procedure disengages agentsthe second is that within discussion groups majorities tendto become larger and opinions more extrememdashnot on thebasis of competing arguments but of the initial balance ofopinions After several decision processes we find opiniondistributions that reflect these observations even though weare not able to precisely coin the set of parameters that yieldssuch distributions (see Figure 7)

Although differences between mixed and dyadic opiniondynamics in terms of the variance of opinions and shiftsare slim the former does much better in terms of judg-ment accuracy In this sense deliberation does moderateopinions but not as much when non-deliberation steps arerelatively frequent throughout the decision-making processIn terms of Moscovici and Doise theory of consensus [15]this makes sense since pair-wise discussion engages or ego-involve agents in the discussion and pushes to convergenceof opinion at the extremes of the opinion distributionHowever what remains unclear is the reason why proceduralparameters that constrain deliberation do not disengageagents from discussion and thus result in a mild consensusas also expressed by Moscovici and Doise [15] A possibleexplanation of this phenomenon may stem from the factthat deliberation in the model is always accompanied by aseries of ego-involving discussions that shadowor counter themoderating effect of deliberation Not only this deliberationcould also be responsible for crystallizing the opinions ofextremists for it favors one or the other part of the distributionof opinions as explained in [14] and to some extent in [13]Furthermore the mixed discussion model gives an exampleof polarization that results from a minority of dissenterson one side of an issue acceding to views of the majorityrsquosside [4] as opposed to the strong polarization observed indyadic interactions Situations in which the status quo orthe undecidability of the proposal argument lingers cannotpossibly yield situations of moderation In cognitive science

24 Complexity

one can be inclined to believe that the longer a deliberationprocess is the more likely individuals will take an immutablestance on the situation In view of the assumption that longdecision-making processes can be more costly for a group orsociety as a whole agents may feel forced to take a stancejust for the sake of ending the process The stance that isadopted would therefore depend on pair-wise interactionsamong individuals and not on deliberation since deliberationtakes too long to be conclusive

442 For Hypothetical Recommendations in Decision-MakingProcess Elaboration On another tune results from themodelsuggest that procedural parameters have marginally decreas-ing positive or negative effects on the metrics of interest Ina hypothetical policy analysis perspective this may indicatethat after choosing a correct number of individuals andimposing the right amount of debates one can attain rea-sonable levels of judgment accuracy coherence and varianceof opinions As accounted in [18] deliberation increasesknowledge (collective knowledge as well) and therefore itheightens the chances of making correct decisions If onebelieves that social welfare is linked to deliberation andeventually to the metrics studied previously then proceduralparameters can both be used as an instrument to attaindesired levels of social welfare and reinforce the notionof legitimacy of collective decisions It can also provide ajustification for choosing a deliberation regime rather thananother on the basis of how important (or urgent) a topic ofdiscussion is Indeed if we adopt the claim that deliberationis a forerunner of welfare then any state in which deliberativecues are at their maximum (highest number of participantsmany deliberation steps before making a decision and asfrequent as possible) has to be mapped to the highest attain-able social welfare In this case is a situation where judgmentaccuracy is at its highest yet the variance of opinions or theproportion of extremists at its lowest an ideal situation Wecannot say for sure since strong opinion consensus and longdecision-making processes in many situations may result ina loss of welfare rather than in a gain of it

Another interesting result in procedure for deliberationis the very similar estimates we find for the effects of theminimum number (119898) of debates necessary to make adecision on a proposal and the proportion (119899119863) of individualsparticipating in deliberation on judgment accuracy In thescope of social welfare this raises the question of whetheradvocating for longer decision processes by demandingmoredeliberation steps (higher 119898) rather than bigger debates(higher 119899119863) may increase welfare Depending on the objec-tives of the decider of the deliberation procedure he or shemay choose to concentrate on one or the other A recom-mendation we can assert from the mixed social interactionsmodel is that if one prefers low to high variance of opinionsandor high to low coherence the decider is better off ifshe focuses on increasing the size of the debate rather thanorganizing debates very frequently One of the reasons forthis might be that when making bigger debates all agents aretruly considered as equals in the decision-making processIn organizing many debates individuals that are chosen toparticipate in many of these will be more influential and will

somehow bias the set of arguments in the deliberation arenaThemajority rulemodeled through120572 also determines the sizeof the shifts and the distribution of opinions Because a bigger120572 is associated with higher levels of extremism and opinionvariance deliberated cues are harder to accept and thereforeit becomesmore difficult for agents to reach concensus andormoderate their opinions

On the other hand and further into the idea of legitimacythe goal of deliberative processes is the agreement amongall participants This agreement stands for the right or bestdecision from a formal or procedural perspective and it is alsothe best from a substantive point of view In other words thelegitimacy of decisions not only derives fromwell-establishedprocedural requirements (respect of the protocol) but alsobegs for the fulfillment of two essential conditions (1) satisfy-ing the procedural requirements for a correct procedure (for-mal legitimacy) and (2) the rational acceptability of the resultsof this procedure (substantive acceptability) Legitimacy canbe then obtained from the coherence statistic scrutinizedin the model insofar as it measures how well agents acceptthe results of a correct procedure (eg deliberation) So fora proposal to be legitimate one has to choose parametersthat would maximize group coherence in decision-makingprocesses

As deliberation may give rise to consensus in the delib-eration arena it may also sow the seeds of dissensus inthe group Our model clearly illustrates this phenomenonthrough the procedural parameters of deliberation and themetrics observed From another perspective in deliberativedemocracy dissensus in collective decision-making may bedesirable According to Landermore and Pagersquos accounts [36]lack of consensus can be perceived as having agents withalternative ideas that combined with other ideas can providea better approximation of the ideal status of an argumentTheauthors in [36] call this dissensus ldquopositive dissensusrdquo andit makes normative sense when solving complex collectivedecision-making problems Our model captures the ideaof having normative requirements to define a good andlegitimate collective decision requirements of high consen-sual deliberative accuracy (judgment accuracy) of legitimacy(coherence) and of diversity (variance of opinions) are alltaken into account and for such we can conclude that ourmodel is successful in describing a sound process of collectivedecision-making with interpretable outcomes

Of course the present simulation results and analysis donot prove that all opinion dynamics can be accounted forby such simple processes like the one presented here Theydo however shed light on the expressiveness of combiningdifferent paradigms to account formore interpretablemodelsand pave the way to creating original models of the sortThe simulations suggest the desirability of discovering theconsequences of relatively simple laws of communication atdifferent levels (micro meso and macro) to determine whatstill needs to be explained

5 Related Work

We see our model as a contribution to the influence andopinion dynamics field in ABM and a pragmatic application

Complexity 25

of abstract argumentation theory To our knowledge we areunaware of existing literature on agent-based modeling thatexplicitly relates collective choice and the notions of deliber-ation and opinion diffusion through abstract argumentationas we have done it

Most models in the literature of opinion diffusion areinterested in opinions because these have an influence incollective decision and in questions of social order Forinstance in [3] the authors are interested in consensus andin how a group collectively chooses among two alternativesIn other models authors are interested in the emergenceof extremism [8] and on the distribution of opinions whenextremists are introduced in the population [5 33 37] whileother authors coin the notion of opinion polarization as anemergent property of the system [8 12] They show usingmodels of ldquobounded confidencerdquo and opinion diffusion withtrust that three different kinds of steady states (unipolarbipolar and central) are possible depending on whetheragents are sufficiently uncertain about their opinions andsufficiently connected andor a certain proportion of indi-viduals are already extreme Recent articles on informationand opinion dynamics stress the importance of governanceand government intervention in the spread of emotions(opinions) in the frangibility of social consensus systemIn [10] the authors show that government intervention inthe spread of negative emotions can lead to an even fasterspread of negative emotions (single-peeked convergence) andin a faster collapse of the social consensus system Opinionas a function of trust is studied in [12] and relates to howindividuals form their opinions on the basis of how muchthey trust agents in their networks

Another stream of discrete opinion dynamics modelsthat explain the emergence of opinion can be found in theliterature Computer multiagent models of attitude changebased on Latanersquos social impact theory are presented in [4] Inthe theory an agent changes her opinion on the basis of theinformational impact she is subject to which depends on thepersuasiveness (strength) supportiveness and immediacy(group structure) of the environment and information shereceives The simulations predict two emergent groups ofphenomena the shifting of attitudes towards incompletelypolarized equilibria and the formation of coherent clusteringof subgroups with deviant attitudes Similarly but in acontinuous more argumentative fashion Mas and Flache[11] present a model of opinion diffusion using argumentsArguments are considered to be for or against a proposal andare given an agent-dependant relevance in dyadic persuasiveinteractions Arguments also determine the agentsrsquo opinionsand their dissimilarities which posits the rules of pair-wiseencounters in theirmodel as agents are assumed to form theiropinions according to the arguments they own They showthat in an argumentative opinion diffusion model the onlyobservable steady states are bipolarization and consensusand that bipolarization can emerge from interactions amongsimilar agents In other words bipolarization of opinionsis possible with homophily and without negative influenceif there is argumentation They subsequently compare theirresults to experimental data In [38] a similar approach toargumentation and opinion formation is taken but for the fact

that they introduce different types of arguments explicitlyapply social judgment theory [22] and use survey data toestablish model-to-real-world comparisons

Closer to opinion formation and abstract argumentationGabbriellini and Torroni [39] are the first to have origi-nally and soundly merged opinion diffusion and abstractargumentation They define an agentrsquos opinion as a functionof the arguments she holds and the attack relation amongthem They devised a focused peer-to-peer dialogue systemof persuasion (NetArg) inspired from Mercier and Sperberrsquosargumentative theory of reasoning [40] which used onlyabstract argumentation to study opinion polarization andopinion dynamics Moreover they considered networks andthe notions of trust and epistemic vigilance to define adynamics for knowledge revision and trust itself whichwe clearly lack in our model They showed that if a con-servative belief operator in argumentation was applied byagents when they reasoned then their dialogue protocoldid not increase polarization among agents That said theyused the model to study the effect of Groenvetterrsquos weaklink theory in the spreading of arguments and were notparticularly interested in notions like judgment accuracy orcoherence nor in tackling questions related to collectivedecision-making procedures Their system is very expressiveand helps position our work in the litterature our work is atthe frontier of pure argumentative opinion diffusion modelsopinion diffusion with arguments and continuous bounded-confidence models of opinion diffusion a la Deffuant

On another note work on collective cognitive conver-gence [41] and opinion sharing [19] shows that consensustowards a certain ldquocorrectrdquo opinion or cognitive state is alwayspossible yet dependent on noise variability and awarenessof agents In [20] the authors show that learning about anexogenous correct state of the world (represented by bits)under confidence was possible but only if the agents werenot too confident In a homogeneous population they showthat the higher the confidence the worse the learningmdashthevery confident agents do not learn the properties of the truestate of the world and disrupt the learning process of the lessconfident ones Collective cognitive convergence can be seenas a result of deliberation in truth-seeking models

When it comes to abstract argumentation theory we takean approach that wires two types of dialogues that are well-studied in the literature persuasion dialogues [42] and delib-eration dialogues [43] Another interesting line of work is theone on mechanism design [44] or the problem of devisingan argumentation protocol where strategic argumentation isnot a liability for success in debates We tackle mechanismdesign in a different way though Instead of consideringstrategy-proofness we are interested in how differences inprotocol can result in epistemologically ldquobetterrdquo collectivechoices and guarantee that opinion distributions are favorablefor deliberation For a survey on persuasion dialogue see[42]

Work on agent-based argumentation usually assumes thatthe semantic relationship between arguments is fixed Inother words if two individuals were to put two argumentsin the public arena such that one logically attacked the otherthen everyone would agree that such attack exists [45 46]

26 Complexity

Other models which do not make this restrictive assumptioncan also be found in the literature [27 47] and derive fromthe class or family of opponent models [27 48] in which twoopposing sides attempt to win a dialogue Ourmodel is in theintersection of these two but the frame combining opiniondiffusion of the kind and argumentation remains new

The idea of mixing interpersonal influence and verticalcommunication however is not original It is described andimplemented in innovation diffusion models such as [5 49]In both vertical communication is modeled as exogenoustransparent informationmdashagents are aware of the existenceof an innovation thus triggering several processes of choiceand stabilization of opinions Also in [50] an Eulerian modelis implemented to show the effect of media and exogenousinformation on opinion distributions With respect to thispoint the originality of our work is in that the informationemitted as vertical communication is endogenous It is issuedfrom a deliberation model that agents shape on the basisof their opinions arguments and behavior In the spirit of[51] where the authors control for the design of verticalcommunication (to whom the vertical communication isaddressed its timing how it influences agents opinions) wecontrol for the process generating the information for ourown set of variables length of debates majority quotas andthe frequency at which discussions take place

6 Conclusion and Perspectives

The main objective of this article was to build a bridgewhere decision-making argumentation and opinion dif-fusion could come together We proposed a model thatcombined abstract argumentation theory and a boundedconfidence opinion diffusion model and showed to whatextent it could explain variance of opinions extremismcoherence in collective decisions and judgment accuracyon arguments given an ideal state of full information Thesecond objective was to show in what way governance pro-viding agents with arenas of discussion and deliberation wasimportant in addressing the success of collective decision-making processes and the quality of their outcomes

The model revealed that allowing for more deliberationtime though in lower frequency and allowing for widerparticipation in deliberation increased the variance of opin-ions and the proportion of extremists in a group Theseobservations are consistent with the combination of resultsfound in [9] and in [13 14] which stress that deliberationmay polarize groups and may have a meager effect on theshifts of opinions and inconsistent with [16] where it isargued that opinions tend to moderate after deliberationIt happens that deliberation alone did moderate opinionyet when integrated into a more complex system in whichindividuals were allowed to interact with one another andnot all that was deliberated was accepted its influence wasshadowed by other more individualistic dynamics

Undeniably the grounded semantics played an importantrole in the weak effect of deliberation since it models askeptical way of reasoning over arguments For agentsaccepting arguments thus updating their opinions accord-ingly happened not too often Nevertheless asking for more

deliberation did increase judgment accuracy as observed in[18] yet in a marginally decreasing fashion We showed thatvoting within the deliberation protocol not only increasedthe proportion of extremists and the variance of opinions ina group but also determined how coherent deliberation andvoting procedures were with each other Lastly we showedthat agents that are focused judged arguments better hadmore stable opinions and constituted a group with higherproportion of extremists than their naive counterparts

In terms of governance the model says that there is per-haps no trade-off between extremismand judgment accuracyInstead it asserts that the higher the variance of opinions thecloser the group gets to the correct decisions This situationmay be interpreted as Landermore and Pagersquos notion of ldquoposi-tive dissensusrdquo (in complex collective decision-making tasks)[36] as opposed to consensus as a normative requirementfor correct collective decision-making If a decider has toorganize deliberation to legitimize a public policy dependingon what kind of world he or she wants different set ofparameters may be chosen to account for different levels oflegitimacy and correctness For instance one may ask for adecision to be legitimate to respect a certain level of coherenceor of judgment accuracy and to be discussed by a sundryof different-opinionated peoples If so deliberation has tobe made more often more participants have to be includedin debates or in deliberation instances and not too manytime steps should be introduced in between two deliberationinstances So the model may capture normative ideas todefine correct and legitimate collective decisions and tomake recommendations on those grounds Requirements ofhigh consensual deliberative accuracy (judgment accuracy)legitimacy (coherence) and diversity (variance of opinions)are all taken into account and for such a reason the model israther successful in describing a sound process of collectivedecision-making with interpretable outcomes

61 Extensions of the Model and Other Ideas Our model canbe extended in many ways Value-based abstract argumen-tation frameworks (VAFs) for example as an extension ofDungrsquos argumentation framework provide a formal descrip-tion of the process of decision-making in which argumentsare given values and audiences (set of agents) ignore attacksbetween arguments on the basis of their preferences over val-ues Social abstract argumentation frameworks (SAFs) alsoprovide an interesting and seemingly convenient formalismfor our model In SAFs agents vote on the acceptability ofarguments and the resulting labeling is a combination of thearguments that are voted for and labeling-based semantics Inour model we apply the same rule yet only for one argumentand given precise agent-based votingmechanisms and votingrules which are eventually parametrized

To consider our model as a specification of a mixed VAFand SAF model may be of great interest for future workThis formalism can provide solid foundations to describeexplain interpret and eventually extend the results foundhere from an argument-based perspective On those groundsfuture work will focus on creating a hybrid argumentationframeworkmdashwith its semanticsmdashthat takes heed of bothvalues or principles and voting

Complexity 27

Concerning other possible extensions of the model anatural reflex would be to argue on the validity and therobustness of the mix between the bounded confidenceopinion diffusion models and deliberation Consequentiallyadding deliberation to different bounded confidence modelsof opinion diffusion such as the Hegelsmann-Krause model[37] or to models of social impact theory like in [4] canbe of great interest for future work We acknowledge that afiner understanding of argumentation frameworks randomlattices and their respective applications for this modelare necessary Testing for different labeling-based semanticsor even trading labeling-based semantics for scoring-basedsemantics (semantics that defines acceptability on scoresgiven to arguments) such as debate semantics and ranking-based semantics (semantics that yield a preorder) may beinteresting to consider in future work

More into modeling deliberation it is not inadequateto believe that agents may also advance proposals duringdeliberation and replace to some extent the proposal forwhich the deliberation is taking place Endogenizing theprocess that produces the type of argument to be deliberatedon may be interesting to explore either by having agentsstrategically replace proposals at certain moments duringthe debate or having the central authority attempt to chooseproposals such that the probability of having them acceptedis high Agents may also be thought to be autoorganizing(independent of a central authority) and concede on the sizeof the instances of deliberation To consider differences insemantics across agents or even endogenizing the semanticson how urgent or important the proposal that is beingdiscussed is can also be an interesting direction to take

Last but not least many nontrivial modifications of ourmodel are possible and most should include better-thoughtdeliberation protocols It may be interesting to design andobserve protocols in which deliberation only affects agentsthat are actually debating for example Trust network effectsmultidimensionality of opinions new processes of argumentexchange or learning among agents are notions to furtherdevelop in order to make the model more realistic and relaxsome unreliable assumptions In sum the model is to berefined and extended with the objective of either studyingconcrete cases of deliberative polling and opinion dynamicsor implementing more intuitive thought experiments

Appendix

We present below some of the pseudocode we used toimplement the model

Algorithms

(a) Algorithm 1 implements the dyadic influence in themodel It describes how two agents interact andupdate their opinions

(b) Algorithm 2 implements the deliberative influence inthe model which describes how agents update theiropinions in sight of the accepted deliberated proposalarguments

(c) Algorithm 3 is used to obtain the grounded labelingof an argumentation framework

(d) Algorithm 4 provides an overview of a run in thesimulations119872 is the number of proposal argumentsto discuss which we set to 100 in the model

Each object in the system is indexed by a119908ℎ119900 number Hencea set of agents is annotated by a set of natural numbers Weuse this fact in the implementation of the algorithm 119874119887119895119890119888119905119894denotes the object of 119908ℎ119900 number 119894 Agentsrsquo opinions (119900) arecalibrated so that 119900 isin [minus1 1]Data Availability

The program and data used to support the findings of thisstudy are available from the corresponding author uponrequest

Disclosure

A preliminary version of this work was presented anddiscussed at the conference Bridging the Gap between FormalArgumentation and Actual Human Reasoning on October 4-5 2018 at BochumGermanyThe authors thank the audiencefor their valuable comments and suggestions

Conflicts of Interest

The authors declare that they have no conflicts of interestregarding the publication of this article

Acknowledgments

The contribution of Gabriella Pigozzi was supported by theDeutsche Forschungsgemeinschaft (DFG) and the CzechScience Foundation (GACR) as part of the joint project FromShared Evidence to Group Attitudes (RO 45486-1)

Endnotes

1 In other words the superiority of the ldquoright answerrdquo orlabeling in our case will appear as such to all

2 All agents hold the same opinion3 By sensitivity we mean how much an agent is ready to

change her opinion on the sole fact that an informationalcue has been debated on

4 The assumption is strong but this trust among agentsmay be rooted from a genuine individual motivation toreach a democratic consensus

5 We adopted a rule of thumbs that says that if an agentis neutral with respect to the principle then she willvote for any proposal that is presented to her duringdeliberation

6 If the argument is not unique she chooses one atrandom

7 If there are more than one candidate the agent will favorthe argument at minimal distance from 119868

28 Complexity

Require Vector of values for parameters (119881 = (120583 119880 119879))Ensure Successful interactions among agents119896 larr997888 0119873119890 larr997888 0

while 119896 le |119873|2 do ⊳Fill119873119890 with |119873|2 agents119896 larr997888 119896 + 1119873119890 larr997888 119877119886119899119889119900119898(|119873| + 1) ⊳Pseudo-random generator Generates integer between 0 and |119873|119873119903 larr997888 0 120575119894119895 larr997888 0 ⊳Variable to store differences in opinionfor all 119894 isin 119873119890 doRandomly pick a number 119895 isin 119873(119873119890 cup 119873119903)119873119903 larr997888 119873119903 cup 119895 120575119894119895 larr997888 |119900119894 minus 119900119895|procedure dyadic-update(119894 119895) ⊳Two agents 119894 and 119895 discuss and update opinions1199001015840 larr997888 119900119894 (120583119895 119880119895 119879119895) larr997888 (120583119894 119880119894 119879119894) larr997888 (120583 119880 119879) ⊳Save agent irsquos opinion before updating and homogenize agents

if 120575119894119895 lt 119880119894 then119900119894 larr997888 120583119894119900119895 + (1 minus 120583119894)119900119894if 120575119894119895 gt 119879119894 then119900119894 larr997888 (1 + 120583119894)119900119894 minus 120583119894119900119895if 120575119894119895 lt 119880119895 then119900119895 larr997888 1205831198951199001015840 + (1 minus 120583119895)119900119895if 120575119894119895 gt 119879119895 then119900119895 larr997888 (1 + 120583119895)119900119895 minus 1205831198951199001015840

Algorithm 1 Dyadic social influence

Require Vector of values for parameters (119881 = (120574 119901119886 119901119903))Ensure Successful interaction between proposal arguments and agents

for all 119894 isin 119873 doprocedure Probability-Update(V119886 119901119886 119901119903) ⊳Update individual probabilities of change due to deliberated cues119909119894 larr997888 119901|V119886minus119900119894 |2119886 119910119894 larr997888 1199012|V119886minus119900119894 |119903119902 larr997888 0if L119886119887(119868) = 119894119899 then

procedure deliberative-update(119894 119886) ⊳Agent i argument 119886120574119894 larr997888 120574 119902 larr997888 119906 sim U(0 1) ⊳ 119906 is a realization of a uniform random variableif 119902 lt 119909119894 then119900119894 larr997888 120574119894119900119894 + (1 minus 120574119894)V119886break

if 119902 lt 119910119894(1 minus 119909119894) then119900119894 larr997888 (1 + 120574119894)119900119894 minus 120574119894V119886Algorithm 2 Deliberation influence

8 In terms of computational complexity naive agentsmaketheirmoves in119874(|A119894| log(|A119894|)) while focused agents doat least as bad

9 Tables have memory Therefore debates are not neces-sarily trees

10 If there are less than (119899119863 times |119873|)2 agents for any of bothgroups the CA summons all of them to the deliberationtable

11 Or careful consideration of the viewpoints of otherswhich implies that citizens keep an open mind and donot reject arguments outright

12 A ldquohealthyrdquo consensus refers to the case in which manyconsensual decisions are taken but opinion-wise agentsdo not agree with each other Quotes are used on theword healthy because some political science theorists

(eg [36]) believe that in certain situations agreement inopinions for deliberation is epistemologically damaging

13 Simulations on groups of |119873| = 800 and |119873| = 400agents shows that scaling the population has an effect onall themetrics but it is confounding depending onwhichcouples of attitudes structures are considered (119880 119879) Westick to |119873| = 400 as in [9] for comparability andsimulation running time

14 By balanced wemean with as many arguments with V119886 lt0 as with V119886 gt 0

15 Experiments on the system suggest that a bigger(smaller) argument pool 119872 = 1000 (119872 = 200) isassociated with a higher (lower) variance of opinions(and extremism) and higher (lower) judgment accuracyFor the rest of the observations 119872 seem to have noeffect We choose a middle value for 119872 Similarly theargument pool chosen to be balanced or uniformly

Complexity 29

Require Argumentation framework 119860119865 = (AR)Ensure Assignment of labels to arguments from the grounded labeling of 119860119865L(A)

L0(A) larr997888 (0 0 0)119896 larr997888 0do119894119899 L119896+1(A) larr997888 119894119899 L119896(A) cup 119886 | 119886 is not labeled 119894119899 L119896(A) 119886119899119889 forall119887 isin A 119904119905 119887 R 119886 then 119887 isin 119900119906119905 L119896(A)119900119906119905 L119896+1(A) larr997888 119900119906119905 L119896(A) cup 119886 | 119886 is not labeled 119894119899 L119896(A) 119886119899119889 exist119887 isin A 119904119905 119887 R 119886 and 119887 isin 119894119899 L119896+1(A)

while L119896+1(A) = L119896(A) = 0Algorithm 3 Solve argumentation frameworks (with grounded semantics)

Require Vector of values for parameters (119881)Ensure Vector of statistics (119880)

procedure Set-parameters(119881) ⊳Initialize parameters for scenarioprocedure Init-objs(119899119898 119896 ) ⊳Create119898 arguments 119899 agents and 119860119865120576procedure Solve-Argumentation-Framework(119860119865) ⊳Assigns the epistemic labelingL120576 toA119894 larr997888 1119905 larr997888 1while 119894 le 119872 do ⊳ 119872 the number of arguments to discuss119868119894 larr997888 119866119890119899 119886119903119892(1) ⊳Generates proposal argument 119868119860 119894119905 larr997888 (119868119894 0)L119886119887(119868119894) larr997888 119906119899119889 ⊳Initialize label for 119868119895 larr997888 1 ⊳Initialize debate counter 119895while ((L119886119887(119868) = 119906119899119889 or 119898 le 119895) and 119895 le 119898) doif 119905 equiv 0(119898119900119889 119905119863) then

procedure build-argumentation-framework(119899119863 119887119890ℎ119886V119894119900119903) ⊳Build the argument graph agents deliberate(119861119894119905 119878119894119905) larr997888 (119883 119884) sub (⋃119897isin119873119899119863 A119897119894119905 ⋃119897isin119873119899119863 R119897119894119905) ⊳ 119873119899119863 sub 119873 are agents sampled by the CA119860 119894119905 larr997888 119889(119860 119894119905 (119861119894119905 119878119894119905))procedure Solve-Argumentation-Framework(A119889) ⊳FindL(A119889)L119886119887(119868119894) larr997888 L119886119887119863(119868119894)119895 larr997888 119895 + 1

else119896 larr997888 1while 119896 le 119905119863 doprocedure dyadic-social-influence(120583 119879 119880) ⊳Agents discuss one-to-one119896 larr997888 119896 + 1119905 larr997888 119905 + 1

if 120572 gt 0 andL119886119887(119868119894) = 119900119906119905 then ⊳Agents vote for the proposalif (|119897 isin 119873|119900119897 times V119868119894 ge 0| ge 120572 times 119899) then

L119886119887(119868119894) larr997888 119894119899else

L119886119887(119868119894) larr997888 119900119906119905if L119886119887(119868119894) = 119894119899 thenprocedure Deliberation-influence(120574 119901119886 119901119903) ⊳Agents are influenced by the result of the decision process119894 larr997888 119894 + 1119905 larr997888 119905 + 1119880 larr997888 119862119900119898119901119906119905119890 119878119905119886119905119894119904119905119894119888119904(119881)

return 119880 ⊳Get the statisticsAlgorithm 4 Run for one scenario

generated does not seem to play a significant explanatoryrole on any of the metrics

16 Experiments show that themaximumnumber of debatesis not an important parameter even if it is an intuitiveone when considering decision-making processes Inpractice debates end after at most 119898 + 1 deliberation

steps and even then it happens rarely However to avoidunpleasant surprises like infinite debates on a completeargument graph containing the proposal argument weadded the constraint 119898 Its value is precisely 7 becausethe biggest minimum number of debates we studyis 6

30 Complexity

17 To ensure comparability with the deliberative andmixedopinion dynamics models on judgment accuracy andcoherence agents vote on proposal arguments every 119905119863times(119898+1) 119905119863 and119898does not explain variability in judgmentaccuracy nor in coherence

18 Student t-tests for independent samples are robust tothe violation of two hypothesis on the distribution ofthe samples normality and homogeneity of variancesas long as the sample size is big enough (central limittheorem) and if the difference in size of the comparedsamples is smallrespectively

19 Studentrsquos test version forwhen variances are not assumedequal and the difference in size of the compared samplesis big

20 We do not correct for homoskedasticity nor normality oferrors in the estimates

21 Balanced as the same number of runs per scenario orsuch that no significant correlations are found betweenparameters

References

[1] R Axelrod ldquoThe dissemination of culture a model withlocal convergence and global polarizationrdquo Journal of ConflictResolution vol 41 no 2 pp 203ndash226 1997

[2] D M Rousseau Reinforcing the MicroMacro Bridge Organiza-tionalThinking and Pluralistic Vehicles SAGEPublications SageLos Angeles Calif USA 2011

[3] S Galam and SMoscovici ldquoTowards a theory of collective phe-nomena consensus and attitude changes in groupsrdquo EuropeanJournal of Social Psychology vol 21 no 1 pp 49ndash74 1991

[4] A Nowak J Szamrej and B Latane ldquoFrom private attitude topublic opinion a dynamic theory of social impactrdquo Psychologi-cal Review vol 97 no 3 pp 362ndash376 1990

[5] G Deffuant S Huet and F Amblard ldquoAn individual-basedmodel of innovation diffusion mixing social value and individ-ual benefitrdquo American Journal of Sociology vol 110 no 4 pp1041ndash1069 2005

[6] G Deffuant D Neau F Amblard and G Weisbuch ldquoMixingbeliefs among interacting agentsrdquoAdvances in Complex Systems(ACS) vol 3 no 01n04 pp 87ndash98 2000

[7] J Rouchier P Tubaro and C Emery ldquoOpinion transmission inorganizations an agent-based modeling approachrdquo Computa-tional and Mathematical OrganizationTheory vol 20 no 3 pp252ndash277 2014

[8] M Meadows and D Cliff ldquoThe relative disagreement model ofopinion dynamics where do extremists come fromrdquo in Pro-ceedings of the International Workshop on Self-Organizing Sys-tems Lecture Notes in Computer Science pp 66ndash77 Springer2013

[9] W Jager and F Amblard ldquoUniformity bipolarization and plu-riformity captured as generic stylized behavior with an agent-based simulation model of attitude changerdquo Computational ampMathematical Organization Theory vol 10 no 4 pp 295ndash3032005

[10] Y-F Zhang H-Y Duan and Z-L Geng ldquoEvolutionary mech-anism of frangibility in social consensus system based onnegative emotions spreadrdquo Complexity vol 2017 Article ID4037049 8 pages 2017

[11] A Flache andMWMacy ldquoSmall worlds and cultural polariza-tionrdquoThe Journal of Mathematical Sociology vol 35 no 1-3 pp146ndash176 2011

[12] A Tsang and K Larson ldquoOpinion dynamics of skepticalagentsrdquo in Proceedings of the 13th International Conference onAutonomous Agents and Multiagent Systems AAMAS 2014 pp277ndash284 International Foundation forAutonomousAgents andMultiagent Systems France May 2014

[13] K M Hansen Deliberative democracy and opinion formationUniversity Press of Southern Denmark Odense 2004

[14] C R Sunstein ldquoThe law of group polarizationrdquo Journal ofPolitical Philosophy vol 10 no 2 pp 175ndash195 2002 WileyOnline Library

[15] S Moscovici and W Doise Dissensions et Consensus unetheorie generale des decisions collectives Presses Universitairesde France-PUF 1992

[16] J S Fishkin and R C Luskin ldquoExperimenting with a demo-cratic ideal deliberative polling and public opinionrdquo ActaPolitica vol 40 no 3 pp 284ndash298 2005

[17] R C Luskin J S Fishkin and S Iyengar Considered Opinionson US Foreign Policy Face-To-Face versus Online Delibera-tive Polling International Communication Association NewOrleans La USA 2004

[18] J Barabas ldquoHowdeliberation affects policy opinionsrdquoAmericanPolitical Science Review vol 98 no 4 pp 687ndash701 2004

[19] O Pryymak A Rogers and N R Jennings ldquoEfficient opinionsharing in large decentralised teamsrdquo in Proceedings of the 11thInternational Conference on Autonomous Agents andMultiagentSystems vol 1 pp 543ndash550 Spain June 2012

[20] J Rouchier and E Tanimura ldquoWhen overconfident agents slowdown collective learningrdquo Simulation vol 88 no 1 pp 33ndash492012

[21] D Miller ldquoDeliberative democracy and social choicerdquo PoliticalStudies vol 40 S1 pp 54ndash67 1992

[22] M Sherif and C I Hovland Social Judgment Assimilation andContrast Effects in Communication and Attitude Change YaleUniversity Press 1961

[23] P M Dung ldquoOn the acceptability of arguments and its funda-mental role in nonmonotonic reasoning logic programmingand n-person gamesrdquo Artificial Intelligence vol 77 no 2 pp321ndash357 1995

[24] M Caminada and G Pigozzi ldquoOn judgment aggregation inabstract argumentationrdquo Autonomous Agents and Multi-AgentSystems vol 22 no 1 pp 64ndash102 2011

[25] P Baroni M Caminada andM Giacomin ldquoAn introduction toargumentation semanticsrdquoThe Knowledge Engineering Reviewvol 26 no 4 pp 365ndash410 2011

[26] S Chambers ldquoDeliberative democratic theoryrdquo Annual Reviewof Political Science vol 6 no 1 pp 307ndash326 2003

[27] E Bonzon andNMaudet ldquoOn the outcomes of multiparty per-suasionrdquo in Proceedings of the 10th International Conference onAutonomous Agents and Multiagent Systems 2011 AAMAS 2011vol 1 pp 47ndash54 International Foundation for AutonomousAgents and Multiagent Systems Taiwan May 2011

[28] J Ranciere Disagreement Politics and Philosophy University ofMinnesota Press 1999

[29] T Bench-Capon and K Atkinson ldquoAbstract argumentation andvaluesrdquo in Argumentation in Artificial Intelligence pp 45ndash64Springer 2009

[30] M Caminada ldquoOn the issue of reinstatement in argumenta-tionrdquo in Proceedings of the European Workshop on Logics in

Complexity 31

Artificial Intelligence vol 4160 of Lecture Notes in ComputerScience pp 111ndash123 Springer Berlin Heidelberg

[31] GAVreeswijk ldquoAn algorithm to computeminimally groundedand admissible defence sets in argument systemsrdquo in Pro-ceedings of the 2006 conference on Computational Models ofArgument Proceedings of COMMA 2006 vol 144 pp 109ndash120IOS Press Amsterdam The Netherlands 2006

[32] C W Sherif M Kelly H L Rodgers et al ldquoPersonal involve-ment social judgment and actionrdquo Journal of Personality andSocial Psychology vol 27 no 3 pp 311ndash328 1973

[33] G Deffuant F Amblard G Weisbuch and T Faure ldquoHow canextremism prevail A study based on the relative agreementinteraction modelrdquo Journal of Artificial Societies and SocialSimulation vol 5 no 4 p 1 2002

[34] E Awad M W Caminada G Pigozzi M Podlaszewski andI Rahwan ldquoPareto optimality and strategy-proofness in groupargument evaluationrdquo Journal of Logic and Computation vol 27no 8 pp 2581ndash2609 2017

[35] D Grossi and G Pigozzi ldquoJudgment aggregation a primerrdquoSynthesis Lectures on Artificial Intelligence and Machine Learn-ing vol 8 no 2 pp 1ndash151 2014

[36] H Landemore and S E Page ldquoDeliberation and disagreementproblem solving prediction and positive dissensusrdquo PoliticsPhilosophy and Economics vol 14 no 3 pp 229ndash254 2015

[37] R Hegselmann and U Krause ldquoOpinion dynamics andbounded confidence models analysis and simulationrdquo Journalof Artificial Societies and Social Simulation vol 5 no 3 2002

[38] A Stefanelli and R Seidl ldquoModerate and polarized opinionsUsing empirical data for an agent-based simulationrdquo inProceed-ings of the Social Simulation Conference 2014

[39] S Gabbriellini and P Torroni ldquoA new framework for ABMsbased on argumentative reasoningrdquo in Advances in SocialSimulation vol 229 pp 25ndash36 Springer Berlin Germany 2014

[40] H Mercier and D Sperber ldquoWhy do humans reason Argu-ments for an argumentative theoryrdquo Behavioral and BrainSciences vol 34 no 2 pp 57ndash74 2011

[41] H V Parunak T C Belding R Hilscher and S BruecknerldquoModeling and managing collective cognitive convergencerdquoin In Proceedings of the 7th iNternational Joint Conference onAutonomous Agents and Multiagent Systems vol 3 pp 1505ndash1508 International Foundation for Autonomous Agents andMultiagent Systems 2008

[42] H Prakken ldquoFormal systems for persuasion dialoguerdquo TheKnowledge Engineering Review vol 21 no 2 pp 163ndash188 2006

[43] K Atkinson T Bench-Capon and P Mcburney ldquoA dialoguegame protocol for multi-agent argument over proposals foractionrdquoAutonomousAgents andMulti-Agent Systems vol 11 no2 pp 153ndash171 2005

[44] M Thimm and A J Garcıa ldquoClassification and strategicalissues of argumentation games on structured argumentationframeworksrdquo in Proceedings of the 9th International Conferenceon Autonomous Agents and Multiagent Systems vol 1 pp 1247ndash1254 International Foundation for Autonomous Agents andMultiagent Systems Canada 2010

[45] I Rahwan and K Larson ldquoMechanism design for abstractargumentationrdquo in Proceedings of the 7th International JointConference on Autonomous Agents and Multiagent SystemsAAMAS 2008 pp 1031ndash1038 International Foundation forAutonomous Agents and Multiagent Systems Portugal May2008

[46] H Prakken R Riveret A Rotolo and G Sartor ldquoHeuristics inArgumentation A Game-Theoretical Investigationrdquo EuropeanUniversity Institute 2008

[47] N Hadidi Y Dimopoulos and P Moraitis ldquoTactics and con-cessions for argumentation-based negotiationrdquo in Proceedingsof the COMMA vol 245 pp 285ndash296 2012

[48] N Oren and T J Norman ldquoArguing using opponent modelsrdquoin Proceedings of the International Workshop on Argumentationin Multi-Agent Systems pp 160ndash174 Springer 2009

[49] S Delre W Jager T Bijmolt and M Janssen ldquoTargeting andtiming promotional activities An agent-based model for thetakeoff of new productsrdquo Journal of Business Research vol 60no 8 pp 826ndash835 2007

[50] A Mirtabatabaei P Jia and F Bullo ldquoEulerian opinion dynam-ics with bounded confidence and exogenous inputsrdquo SIAMJournal on Applied Dynamical Systems vol 13 no 1 pp 425ndash446 2014

[51] V Barbet N Guiraud V Laperriere and J Rouchier Hagglingon Values Towards Consensus or Trouble 2019

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Submit your manuscripts atwwwhindawicom

Complexity 15

() (U T) = (02 18) (b) ( pa m nD) = (005 05 00 2 02) (＝) ０ＬＧＮＬＭ ＣＨ (＜) (＝) Ｈ＞ tD = 3

(d) (５ ４) = (02 14) (e) ( pa m nD) = (01 03 05 3 02) () ０ＬＧＮＬＭ ＣＨ (＞) () Ｈ＞ tD = 6

Figure 7 From left to right common opinion trajectories and distributions for dyadic deliberative and mixed opinion dynamics within thestructure of different decision-making processes

small enough to detect significant differences in the estimatesand (2) meaningful for interpretation For the fixed one-valued parameters in Table 2 we use Table 1 to set them totheir minimum mean or median values

41 Comments on Dyadic and Deliberative Opinions Dynam-ics In this subsection we present our first results regardingthe deliberative opinion dynamics described in Equation (6)and its differences with the dyadic dynamics (Equation (5))in terms of our observations

411 Qualitative Analysis of Pair-Wise Opinion DynamicsPair-wise opinion dynamics alone produces multiclusterstable convergencewith variable cluster sizes (see Figures 7(a)and 7(d)) Insight on the complexity of these dynamics wasdiscussed in Section 321

In Figures 7(a) and 7(d) agents discuss randomly withone another in pairs Clusters in the extremes form quicklysince either agents that are close to the extremes attractone another or are convinced to stay close to the extremesby interacting with other agents they disagree with Otheragents with near-extreme positions may be simply attractedto one of the several moderate foci of agents in the opiniondistribution On the other hand moderate agents are eitherattracted to the closest opinion focus or pushed towards theextremes of the distribution Over time agents in the centralfocus either attract other agents into it or ignore the opinions

of the extreme agents that interact with them The result ofthese dynamics yields amulticlustered convergencewhere theopinion foci are at least at distance 119880 from one another andonly the extreme foci (clusters of agents with |119900119894| ge 075) areat a distance greater than or equal to 119879 from one anotherThefoci at the bounds of the opinion distribution are very stableand in analogy to the (12119889)-rule in assimilation bounded-confidence models [33] the number of clusters that form isroughly equal to 119880minus1412 Qualitative Analysis of Deliberative Opinion DynamicsQualitatively deliberative opinion dynamics yield a looseunipolar convergence of opinions near the center (centralconvergence see Figure 7(b)) near the center of either the leftor right portions of the distribution (group polarization seeFigure 7(e)) or to a sparse bipolar opinion distribution withclusters at the center and at the extremes (see Figure 7(c)) ifmixed with pair-wise interactions This is probably a reasonto expect the variance of opinions and the proportion ofextremists (almost perfectly correlated 120588 = 089 119901 lt 0001)to be low for these scenarios The side towards which theopinion cluster skews depends essentially on the valence ofthe first argument that is collectively accepted

In Figures 7(e) and 7(b) agents that update their opin-ions do so at the same time and only when deliberationis successful In these scenarios every time a decision iscollectively accepted agents update their opinions towards

16 Complexity

Table 3 Ordinary least squares estimates for pair-wise interactions scenarios for most of the observed metrics

Dependent variable119881119886119903(119900) 119878ℎ 119890119888 119894119903(1) (2) (3) (4)Constant 0610lowastlowastlowast 4429lowastlowastlowast 0472lowastlowastlowast 0000

(0002) (0206) (0003) (0000)

119880 = 06 (119880 = 02) minus0012lowastlowastlowast 2981lowastlowastlowast 0061lowastlowastlowast 0000(0002) (0260) (0004) (0000)

119879 = 18 (ref 119879 = 14) minus0264lowastlowastlowast minus3261lowastlowastlowast 0004 0000(0002) (0260) (0004) (0000)

120572 = 066 (ref 120572 = 05) 0002 minus0059 minus0100lowastlowastlowast 0000(0002) (0184) (0003) (0000)

(119880 119879) = (06 18) (ref (119880 119879) = (02 14)) minus0324lowastlowastlowast minus3728lowastlowastlowast minus0033lowastlowastlowast 0000(0003) (0368) (0006) (0000)

Observations 1440 1440 1440 1440R2 0983 0389 0529Adjusted R2 0983 0388 0527Residual Std Error (df = 1435) 0032 3492 0053 0000F Statistic (df = 4 1435) 21 272740lowastlowastlowast 228686lowastlowastlowast 402326lowastlowastlowastNote lowastplt01 lowastlowastplt005 lowastlowastlowastplt001

the argumentrsquos defended position Hence if there is votingand if agents happen to obtain 119894119899 proposal arguments theyinexorably cluster towards one or the other side of the opiniondistribution resulting in opinion convergence remeniscent ofgroup polarization (see Figure 7(e)) Furthermore as agentscluster in the same side of the opinion spectrum the argu-ments that may be advanced in deliberation are fewer andmore skewed towards that one side of the opinion spectrumAgents become more easily persuaded in deliberation andonly by arguments supporting one side of the spectrumConvergence of opinions towards one loose opinion focusbecomes faster and more certain If there is no vote (seeFigure 7(b)) the dynamics are similar but for the fact thatit is the uniform randomness of the proposal argument thatguarantees a central convergence of opinions

413 Quantitative Analysis of Pair-Wise Opinion DynamicsThe three parameters that allow for quantitative analysis are119880 and119879 the agentsrsquo attitude structure and themajority quotavoting rule 120572 Table 3 supports the claim that going from 120572 =05 to 120572 = 066 does not affect the variance nor the shifts ofopinions However judgment accuracy is significantly higherwhen accepting proposals becomes more difficult This israther surprising but also intuitive since decisions that aretaken using more restrictive methods of scrutiny should bemore accurate

Concerning the attitude structures the interaction termbetween 119880 and 119879 shows that 119880 has a small effect on thevariance of opinions given that 119879 is low This shows thateven when the assimilation threshold is high a sufficientlysmall rejection threshold can make its one-way effect vanishHowever high levels of both 119880 and 119879 reduce the variance ofopinions dramatically In contrast when it comes to overallchanges in opinion the magnitude of 119880rsquos one-way marginal

effect increases the shifts of opinions whereas 119879 lowers itThis can be interpreted as having the rejection dynamics pushmoderate-to-extreme agents to positions that are close tothe extremes and very moderate agents to the center of thedistribution Not too many quantitative changes (on average)can therefore be observedThe fact that higher119880 is associatedwith faster convergence of opinions seems to imply lowerjudgment accuracy At some point voting for the same type ofproposals for too longmay result inmany collectivemistakesThis is an interesting conclusion to consider in the analysis ofthe mixed model Finally the interaction term shows that 119879has no strong impact on the metric

414 Quantitative Analysis of Deliberative OpinionDynamicsSeven parameters can be considered for quantitative anal-ysis in this dynamics the parameters of interest recalledin Section 312 the size of the argument sacks (119896) agentbehavior during deliberation the strength of the deliberationdynamics (120574) and the measure sensitivity to deliberatedresults (119901119886) The most ldquounexpectedrdquo result obtained fromthe deliberation dynamics is that it shows how deliberationcontributes to the proportion of extremists and the varianceof opinions It happens that having more agents discussproposals and having more of these discussions increasesthe variance of opinions and the proportion of extremistswhile lowering judgment inaccuracy and coherence For theformer the fact that far too many arguments are advancedduring deliberation plummets the chances that the centralargument is accepted hence deliberation does not moveopinions too often For the latter we observe that only ldquobigrdquodifferences (going from the lowest value of a parameter to thehighest) in protocol actually prove to have a preponderantimpact on the metrics These observations hint at a possibletrade-off between judgment accuracy and the variance of

Complexity 17

Table 4 Ordinary least squares estimates for the deliberative interactions scenarios for most of the observed metrics

Dependent variable119881119886119903(119900) 119878ℎ 119890119888 119894119903(1) (2) (3) (4)Constant 0032lowastlowastlowast 13092lowastlowastlowast 0370lowastlowastlowast 1007lowastlowastlowast

(0001) (0406) (0001) (0001)

119898 = 3 (ref119898 = 1) 0012lowastlowastlowast minus2062lowastlowastlowast minus0012lowastlowastlowast minus0001(0001) (0276) (0001) (0001)

119898 = 6 (ref119898 = 1) 0023lowastlowastlowast minus3285lowastlowastlowast minus0018lowastlowastlowast minus0004lowastlowastlowast(0001) (0276) (0001) (0001)

119899119863 = 002 (ref 119899119863 = 001) 0007lowastlowastlowast minus0965lowastlowastlowast minus0006lowastlowastlowast minus0002lowastlowast(0001) (0276) (0001) (0001)

119899119863 = 005 (ref 119899119863 = 001) 0020lowastlowastlowast minus3316lowastlowastlowast minus0015lowastlowastlowast minus0004lowastlowastlowast(0001) (0276) (0001) (0001)

120572 = 05 (ref 120572 = 0) 0146lowastlowastlowast 68932lowastlowastlowast 0068lowastlowastlowast minus0507lowastlowastlowast(0001) (0276) (0001) (0001)

120572 = 066 (ref 120572 = 0) 0291lowastlowastlowast minus13842lowastlowastlowast 00004 minus0000(0001) (0276) (0001) (0001)

119896 = 8 (ref 119896 = 4) 0004lowastlowastlowast minus0815lowastlowastlowast minus0002lowastlowastlowast minus0001(0001) (0276) (0001) (0001)

119896 = 16 (ref 119896 = 4) 0008lowastlowastlowast minus1118lowastlowastlowast minus0004lowastlowastlowast minus0003lowastlowastlowast(0001) (0276) (0001) (0001)

119861119890ℎ119886V119894119900119903 = ldquoFocusedrdquo (ref ldquoNaiverdquo) 0015lowastlowastlowast minus1980lowastlowastlowast minus0012lowastlowastlowast minus0004lowastlowastlowast(00005) (0225) (00004) (0001)

120574 = 01 (ref 120574 = 005) minus0020lowastlowastlowast 5762lowastlowastlowast minus00003 minus0001(0001) (0276) (0001) (0001)

120574 = 02 (ref 120574 = 005) minus0025lowastlowastlowast 8648lowastlowastlowast minus00003 minus0001(0001) (0276) (0001) (0001)

119901119886 = 05 (ref 119901119886 = 03) minus0013lowastlowastlowast 1581lowastlowastlowast 00004 00003(00005) (0225) (00004) (0001)

Observations 19440 19440 19440 19440R2 0931 0844 0562 0958Adjusted R2 0931 0844 0562 0958Residual Std Error (df = 19427) 0033 15701 0030 0050F Statistic (df = 12 19427) 21 844790lowastlowastlowast 8 733563lowastlowastlowast 2 079585lowastlowastlowast 36 761250lowastlowastlowastNote lowastplt01 lowastlowastplt005 lowastlowastlowastplt001

opinions that may be of interest in the mixed discussionmodel

Group coherence shifts of opinions and the varianceof opinions are highly affected by the voting thresholdthe coherence metric is reduced by half when allowing forclassic majority voting shifts explode and the variance ofopinions increases dramatically This is not surprising sincethe majority quota rule works as a buffer to the success ofdecision-making processes and in consequence it hampersthe deliberative opinion updates For the shifts of opinions itis clear that the two-third majority voting rule results in littleopinion change and that the simple majority rule inducesgroup polarization As expected there are less extremists inscenarios where there is no voting and the shifts of opinionare more likely to happen when agents are more sensitiveto deliberation When we include voting in the system

the grouprsquos judgment accuracy decreases due to the correctdeliberated decisions that are vetoed by a salient opinion-agent majority Strangely the effect only holds for the simplemajority decision rule Lastly the proportion of individualsthat participate in the debate and the minimal amount ofdebates have similar marginal effects on all the metrics (seeTable 4)

415 Comparing Deliberative and Pair-Wise Opinion Dynam-ics in a Nuthshell Qualitatively both models yield similaroutcomes but for the fact that deliberation never producesopinion polarization with both types of extremists (seeFigure 7) Furthermore clusters in the deliberative interac-tions model are less homogeneous in size and multiclusterconvergence occurs only when deliberation is rarely suc-cessful which happens sporadically given the calibration of

18 Complexity

Table 5 Mean difference 095 confidence intervals for metrics by model comparison

Model Metric119881119886119903(119900) 119901119903119900119901119890119909 119878ℎ 119890119888 119894119903Dyadic vs Deliberative [0191 0216] [0170 0197] [-2948 -2828] [0074 0082] [0165 0172]Mixed vs Deliberative [0142 0147] [0150 0155] [-2085 -1976] [0003 0002] [-0038 -0030]Mixed vs Dyadic [-0072 -0046] [-0045 -0018] [8332 8859] [-0079 -0071] [-0205 -0201]the model On the other hand pair-wise opinion dynamicsproduces multicluster convergence most of the time andcentral convergence only in the infrequent cases where thelatitude of acceptance is large

The comparison of these two scenarios in terms ofthe metrics is straightforward Whilst dyadic interactionsallow for more variability in opinions through pair-wiseinteractions and more mistakes in judging proposals delib-erative updates simply act as an antagonist force On aver-age the deliberative interaction model produces opiniondistributions with smaller variance and yields steady stateswith higher judgment accuracy a higher shift statistic andnaturally lower group coherence since only proposals that aredeliberated 119894119899 are eligible for voting (see Table 5)

42 Articulating Dyadic and Deliberative Social Interactionsin Opinion Dynamics So far we have considered scenarioswhere argumentation had no place and where individualsmade decisions according to opinions that were purelyconstructed from pair-wise discussions We also observedcases in which agents formed their opinions only by inte-grating deliberated proposals into their opinion updates Thescenarios we present in this section combine the two pre-ceding scenarios to account for both the effects of pair-wisediscussion and the effects of deliberation on opinions judg-ment accuracy and group coherence From the precedingobservations we conclude the following deliberative opiniondynamics reduces the variance and the proportion of extrem-ists because it shifts opinions greatly and polarizes groupsby eliminating at least one type of extremist the one whoseopinions are initially opposed to the deliberated resultsor it brings opinions close to neutrality (see Figure 7(b))This may not be desired in a group since the diversity andstability of opinions are necessary for deliberation to makesense and lead to the collective acceptance of controversialproposals Dyadic opinion dynamics on the other handtend to increase the variance of opinions and the numberof extremists while having a lesser aggregated effect on theshifts of opinion This is an immediate result of the pair-wiseinteraction dynamics inasmuch as it polarizes individualsvery quickly

To sum up in the mixed interactions model deliberationhinders the effect of voting on opinions by making someproposals that would normally be submitted for voting noteligible for voting It may also undo opinion changes dueto pair-wise interactions The mixed model can be seen asthe deliberation opinion dynamics model in which pair-wiseinteractions occur in between collective discussion and mayaffect which arguments are advanced during deliberationEquivalently it can be seen as the dyadic opinion dynamics

model in which deliberation accounts for endogenousldquoshocksrdquo on the opinion distribution

421 Qualitative Analysis of the Mixed Dynamics In Figures7(c) and 7(f) agents update their opinions through delibera-tion and pair-wise interactions Opinion trajectories visualsshow a combination of both previously described opiniontrajectories Over time as agents discuss with one anotheropinion clusters form as in the pair-wise or dyadic opiniondynamics Deliberation may disrupt the formation of theseclusters and three different situations may occur

(i) Deliberation disrupts the formation of the opinionclusters but not sufficiently to jeopardize pair-wiseopinion cluster formation (eg for high values 119905119863)Two phenomena may be responsible for it (1) thevariability of the proposal argumentrsquos support for theprinciple P and (2) too few (successive) deliberationsteps that result in opinion updates Once the opinionclusters are formed successful deliberation foreshad-ows the merging of the previously formed clustersmay bring extremists closer to moderate agents andthanks to the assimilation component of the pair-wisedynamics allow bigger and more moderate opinionclusters to form (refer to Figure 7(c))

(ii) Deliberation significantly disrupts the pair-wisedynamic formation of clusters This may happenwhen the requirements for accepting proposals areweak in the decision-making process (eg low 120572) Inthis case one has either central convergence sinceall agents follow the deliberated results in the samemanner or 3-cluster convergence with two extremeopinion foci In the latter situation the extremegroups form quickly (like in the pair-wise model) anddeliberation cannot pull extreme agents away fromthe extreme foci they are in The main reason for thisis that immediately after an extreme agent attempts toleave her foci through deliberation updates she getspulled or pushed back into it by means of a dyadicinteraction with another extreme agent

(iii) Deliberation does not disrupt the formation of theopinions cluster This is the rare case scenario forvery demanding decision-making processes in termsof inclusion (eg 119899119863) and argument acceptability (eg120572) Opinions converge as in the pair-wise opiniondynamics model (refer to Figures 7(a) and 7(d))

Last when there is voting (120572 = 0) the position ofthe moderate opinion cluster is determined by the first

Complexity 19

minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

Var(o)

sh

ec

ir

Var(o)

propex

sh

ec

ir

1

098

minus02

minus004

001

1

minus014

minus001

minus003

1

026

minus028

1

minus057 1

ＪＬＩＪ

Ｒ

Figure 8 Significant correlation estimates for themetrics of interestin the mixed opinion dyamics scenarios

accepted proposal argument andor the opinionmajority thatis formed from previous dyadic interactions (see Figure 7(f))

422 Relating the Metrics of Interest in Mixed OpinionDynamics The first thing to notice is that for all of thesescenarios all the metrics are significantly correlated (seeFigure 8) The variance of opinions and the proportion ofextremists are almost perfectly correlated To some extentthis means that a high variance of opinions known thatthe mean of opinions is statistically null equates to havingmany extremists in the population be towards one or theother side of the opinion distribution (eg extreme bipolar-ization) Similarly we find a negative and strong correlationbetween the judgment inaccuracymeasure and the coherencecoefficient (see Figure 8)mdashwhen judgment inaccuracy is high(low) coherence to deliberation is low (high) Cases likethese are telling since they may be related to sequences ofcollective decisions in which agents do not make too manymistakes when judging proposal arguments and still makedeliberated decisions that are on average in line with theirprinciples

The shift statistic interestingly is negatively correlatedwith the variance of opinions with judgment accuracy andwith the coherence coefficient In a way this means thathaving more moderate agents conduces on average to moremistakes in judging arguments and to less coherent collectivedecisions An explanation for this phenomenon could be thatas agents reach consensus in opinion important (possiblyextreme) arguments for correctly judging argument propos-als are not advanced in the deliberation arenawhich results inmore mistakes and in possibly accepting proposal argumentsthat after voting are not meant to be accepted Indeed the

negative correlation between the variance of opinions andjudgment inaccuracy points towards this direction since onaverage the more shifts there are the smaller the variance ofopinions is (see Figure 8)

423 Comparing Mixed Dynamics with Other MonolithicDynamics of Interactions Qualitatively opinion trajectoriesin the mixed model are more chaotic than in the pair-wise and deliberative interaction models (see Figure 7) Asanalyzed previously deliberation dynamics disrupt the pair-wise dynamics and create situations in which central andmulticlustered convergence (with two clusters at the extremesof the distribution) are possible at the expense of more(pair-wise dynamics) or less (deliberative dynamics) clustersIt tolerates the survival of very small isolated group ofagents that the previous models rarely tolerate in a hundredcollective decision-making steps (see Figure 7(c))

One of the most interesting observations one can makeof this family of models regards how groups change (in termsof our metrics) when dyadic social interactions are addedto deliberative interactions and vice-versa When comparingthe mixed interactions with only deliberative ones shiftsagentsrsquo aggregated change in opinions are on average lesspronounced The variance of opinions and the proportionof extremists are significantly higher for the former than forthe latter group coherence in contrast is lower with respectto the scenarios where opinions are formed only throughdeliberation Judgment accuracy follows that same trendA reason for this might be that mixed dynamics are fasterto yield semistable clusters andor opinion convergence Inconsequence the potential argument pool for deliberativeexchange shrinks faster and deliberation becomes less effec-tive over time which results in more collective mistakes Theoutcomeon coherence is predictable inasmuch as deliberativeinteractions produce group polarization If a group agrees inopinion debates are sterile because themembers of the groupwill most likely refrain from attacking the proposal argu-ments they all support and will work together to disparagethe proposal arguments they do not support (see Table 5)

In the same manner we report that the proportion ofextremists and the variance of opinions are barely lower inthe pair-wise interaction model when including deliberationin the decision-making process Shifts of opinions are alsobarely lower when there is no deliberation This can indicatethat deliberation does not undo the changes in opiniondue to pair-wise interactions nor does it amplifies them Incontrast when we consider the labeling-based metrics themixed interaction model ensures lower judgment inaccuracybecause of deliberation and lower group coherence (seeTable 5)

In sum the mixed discussion model is a compromisebetween the pair-wise interaction and deliberative opiniondynamic models in that it offers some guarantees in termsof judgment accuracy and coherence while allowing forldquoreasonablerdquo variance in the opinion pool It establishes somekind of trade-off between labeling-based metrics and thevariance of opinions that may be of interest for decision-making process design

20 Complexity

Table 6 Summary of sign or relative marginal effects for all varying parameters on dyadic deliberative and mixed interactions models inand for all experiments

Metrics Parameters119905119863 119898 119899119863 120572 119879 119880 119901119886 120574 119896 119861119890ℎ119886V119894119900119903119881119886119903(119900) XX X X O X X X X X X119901119903119900119901119890119909 XX X X O X X X X X X119878ℎ XX X X O X X X X X119890119888 XX X X XO X XOO XOO X X119894119903 XX X X O XO O X OOX XOO X XNota gray symbols correspond to dyadic dynamics light gray to deliberative dynamics and bold black to mixed dynamics Given a scenario the symbol 119883stands for constant parameter and 119874 for insignificant marginal effect and as parameters grow for positive (negative) marginal effect for the parameterFrom left to right at the parameter level the leftmost arrow corresponds to the scenario with the strongest (in absolute value) marginal effect of the parameter(with respect to to its factors) Example Let 119901 be a parameter and 119874 stands for ldquo119901 has a significant positive effect in dyadic discussion scenario negativein the deliberative one and insignificant in mixed discussion The positive effect in dyadic discussion is greater than the absolute value of the negative one inthe purely deliberative scenariordquo

424 Comments on the Differences between the Three Modelsfor OLS Regression Estimates The significance of most of theregression parameters estimated for the mixed interactionsscenarios are similar to those estimated for the pure dyadicanddeliberative scenariosNevertheless themagnitude of theeffects is different insomuch that the dynamics themselvesare different So instead of commenting the linear regressionestimates of the mixed model we compare when possiblethe standardized significant coefficients of the regressors ofthe mixed model with those of the pair-wise and deliberativeinteractionmodels (see Table 6 for a summary of the results)

In mixed interactions for instance the influence of119898 onthe opinion distribution is much more preponderant than inthe deliberative dynamics scenarios Going from 119898 = 1 to119898 = 3 increases the standard variation units of the variance ofopinions by twice as much as the one in the deliberative sce-nariosThis in particular is due to the design of the decision-making process If 119898 is higher then the number of socialinfluence steps (119905119863) in the decision process is also higher(compared to 119905119863 = 0 in the deliberative case) If we decideto consider these steps as contributing to the variance ofopinions then the effect of119898 on the variance of opinions hasto be stronger in the mixed interactions model In contrastthe number of agents participating in the debate has a weakereffect on the variance of opinions in the mixed interactionsmodel than in the deliberative dynamics specially for 119899119863 =005 (more than twice as much) The difference may comefrom the fact that in mixed discussions the part of theargument pool used in deliberation is more restrictive thanin the deliberative case We can argue that since pair-wisedynamics may get individuals closer to one another aftersome deliberation the same happens to the arguments thatindividuals will most likely use in deliberation Taking thisinto account means that deliberation will at many instancesbe unfruitful and have no effect on opinions thus failing tomoderate extreme opinions andor polarize the group In thesame direction a high number of participants in deliberationcan make the deliberation dynamics stiff In this case thedistribution of opinion is close to the uniform distributionwhich has a relatively high-to-moderate variancewith respectto the reference results

The strength of the deliberative dynamics has a positivestronger effect on the variance of opinions in themixedmodelthan in the deliberative model when 120574 = 02 and a negativeweaker one when 120574 = 01 A sound interpretation of thisresult is that when deliberation is successful and its effect onopinions does not get agents to move sufficiently away fromtheir current positions agents are pulled or pushed back totheir previously held opinions the variance of opinions doesnot changemuchOn the other hand if the effect of successfuldeliberation is strong opinions can change sufficiently tomake agents much more prone to assimilation into moderateopinion foci the variance of opinions falls

The deliberative opinion dynamics model is much moresensitive to 120572 than the mixed discussion model Differ-ences in marginal effects can double The probability ofattraction to deliberated results (119901119886) for instance has astrongmoderating effect (negative) on agentsrsquo opinions in thedeliberative opinion dynamics but a meager one (two-fold)in the mixed and dyadic dynamic models This is becauseopinion updates in the deliberative model depend only onthat parameter whereas in the mixed model more numeroustypes of interactions make opinion updates possible

In terms of judgment accuracy all parameters except thesize of the argument sacks (119896) have a significant slightlystronger effect on the deliberative dynamics than in themixedmodel In effect the difference may be attributed to theupshot of pair-wise interactions on the distribution of opin-ions and on the arguments that would be advanced duringdeliberation The result respecting parameter 119896 however iscounterintuitive Since argument sacks are static scenarioswhere shifts are higher should make 119896 more important fordeliberation to succeed and it is precisely in the scenarios ofpure deliberative dynamics that we find the biggest opinionshifts This may point to the hypothesis that in deliberativeinteractions agents change opinions substantially but do notoften change their adherence to the principle (go from anegative opinion to a positive one or vice versa) when thereis no group polarization

43 Sensitivity Analysis for the Procedural Parameters ofInterest Weperform a sensitivity analysis of the observationson the parameters of interest in themixed interactionsmodel

Complexity 21

We extend the parameter domains of the pure dyadic anddeliberative scenarios as we make it explicit in Table 2

431 Minimum Number of Debates (119898) We observe that theminimum number of debates has a significant well-observedeffect on all of our metrics excluding group coherence Forthe variance of opinions we can observe that the moredebates there are the bigger the value of the metric isalthough the higher 119899119863 and 119905119863 are the weaker the overalleffect of119898 becomes (Figures 9(d) and 9(a)) Furthermore themarginal effect of increasing minimal debates is decreasingan additional deliberation step requirement increases thevariance of opinions but less and less as it grows 119898 coupledwith 119899119863 has a shy S-shaped effect (see Figure 9(d)) on thevariance of opinions meaning that it has a stronger effectwhen 119899119863 is low which suggests a possible trade-off betweenthe number of arguments accepted in the debate and thenumber of debates required before accepting a proposal

In contrast the shifts in opinion are less and less likelyas 119898 grows and this is independent of the variations of theother parameters Again the effect is marginally decreasingand is only truly significant when 120572 = 05 An explanationto these effects may be linked to the design of the systemFirst the variance of opinions is higher when deliberation isasked for because the more deliberation steps there are thehigher the chances are that the central proposal argumentis deemed unacceptable specially when agents are focusedFurthermoremechanically speaking increasing theminimalamount of debates implies that whenever a decision is to betaken at least 119898 times 119905119863 time steps have to take place and if atany moment a proposal argument is considered undecided119905119863 time steps are added to the process So unless the debatedoes not yield 119906119899119889 labels for proposal arguments (highlyunlikely considering that 120590 = grounded semantics) the morenondeliberation steps there are in the decision process thehigher the variance of opinions is Concerning the shiftswhen 120572 = 05 either the system is too stiff to accept anyproposal argument and opinions do not changemuch or theeffects of pair-wise discussion and deliberation cancel out insuch a way that individual shifts are minimal (Figure 9(b))

On the side of labeling-based metrics the more debatesare asked for the more accurate a group is in its judgmentthe effect being smaller as 119898 grows When agents are naivethe effect is quasilinear while when they are focused thestrongest effects of acquiring more deliberation are foundwhen levels of deliberation are already low (Figure 9(f))Thiscan be explained by the fact that the more debates thereare in the decision process the closer one gets to the idealargumentation framework

432 Proportion of the Population in Deliberation Steps (119899119863)Like for 119898 119899119863 has a significant effect on the proportionof extremists and on the shifts and variance of opinions(Figure 9(e)) This may result from the fact that being ableto put more arguments in play at the same debate step mayincrease the odds of revealing cycles around the proposalargument in119860119865120576 Given that we use grounded semantics thearguments in the cycles are labeled 119906119899119889 and in consequence

debates are more often postponed when 119899119863 is higher Post-poning debates in turn increases the number of nondebatessteps in the decision process which increases the varianceof opinions and limits over time the moderating effect ofdeliberation Moreover the effect of this parameter is verydependent on the value of 120572 (Figure 9(e)) For shifts forinstance 120572 = 05 makes the effect of 119899119863 negative while120572 = 0 makes it positive but to a lesser extent Furthermorefor higher requirements of deliberation (119898) adding moreindividuals to the deliberation process has weaker effectson the variance of opinions and on the other metrics It isalso quite interesting to notice that it has the same effect onagents independently of agent behavior during deliberationThis is a surprising result as one would have expected morefocused agents in a deliberation arena to cause a raise inthe proportion of extremists as they play to knock outopposing proposal arguments and thereof reducing the effectof deliberation (lowers variance) on the whole population

Finally similar to119898 addingmore people into the deliber-ation process increases judgment accuracy (see Figure 9(c))and has no clear effect if any on group coherence (seeFigure 9(i))

433 Steps between Deliberation Steps (119905119863) In all configu-rations 119905119863 increases the proportion of extremists (variance)and decreases the shifts of opinions among agents Theshifts and the effect on the variance of opinions are presentfor 120572 = 05 insignificant otherwise (Figure 9(b)) 119905119863 ishighly linked to 119898 by construction When 119898 = 1 thecurve linking the variance of opinions and 119905119863 is convex As119898 increases the curve becomes more and more concavewhich means that 119905119863 has a more important effect on theopinion distribution as collective decisions take longer to beachieved This seems counterintuitive but in reality it reflectsthemultiplicative relationship between deliberation and pair-wise interactions and the semantics in the model If 119898 is lowand 119905119863 is high the effective number of pair-wise interactionsis on average smaller in the deliberation process Hencethe structure of the decision-making process hinders anyincrease in the variance of opinions If119898 is high the oppositeeffect is observed Additionally since the grounded semanticsyields few 119894119899 arguments with respect to other admissibility-based semantics getting closer to the ideal argumentationframeworkmaymake it difficult to obtain the expected effectsof deliberative discussion on agents Lower 119898 constrains thevariance of opinions by making deliberation more influentialon opinions

In terms of judgment accuracy and coherence we observethat 119905119863 does not explain coherence yet it tends to decreasejudgment inaccuracy This may be due to a more variedargument pool in the debate which in turn may be aconsequence of a higher variance of opinions or vice versa

434 Acceptability Voting Quota (120572) By far it is the mostinfluential parameter in our study It changes the directionand the intensity of the effect of all the other proceduralparameters and by construction heavily constrains the roadto accepting a proposal (see Figure 9)

22 Complexity

015

020

025

030

035

040

1 2 3 4 5 6

12345

Values of Ｇ

(a) 119905119863 times 119901119903119900119901119890119909 by119898

0

10

20

30

40

50

1 2 3 4 5 6

005066

Values of

(b) 119905119863 times 119878ℎ by 120572

040

045

050

055

060

1 2 3 4 5

001002003004005

Values of Ｈ＄

(c) 119898 times 119890119888 by 119899119863

015

020

025

030

035

040

1 2 3 4 5

001002003004005

Values of Ｈ＄

(d) 119898 times 119901119903119900119901119890119909 by 119899119863

0

10

20

30

40

50

001 002 003 004 005

00005066

Values of

(e) 119899119863 times 119878ℎ by 120572

040

045

050

055

060

1 2 3 4 5

BehaviorNaiveFocused

(f) 119898 times 119890119888 by 119887119890ℎ119886V119894119900119903

015

020

025

030

035

040

0 05 066Behavior

NaiveFocused

(g) 120572 times 119901119903119900119901119890119909 by 119887119890ℎ119886V119894119900119903

05

06

07

08

09

10

1 2 3 4 5 6

00005066

Values of

(h) 119905119863 times 119894119903 by 120572

08700

08725

08750

08775

08800

001 002 003 004 005

BehaviorNaiveFocused

(i) 119899119863 times 119894119903 by 119887119890ℎ119886V119894119900119903

Figure 9 Curves of mean observations for different parameters and metrics for 36000 runs Bars are 095 confidence intervals Plots withthe same observables (ordinate) except for 119894119903 are on the same scale

Complexity 23

The higher the requirement for accepting the proposal isthe higher the proportion of extremists and the variance ofopinions are and the lower the shift statistic is In few words120572 constrains the world to dyadic discussion or throws it intoa process in which deliberation is much more importantthan dyadic interactions This is why for any 120572 = 05 oneeither gives too much weight to deliberated results whichshadows pair-wise interactions or too little weight in thesense that no deliberated result is ever accepted and thusnever integrated into the opinions of the agents that havevoted for it The reason for the latter is that the latitudeacceptance is too low and thereafter pair-wise interactionsare not enough to unevenly polarize the population in such away that deliberated proposal arguments are voted favorablyby a non-negligeable majority Shifts for 120572 = 0 are stable fordifferent levels of 119905119863 which implies that deliberation keeps incheck all the shifts of opinions related to an increase in thenumber of pair-wise interactions For 120572 = 05 deliberationis less likely to be successful and shifts tend towards the shiftlevels for 120572 = 0 (see Figure 9(b))

Concerning labeling-basedmetrics120572 entirely determinesthe coherence statistic For 120572 = 05 there is no difference incoherence because of how coherence is defined it is maximalwhen 120572 = 0 and strangely maximal when 120572 = 066(see Figure 9(h)) Surprisingly 120572 has no effect on judgmentaccuracy One would have expected a higher 120572 to increasejudgment accuracy since agents would hardly collectivelyaccept a proposal argument discussed during deliberationProposal arguments have given how the argument lattice isgenerated a higher probability of being rejected than of beingaccepted in the ideal argumentation framework

435 A Word on Focused and Naive Agents Focused andnaive specifications for agent behavior are an importantparameter in the decision process since they model howagents choose arguments in the deliberation arena In the sce-narios where agents are focused agents show equal or highervariance of opinions and are therefore more extreme (seeFigure 9(g)) independently of the voting requirements Shiftsof opinion are less likely as well We can see this happeningbecause focused agents knock out proposal arguments moreoften than naive agents provided that 119899119863 is low

One could have expected naive agents to yield morecoherent decisions since they argue sincerely However thisseems not to be the case (see Figure 9(i)) Agents are unable todeliberate in away thatmakes deliberation reflect their votingintentions either because they do not have the necessaryarguments to do so or because such arguments do not exist

In terms of judgment accuracy focused agents do betterthan naive agents in finding the ideal labels for the proposalarguments This is likely because when reconstructing theframework focused agents take into account the deliberatedproposal argument and choose themost pertinent argumentsin their sacks that relate to the state of affairs at the deliber-ation arena In fine debates result in a better approximationofL119886119887120576(119868) even though one could have believed the oppositethat is focused agents are ready to sacrifice a correct collectivelabel of a proposal argument in the goal of defending theirpositions

44 Points of Discussion In this subsection we discuss theresults obtained from the simulations We attempt to makesense of them from a social sciences perspective

441 Opinion Consensus and Dissensus The deliberativeopinion dynamics corresponds to the simulated scenarios inwhich the variance of opinions is at its lowest and judgmentaccuracy at its highest Dyadic opinion dynamics are associ-ated with the scenarios in which the variance of opinions is atits highest judgment accuracy at its lowest and coherence atitsmaximumpossible levelThemixed interactions dynamicsis a combination of these aspects for all the metrics exceptfor coherence at which it is the worsemdashdeliberation leadsto some kind of loose opinion consensus that is reinforcedby pair-wise interactions Likewise the mixed discussionmodel confirms some results exposed in [16] expressingthat opinions and voting intentions in a deliberative contextldquooften changerdquo That being said and more particularly indeliberative opinion dynamics outputs align well with twointeresting results on consensus formation given by aninterpretation of Moscovici and Doisersquos [15] work Sunsteinrsquos[14] account on deliberation and group polarization andsocial impact theory [4] The first states that mild consensusis reached in deliberation as procedure disengages agentsthe second is that within discussion groups majorities tendto become larger and opinions more extrememdashnot on thebasis of competing arguments but of the initial balance ofopinions After several decision processes we find opiniondistributions that reflect these observations even though weare not able to precisely coin the set of parameters that yieldssuch distributions (see Figure 7)

Although differences between mixed and dyadic opiniondynamics in terms of the variance of opinions and shiftsare slim the former does much better in terms of judg-ment accuracy In this sense deliberation does moderateopinions but not as much when non-deliberation steps arerelatively frequent throughout the decision-making processIn terms of Moscovici and Doise theory of consensus [15]this makes sense since pair-wise discussion engages or ego-involve agents in the discussion and pushes to convergenceof opinion at the extremes of the opinion distributionHowever what remains unclear is the reason why proceduralparameters that constrain deliberation do not disengageagents from discussion and thus result in a mild consensusas also expressed by Moscovici and Doise [15] A possibleexplanation of this phenomenon may stem from the factthat deliberation in the model is always accompanied by aseries of ego-involving discussions that shadowor counter themoderating effect of deliberation Not only this deliberationcould also be responsible for crystallizing the opinions ofextremists for it favors one or the other part of the distributionof opinions as explained in [14] and to some extent in [13]Furthermore the mixed discussion model gives an exampleof polarization that results from a minority of dissenterson one side of an issue acceding to views of the majorityrsquosside [4] as opposed to the strong polarization observed indyadic interactions Situations in which the status quo orthe undecidability of the proposal argument lingers cannotpossibly yield situations of moderation In cognitive science

24 Complexity

one can be inclined to believe that the longer a deliberationprocess is the more likely individuals will take an immutablestance on the situation In view of the assumption that longdecision-making processes can be more costly for a group orsociety as a whole agents may feel forced to take a stancejust for the sake of ending the process The stance that isadopted would therefore depend on pair-wise interactionsamong individuals and not on deliberation since deliberationtakes too long to be conclusive

442 For Hypothetical Recommendations in Decision-MakingProcess Elaboration On another tune results from themodelsuggest that procedural parameters have marginally decreas-ing positive or negative effects on the metrics of interest Ina hypothetical policy analysis perspective this may indicatethat after choosing a correct number of individuals andimposing the right amount of debates one can attain rea-sonable levels of judgment accuracy coherence and varianceof opinions As accounted in [18] deliberation increasesknowledge (collective knowledge as well) and therefore itheightens the chances of making correct decisions If onebelieves that social welfare is linked to deliberation andeventually to the metrics studied previously then proceduralparameters can both be used as an instrument to attaindesired levels of social welfare and reinforce the notionof legitimacy of collective decisions It can also provide ajustification for choosing a deliberation regime rather thananother on the basis of how important (or urgent) a topic ofdiscussion is Indeed if we adopt the claim that deliberationis a forerunner of welfare then any state in which deliberativecues are at their maximum (highest number of participantsmany deliberation steps before making a decision and asfrequent as possible) has to be mapped to the highest attain-able social welfare In this case is a situation where judgmentaccuracy is at its highest yet the variance of opinions or theproportion of extremists at its lowest an ideal situation Wecannot say for sure since strong opinion consensus and longdecision-making processes in many situations may result ina loss of welfare rather than in a gain of it

Another interesting result in procedure for deliberationis the very similar estimates we find for the effects of theminimum number (119898) of debates necessary to make adecision on a proposal and the proportion (119899119863) of individualsparticipating in deliberation on judgment accuracy In thescope of social welfare this raises the question of whetheradvocating for longer decision processes by demandingmoredeliberation steps (higher 119898) rather than bigger debates(higher 119899119863) may increase welfare Depending on the objec-tives of the decider of the deliberation procedure he or shemay choose to concentrate on one or the other A recom-mendation we can assert from the mixed social interactionsmodel is that if one prefers low to high variance of opinionsandor high to low coherence the decider is better off ifshe focuses on increasing the size of the debate rather thanorganizing debates very frequently One of the reasons forthis might be that when making bigger debates all agents aretruly considered as equals in the decision-making processIn organizing many debates individuals that are chosen toparticipate in many of these will be more influential and will

somehow bias the set of arguments in the deliberation arenaThemajority rulemodeled through120572 also determines the sizeof the shifts and the distribution of opinions Because a bigger120572 is associated with higher levels of extremism and opinionvariance deliberated cues are harder to accept and thereforeit becomesmore difficult for agents to reach concensus andormoderate their opinions

On the other hand and further into the idea of legitimacythe goal of deliberative processes is the agreement amongall participants This agreement stands for the right or bestdecision from a formal or procedural perspective and it is alsothe best from a substantive point of view In other words thelegitimacy of decisions not only derives fromwell-establishedprocedural requirements (respect of the protocol) but alsobegs for the fulfillment of two essential conditions (1) satisfy-ing the procedural requirements for a correct procedure (for-mal legitimacy) and (2) the rational acceptability of the resultsof this procedure (substantive acceptability) Legitimacy canbe then obtained from the coherence statistic scrutinizedin the model insofar as it measures how well agents acceptthe results of a correct procedure (eg deliberation) So fora proposal to be legitimate one has to choose parametersthat would maximize group coherence in decision-makingprocesses

As deliberation may give rise to consensus in the delib-eration arena it may also sow the seeds of dissensus inthe group Our model clearly illustrates this phenomenonthrough the procedural parameters of deliberation and themetrics observed From another perspective in deliberativedemocracy dissensus in collective decision-making may bedesirable According to Landermore and Pagersquos accounts [36]lack of consensus can be perceived as having agents withalternative ideas that combined with other ideas can providea better approximation of the ideal status of an argumentTheauthors in [36] call this dissensus ldquopositive dissensusrdquo andit makes normative sense when solving complex collectivedecision-making problems Our model captures the ideaof having normative requirements to define a good andlegitimate collective decision requirements of high consen-sual deliberative accuracy (judgment accuracy) of legitimacy(coherence) and of diversity (variance of opinions) are alltaken into account and for such we can conclude that ourmodel is successful in describing a sound process of collectivedecision-making with interpretable outcomes

Of course the present simulation results and analysis donot prove that all opinion dynamics can be accounted forby such simple processes like the one presented here Theydo however shed light on the expressiveness of combiningdifferent paradigms to account formore interpretablemodelsand pave the way to creating original models of the sortThe simulations suggest the desirability of discovering theconsequences of relatively simple laws of communication atdifferent levels (micro meso and macro) to determine whatstill needs to be explained

5 Related Work

We see our model as a contribution to the influence andopinion dynamics field in ABM and a pragmatic application

Complexity 25

of abstract argumentation theory To our knowledge we areunaware of existing literature on agent-based modeling thatexplicitly relates collective choice and the notions of deliber-ation and opinion diffusion through abstract argumentationas we have done it

Most models in the literature of opinion diffusion areinterested in opinions because these have an influence incollective decision and in questions of social order Forinstance in [3] the authors are interested in consensus andin how a group collectively chooses among two alternativesIn other models authors are interested in the emergenceof extremism [8] and on the distribution of opinions whenextremists are introduced in the population [5 33 37] whileother authors coin the notion of opinion polarization as anemergent property of the system [8 12] They show usingmodels of ldquobounded confidencerdquo and opinion diffusion withtrust that three different kinds of steady states (unipolarbipolar and central) are possible depending on whetheragents are sufficiently uncertain about their opinions andsufficiently connected andor a certain proportion of indi-viduals are already extreme Recent articles on informationand opinion dynamics stress the importance of governanceand government intervention in the spread of emotions(opinions) in the frangibility of social consensus systemIn [10] the authors show that government intervention inthe spread of negative emotions can lead to an even fasterspread of negative emotions (single-peeked convergence) andin a faster collapse of the social consensus system Opinionas a function of trust is studied in [12] and relates to howindividuals form their opinions on the basis of how muchthey trust agents in their networks

Another stream of discrete opinion dynamics modelsthat explain the emergence of opinion can be found in theliterature Computer multiagent models of attitude changebased on Latanersquos social impact theory are presented in [4] Inthe theory an agent changes her opinion on the basis of theinformational impact she is subject to which depends on thepersuasiveness (strength) supportiveness and immediacy(group structure) of the environment and information shereceives The simulations predict two emergent groups ofphenomena the shifting of attitudes towards incompletelypolarized equilibria and the formation of coherent clusteringof subgroups with deviant attitudes Similarly but in acontinuous more argumentative fashion Mas and Flache[11] present a model of opinion diffusion using argumentsArguments are considered to be for or against a proposal andare given an agent-dependant relevance in dyadic persuasiveinteractions Arguments also determine the agentsrsquo opinionsand their dissimilarities which posits the rules of pair-wiseencounters in theirmodel as agents are assumed to form theiropinions according to the arguments they own They showthat in an argumentative opinion diffusion model the onlyobservable steady states are bipolarization and consensusand that bipolarization can emerge from interactions amongsimilar agents In other words bipolarization of opinionsis possible with homophily and without negative influenceif there is argumentation They subsequently compare theirresults to experimental data In [38] a similar approach toargumentation and opinion formation is taken but for the fact

that they introduce different types of arguments explicitlyapply social judgment theory [22] and use survey data toestablish model-to-real-world comparisons

Closer to opinion formation and abstract argumentationGabbriellini and Torroni [39] are the first to have origi-nally and soundly merged opinion diffusion and abstractargumentation They define an agentrsquos opinion as a functionof the arguments she holds and the attack relation amongthem They devised a focused peer-to-peer dialogue systemof persuasion (NetArg) inspired from Mercier and Sperberrsquosargumentative theory of reasoning [40] which used onlyabstract argumentation to study opinion polarization andopinion dynamics Moreover they considered networks andthe notions of trust and epistemic vigilance to define adynamics for knowledge revision and trust itself whichwe clearly lack in our model They showed that if a con-servative belief operator in argumentation was applied byagents when they reasoned then their dialogue protocoldid not increase polarization among agents That said theyused the model to study the effect of Groenvetterrsquos weaklink theory in the spreading of arguments and were notparticularly interested in notions like judgment accuracy orcoherence nor in tackling questions related to collectivedecision-making procedures Their system is very expressiveand helps position our work in the litterature our work is atthe frontier of pure argumentative opinion diffusion modelsopinion diffusion with arguments and continuous bounded-confidence models of opinion diffusion a la Deffuant

On another note work on collective cognitive conver-gence [41] and opinion sharing [19] shows that consensustowards a certain ldquocorrectrdquo opinion or cognitive state is alwayspossible yet dependent on noise variability and awarenessof agents In [20] the authors show that learning about anexogenous correct state of the world (represented by bits)under confidence was possible but only if the agents werenot too confident In a homogeneous population they showthat the higher the confidence the worse the learningmdashthevery confident agents do not learn the properties of the truestate of the world and disrupt the learning process of the lessconfident ones Collective cognitive convergence can be seenas a result of deliberation in truth-seeking models

When it comes to abstract argumentation theory we takean approach that wires two types of dialogues that are well-studied in the literature persuasion dialogues [42] and delib-eration dialogues [43] Another interesting line of work is theone on mechanism design [44] or the problem of devisingan argumentation protocol where strategic argumentation isnot a liability for success in debates We tackle mechanismdesign in a different way though Instead of consideringstrategy-proofness we are interested in how differences inprotocol can result in epistemologically ldquobetterrdquo collectivechoices and guarantee that opinion distributions are favorablefor deliberation For a survey on persuasion dialogue see[42]

Work on agent-based argumentation usually assumes thatthe semantic relationship between arguments is fixed Inother words if two individuals were to put two argumentsin the public arena such that one logically attacked the otherthen everyone would agree that such attack exists [45 46]

26 Complexity

Other models which do not make this restrictive assumptioncan also be found in the literature [27 47] and derive fromthe class or family of opponent models [27 48] in which twoopposing sides attempt to win a dialogue Ourmodel is in theintersection of these two but the frame combining opiniondiffusion of the kind and argumentation remains new

The idea of mixing interpersonal influence and verticalcommunication however is not original It is described andimplemented in innovation diffusion models such as [5 49]In both vertical communication is modeled as exogenoustransparent informationmdashagents are aware of the existenceof an innovation thus triggering several processes of choiceand stabilization of opinions Also in [50] an Eulerian modelis implemented to show the effect of media and exogenousinformation on opinion distributions With respect to thispoint the originality of our work is in that the informationemitted as vertical communication is endogenous It is issuedfrom a deliberation model that agents shape on the basisof their opinions arguments and behavior In the spirit of[51] where the authors control for the design of verticalcommunication (to whom the vertical communication isaddressed its timing how it influences agents opinions) wecontrol for the process generating the information for ourown set of variables length of debates majority quotas andthe frequency at which discussions take place

6 Conclusion and Perspectives

The main objective of this article was to build a bridgewhere decision-making argumentation and opinion dif-fusion could come together We proposed a model thatcombined abstract argumentation theory and a boundedconfidence opinion diffusion model and showed to whatextent it could explain variance of opinions extremismcoherence in collective decisions and judgment accuracyon arguments given an ideal state of full information Thesecond objective was to show in what way governance pro-viding agents with arenas of discussion and deliberation wasimportant in addressing the success of collective decision-making processes and the quality of their outcomes

The model revealed that allowing for more deliberationtime though in lower frequency and allowing for widerparticipation in deliberation increased the variance of opin-ions and the proportion of extremists in a group Theseobservations are consistent with the combination of resultsfound in [9] and in [13 14] which stress that deliberationmay polarize groups and may have a meager effect on theshifts of opinions and inconsistent with [16] where it isargued that opinions tend to moderate after deliberationIt happens that deliberation alone did moderate opinionyet when integrated into a more complex system in whichindividuals were allowed to interact with one another andnot all that was deliberated was accepted its influence wasshadowed by other more individualistic dynamics

Undeniably the grounded semantics played an importantrole in the weak effect of deliberation since it models askeptical way of reasoning over arguments For agentsaccepting arguments thus updating their opinions accord-ingly happened not too often Nevertheless asking for more

deliberation did increase judgment accuracy as observed in[18] yet in a marginally decreasing fashion We showed thatvoting within the deliberation protocol not only increasedthe proportion of extremists and the variance of opinions ina group but also determined how coherent deliberation andvoting procedures were with each other Lastly we showedthat agents that are focused judged arguments better hadmore stable opinions and constituted a group with higherproportion of extremists than their naive counterparts

In terms of governance the model says that there is per-haps no trade-off between extremismand judgment accuracyInstead it asserts that the higher the variance of opinions thecloser the group gets to the correct decisions This situationmay be interpreted as Landermore and Pagersquos notion of ldquoposi-tive dissensusrdquo (in complex collective decision-making tasks)[36] as opposed to consensus as a normative requirementfor correct collective decision-making If a decider has toorganize deliberation to legitimize a public policy dependingon what kind of world he or she wants different set ofparameters may be chosen to account for different levels oflegitimacy and correctness For instance one may ask for adecision to be legitimate to respect a certain level of coherenceor of judgment accuracy and to be discussed by a sundryof different-opinionated peoples If so deliberation has tobe made more often more participants have to be includedin debates or in deliberation instances and not too manytime steps should be introduced in between two deliberationinstances So the model may capture normative ideas todefine correct and legitimate collective decisions and tomake recommendations on those grounds Requirements ofhigh consensual deliberative accuracy (judgment accuracy)legitimacy (coherence) and diversity (variance of opinions)are all taken into account and for such a reason the model israther successful in describing a sound process of collectivedecision-making with interpretable outcomes

61 Extensions of the Model and Other Ideas Our model canbe extended in many ways Value-based abstract argumen-tation frameworks (VAFs) for example as an extension ofDungrsquos argumentation framework provide a formal descrip-tion of the process of decision-making in which argumentsare given values and audiences (set of agents) ignore attacksbetween arguments on the basis of their preferences over val-ues Social abstract argumentation frameworks (SAFs) alsoprovide an interesting and seemingly convenient formalismfor our model In SAFs agents vote on the acceptability ofarguments and the resulting labeling is a combination of thearguments that are voted for and labeling-based semantics Inour model we apply the same rule yet only for one argumentand given precise agent-based votingmechanisms and votingrules which are eventually parametrized

To consider our model as a specification of a mixed VAFand SAF model may be of great interest for future workThis formalism can provide solid foundations to describeexplain interpret and eventually extend the results foundhere from an argument-based perspective On those groundsfuture work will focus on creating a hybrid argumentationframeworkmdashwith its semanticsmdashthat takes heed of bothvalues or principles and voting

Complexity 27

Concerning other possible extensions of the model anatural reflex would be to argue on the validity and therobustness of the mix between the bounded confidenceopinion diffusion models and deliberation Consequentiallyadding deliberation to different bounded confidence modelsof opinion diffusion such as the Hegelsmann-Krause model[37] or to models of social impact theory like in [4] canbe of great interest for future work We acknowledge that afiner understanding of argumentation frameworks randomlattices and their respective applications for this modelare necessary Testing for different labeling-based semanticsor even trading labeling-based semantics for scoring-basedsemantics (semantics that defines acceptability on scoresgiven to arguments) such as debate semantics and ranking-based semantics (semantics that yield a preorder) may beinteresting to consider in future work

More into modeling deliberation it is not inadequateto believe that agents may also advance proposals duringdeliberation and replace to some extent the proposal forwhich the deliberation is taking place Endogenizing theprocess that produces the type of argument to be deliberatedon may be interesting to explore either by having agentsstrategically replace proposals at certain moments duringthe debate or having the central authority attempt to chooseproposals such that the probability of having them acceptedis high Agents may also be thought to be autoorganizing(independent of a central authority) and concede on the sizeof the instances of deliberation To consider differences insemantics across agents or even endogenizing the semanticson how urgent or important the proposal that is beingdiscussed is can also be an interesting direction to take

Last but not least many nontrivial modifications of ourmodel are possible and most should include better-thoughtdeliberation protocols It may be interesting to design andobserve protocols in which deliberation only affects agentsthat are actually debating for example Trust network effectsmultidimensionality of opinions new processes of argumentexchange or learning among agents are notions to furtherdevelop in order to make the model more realistic and relaxsome unreliable assumptions In sum the model is to berefined and extended with the objective of either studyingconcrete cases of deliberative polling and opinion dynamicsor implementing more intuitive thought experiments

Appendix

We present below some of the pseudocode we used toimplement the model

Algorithms

(a) Algorithm 1 implements the dyadic influence in themodel It describes how two agents interact andupdate their opinions

(b) Algorithm 2 implements the deliberative influence inthe model which describes how agents update theiropinions in sight of the accepted deliberated proposalarguments

(c) Algorithm 3 is used to obtain the grounded labelingof an argumentation framework

(d) Algorithm 4 provides an overview of a run in thesimulations119872 is the number of proposal argumentsto discuss which we set to 100 in the model

Each object in the system is indexed by a119908ℎ119900 number Hencea set of agents is annotated by a set of natural numbers Weuse this fact in the implementation of the algorithm 119874119887119895119890119888119905119894denotes the object of 119908ℎ119900 number 119894 Agentsrsquo opinions (119900) arecalibrated so that 119900 isin [minus1 1]Data Availability

The program and data used to support the findings of thisstudy are available from the corresponding author uponrequest

Disclosure

A preliminary version of this work was presented anddiscussed at the conference Bridging the Gap between FormalArgumentation and Actual Human Reasoning on October 4-5 2018 at BochumGermanyThe authors thank the audiencefor their valuable comments and suggestions

Conflicts of Interest

The authors declare that they have no conflicts of interestregarding the publication of this article

Acknowledgments

The contribution of Gabriella Pigozzi was supported by theDeutsche Forschungsgemeinschaft (DFG) and the CzechScience Foundation (GACR) as part of the joint project FromShared Evidence to Group Attitudes (RO 45486-1)

Endnotes

1 In other words the superiority of the ldquoright answerrdquo orlabeling in our case will appear as such to all

2 All agents hold the same opinion3 By sensitivity we mean how much an agent is ready to

change her opinion on the sole fact that an informationalcue has been debated on

4 The assumption is strong but this trust among agentsmay be rooted from a genuine individual motivation toreach a democratic consensus

5 We adopted a rule of thumbs that says that if an agentis neutral with respect to the principle then she willvote for any proposal that is presented to her duringdeliberation

6 If the argument is not unique she chooses one atrandom

7 If there are more than one candidate the agent will favorthe argument at minimal distance from 119868

28 Complexity

Require Vector of values for parameters (119881 = (120583 119880 119879))Ensure Successful interactions among agents119896 larr997888 0119873119890 larr997888 0

while 119896 le |119873|2 do ⊳Fill119873119890 with |119873|2 agents119896 larr997888 119896 + 1119873119890 larr997888 119877119886119899119889119900119898(|119873| + 1) ⊳Pseudo-random generator Generates integer between 0 and |119873|119873119903 larr997888 0 120575119894119895 larr997888 0 ⊳Variable to store differences in opinionfor all 119894 isin 119873119890 doRandomly pick a number 119895 isin 119873(119873119890 cup 119873119903)119873119903 larr997888 119873119903 cup 119895 120575119894119895 larr997888 |119900119894 minus 119900119895|procedure dyadic-update(119894 119895) ⊳Two agents 119894 and 119895 discuss and update opinions1199001015840 larr997888 119900119894 (120583119895 119880119895 119879119895) larr997888 (120583119894 119880119894 119879119894) larr997888 (120583 119880 119879) ⊳Save agent irsquos opinion before updating and homogenize agents

if 120575119894119895 lt 119880119894 then119900119894 larr997888 120583119894119900119895 + (1 minus 120583119894)119900119894if 120575119894119895 gt 119879119894 then119900119894 larr997888 (1 + 120583119894)119900119894 minus 120583119894119900119895if 120575119894119895 lt 119880119895 then119900119895 larr997888 1205831198951199001015840 + (1 minus 120583119895)119900119895if 120575119894119895 gt 119879119895 then119900119895 larr997888 (1 + 120583119895)119900119895 minus 1205831198951199001015840

Algorithm 1 Dyadic social influence

Require Vector of values for parameters (119881 = (120574 119901119886 119901119903))Ensure Successful interaction between proposal arguments and agents

for all 119894 isin 119873 doprocedure Probability-Update(V119886 119901119886 119901119903) ⊳Update individual probabilities of change due to deliberated cues119909119894 larr997888 119901|V119886minus119900119894 |2119886 119910119894 larr997888 1199012|V119886minus119900119894 |119903119902 larr997888 0if L119886119887(119868) = 119894119899 then

procedure deliberative-update(119894 119886) ⊳Agent i argument 119886120574119894 larr997888 120574 119902 larr997888 119906 sim U(0 1) ⊳ 119906 is a realization of a uniform random variableif 119902 lt 119909119894 then119900119894 larr997888 120574119894119900119894 + (1 minus 120574119894)V119886break

if 119902 lt 119910119894(1 minus 119909119894) then119900119894 larr997888 (1 + 120574119894)119900119894 minus 120574119894V119886Algorithm 2 Deliberation influence

8 In terms of computational complexity naive agentsmaketheirmoves in119874(|A119894| log(|A119894|)) while focused agents doat least as bad

9 Tables have memory Therefore debates are not neces-sarily trees

10 If there are less than (119899119863 times |119873|)2 agents for any of bothgroups the CA summons all of them to the deliberationtable

11 Or careful consideration of the viewpoints of otherswhich implies that citizens keep an open mind and donot reject arguments outright

12 A ldquohealthyrdquo consensus refers to the case in which manyconsensual decisions are taken but opinion-wise agentsdo not agree with each other Quotes are used on theword healthy because some political science theorists

(eg [36]) believe that in certain situations agreement inopinions for deliberation is epistemologically damaging

13 Simulations on groups of |119873| = 800 and |119873| = 400agents shows that scaling the population has an effect onall themetrics but it is confounding depending onwhichcouples of attitudes structures are considered (119880 119879) Westick to |119873| = 400 as in [9] for comparability andsimulation running time

14 By balanced wemean with as many arguments with V119886 lt0 as with V119886 gt 0

15 Experiments on the system suggest that a bigger(smaller) argument pool 119872 = 1000 (119872 = 200) isassociated with a higher (lower) variance of opinions(and extremism) and higher (lower) judgment accuracyFor the rest of the observations 119872 seem to have noeffect We choose a middle value for 119872 Similarly theargument pool chosen to be balanced or uniformly

Complexity 29

Require Argumentation framework 119860119865 = (AR)Ensure Assignment of labels to arguments from the grounded labeling of 119860119865L(A)

L0(A) larr997888 (0 0 0)119896 larr997888 0do119894119899 L119896+1(A) larr997888 119894119899 L119896(A) cup 119886 | 119886 is not labeled 119894119899 L119896(A) 119886119899119889 forall119887 isin A 119904119905 119887 R 119886 then 119887 isin 119900119906119905 L119896(A)119900119906119905 L119896+1(A) larr997888 119900119906119905 L119896(A) cup 119886 | 119886 is not labeled 119894119899 L119896(A) 119886119899119889 exist119887 isin A 119904119905 119887 R 119886 and 119887 isin 119894119899 L119896+1(A)

while L119896+1(A) = L119896(A) = 0Algorithm 3 Solve argumentation frameworks (with grounded semantics)

Require Vector of values for parameters (119881)Ensure Vector of statistics (119880)

procedure Set-parameters(119881) ⊳Initialize parameters for scenarioprocedure Init-objs(119899119898 119896 ) ⊳Create119898 arguments 119899 agents and 119860119865120576procedure Solve-Argumentation-Framework(119860119865) ⊳Assigns the epistemic labelingL120576 toA119894 larr997888 1119905 larr997888 1while 119894 le 119872 do ⊳ 119872 the number of arguments to discuss119868119894 larr997888 119866119890119899 119886119903119892(1) ⊳Generates proposal argument 119868119860 119894119905 larr997888 (119868119894 0)L119886119887(119868119894) larr997888 119906119899119889 ⊳Initialize label for 119868119895 larr997888 1 ⊳Initialize debate counter 119895while ((L119886119887(119868) = 119906119899119889 or 119898 le 119895) and 119895 le 119898) doif 119905 equiv 0(119898119900119889 119905119863) then

procedure build-argumentation-framework(119899119863 119887119890ℎ119886V119894119900119903) ⊳Build the argument graph agents deliberate(119861119894119905 119878119894119905) larr997888 (119883 119884) sub (⋃119897isin119873119899119863 A119897119894119905 ⋃119897isin119873119899119863 R119897119894119905) ⊳ 119873119899119863 sub 119873 are agents sampled by the CA119860 119894119905 larr997888 119889(119860 119894119905 (119861119894119905 119878119894119905))procedure Solve-Argumentation-Framework(A119889) ⊳FindL(A119889)L119886119887(119868119894) larr997888 L119886119887119863(119868119894)119895 larr997888 119895 + 1

else119896 larr997888 1while 119896 le 119905119863 doprocedure dyadic-social-influence(120583 119879 119880) ⊳Agents discuss one-to-one119896 larr997888 119896 + 1119905 larr997888 119905 + 1

if 120572 gt 0 andL119886119887(119868119894) = 119900119906119905 then ⊳Agents vote for the proposalif (|119897 isin 119873|119900119897 times V119868119894 ge 0| ge 120572 times 119899) then

L119886119887(119868119894) larr997888 119894119899else

L119886119887(119868119894) larr997888 119900119906119905if L119886119887(119868119894) = 119894119899 thenprocedure Deliberation-influence(120574 119901119886 119901119903) ⊳Agents are influenced by the result of the decision process119894 larr997888 119894 + 1119905 larr997888 119905 + 1119880 larr997888 119862119900119898119901119906119905119890 119878119905119886119905119894119904119905119894119888119904(119881)

return 119880 ⊳Get the statisticsAlgorithm 4 Run for one scenario

generated does not seem to play a significant explanatoryrole on any of the metrics

16 Experiments show that themaximumnumber of debatesis not an important parameter even if it is an intuitiveone when considering decision-making processes Inpractice debates end after at most 119898 + 1 deliberation

steps and even then it happens rarely However to avoidunpleasant surprises like infinite debates on a completeargument graph containing the proposal argument weadded the constraint 119898 Its value is precisely 7 becausethe biggest minimum number of debates we studyis 6

30 Complexity

17 To ensure comparability with the deliberative andmixedopinion dynamics models on judgment accuracy andcoherence agents vote on proposal arguments every 119905119863times(119898+1) 119905119863 and119898does not explain variability in judgmentaccuracy nor in coherence

18 Student t-tests for independent samples are robust tothe violation of two hypothesis on the distribution ofthe samples normality and homogeneity of variancesas long as the sample size is big enough (central limittheorem) and if the difference in size of the comparedsamples is smallrespectively

19 Studentrsquos test version forwhen variances are not assumedequal and the difference in size of the compared samplesis big

20 We do not correct for homoskedasticity nor normality oferrors in the estimates

21 Balanced as the same number of runs per scenario orsuch that no significant correlations are found betweenparameters

References

[1] R Axelrod ldquoThe dissemination of culture a model withlocal convergence and global polarizationrdquo Journal of ConflictResolution vol 41 no 2 pp 203ndash226 1997

[2] D M Rousseau Reinforcing the MicroMacro Bridge Organiza-tionalThinking and Pluralistic Vehicles SAGEPublications SageLos Angeles Calif USA 2011

[3] S Galam and SMoscovici ldquoTowards a theory of collective phe-nomena consensus and attitude changes in groupsrdquo EuropeanJournal of Social Psychology vol 21 no 1 pp 49ndash74 1991

[4] A Nowak J Szamrej and B Latane ldquoFrom private attitude topublic opinion a dynamic theory of social impactrdquo Psychologi-cal Review vol 97 no 3 pp 362ndash376 1990

[5] G Deffuant S Huet and F Amblard ldquoAn individual-basedmodel of innovation diffusion mixing social value and individ-ual benefitrdquo American Journal of Sociology vol 110 no 4 pp1041ndash1069 2005

[6] G Deffuant D Neau F Amblard and G Weisbuch ldquoMixingbeliefs among interacting agentsrdquoAdvances in Complex Systems(ACS) vol 3 no 01n04 pp 87ndash98 2000

[7] J Rouchier P Tubaro and C Emery ldquoOpinion transmission inorganizations an agent-based modeling approachrdquo Computa-tional and Mathematical OrganizationTheory vol 20 no 3 pp252ndash277 2014

[8] M Meadows and D Cliff ldquoThe relative disagreement model ofopinion dynamics where do extremists come fromrdquo in Pro-ceedings of the International Workshop on Self-Organizing Sys-tems Lecture Notes in Computer Science pp 66ndash77 Springer2013

[9] W Jager and F Amblard ldquoUniformity bipolarization and plu-riformity captured as generic stylized behavior with an agent-based simulation model of attitude changerdquo Computational ampMathematical Organization Theory vol 10 no 4 pp 295ndash3032005

[10] Y-F Zhang H-Y Duan and Z-L Geng ldquoEvolutionary mech-anism of frangibility in social consensus system based onnegative emotions spreadrdquo Complexity vol 2017 Article ID4037049 8 pages 2017

[11] A Flache andMWMacy ldquoSmall worlds and cultural polariza-tionrdquoThe Journal of Mathematical Sociology vol 35 no 1-3 pp146ndash176 2011

[12] A Tsang and K Larson ldquoOpinion dynamics of skepticalagentsrdquo in Proceedings of the 13th International Conference onAutonomous Agents and Multiagent Systems AAMAS 2014 pp277ndash284 International Foundation forAutonomousAgents andMultiagent Systems France May 2014

[13] K M Hansen Deliberative democracy and opinion formationUniversity Press of Southern Denmark Odense 2004

[14] C R Sunstein ldquoThe law of group polarizationrdquo Journal ofPolitical Philosophy vol 10 no 2 pp 175ndash195 2002 WileyOnline Library

[15] S Moscovici and W Doise Dissensions et Consensus unetheorie generale des decisions collectives Presses Universitairesde France-PUF 1992

[16] J S Fishkin and R C Luskin ldquoExperimenting with a demo-cratic ideal deliberative polling and public opinionrdquo ActaPolitica vol 40 no 3 pp 284ndash298 2005

[17] R C Luskin J S Fishkin and S Iyengar Considered Opinionson US Foreign Policy Face-To-Face versus Online Delibera-tive Polling International Communication Association NewOrleans La USA 2004

[18] J Barabas ldquoHowdeliberation affects policy opinionsrdquoAmericanPolitical Science Review vol 98 no 4 pp 687ndash701 2004

[19] O Pryymak A Rogers and N R Jennings ldquoEfficient opinionsharing in large decentralised teamsrdquo in Proceedings of the 11thInternational Conference on Autonomous Agents andMultiagentSystems vol 1 pp 543ndash550 Spain June 2012

[20] J Rouchier and E Tanimura ldquoWhen overconfident agents slowdown collective learningrdquo Simulation vol 88 no 1 pp 33ndash492012

[21] D Miller ldquoDeliberative democracy and social choicerdquo PoliticalStudies vol 40 S1 pp 54ndash67 1992

[22] M Sherif and C I Hovland Social Judgment Assimilation andContrast Effects in Communication and Attitude Change YaleUniversity Press 1961

[23] P M Dung ldquoOn the acceptability of arguments and its funda-mental role in nonmonotonic reasoning logic programmingand n-person gamesrdquo Artificial Intelligence vol 77 no 2 pp321ndash357 1995

[24] M Caminada and G Pigozzi ldquoOn judgment aggregation inabstract argumentationrdquo Autonomous Agents and Multi-AgentSystems vol 22 no 1 pp 64ndash102 2011

[25] P Baroni M Caminada andM Giacomin ldquoAn introduction toargumentation semanticsrdquoThe Knowledge Engineering Reviewvol 26 no 4 pp 365ndash410 2011

[26] S Chambers ldquoDeliberative democratic theoryrdquo Annual Reviewof Political Science vol 6 no 1 pp 307ndash326 2003

[27] E Bonzon andNMaudet ldquoOn the outcomes of multiparty per-suasionrdquo in Proceedings of the 10th International Conference onAutonomous Agents and Multiagent Systems 2011 AAMAS 2011vol 1 pp 47ndash54 International Foundation for AutonomousAgents and Multiagent Systems Taiwan May 2011

[28] J Ranciere Disagreement Politics and Philosophy University ofMinnesota Press 1999

[29] T Bench-Capon and K Atkinson ldquoAbstract argumentation andvaluesrdquo in Argumentation in Artificial Intelligence pp 45ndash64Springer 2009

[30] M Caminada ldquoOn the issue of reinstatement in argumenta-tionrdquo in Proceedings of the European Workshop on Logics in

Complexity 31

Artificial Intelligence vol 4160 of Lecture Notes in ComputerScience pp 111ndash123 Springer Berlin Heidelberg

[31] GAVreeswijk ldquoAn algorithm to computeminimally groundedand admissible defence sets in argument systemsrdquo in Pro-ceedings of the 2006 conference on Computational Models ofArgument Proceedings of COMMA 2006 vol 144 pp 109ndash120IOS Press Amsterdam The Netherlands 2006

[32] C W Sherif M Kelly H L Rodgers et al ldquoPersonal involve-ment social judgment and actionrdquo Journal of Personality andSocial Psychology vol 27 no 3 pp 311ndash328 1973

[33] G Deffuant F Amblard G Weisbuch and T Faure ldquoHow canextremism prevail A study based on the relative agreementinteraction modelrdquo Journal of Artificial Societies and SocialSimulation vol 5 no 4 p 1 2002

[34] E Awad M W Caminada G Pigozzi M Podlaszewski andI Rahwan ldquoPareto optimality and strategy-proofness in groupargument evaluationrdquo Journal of Logic and Computation vol 27no 8 pp 2581ndash2609 2017

[35] D Grossi and G Pigozzi ldquoJudgment aggregation a primerrdquoSynthesis Lectures on Artificial Intelligence and Machine Learn-ing vol 8 no 2 pp 1ndash151 2014

[36] H Landemore and S E Page ldquoDeliberation and disagreementproblem solving prediction and positive dissensusrdquo PoliticsPhilosophy and Economics vol 14 no 3 pp 229ndash254 2015

[37] R Hegselmann and U Krause ldquoOpinion dynamics andbounded confidence models analysis and simulationrdquo Journalof Artificial Societies and Social Simulation vol 5 no 3 2002

[38] A Stefanelli and R Seidl ldquoModerate and polarized opinionsUsing empirical data for an agent-based simulationrdquo inProceed-ings of the Social Simulation Conference 2014

[39] S Gabbriellini and P Torroni ldquoA new framework for ABMsbased on argumentative reasoningrdquo in Advances in SocialSimulation vol 229 pp 25ndash36 Springer Berlin Germany 2014

[40] H Mercier and D Sperber ldquoWhy do humans reason Argu-ments for an argumentative theoryrdquo Behavioral and BrainSciences vol 34 no 2 pp 57ndash74 2011

[41] H V Parunak T C Belding R Hilscher and S BruecknerldquoModeling and managing collective cognitive convergencerdquoin In Proceedings of the 7th iNternational Joint Conference onAutonomous Agents and Multiagent Systems vol 3 pp 1505ndash1508 International Foundation for Autonomous Agents andMultiagent Systems 2008

[42] H Prakken ldquoFormal systems for persuasion dialoguerdquo TheKnowledge Engineering Review vol 21 no 2 pp 163ndash188 2006

[43] K Atkinson T Bench-Capon and P Mcburney ldquoA dialoguegame protocol for multi-agent argument over proposals foractionrdquoAutonomousAgents andMulti-Agent Systems vol 11 no2 pp 153ndash171 2005

[44] M Thimm and A J Garcıa ldquoClassification and strategicalissues of argumentation games on structured argumentationframeworksrdquo in Proceedings of the 9th International Conferenceon Autonomous Agents and Multiagent Systems vol 1 pp 1247ndash1254 International Foundation for Autonomous Agents andMultiagent Systems Canada 2010

[45] I Rahwan and K Larson ldquoMechanism design for abstractargumentationrdquo in Proceedings of the 7th International JointConference on Autonomous Agents and Multiagent SystemsAAMAS 2008 pp 1031ndash1038 International Foundation forAutonomous Agents and Multiagent Systems Portugal May2008

[46] H Prakken R Riveret A Rotolo and G Sartor ldquoHeuristics inArgumentation A Game-Theoretical Investigationrdquo EuropeanUniversity Institute 2008

[47] N Hadidi Y Dimopoulos and P Moraitis ldquoTactics and con-cessions for argumentation-based negotiationrdquo in Proceedingsof the COMMA vol 245 pp 285ndash296 2012

[48] N Oren and T J Norman ldquoArguing using opponent modelsrdquoin Proceedings of the International Workshop on Argumentationin Multi-Agent Systems pp 160ndash174 Springer 2009

[49] S Delre W Jager T Bijmolt and M Janssen ldquoTargeting andtiming promotional activities An agent-based model for thetakeoff of new productsrdquo Journal of Business Research vol 60no 8 pp 826ndash835 2007

[50] A Mirtabatabaei P Jia and F Bullo ldquoEulerian opinion dynam-ics with bounded confidence and exogenous inputsrdquo SIAMJournal on Applied Dynamical Systems vol 13 no 1 pp 425ndash446 2014

[51] V Barbet N Guiraud V Laperriere and J Rouchier Hagglingon Values Towards Consensus or Trouble 2019

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16 Complexity

Table 3 Ordinary least squares estimates for pair-wise interactions scenarios for most of the observed metrics

Dependent variable119881119886119903(119900) 119878ℎ 119890119888 119894119903(1) (2) (3) (4)Constant 0610lowastlowastlowast 4429lowastlowastlowast 0472lowastlowastlowast 0000

(0002) (0206) (0003) (0000)

119880 = 06 (119880 = 02) minus0012lowastlowastlowast 2981lowastlowastlowast 0061lowastlowastlowast 0000(0002) (0260) (0004) (0000)

119879 = 18 (ref 119879 = 14) minus0264lowastlowastlowast minus3261lowastlowastlowast 0004 0000(0002) (0260) (0004) (0000)

120572 = 066 (ref 120572 = 05) 0002 minus0059 minus0100lowastlowastlowast 0000(0002) (0184) (0003) (0000)

(119880 119879) = (06 18) (ref (119880 119879) = (02 14)) minus0324lowastlowastlowast minus3728lowastlowastlowast minus0033lowastlowastlowast 0000(0003) (0368) (0006) (0000)

Observations 1440 1440 1440 1440R2 0983 0389 0529Adjusted R2 0983 0388 0527Residual Std Error (df = 1435) 0032 3492 0053 0000F Statistic (df = 4 1435) 21 272740lowastlowastlowast 228686lowastlowastlowast 402326lowastlowastlowastNote lowastplt01 lowastlowastplt005 lowastlowastlowastplt001

the argumentrsquos defended position Hence if there is votingand if agents happen to obtain 119894119899 proposal arguments theyinexorably cluster towards one or the other side of the opiniondistribution resulting in opinion convergence remeniscent ofgroup polarization (see Figure 7(e)) Furthermore as agentscluster in the same side of the opinion spectrum the argu-ments that may be advanced in deliberation are fewer andmore skewed towards that one side of the opinion spectrumAgents become more easily persuaded in deliberation andonly by arguments supporting one side of the spectrumConvergence of opinions towards one loose opinion focusbecomes faster and more certain If there is no vote (seeFigure 7(b)) the dynamics are similar but for the fact thatit is the uniform randomness of the proposal argument thatguarantees a central convergence of opinions

413 Quantitative Analysis of Pair-Wise Opinion DynamicsThe three parameters that allow for quantitative analysis are119880 and119879 the agentsrsquo attitude structure and themajority quotavoting rule 120572 Table 3 supports the claim that going from 120572 =05 to 120572 = 066 does not affect the variance nor the shifts ofopinions However judgment accuracy is significantly higherwhen accepting proposals becomes more difficult This israther surprising but also intuitive since decisions that aretaken using more restrictive methods of scrutiny should bemore accurate

Concerning the attitude structures the interaction termbetween 119880 and 119879 shows that 119880 has a small effect on thevariance of opinions given that 119879 is low This shows thateven when the assimilation threshold is high a sufficientlysmall rejection threshold can make its one-way effect vanishHowever high levels of both 119880 and 119879 reduce the variance ofopinions dramatically In contrast when it comes to overallchanges in opinion the magnitude of 119880rsquos one-way marginal

effect increases the shifts of opinions whereas 119879 lowers itThis can be interpreted as having the rejection dynamics pushmoderate-to-extreme agents to positions that are close tothe extremes and very moderate agents to the center of thedistribution Not too many quantitative changes (on average)can therefore be observedThe fact that higher119880 is associatedwith faster convergence of opinions seems to imply lowerjudgment accuracy At some point voting for the same type ofproposals for too longmay result inmany collectivemistakesThis is an interesting conclusion to consider in the analysis ofthe mixed model Finally the interaction term shows that 119879has no strong impact on the metric

414 Quantitative Analysis of Deliberative OpinionDynamicsSeven parameters can be considered for quantitative anal-ysis in this dynamics the parameters of interest recalledin Section 312 the size of the argument sacks (119896) agentbehavior during deliberation the strength of the deliberationdynamics (120574) and the measure sensitivity to deliberatedresults (119901119886) The most ldquounexpectedrdquo result obtained fromthe deliberation dynamics is that it shows how deliberationcontributes to the proportion of extremists and the varianceof opinions It happens that having more agents discussproposals and having more of these discussions increasesthe variance of opinions and the proportion of extremistswhile lowering judgment inaccuracy and coherence For theformer the fact that far too many arguments are advancedduring deliberation plummets the chances that the centralargument is accepted hence deliberation does not moveopinions too often For the latter we observe that only ldquobigrdquodifferences (going from the lowest value of a parameter to thehighest) in protocol actually prove to have a preponderantimpact on the metrics These observations hint at a possibletrade-off between judgment accuracy and the variance of

Complexity 17

Table 4 Ordinary least squares estimates for the deliberative interactions scenarios for most of the observed metrics

Dependent variable119881119886119903(119900) 119878ℎ 119890119888 119894119903(1) (2) (3) (4)Constant 0032lowastlowastlowast 13092lowastlowastlowast 0370lowastlowastlowast 1007lowastlowastlowast

(0001) (0406) (0001) (0001)

119898 = 3 (ref119898 = 1) 0012lowastlowastlowast minus2062lowastlowastlowast minus0012lowastlowastlowast minus0001(0001) (0276) (0001) (0001)

119898 = 6 (ref119898 = 1) 0023lowastlowastlowast minus3285lowastlowastlowast minus0018lowastlowastlowast minus0004lowastlowastlowast(0001) (0276) (0001) (0001)

119899119863 = 002 (ref 119899119863 = 001) 0007lowastlowastlowast minus0965lowastlowastlowast minus0006lowastlowastlowast minus0002lowastlowast(0001) (0276) (0001) (0001)

119899119863 = 005 (ref 119899119863 = 001) 0020lowastlowastlowast minus3316lowastlowastlowast minus0015lowastlowastlowast minus0004lowastlowastlowast(0001) (0276) (0001) (0001)

120572 = 05 (ref 120572 = 0) 0146lowastlowastlowast 68932lowastlowastlowast 0068lowastlowastlowast minus0507lowastlowastlowast(0001) (0276) (0001) (0001)

120572 = 066 (ref 120572 = 0) 0291lowastlowastlowast minus13842lowastlowastlowast 00004 minus0000(0001) (0276) (0001) (0001)

119896 = 8 (ref 119896 = 4) 0004lowastlowastlowast minus0815lowastlowastlowast minus0002lowastlowastlowast minus0001(0001) (0276) (0001) (0001)

119896 = 16 (ref 119896 = 4) 0008lowastlowastlowast minus1118lowastlowastlowast minus0004lowastlowastlowast minus0003lowastlowastlowast(0001) (0276) (0001) (0001)

119861119890ℎ119886V119894119900119903 = ldquoFocusedrdquo (ref ldquoNaiverdquo) 0015lowastlowastlowast minus1980lowastlowastlowast minus0012lowastlowastlowast minus0004lowastlowastlowast(00005) (0225) (00004) (0001)

120574 = 01 (ref 120574 = 005) minus0020lowastlowastlowast 5762lowastlowastlowast minus00003 minus0001(0001) (0276) (0001) (0001)

120574 = 02 (ref 120574 = 005) minus0025lowastlowastlowast 8648lowastlowastlowast minus00003 minus0001(0001) (0276) (0001) (0001)

119901119886 = 05 (ref 119901119886 = 03) minus0013lowastlowastlowast 1581lowastlowastlowast 00004 00003(00005) (0225) (00004) (0001)

Observations 19440 19440 19440 19440R2 0931 0844 0562 0958Adjusted R2 0931 0844 0562 0958Residual Std Error (df = 19427) 0033 15701 0030 0050F Statistic (df = 12 19427) 21 844790lowastlowastlowast 8 733563lowastlowastlowast 2 079585lowastlowastlowast 36 761250lowastlowastlowastNote lowastplt01 lowastlowastplt005 lowastlowastlowastplt001

opinions that may be of interest in the mixed discussionmodel

Group coherence shifts of opinions and the varianceof opinions are highly affected by the voting thresholdthe coherence metric is reduced by half when allowing forclassic majority voting shifts explode and the variance ofopinions increases dramatically This is not surprising sincethe majority quota rule works as a buffer to the success ofdecision-making processes and in consequence it hampersthe deliberative opinion updates For the shifts of opinions itis clear that the two-third majority voting rule results in littleopinion change and that the simple majority rule inducesgroup polarization As expected there are less extremists inscenarios where there is no voting and the shifts of opinionare more likely to happen when agents are more sensitiveto deliberation When we include voting in the system

the grouprsquos judgment accuracy decreases due to the correctdeliberated decisions that are vetoed by a salient opinion-agent majority Strangely the effect only holds for the simplemajority decision rule Lastly the proportion of individualsthat participate in the debate and the minimal amount ofdebates have similar marginal effects on all the metrics (seeTable 4)

415 Comparing Deliberative and Pair-Wise Opinion Dynam-ics in a Nuthshell Qualitatively both models yield similaroutcomes but for the fact that deliberation never producesopinion polarization with both types of extremists (seeFigure 7) Furthermore clusters in the deliberative interac-tions model are less homogeneous in size and multiclusterconvergence occurs only when deliberation is rarely suc-cessful which happens sporadically given the calibration of

18 Complexity

Table 5 Mean difference 095 confidence intervals for metrics by model comparison

Model Metric119881119886119903(119900) 119901119903119900119901119890119909 119878ℎ 119890119888 119894119903Dyadic vs Deliberative [0191 0216] [0170 0197] [-2948 -2828] [0074 0082] [0165 0172]Mixed vs Deliberative [0142 0147] [0150 0155] [-2085 -1976] [0003 0002] [-0038 -0030]Mixed vs Dyadic [-0072 -0046] [-0045 -0018] [8332 8859] [-0079 -0071] [-0205 -0201]the model On the other hand pair-wise opinion dynamicsproduces multicluster convergence most of the time andcentral convergence only in the infrequent cases where thelatitude of acceptance is large

The comparison of these two scenarios in terms ofthe metrics is straightforward Whilst dyadic interactionsallow for more variability in opinions through pair-wiseinteractions and more mistakes in judging proposals delib-erative updates simply act as an antagonist force On aver-age the deliberative interaction model produces opiniondistributions with smaller variance and yields steady stateswith higher judgment accuracy a higher shift statistic andnaturally lower group coherence since only proposals that aredeliberated 119894119899 are eligible for voting (see Table 5)

42 Articulating Dyadic and Deliberative Social Interactionsin Opinion Dynamics So far we have considered scenarioswhere argumentation had no place and where individualsmade decisions according to opinions that were purelyconstructed from pair-wise discussions We also observedcases in which agents formed their opinions only by inte-grating deliberated proposals into their opinion updates Thescenarios we present in this section combine the two pre-ceding scenarios to account for both the effects of pair-wisediscussion and the effects of deliberation on opinions judg-ment accuracy and group coherence From the precedingobservations we conclude the following deliberative opiniondynamics reduces the variance and the proportion of extrem-ists because it shifts opinions greatly and polarizes groupsby eliminating at least one type of extremist the one whoseopinions are initially opposed to the deliberated resultsor it brings opinions close to neutrality (see Figure 7(b))This may not be desired in a group since the diversity andstability of opinions are necessary for deliberation to makesense and lead to the collective acceptance of controversialproposals Dyadic opinion dynamics on the other handtend to increase the variance of opinions and the numberof extremists while having a lesser aggregated effect on theshifts of opinion This is an immediate result of the pair-wiseinteraction dynamics inasmuch as it polarizes individualsvery quickly

To sum up in the mixed interactions model deliberationhinders the effect of voting on opinions by making someproposals that would normally be submitted for voting noteligible for voting It may also undo opinion changes dueto pair-wise interactions The mixed model can be seen asthe deliberation opinion dynamics model in which pair-wiseinteractions occur in between collective discussion and mayaffect which arguments are advanced during deliberationEquivalently it can be seen as the dyadic opinion dynamics

model in which deliberation accounts for endogenousldquoshocksrdquo on the opinion distribution

421 Qualitative Analysis of the Mixed Dynamics In Figures7(c) and 7(f) agents update their opinions through delibera-tion and pair-wise interactions Opinion trajectories visualsshow a combination of both previously described opiniontrajectories Over time as agents discuss with one anotheropinion clusters form as in the pair-wise or dyadic opiniondynamics Deliberation may disrupt the formation of theseclusters and three different situations may occur

(i) Deliberation disrupts the formation of the opinionclusters but not sufficiently to jeopardize pair-wiseopinion cluster formation (eg for high values 119905119863)Two phenomena may be responsible for it (1) thevariability of the proposal argumentrsquos support for theprinciple P and (2) too few (successive) deliberationsteps that result in opinion updates Once the opinionclusters are formed successful deliberation foreshad-ows the merging of the previously formed clustersmay bring extremists closer to moderate agents andthanks to the assimilation component of the pair-wisedynamics allow bigger and more moderate opinionclusters to form (refer to Figure 7(c))

(ii) Deliberation significantly disrupts the pair-wisedynamic formation of clusters This may happenwhen the requirements for accepting proposals areweak in the decision-making process (eg low 120572) Inthis case one has either central convergence sinceall agents follow the deliberated results in the samemanner or 3-cluster convergence with two extremeopinion foci In the latter situation the extremegroups form quickly (like in the pair-wise model) anddeliberation cannot pull extreme agents away fromthe extreme foci they are in The main reason for thisis that immediately after an extreme agent attempts toleave her foci through deliberation updates she getspulled or pushed back into it by means of a dyadicinteraction with another extreme agent

(iii) Deliberation does not disrupt the formation of theopinions cluster This is the rare case scenario forvery demanding decision-making processes in termsof inclusion (eg 119899119863) and argument acceptability (eg120572) Opinions converge as in the pair-wise opiniondynamics model (refer to Figures 7(a) and 7(d))

Last when there is voting (120572 = 0) the position ofthe moderate opinion cluster is determined by the first

Complexity 19

minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

Var(o)

sh

ec

ir

Var(o)

propex

sh

ec

ir

1

098

minus02

minus004

001

1

minus014

minus001

minus003

1

026

minus028

1

minus057 1

ＪＬＩＪ

Ｒ

Figure 8 Significant correlation estimates for themetrics of interestin the mixed opinion dyamics scenarios

accepted proposal argument andor the opinionmajority thatis formed from previous dyadic interactions (see Figure 7(f))

422 Relating the Metrics of Interest in Mixed OpinionDynamics The first thing to notice is that for all of thesescenarios all the metrics are significantly correlated (seeFigure 8) The variance of opinions and the proportion ofextremists are almost perfectly correlated To some extentthis means that a high variance of opinions known thatthe mean of opinions is statistically null equates to havingmany extremists in the population be towards one or theother side of the opinion distribution (eg extreme bipolar-ization) Similarly we find a negative and strong correlationbetween the judgment inaccuracymeasure and the coherencecoefficient (see Figure 8)mdashwhen judgment inaccuracy is high(low) coherence to deliberation is low (high) Cases likethese are telling since they may be related to sequences ofcollective decisions in which agents do not make too manymistakes when judging proposal arguments and still makedeliberated decisions that are on average in line with theirprinciples

The shift statistic interestingly is negatively correlatedwith the variance of opinions with judgment accuracy andwith the coherence coefficient In a way this means thathaving more moderate agents conduces on average to moremistakes in judging arguments and to less coherent collectivedecisions An explanation for this phenomenon could be thatas agents reach consensus in opinion important (possiblyextreme) arguments for correctly judging argument propos-als are not advanced in the deliberation arenawhich results inmore mistakes and in possibly accepting proposal argumentsthat after voting are not meant to be accepted Indeed the

negative correlation between the variance of opinions andjudgment inaccuracy points towards this direction since onaverage the more shifts there are the smaller the variance ofopinions is (see Figure 8)

423 Comparing Mixed Dynamics with Other MonolithicDynamics of Interactions Qualitatively opinion trajectoriesin the mixed model are more chaotic than in the pair-wise and deliberative interaction models (see Figure 7) Asanalyzed previously deliberation dynamics disrupt the pair-wise dynamics and create situations in which central andmulticlustered convergence (with two clusters at the extremesof the distribution) are possible at the expense of more(pair-wise dynamics) or less (deliberative dynamics) clustersIt tolerates the survival of very small isolated group ofagents that the previous models rarely tolerate in a hundredcollective decision-making steps (see Figure 7(c))

One of the most interesting observations one can makeof this family of models regards how groups change (in termsof our metrics) when dyadic social interactions are addedto deliberative interactions and vice-versa When comparingthe mixed interactions with only deliberative ones shiftsagentsrsquo aggregated change in opinions are on average lesspronounced The variance of opinions and the proportionof extremists are significantly higher for the former than forthe latter group coherence in contrast is lower with respectto the scenarios where opinions are formed only throughdeliberation Judgment accuracy follows that same trendA reason for this might be that mixed dynamics are fasterto yield semistable clusters andor opinion convergence Inconsequence the potential argument pool for deliberativeexchange shrinks faster and deliberation becomes less effec-tive over time which results in more collective mistakes Theoutcomeon coherence is predictable inasmuch as deliberativeinteractions produce group polarization If a group agrees inopinion debates are sterile because themembers of the groupwill most likely refrain from attacking the proposal argu-ments they all support and will work together to disparagethe proposal arguments they do not support (see Table 5)

In the same manner we report that the proportion ofextremists and the variance of opinions are barely lower inthe pair-wise interaction model when including deliberationin the decision-making process Shifts of opinions are alsobarely lower when there is no deliberation This can indicatethat deliberation does not undo the changes in opiniondue to pair-wise interactions nor does it amplifies them Incontrast when we consider the labeling-based metrics themixed interaction model ensures lower judgment inaccuracybecause of deliberation and lower group coherence (seeTable 5)

In sum the mixed discussion model is a compromisebetween the pair-wise interaction and deliberative opiniondynamic models in that it offers some guarantees in termsof judgment accuracy and coherence while allowing forldquoreasonablerdquo variance in the opinion pool It establishes somekind of trade-off between labeling-based metrics and thevariance of opinions that may be of interest for decision-making process design

20 Complexity

Table 6 Summary of sign or relative marginal effects for all varying parameters on dyadic deliberative and mixed interactions models inand for all experiments

Metrics Parameters119905119863 119898 119899119863 120572 119879 119880 119901119886 120574 119896 119861119890ℎ119886V119894119900119903119881119886119903(119900) XX X X O X X X X X X119901119903119900119901119890119909 XX X X O X X X X X X119878ℎ XX X X O X X X X X119890119888 XX X X XO X XOO XOO X X119894119903 XX X X O XO O X OOX XOO X XNota gray symbols correspond to dyadic dynamics light gray to deliberative dynamics and bold black to mixed dynamics Given a scenario the symbol 119883stands for constant parameter and 119874 for insignificant marginal effect and as parameters grow for positive (negative) marginal effect for the parameterFrom left to right at the parameter level the leftmost arrow corresponds to the scenario with the strongest (in absolute value) marginal effect of the parameter(with respect to to its factors) Example Let 119901 be a parameter and 119874 stands for ldquo119901 has a significant positive effect in dyadic discussion scenario negativein the deliberative one and insignificant in mixed discussion The positive effect in dyadic discussion is greater than the absolute value of the negative one inthe purely deliberative scenariordquo

424 Comments on the Differences between the Three Modelsfor OLS Regression Estimates The significance of most of theregression parameters estimated for the mixed interactionsscenarios are similar to those estimated for the pure dyadicanddeliberative scenariosNevertheless themagnitude of theeffects is different insomuch that the dynamics themselvesare different So instead of commenting the linear regressionestimates of the mixed model we compare when possiblethe standardized significant coefficients of the regressors ofthe mixed model with those of the pair-wise and deliberativeinteractionmodels (see Table 6 for a summary of the results)

In mixed interactions for instance the influence of119898 onthe opinion distribution is much more preponderant than inthe deliberative dynamics scenarios Going from 119898 = 1 to119898 = 3 increases the standard variation units of the variance ofopinions by twice as much as the one in the deliberative sce-nariosThis in particular is due to the design of the decision-making process If 119898 is higher then the number of socialinfluence steps (119905119863) in the decision process is also higher(compared to 119905119863 = 0 in the deliberative case) If we decideto consider these steps as contributing to the variance ofopinions then the effect of119898 on the variance of opinions hasto be stronger in the mixed interactions model In contrastthe number of agents participating in the debate has a weakereffect on the variance of opinions in the mixed interactionsmodel than in the deliberative dynamics specially for 119899119863 =005 (more than twice as much) The difference may comefrom the fact that in mixed discussions the part of theargument pool used in deliberation is more restrictive thanin the deliberative case We can argue that since pair-wisedynamics may get individuals closer to one another aftersome deliberation the same happens to the arguments thatindividuals will most likely use in deliberation Taking thisinto account means that deliberation will at many instancesbe unfruitful and have no effect on opinions thus failing tomoderate extreme opinions andor polarize the group In thesame direction a high number of participants in deliberationcan make the deliberation dynamics stiff In this case thedistribution of opinion is close to the uniform distributionwhich has a relatively high-to-moderate variancewith respectto the reference results

The strength of the deliberative dynamics has a positivestronger effect on the variance of opinions in themixedmodelthan in the deliberative model when 120574 = 02 and a negativeweaker one when 120574 = 01 A sound interpretation of thisresult is that when deliberation is successful and its effect onopinions does not get agents to move sufficiently away fromtheir current positions agents are pulled or pushed back totheir previously held opinions the variance of opinions doesnot changemuchOn the other hand if the effect of successfuldeliberation is strong opinions can change sufficiently tomake agents much more prone to assimilation into moderateopinion foci the variance of opinions falls

The deliberative opinion dynamics model is much moresensitive to 120572 than the mixed discussion model Differ-ences in marginal effects can double The probability ofattraction to deliberated results (119901119886) for instance has astrongmoderating effect (negative) on agentsrsquo opinions in thedeliberative opinion dynamics but a meager one (two-fold)in the mixed and dyadic dynamic models This is becauseopinion updates in the deliberative model depend only onthat parameter whereas in the mixed model more numeroustypes of interactions make opinion updates possible

In terms of judgment accuracy all parameters except thesize of the argument sacks (119896) have a significant slightlystronger effect on the deliberative dynamics than in themixedmodel In effect the difference may be attributed to theupshot of pair-wise interactions on the distribution of opin-ions and on the arguments that would be advanced duringdeliberation The result respecting parameter 119896 however iscounterintuitive Since argument sacks are static scenarioswhere shifts are higher should make 119896 more important fordeliberation to succeed and it is precisely in the scenarios ofpure deliberative dynamics that we find the biggest opinionshifts This may point to the hypothesis that in deliberativeinteractions agents change opinions substantially but do notoften change their adherence to the principle (go from anegative opinion to a positive one or vice versa) when thereis no group polarization

43 Sensitivity Analysis for the Procedural Parameters ofInterest Weperform a sensitivity analysis of the observationson the parameters of interest in themixed interactionsmodel

Complexity 21

We extend the parameter domains of the pure dyadic anddeliberative scenarios as we make it explicit in Table 2

431 Minimum Number of Debates (119898) We observe that theminimum number of debates has a significant well-observedeffect on all of our metrics excluding group coherence Forthe variance of opinions we can observe that the moredebates there are the bigger the value of the metric isalthough the higher 119899119863 and 119905119863 are the weaker the overalleffect of119898 becomes (Figures 9(d) and 9(a)) Furthermore themarginal effect of increasing minimal debates is decreasingan additional deliberation step requirement increases thevariance of opinions but less and less as it grows 119898 coupledwith 119899119863 has a shy S-shaped effect (see Figure 9(d)) on thevariance of opinions meaning that it has a stronger effectwhen 119899119863 is low which suggests a possible trade-off betweenthe number of arguments accepted in the debate and thenumber of debates required before accepting a proposal

In contrast the shifts in opinion are less and less likelyas 119898 grows and this is independent of the variations of theother parameters Again the effect is marginally decreasingand is only truly significant when 120572 = 05 An explanationto these effects may be linked to the design of the systemFirst the variance of opinions is higher when deliberation isasked for because the more deliberation steps there are thehigher the chances are that the central proposal argumentis deemed unacceptable specially when agents are focusedFurthermoremechanically speaking increasing theminimalamount of debates implies that whenever a decision is to betaken at least 119898 times 119905119863 time steps have to take place and if atany moment a proposal argument is considered undecided119905119863 time steps are added to the process So unless the debatedoes not yield 119906119899119889 labels for proposal arguments (highlyunlikely considering that 120590 = grounded semantics) the morenondeliberation steps there are in the decision process thehigher the variance of opinions is Concerning the shiftswhen 120572 = 05 either the system is too stiff to accept anyproposal argument and opinions do not changemuch or theeffects of pair-wise discussion and deliberation cancel out insuch a way that individual shifts are minimal (Figure 9(b))

On the side of labeling-based metrics the more debatesare asked for the more accurate a group is in its judgmentthe effect being smaller as 119898 grows When agents are naivethe effect is quasilinear while when they are focused thestrongest effects of acquiring more deliberation are foundwhen levels of deliberation are already low (Figure 9(f))Thiscan be explained by the fact that the more debates thereare in the decision process the closer one gets to the idealargumentation framework

432 Proportion of the Population in Deliberation Steps (119899119863)Like for 119898 119899119863 has a significant effect on the proportionof extremists and on the shifts and variance of opinions(Figure 9(e)) This may result from the fact that being ableto put more arguments in play at the same debate step mayincrease the odds of revealing cycles around the proposalargument in119860119865120576 Given that we use grounded semantics thearguments in the cycles are labeled 119906119899119889 and in consequence

debates are more often postponed when 119899119863 is higher Post-poning debates in turn increases the number of nondebatessteps in the decision process which increases the varianceof opinions and limits over time the moderating effect ofdeliberation Moreover the effect of this parameter is verydependent on the value of 120572 (Figure 9(e)) For shifts forinstance 120572 = 05 makes the effect of 119899119863 negative while120572 = 0 makes it positive but to a lesser extent Furthermorefor higher requirements of deliberation (119898) adding moreindividuals to the deliberation process has weaker effectson the variance of opinions and on the other metrics It isalso quite interesting to notice that it has the same effect onagents independently of agent behavior during deliberationThis is a surprising result as one would have expected morefocused agents in a deliberation arena to cause a raise inthe proportion of extremists as they play to knock outopposing proposal arguments and thereof reducing the effectof deliberation (lowers variance) on the whole population

Finally similar to119898 addingmore people into the deliber-ation process increases judgment accuracy (see Figure 9(c))and has no clear effect if any on group coherence (seeFigure 9(i))

433 Steps between Deliberation Steps (119905119863) In all configu-rations 119905119863 increases the proportion of extremists (variance)and decreases the shifts of opinions among agents Theshifts and the effect on the variance of opinions are presentfor 120572 = 05 insignificant otherwise (Figure 9(b)) 119905119863 ishighly linked to 119898 by construction When 119898 = 1 thecurve linking the variance of opinions and 119905119863 is convex As119898 increases the curve becomes more and more concavewhich means that 119905119863 has a more important effect on theopinion distribution as collective decisions take longer to beachieved This seems counterintuitive but in reality it reflectsthemultiplicative relationship between deliberation and pair-wise interactions and the semantics in the model If 119898 is lowand 119905119863 is high the effective number of pair-wise interactionsis on average smaller in the deliberation process Hencethe structure of the decision-making process hinders anyincrease in the variance of opinions If119898 is high the oppositeeffect is observed Additionally since the grounded semanticsyields few 119894119899 arguments with respect to other admissibility-based semantics getting closer to the ideal argumentationframeworkmaymake it difficult to obtain the expected effectsof deliberative discussion on agents Lower 119898 constrains thevariance of opinions by making deliberation more influentialon opinions

In terms of judgment accuracy and coherence we observethat 119905119863 does not explain coherence yet it tends to decreasejudgment inaccuracy This may be due to a more variedargument pool in the debate which in turn may be aconsequence of a higher variance of opinions or vice versa

434 Acceptability Voting Quota (120572) By far it is the mostinfluential parameter in our study It changes the directionand the intensity of the effect of all the other proceduralparameters and by construction heavily constrains the roadto accepting a proposal (see Figure 9)

22 Complexity

015

020

025

030

035

040

1 2 3 4 5 6

12345

Values of Ｇ

(a) 119905119863 times 119901119903119900119901119890119909 by119898

0

10

20

30

40

50

1 2 3 4 5 6

005066

Values of

(b) 119905119863 times 119878ℎ by 120572

040

045

050

055

060

1 2 3 4 5

001002003004005

Values of Ｈ＄

(c) 119898 times 119890119888 by 119899119863

015

020

025

030

035

040

1 2 3 4 5

001002003004005

Values of Ｈ＄

(d) 119898 times 119901119903119900119901119890119909 by 119899119863

0

10

20

30

40

50

001 002 003 004 005

00005066

Values of

(e) 119899119863 times 119878ℎ by 120572

040

045

050

055

060

1 2 3 4 5

BehaviorNaiveFocused

(f) 119898 times 119890119888 by 119887119890ℎ119886V119894119900119903

015

020

025

030

035

040

0 05 066Behavior

NaiveFocused

(g) 120572 times 119901119903119900119901119890119909 by 119887119890ℎ119886V119894119900119903

05

06

07

08

09

10

1 2 3 4 5 6

00005066

Values of

(h) 119905119863 times 119894119903 by 120572

08700

08725

08750

08775

08800

001 002 003 004 005

BehaviorNaiveFocused

(i) 119899119863 times 119894119903 by 119887119890ℎ119886V119894119900119903

Figure 9 Curves of mean observations for different parameters and metrics for 36000 runs Bars are 095 confidence intervals Plots withthe same observables (ordinate) except for 119894119903 are on the same scale

Complexity 23

The higher the requirement for accepting the proposal isthe higher the proportion of extremists and the variance ofopinions are and the lower the shift statistic is In few words120572 constrains the world to dyadic discussion or throws it intoa process in which deliberation is much more importantthan dyadic interactions This is why for any 120572 = 05 oneeither gives too much weight to deliberated results whichshadows pair-wise interactions or too little weight in thesense that no deliberated result is ever accepted and thusnever integrated into the opinions of the agents that havevoted for it The reason for the latter is that the latitudeacceptance is too low and thereafter pair-wise interactionsare not enough to unevenly polarize the population in such away that deliberated proposal arguments are voted favorablyby a non-negligeable majority Shifts for 120572 = 0 are stable fordifferent levels of 119905119863 which implies that deliberation keeps incheck all the shifts of opinions related to an increase in thenumber of pair-wise interactions For 120572 = 05 deliberationis less likely to be successful and shifts tend towards the shiftlevels for 120572 = 0 (see Figure 9(b))

Concerning labeling-basedmetrics120572 entirely determinesthe coherence statistic For 120572 = 05 there is no difference incoherence because of how coherence is defined it is maximalwhen 120572 = 0 and strangely maximal when 120572 = 066(see Figure 9(h)) Surprisingly 120572 has no effect on judgmentaccuracy One would have expected a higher 120572 to increasejudgment accuracy since agents would hardly collectivelyaccept a proposal argument discussed during deliberationProposal arguments have given how the argument lattice isgenerated a higher probability of being rejected than of beingaccepted in the ideal argumentation framework

435 A Word on Focused and Naive Agents Focused andnaive specifications for agent behavior are an importantparameter in the decision process since they model howagents choose arguments in the deliberation arena In the sce-narios where agents are focused agents show equal or highervariance of opinions and are therefore more extreme (seeFigure 9(g)) independently of the voting requirements Shiftsof opinion are less likely as well We can see this happeningbecause focused agents knock out proposal arguments moreoften than naive agents provided that 119899119863 is low

One could have expected naive agents to yield morecoherent decisions since they argue sincerely However thisseems not to be the case (see Figure 9(i)) Agents are unable todeliberate in away thatmakes deliberation reflect their votingintentions either because they do not have the necessaryarguments to do so or because such arguments do not exist

In terms of judgment accuracy focused agents do betterthan naive agents in finding the ideal labels for the proposalarguments This is likely because when reconstructing theframework focused agents take into account the deliberatedproposal argument and choose themost pertinent argumentsin their sacks that relate to the state of affairs at the deliber-ation arena In fine debates result in a better approximationofL119886119887120576(119868) even though one could have believed the oppositethat is focused agents are ready to sacrifice a correct collectivelabel of a proposal argument in the goal of defending theirpositions

44 Points of Discussion In this subsection we discuss theresults obtained from the simulations We attempt to makesense of them from a social sciences perspective

441 Opinion Consensus and Dissensus The deliberativeopinion dynamics corresponds to the simulated scenarios inwhich the variance of opinions is at its lowest and judgmentaccuracy at its highest Dyadic opinion dynamics are associ-ated with the scenarios in which the variance of opinions is atits highest judgment accuracy at its lowest and coherence atitsmaximumpossible levelThemixed interactions dynamicsis a combination of these aspects for all the metrics exceptfor coherence at which it is the worsemdashdeliberation leadsto some kind of loose opinion consensus that is reinforcedby pair-wise interactions Likewise the mixed discussionmodel confirms some results exposed in [16] expressingthat opinions and voting intentions in a deliberative contextldquooften changerdquo That being said and more particularly indeliberative opinion dynamics outputs align well with twointeresting results on consensus formation given by aninterpretation of Moscovici and Doisersquos [15] work Sunsteinrsquos[14] account on deliberation and group polarization andsocial impact theory [4] The first states that mild consensusis reached in deliberation as procedure disengages agentsthe second is that within discussion groups majorities tendto become larger and opinions more extrememdashnot on thebasis of competing arguments but of the initial balance ofopinions After several decision processes we find opiniondistributions that reflect these observations even though weare not able to precisely coin the set of parameters that yieldssuch distributions (see Figure 7)

Although differences between mixed and dyadic opiniondynamics in terms of the variance of opinions and shiftsare slim the former does much better in terms of judg-ment accuracy In this sense deliberation does moderateopinions but not as much when non-deliberation steps arerelatively frequent throughout the decision-making processIn terms of Moscovici and Doise theory of consensus [15]this makes sense since pair-wise discussion engages or ego-involve agents in the discussion and pushes to convergenceof opinion at the extremes of the opinion distributionHowever what remains unclear is the reason why proceduralparameters that constrain deliberation do not disengageagents from discussion and thus result in a mild consensusas also expressed by Moscovici and Doise [15] A possibleexplanation of this phenomenon may stem from the factthat deliberation in the model is always accompanied by aseries of ego-involving discussions that shadowor counter themoderating effect of deliberation Not only this deliberationcould also be responsible for crystallizing the opinions ofextremists for it favors one or the other part of the distributionof opinions as explained in [14] and to some extent in [13]Furthermore the mixed discussion model gives an exampleof polarization that results from a minority of dissenterson one side of an issue acceding to views of the majorityrsquosside [4] as opposed to the strong polarization observed indyadic interactions Situations in which the status quo orthe undecidability of the proposal argument lingers cannotpossibly yield situations of moderation In cognitive science

24 Complexity

one can be inclined to believe that the longer a deliberationprocess is the more likely individuals will take an immutablestance on the situation In view of the assumption that longdecision-making processes can be more costly for a group orsociety as a whole agents may feel forced to take a stancejust for the sake of ending the process The stance that isadopted would therefore depend on pair-wise interactionsamong individuals and not on deliberation since deliberationtakes too long to be conclusive

442 For Hypothetical Recommendations in Decision-MakingProcess Elaboration On another tune results from themodelsuggest that procedural parameters have marginally decreas-ing positive or negative effects on the metrics of interest Ina hypothetical policy analysis perspective this may indicatethat after choosing a correct number of individuals andimposing the right amount of debates one can attain rea-sonable levels of judgment accuracy coherence and varianceof opinions As accounted in [18] deliberation increasesknowledge (collective knowledge as well) and therefore itheightens the chances of making correct decisions If onebelieves that social welfare is linked to deliberation andeventually to the metrics studied previously then proceduralparameters can both be used as an instrument to attaindesired levels of social welfare and reinforce the notionof legitimacy of collective decisions It can also provide ajustification for choosing a deliberation regime rather thananother on the basis of how important (or urgent) a topic ofdiscussion is Indeed if we adopt the claim that deliberationis a forerunner of welfare then any state in which deliberativecues are at their maximum (highest number of participantsmany deliberation steps before making a decision and asfrequent as possible) has to be mapped to the highest attain-able social welfare In this case is a situation where judgmentaccuracy is at its highest yet the variance of opinions or theproportion of extremists at its lowest an ideal situation Wecannot say for sure since strong opinion consensus and longdecision-making processes in many situations may result ina loss of welfare rather than in a gain of it

Another interesting result in procedure for deliberationis the very similar estimates we find for the effects of theminimum number (119898) of debates necessary to make adecision on a proposal and the proportion (119899119863) of individualsparticipating in deliberation on judgment accuracy In thescope of social welfare this raises the question of whetheradvocating for longer decision processes by demandingmoredeliberation steps (higher 119898) rather than bigger debates(higher 119899119863) may increase welfare Depending on the objec-tives of the decider of the deliberation procedure he or shemay choose to concentrate on one or the other A recom-mendation we can assert from the mixed social interactionsmodel is that if one prefers low to high variance of opinionsandor high to low coherence the decider is better off ifshe focuses on increasing the size of the debate rather thanorganizing debates very frequently One of the reasons forthis might be that when making bigger debates all agents aretruly considered as equals in the decision-making processIn organizing many debates individuals that are chosen toparticipate in many of these will be more influential and will

somehow bias the set of arguments in the deliberation arenaThemajority rulemodeled through120572 also determines the sizeof the shifts and the distribution of opinions Because a bigger120572 is associated with higher levels of extremism and opinionvariance deliberated cues are harder to accept and thereforeit becomesmore difficult for agents to reach concensus andormoderate their opinions

On the other hand and further into the idea of legitimacythe goal of deliberative processes is the agreement amongall participants This agreement stands for the right or bestdecision from a formal or procedural perspective and it is alsothe best from a substantive point of view In other words thelegitimacy of decisions not only derives fromwell-establishedprocedural requirements (respect of the protocol) but alsobegs for the fulfillment of two essential conditions (1) satisfy-ing the procedural requirements for a correct procedure (for-mal legitimacy) and (2) the rational acceptability of the resultsof this procedure (substantive acceptability) Legitimacy canbe then obtained from the coherence statistic scrutinizedin the model insofar as it measures how well agents acceptthe results of a correct procedure (eg deliberation) So fora proposal to be legitimate one has to choose parametersthat would maximize group coherence in decision-makingprocesses

As deliberation may give rise to consensus in the delib-eration arena it may also sow the seeds of dissensus inthe group Our model clearly illustrates this phenomenonthrough the procedural parameters of deliberation and themetrics observed From another perspective in deliberativedemocracy dissensus in collective decision-making may bedesirable According to Landermore and Pagersquos accounts [36]lack of consensus can be perceived as having agents withalternative ideas that combined with other ideas can providea better approximation of the ideal status of an argumentTheauthors in [36] call this dissensus ldquopositive dissensusrdquo andit makes normative sense when solving complex collectivedecision-making problems Our model captures the ideaof having normative requirements to define a good andlegitimate collective decision requirements of high consen-sual deliberative accuracy (judgment accuracy) of legitimacy(coherence) and of diversity (variance of opinions) are alltaken into account and for such we can conclude that ourmodel is successful in describing a sound process of collectivedecision-making with interpretable outcomes

Of course the present simulation results and analysis donot prove that all opinion dynamics can be accounted forby such simple processes like the one presented here Theydo however shed light on the expressiveness of combiningdifferent paradigms to account formore interpretablemodelsand pave the way to creating original models of the sortThe simulations suggest the desirability of discovering theconsequences of relatively simple laws of communication atdifferent levels (micro meso and macro) to determine whatstill needs to be explained

5 Related Work

We see our model as a contribution to the influence andopinion dynamics field in ABM and a pragmatic application

Complexity 25

of abstract argumentation theory To our knowledge we areunaware of existing literature on agent-based modeling thatexplicitly relates collective choice and the notions of deliber-ation and opinion diffusion through abstract argumentationas we have done it

Most models in the literature of opinion diffusion areinterested in opinions because these have an influence incollective decision and in questions of social order Forinstance in [3] the authors are interested in consensus andin how a group collectively chooses among two alternativesIn other models authors are interested in the emergenceof extremism [8] and on the distribution of opinions whenextremists are introduced in the population [5 33 37] whileother authors coin the notion of opinion polarization as anemergent property of the system [8 12] They show usingmodels of ldquobounded confidencerdquo and opinion diffusion withtrust that three different kinds of steady states (unipolarbipolar and central) are possible depending on whetheragents are sufficiently uncertain about their opinions andsufficiently connected andor a certain proportion of indi-viduals are already extreme Recent articles on informationand opinion dynamics stress the importance of governanceand government intervention in the spread of emotions(opinions) in the frangibility of social consensus systemIn [10] the authors show that government intervention inthe spread of negative emotions can lead to an even fasterspread of negative emotions (single-peeked convergence) andin a faster collapse of the social consensus system Opinionas a function of trust is studied in [12] and relates to howindividuals form their opinions on the basis of how muchthey trust agents in their networks

Another stream of discrete opinion dynamics modelsthat explain the emergence of opinion can be found in theliterature Computer multiagent models of attitude changebased on Latanersquos social impact theory are presented in [4] Inthe theory an agent changes her opinion on the basis of theinformational impact she is subject to which depends on thepersuasiveness (strength) supportiveness and immediacy(group structure) of the environment and information shereceives The simulations predict two emergent groups ofphenomena the shifting of attitudes towards incompletelypolarized equilibria and the formation of coherent clusteringof subgroups with deviant attitudes Similarly but in acontinuous more argumentative fashion Mas and Flache[11] present a model of opinion diffusion using argumentsArguments are considered to be for or against a proposal andare given an agent-dependant relevance in dyadic persuasiveinteractions Arguments also determine the agentsrsquo opinionsand their dissimilarities which posits the rules of pair-wiseencounters in theirmodel as agents are assumed to form theiropinions according to the arguments they own They showthat in an argumentative opinion diffusion model the onlyobservable steady states are bipolarization and consensusand that bipolarization can emerge from interactions amongsimilar agents In other words bipolarization of opinionsis possible with homophily and without negative influenceif there is argumentation They subsequently compare theirresults to experimental data In [38] a similar approach toargumentation and opinion formation is taken but for the fact

that they introduce different types of arguments explicitlyapply social judgment theory [22] and use survey data toestablish model-to-real-world comparisons

Closer to opinion formation and abstract argumentationGabbriellini and Torroni [39] are the first to have origi-nally and soundly merged opinion diffusion and abstractargumentation They define an agentrsquos opinion as a functionof the arguments she holds and the attack relation amongthem They devised a focused peer-to-peer dialogue systemof persuasion (NetArg) inspired from Mercier and Sperberrsquosargumentative theory of reasoning [40] which used onlyabstract argumentation to study opinion polarization andopinion dynamics Moreover they considered networks andthe notions of trust and epistemic vigilance to define adynamics for knowledge revision and trust itself whichwe clearly lack in our model They showed that if a con-servative belief operator in argumentation was applied byagents when they reasoned then their dialogue protocoldid not increase polarization among agents That said theyused the model to study the effect of Groenvetterrsquos weaklink theory in the spreading of arguments and were notparticularly interested in notions like judgment accuracy orcoherence nor in tackling questions related to collectivedecision-making procedures Their system is very expressiveand helps position our work in the litterature our work is atthe frontier of pure argumentative opinion diffusion modelsopinion diffusion with arguments and continuous bounded-confidence models of opinion diffusion a la Deffuant

On another note work on collective cognitive conver-gence [41] and opinion sharing [19] shows that consensustowards a certain ldquocorrectrdquo opinion or cognitive state is alwayspossible yet dependent on noise variability and awarenessof agents In [20] the authors show that learning about anexogenous correct state of the world (represented by bits)under confidence was possible but only if the agents werenot too confident In a homogeneous population they showthat the higher the confidence the worse the learningmdashthevery confident agents do not learn the properties of the truestate of the world and disrupt the learning process of the lessconfident ones Collective cognitive convergence can be seenas a result of deliberation in truth-seeking models

When it comes to abstract argumentation theory we takean approach that wires two types of dialogues that are well-studied in the literature persuasion dialogues [42] and delib-eration dialogues [43] Another interesting line of work is theone on mechanism design [44] or the problem of devisingan argumentation protocol where strategic argumentation isnot a liability for success in debates We tackle mechanismdesign in a different way though Instead of consideringstrategy-proofness we are interested in how differences inprotocol can result in epistemologically ldquobetterrdquo collectivechoices and guarantee that opinion distributions are favorablefor deliberation For a survey on persuasion dialogue see[42]

Work on agent-based argumentation usually assumes thatthe semantic relationship between arguments is fixed Inother words if two individuals were to put two argumentsin the public arena such that one logically attacked the otherthen everyone would agree that such attack exists [45 46]

26 Complexity

Other models which do not make this restrictive assumptioncan also be found in the literature [27 47] and derive fromthe class or family of opponent models [27 48] in which twoopposing sides attempt to win a dialogue Ourmodel is in theintersection of these two but the frame combining opiniondiffusion of the kind and argumentation remains new

The idea of mixing interpersonal influence and verticalcommunication however is not original It is described andimplemented in innovation diffusion models such as [5 49]In both vertical communication is modeled as exogenoustransparent informationmdashagents are aware of the existenceof an innovation thus triggering several processes of choiceand stabilization of opinions Also in [50] an Eulerian modelis implemented to show the effect of media and exogenousinformation on opinion distributions With respect to thispoint the originality of our work is in that the informationemitted as vertical communication is endogenous It is issuedfrom a deliberation model that agents shape on the basisof their opinions arguments and behavior In the spirit of[51] where the authors control for the design of verticalcommunication (to whom the vertical communication isaddressed its timing how it influences agents opinions) wecontrol for the process generating the information for ourown set of variables length of debates majority quotas andthe frequency at which discussions take place

6 Conclusion and Perspectives

The main objective of this article was to build a bridgewhere decision-making argumentation and opinion dif-fusion could come together We proposed a model thatcombined abstract argumentation theory and a boundedconfidence opinion diffusion model and showed to whatextent it could explain variance of opinions extremismcoherence in collective decisions and judgment accuracyon arguments given an ideal state of full information Thesecond objective was to show in what way governance pro-viding agents with arenas of discussion and deliberation wasimportant in addressing the success of collective decision-making processes and the quality of their outcomes

The model revealed that allowing for more deliberationtime though in lower frequency and allowing for widerparticipation in deliberation increased the variance of opin-ions and the proportion of extremists in a group Theseobservations are consistent with the combination of resultsfound in [9] and in [13 14] which stress that deliberationmay polarize groups and may have a meager effect on theshifts of opinions and inconsistent with [16] where it isargued that opinions tend to moderate after deliberationIt happens that deliberation alone did moderate opinionyet when integrated into a more complex system in whichindividuals were allowed to interact with one another andnot all that was deliberated was accepted its influence wasshadowed by other more individualistic dynamics

Undeniably the grounded semantics played an importantrole in the weak effect of deliberation since it models askeptical way of reasoning over arguments For agentsaccepting arguments thus updating their opinions accord-ingly happened not too often Nevertheless asking for more

deliberation did increase judgment accuracy as observed in[18] yet in a marginally decreasing fashion We showed thatvoting within the deliberation protocol not only increasedthe proportion of extremists and the variance of opinions ina group but also determined how coherent deliberation andvoting procedures were with each other Lastly we showedthat agents that are focused judged arguments better hadmore stable opinions and constituted a group with higherproportion of extremists than their naive counterparts

In terms of governance the model says that there is per-haps no trade-off between extremismand judgment accuracyInstead it asserts that the higher the variance of opinions thecloser the group gets to the correct decisions This situationmay be interpreted as Landermore and Pagersquos notion of ldquoposi-tive dissensusrdquo (in complex collective decision-making tasks)[36] as opposed to consensus as a normative requirementfor correct collective decision-making If a decider has toorganize deliberation to legitimize a public policy dependingon what kind of world he or she wants different set ofparameters may be chosen to account for different levels oflegitimacy and correctness For instance one may ask for adecision to be legitimate to respect a certain level of coherenceor of judgment accuracy and to be discussed by a sundryof different-opinionated peoples If so deliberation has tobe made more often more participants have to be includedin debates or in deliberation instances and not too manytime steps should be introduced in between two deliberationinstances So the model may capture normative ideas todefine correct and legitimate collective decisions and tomake recommendations on those grounds Requirements ofhigh consensual deliberative accuracy (judgment accuracy)legitimacy (coherence) and diversity (variance of opinions)are all taken into account and for such a reason the model israther successful in describing a sound process of collectivedecision-making with interpretable outcomes

61 Extensions of the Model and Other Ideas Our model canbe extended in many ways Value-based abstract argumen-tation frameworks (VAFs) for example as an extension ofDungrsquos argumentation framework provide a formal descrip-tion of the process of decision-making in which argumentsare given values and audiences (set of agents) ignore attacksbetween arguments on the basis of their preferences over val-ues Social abstract argumentation frameworks (SAFs) alsoprovide an interesting and seemingly convenient formalismfor our model In SAFs agents vote on the acceptability ofarguments and the resulting labeling is a combination of thearguments that are voted for and labeling-based semantics Inour model we apply the same rule yet only for one argumentand given precise agent-based votingmechanisms and votingrules which are eventually parametrized

To consider our model as a specification of a mixed VAFand SAF model may be of great interest for future workThis formalism can provide solid foundations to describeexplain interpret and eventually extend the results foundhere from an argument-based perspective On those groundsfuture work will focus on creating a hybrid argumentationframeworkmdashwith its semanticsmdashthat takes heed of bothvalues or principles and voting

Complexity 27

Concerning other possible extensions of the model anatural reflex would be to argue on the validity and therobustness of the mix between the bounded confidenceopinion diffusion models and deliberation Consequentiallyadding deliberation to different bounded confidence modelsof opinion diffusion such as the Hegelsmann-Krause model[37] or to models of social impact theory like in [4] canbe of great interest for future work We acknowledge that afiner understanding of argumentation frameworks randomlattices and their respective applications for this modelare necessary Testing for different labeling-based semanticsor even trading labeling-based semantics for scoring-basedsemantics (semantics that defines acceptability on scoresgiven to arguments) such as debate semantics and ranking-based semantics (semantics that yield a preorder) may beinteresting to consider in future work

More into modeling deliberation it is not inadequateto believe that agents may also advance proposals duringdeliberation and replace to some extent the proposal forwhich the deliberation is taking place Endogenizing theprocess that produces the type of argument to be deliberatedon may be interesting to explore either by having agentsstrategically replace proposals at certain moments duringthe debate or having the central authority attempt to chooseproposals such that the probability of having them acceptedis high Agents may also be thought to be autoorganizing(independent of a central authority) and concede on the sizeof the instances of deliberation To consider differences insemantics across agents or even endogenizing the semanticson how urgent or important the proposal that is beingdiscussed is can also be an interesting direction to take

Last but not least many nontrivial modifications of ourmodel are possible and most should include better-thoughtdeliberation protocols It may be interesting to design andobserve protocols in which deliberation only affects agentsthat are actually debating for example Trust network effectsmultidimensionality of opinions new processes of argumentexchange or learning among agents are notions to furtherdevelop in order to make the model more realistic and relaxsome unreliable assumptions In sum the model is to berefined and extended with the objective of either studyingconcrete cases of deliberative polling and opinion dynamicsor implementing more intuitive thought experiments

Appendix

We present below some of the pseudocode we used toimplement the model

Algorithms

(a) Algorithm 1 implements the dyadic influence in themodel It describes how two agents interact andupdate their opinions

(b) Algorithm 2 implements the deliberative influence inthe model which describes how agents update theiropinions in sight of the accepted deliberated proposalarguments

(c) Algorithm 3 is used to obtain the grounded labelingof an argumentation framework

(d) Algorithm 4 provides an overview of a run in thesimulations119872 is the number of proposal argumentsto discuss which we set to 100 in the model

Each object in the system is indexed by a119908ℎ119900 number Hencea set of agents is annotated by a set of natural numbers Weuse this fact in the implementation of the algorithm 119874119887119895119890119888119905119894denotes the object of 119908ℎ119900 number 119894 Agentsrsquo opinions (119900) arecalibrated so that 119900 isin [minus1 1]Data Availability

The program and data used to support the findings of thisstudy are available from the corresponding author uponrequest

Disclosure

A preliminary version of this work was presented anddiscussed at the conference Bridging the Gap between FormalArgumentation and Actual Human Reasoning on October 4-5 2018 at BochumGermanyThe authors thank the audiencefor their valuable comments and suggestions

Conflicts of Interest

The authors declare that they have no conflicts of interestregarding the publication of this article

Acknowledgments

The contribution of Gabriella Pigozzi was supported by theDeutsche Forschungsgemeinschaft (DFG) and the CzechScience Foundation (GACR) as part of the joint project FromShared Evidence to Group Attitudes (RO 45486-1)

Endnotes

1 In other words the superiority of the ldquoright answerrdquo orlabeling in our case will appear as such to all

2 All agents hold the same opinion3 By sensitivity we mean how much an agent is ready to

change her opinion on the sole fact that an informationalcue has been debated on

4 The assumption is strong but this trust among agentsmay be rooted from a genuine individual motivation toreach a democratic consensus

5 We adopted a rule of thumbs that says that if an agentis neutral with respect to the principle then she willvote for any proposal that is presented to her duringdeliberation

6 If the argument is not unique she chooses one atrandom

7 If there are more than one candidate the agent will favorthe argument at minimal distance from 119868

28 Complexity

Require Vector of values for parameters (119881 = (120583 119880 119879))Ensure Successful interactions among agents119896 larr997888 0119873119890 larr997888 0

while 119896 le |119873|2 do ⊳Fill119873119890 with |119873|2 agents119896 larr997888 119896 + 1119873119890 larr997888 119877119886119899119889119900119898(|119873| + 1) ⊳Pseudo-random generator Generates integer between 0 and |119873|119873119903 larr997888 0 120575119894119895 larr997888 0 ⊳Variable to store differences in opinionfor all 119894 isin 119873119890 doRandomly pick a number 119895 isin 119873(119873119890 cup 119873119903)119873119903 larr997888 119873119903 cup 119895 120575119894119895 larr997888 |119900119894 minus 119900119895|procedure dyadic-update(119894 119895) ⊳Two agents 119894 and 119895 discuss and update opinions1199001015840 larr997888 119900119894 (120583119895 119880119895 119879119895) larr997888 (120583119894 119880119894 119879119894) larr997888 (120583 119880 119879) ⊳Save agent irsquos opinion before updating and homogenize agents

if 120575119894119895 lt 119880119894 then119900119894 larr997888 120583119894119900119895 + (1 minus 120583119894)119900119894if 120575119894119895 gt 119879119894 then119900119894 larr997888 (1 + 120583119894)119900119894 minus 120583119894119900119895if 120575119894119895 lt 119880119895 then119900119895 larr997888 1205831198951199001015840 + (1 minus 120583119895)119900119895if 120575119894119895 gt 119879119895 then119900119895 larr997888 (1 + 120583119895)119900119895 minus 1205831198951199001015840

Algorithm 1 Dyadic social influence

Require Vector of values for parameters (119881 = (120574 119901119886 119901119903))Ensure Successful interaction between proposal arguments and agents

for all 119894 isin 119873 doprocedure Probability-Update(V119886 119901119886 119901119903) ⊳Update individual probabilities of change due to deliberated cues119909119894 larr997888 119901|V119886minus119900119894 |2119886 119910119894 larr997888 1199012|V119886minus119900119894 |119903119902 larr997888 0if L119886119887(119868) = 119894119899 then

procedure deliberative-update(119894 119886) ⊳Agent i argument 119886120574119894 larr997888 120574 119902 larr997888 119906 sim U(0 1) ⊳ 119906 is a realization of a uniform random variableif 119902 lt 119909119894 then119900119894 larr997888 120574119894119900119894 + (1 minus 120574119894)V119886break

if 119902 lt 119910119894(1 minus 119909119894) then119900119894 larr997888 (1 + 120574119894)119900119894 minus 120574119894V119886Algorithm 2 Deliberation influence

8 In terms of computational complexity naive agentsmaketheirmoves in119874(|A119894| log(|A119894|)) while focused agents doat least as bad

9 Tables have memory Therefore debates are not neces-sarily trees

10 If there are less than (119899119863 times |119873|)2 agents for any of bothgroups the CA summons all of them to the deliberationtable

11 Or careful consideration of the viewpoints of otherswhich implies that citizens keep an open mind and donot reject arguments outright

12 A ldquohealthyrdquo consensus refers to the case in which manyconsensual decisions are taken but opinion-wise agentsdo not agree with each other Quotes are used on theword healthy because some political science theorists

(eg [36]) believe that in certain situations agreement inopinions for deliberation is epistemologically damaging

13 Simulations on groups of |119873| = 800 and |119873| = 400agents shows that scaling the population has an effect onall themetrics but it is confounding depending onwhichcouples of attitudes structures are considered (119880 119879) Westick to |119873| = 400 as in [9] for comparability andsimulation running time

14 By balanced wemean with as many arguments with V119886 lt0 as with V119886 gt 0

15 Experiments on the system suggest that a bigger(smaller) argument pool 119872 = 1000 (119872 = 200) isassociated with a higher (lower) variance of opinions(and extremism) and higher (lower) judgment accuracyFor the rest of the observations 119872 seem to have noeffect We choose a middle value for 119872 Similarly theargument pool chosen to be balanced or uniformly

Complexity 29

Require Argumentation framework 119860119865 = (AR)Ensure Assignment of labels to arguments from the grounded labeling of 119860119865L(A)

L0(A) larr997888 (0 0 0)119896 larr997888 0do119894119899 L119896+1(A) larr997888 119894119899 L119896(A) cup 119886 | 119886 is not labeled 119894119899 L119896(A) 119886119899119889 forall119887 isin A 119904119905 119887 R 119886 then 119887 isin 119900119906119905 L119896(A)119900119906119905 L119896+1(A) larr997888 119900119906119905 L119896(A) cup 119886 | 119886 is not labeled 119894119899 L119896(A) 119886119899119889 exist119887 isin A 119904119905 119887 R 119886 and 119887 isin 119894119899 L119896+1(A)

while L119896+1(A) = L119896(A) = 0Algorithm 3 Solve argumentation frameworks (with grounded semantics)

Require Vector of values for parameters (119881)Ensure Vector of statistics (119880)

procedure Set-parameters(119881) ⊳Initialize parameters for scenarioprocedure Init-objs(119899119898 119896 ) ⊳Create119898 arguments 119899 agents and 119860119865120576procedure Solve-Argumentation-Framework(119860119865) ⊳Assigns the epistemic labelingL120576 toA119894 larr997888 1119905 larr997888 1while 119894 le 119872 do ⊳ 119872 the number of arguments to discuss119868119894 larr997888 119866119890119899 119886119903119892(1) ⊳Generates proposal argument 119868119860 119894119905 larr997888 (119868119894 0)L119886119887(119868119894) larr997888 119906119899119889 ⊳Initialize label for 119868119895 larr997888 1 ⊳Initialize debate counter 119895while ((L119886119887(119868) = 119906119899119889 or 119898 le 119895) and 119895 le 119898) doif 119905 equiv 0(119898119900119889 119905119863) then

procedure build-argumentation-framework(119899119863 119887119890ℎ119886V119894119900119903) ⊳Build the argument graph agents deliberate(119861119894119905 119878119894119905) larr997888 (119883 119884) sub (⋃119897isin119873119899119863 A119897119894119905 ⋃119897isin119873119899119863 R119897119894119905) ⊳ 119873119899119863 sub 119873 are agents sampled by the CA119860 119894119905 larr997888 119889(119860 119894119905 (119861119894119905 119878119894119905))procedure Solve-Argumentation-Framework(A119889) ⊳FindL(A119889)L119886119887(119868119894) larr997888 L119886119887119863(119868119894)119895 larr997888 119895 + 1

else119896 larr997888 1while 119896 le 119905119863 doprocedure dyadic-social-influence(120583 119879 119880) ⊳Agents discuss one-to-one119896 larr997888 119896 + 1119905 larr997888 119905 + 1

if 120572 gt 0 andL119886119887(119868119894) = 119900119906119905 then ⊳Agents vote for the proposalif (|119897 isin 119873|119900119897 times V119868119894 ge 0| ge 120572 times 119899) then

L119886119887(119868119894) larr997888 119894119899else

L119886119887(119868119894) larr997888 119900119906119905if L119886119887(119868119894) = 119894119899 thenprocedure Deliberation-influence(120574 119901119886 119901119903) ⊳Agents are influenced by the result of the decision process119894 larr997888 119894 + 1119905 larr997888 119905 + 1119880 larr997888 119862119900119898119901119906119905119890 119878119905119886119905119894119904119905119894119888119904(119881)

return 119880 ⊳Get the statisticsAlgorithm 4 Run for one scenario

generated does not seem to play a significant explanatoryrole on any of the metrics

16 Experiments show that themaximumnumber of debatesis not an important parameter even if it is an intuitiveone when considering decision-making processes Inpractice debates end after at most 119898 + 1 deliberation

steps and even then it happens rarely However to avoidunpleasant surprises like infinite debates on a completeargument graph containing the proposal argument weadded the constraint 119898 Its value is precisely 7 becausethe biggest minimum number of debates we studyis 6

30 Complexity

17 To ensure comparability with the deliberative andmixedopinion dynamics models on judgment accuracy andcoherence agents vote on proposal arguments every 119905119863times(119898+1) 119905119863 and119898does not explain variability in judgmentaccuracy nor in coherence

18 Student t-tests for independent samples are robust tothe violation of two hypothesis on the distribution ofthe samples normality and homogeneity of variancesas long as the sample size is big enough (central limittheorem) and if the difference in size of the comparedsamples is smallrespectively

19 Studentrsquos test version forwhen variances are not assumedequal and the difference in size of the compared samplesis big

20 We do not correct for homoskedasticity nor normality oferrors in the estimates

21 Balanced as the same number of runs per scenario orsuch that no significant correlations are found betweenparameters

References

[1] R Axelrod ldquoThe dissemination of culture a model withlocal convergence and global polarizationrdquo Journal of ConflictResolution vol 41 no 2 pp 203ndash226 1997

[2] D M Rousseau Reinforcing the MicroMacro Bridge Organiza-tionalThinking and Pluralistic Vehicles SAGEPublications SageLos Angeles Calif USA 2011

[3] S Galam and SMoscovici ldquoTowards a theory of collective phe-nomena consensus and attitude changes in groupsrdquo EuropeanJournal of Social Psychology vol 21 no 1 pp 49ndash74 1991

[4] A Nowak J Szamrej and B Latane ldquoFrom private attitude topublic opinion a dynamic theory of social impactrdquo Psychologi-cal Review vol 97 no 3 pp 362ndash376 1990

[5] G Deffuant S Huet and F Amblard ldquoAn individual-basedmodel of innovation diffusion mixing social value and individ-ual benefitrdquo American Journal of Sociology vol 110 no 4 pp1041ndash1069 2005

[6] G Deffuant D Neau F Amblard and G Weisbuch ldquoMixingbeliefs among interacting agentsrdquoAdvances in Complex Systems(ACS) vol 3 no 01n04 pp 87ndash98 2000

[7] J Rouchier P Tubaro and C Emery ldquoOpinion transmission inorganizations an agent-based modeling approachrdquo Computa-tional and Mathematical OrganizationTheory vol 20 no 3 pp252ndash277 2014

[8] M Meadows and D Cliff ldquoThe relative disagreement model ofopinion dynamics where do extremists come fromrdquo in Pro-ceedings of the International Workshop on Self-Organizing Sys-tems Lecture Notes in Computer Science pp 66ndash77 Springer2013

[9] W Jager and F Amblard ldquoUniformity bipolarization and plu-riformity captured as generic stylized behavior with an agent-based simulation model of attitude changerdquo Computational ampMathematical Organization Theory vol 10 no 4 pp 295ndash3032005

[10] Y-F Zhang H-Y Duan and Z-L Geng ldquoEvolutionary mech-anism of frangibility in social consensus system based onnegative emotions spreadrdquo Complexity vol 2017 Article ID4037049 8 pages 2017

[11] A Flache andMWMacy ldquoSmall worlds and cultural polariza-tionrdquoThe Journal of Mathematical Sociology vol 35 no 1-3 pp146ndash176 2011

[12] A Tsang and K Larson ldquoOpinion dynamics of skepticalagentsrdquo in Proceedings of the 13th International Conference onAutonomous Agents and Multiagent Systems AAMAS 2014 pp277ndash284 International Foundation forAutonomousAgents andMultiagent Systems France May 2014

[13] K M Hansen Deliberative democracy and opinion formationUniversity Press of Southern Denmark Odense 2004

[14] C R Sunstein ldquoThe law of group polarizationrdquo Journal ofPolitical Philosophy vol 10 no 2 pp 175ndash195 2002 WileyOnline Library

[15] S Moscovici and W Doise Dissensions et Consensus unetheorie generale des decisions collectives Presses Universitairesde France-PUF 1992

[16] J S Fishkin and R C Luskin ldquoExperimenting with a demo-cratic ideal deliberative polling and public opinionrdquo ActaPolitica vol 40 no 3 pp 284ndash298 2005

[17] R C Luskin J S Fishkin and S Iyengar Considered Opinionson US Foreign Policy Face-To-Face versus Online Delibera-tive Polling International Communication Association NewOrleans La USA 2004

[18] J Barabas ldquoHowdeliberation affects policy opinionsrdquoAmericanPolitical Science Review vol 98 no 4 pp 687ndash701 2004

[19] O Pryymak A Rogers and N R Jennings ldquoEfficient opinionsharing in large decentralised teamsrdquo in Proceedings of the 11thInternational Conference on Autonomous Agents andMultiagentSystems vol 1 pp 543ndash550 Spain June 2012

[20] J Rouchier and E Tanimura ldquoWhen overconfident agents slowdown collective learningrdquo Simulation vol 88 no 1 pp 33ndash492012

[21] D Miller ldquoDeliberative democracy and social choicerdquo PoliticalStudies vol 40 S1 pp 54ndash67 1992

[22] M Sherif and C I Hovland Social Judgment Assimilation andContrast Effects in Communication and Attitude Change YaleUniversity Press 1961

[23] P M Dung ldquoOn the acceptability of arguments and its funda-mental role in nonmonotonic reasoning logic programmingand n-person gamesrdquo Artificial Intelligence vol 77 no 2 pp321ndash357 1995

[24] M Caminada and G Pigozzi ldquoOn judgment aggregation inabstract argumentationrdquo Autonomous Agents and Multi-AgentSystems vol 22 no 1 pp 64ndash102 2011

[25] P Baroni M Caminada andM Giacomin ldquoAn introduction toargumentation semanticsrdquoThe Knowledge Engineering Reviewvol 26 no 4 pp 365ndash410 2011

[26] S Chambers ldquoDeliberative democratic theoryrdquo Annual Reviewof Political Science vol 6 no 1 pp 307ndash326 2003

[27] E Bonzon andNMaudet ldquoOn the outcomes of multiparty per-suasionrdquo in Proceedings of the 10th International Conference onAutonomous Agents and Multiagent Systems 2011 AAMAS 2011vol 1 pp 47ndash54 International Foundation for AutonomousAgents and Multiagent Systems Taiwan May 2011

[28] J Ranciere Disagreement Politics and Philosophy University ofMinnesota Press 1999

[29] T Bench-Capon and K Atkinson ldquoAbstract argumentation andvaluesrdquo in Argumentation in Artificial Intelligence pp 45ndash64Springer 2009

[30] M Caminada ldquoOn the issue of reinstatement in argumenta-tionrdquo in Proceedings of the European Workshop on Logics in

Complexity 31

Artificial Intelligence vol 4160 of Lecture Notes in ComputerScience pp 111ndash123 Springer Berlin Heidelberg

[31] GAVreeswijk ldquoAn algorithm to computeminimally groundedand admissible defence sets in argument systemsrdquo in Pro-ceedings of the 2006 conference on Computational Models ofArgument Proceedings of COMMA 2006 vol 144 pp 109ndash120IOS Press Amsterdam The Netherlands 2006

[32] C W Sherif M Kelly H L Rodgers et al ldquoPersonal involve-ment social judgment and actionrdquo Journal of Personality andSocial Psychology vol 27 no 3 pp 311ndash328 1973

[33] G Deffuant F Amblard G Weisbuch and T Faure ldquoHow canextremism prevail A study based on the relative agreementinteraction modelrdquo Journal of Artificial Societies and SocialSimulation vol 5 no 4 p 1 2002

[34] E Awad M W Caminada G Pigozzi M Podlaszewski andI Rahwan ldquoPareto optimality and strategy-proofness in groupargument evaluationrdquo Journal of Logic and Computation vol 27no 8 pp 2581ndash2609 2017

[35] D Grossi and G Pigozzi ldquoJudgment aggregation a primerrdquoSynthesis Lectures on Artificial Intelligence and Machine Learn-ing vol 8 no 2 pp 1ndash151 2014

[36] H Landemore and S E Page ldquoDeliberation and disagreementproblem solving prediction and positive dissensusrdquo PoliticsPhilosophy and Economics vol 14 no 3 pp 229ndash254 2015

[37] R Hegselmann and U Krause ldquoOpinion dynamics andbounded confidence models analysis and simulationrdquo Journalof Artificial Societies and Social Simulation vol 5 no 3 2002

[38] A Stefanelli and R Seidl ldquoModerate and polarized opinionsUsing empirical data for an agent-based simulationrdquo inProceed-ings of the Social Simulation Conference 2014

[39] S Gabbriellini and P Torroni ldquoA new framework for ABMsbased on argumentative reasoningrdquo in Advances in SocialSimulation vol 229 pp 25ndash36 Springer Berlin Germany 2014

[40] H Mercier and D Sperber ldquoWhy do humans reason Argu-ments for an argumentative theoryrdquo Behavioral and BrainSciences vol 34 no 2 pp 57ndash74 2011

[41] H V Parunak T C Belding R Hilscher and S BruecknerldquoModeling and managing collective cognitive convergencerdquoin In Proceedings of the 7th iNternational Joint Conference onAutonomous Agents and Multiagent Systems vol 3 pp 1505ndash1508 International Foundation for Autonomous Agents andMultiagent Systems 2008

[42] H Prakken ldquoFormal systems for persuasion dialoguerdquo TheKnowledge Engineering Review vol 21 no 2 pp 163ndash188 2006

[43] K Atkinson T Bench-Capon and P Mcburney ldquoA dialoguegame protocol for multi-agent argument over proposals foractionrdquoAutonomousAgents andMulti-Agent Systems vol 11 no2 pp 153ndash171 2005

[44] M Thimm and A J Garcıa ldquoClassification and strategicalissues of argumentation games on structured argumentationframeworksrdquo in Proceedings of the 9th International Conferenceon Autonomous Agents and Multiagent Systems vol 1 pp 1247ndash1254 International Foundation for Autonomous Agents andMultiagent Systems Canada 2010

[45] I Rahwan and K Larson ldquoMechanism design for abstractargumentationrdquo in Proceedings of the 7th International JointConference on Autonomous Agents and Multiagent SystemsAAMAS 2008 pp 1031ndash1038 International Foundation forAutonomous Agents and Multiagent Systems Portugal May2008

[46] H Prakken R Riveret A Rotolo and G Sartor ldquoHeuristics inArgumentation A Game-Theoretical Investigationrdquo EuropeanUniversity Institute 2008

[47] N Hadidi Y Dimopoulos and P Moraitis ldquoTactics and con-cessions for argumentation-based negotiationrdquo in Proceedingsof the COMMA vol 245 pp 285ndash296 2012

[48] N Oren and T J Norman ldquoArguing using opponent modelsrdquoin Proceedings of the International Workshop on Argumentationin Multi-Agent Systems pp 160ndash174 Springer 2009

[49] S Delre W Jager T Bijmolt and M Janssen ldquoTargeting andtiming promotional activities An agent-based model for thetakeoff of new productsrdquo Journal of Business Research vol 60no 8 pp 826ndash835 2007

[50] A Mirtabatabaei P Jia and F Bullo ldquoEulerian opinion dynam-ics with bounded confidence and exogenous inputsrdquo SIAMJournal on Applied Dynamical Systems vol 13 no 1 pp 425ndash446 2014

[51] V Barbet N Guiraud V Laperriere and J Rouchier Hagglingon Values Towards Consensus or Trouble 2019

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Submit your manuscripts atwwwhindawicom

Complexity 17

Table 4 Ordinary least squares estimates for the deliberative interactions scenarios for most of the observed metrics

Dependent variable119881119886119903(119900) 119878ℎ 119890119888 119894119903(1) (2) (3) (4)Constant 0032lowastlowastlowast 13092lowastlowastlowast 0370lowastlowastlowast 1007lowastlowastlowast

(0001) (0406) (0001) (0001)

119898 = 3 (ref119898 = 1) 0012lowastlowastlowast minus2062lowastlowastlowast minus0012lowastlowastlowast minus0001(0001) (0276) (0001) (0001)

119898 = 6 (ref119898 = 1) 0023lowastlowastlowast minus3285lowastlowastlowast minus0018lowastlowastlowast minus0004lowastlowastlowast(0001) (0276) (0001) (0001)

119899119863 = 002 (ref 119899119863 = 001) 0007lowastlowastlowast minus0965lowastlowastlowast minus0006lowastlowastlowast minus0002lowastlowast(0001) (0276) (0001) (0001)

119899119863 = 005 (ref 119899119863 = 001) 0020lowastlowastlowast minus3316lowastlowastlowast minus0015lowastlowastlowast minus0004lowastlowastlowast(0001) (0276) (0001) (0001)

120572 = 05 (ref 120572 = 0) 0146lowastlowastlowast 68932lowastlowastlowast 0068lowastlowastlowast minus0507lowastlowastlowast(0001) (0276) (0001) (0001)

120572 = 066 (ref 120572 = 0) 0291lowastlowastlowast minus13842lowastlowastlowast 00004 minus0000(0001) (0276) (0001) (0001)

119896 = 8 (ref 119896 = 4) 0004lowastlowastlowast minus0815lowastlowastlowast minus0002lowastlowastlowast minus0001(0001) (0276) (0001) (0001)

119896 = 16 (ref 119896 = 4) 0008lowastlowastlowast minus1118lowastlowastlowast minus0004lowastlowastlowast minus0003lowastlowastlowast(0001) (0276) (0001) (0001)

119861119890ℎ119886V119894119900119903 = ldquoFocusedrdquo (ref ldquoNaiverdquo) 0015lowastlowastlowast minus1980lowastlowastlowast minus0012lowastlowastlowast minus0004lowastlowastlowast(00005) (0225) (00004) (0001)

120574 = 01 (ref 120574 = 005) minus0020lowastlowastlowast 5762lowastlowastlowast minus00003 minus0001(0001) (0276) (0001) (0001)

120574 = 02 (ref 120574 = 005) minus0025lowastlowastlowast 8648lowastlowastlowast minus00003 minus0001(0001) (0276) (0001) (0001)

119901119886 = 05 (ref 119901119886 = 03) minus0013lowastlowastlowast 1581lowastlowastlowast 00004 00003(00005) (0225) (00004) (0001)

Observations 19440 19440 19440 19440R2 0931 0844 0562 0958Adjusted R2 0931 0844 0562 0958Residual Std Error (df = 19427) 0033 15701 0030 0050F Statistic (df = 12 19427) 21 844790lowastlowastlowast 8 733563lowastlowastlowast 2 079585lowastlowastlowast 36 761250lowastlowastlowastNote lowastplt01 lowastlowastplt005 lowastlowastlowastplt001

opinions that may be of interest in the mixed discussionmodel

Group coherence shifts of opinions and the varianceof opinions are highly affected by the voting thresholdthe coherence metric is reduced by half when allowing forclassic majority voting shifts explode and the variance ofopinions increases dramatically This is not surprising sincethe majority quota rule works as a buffer to the success ofdecision-making processes and in consequence it hampersthe deliberative opinion updates For the shifts of opinions itis clear that the two-third majority voting rule results in littleopinion change and that the simple majority rule inducesgroup polarization As expected there are less extremists inscenarios where there is no voting and the shifts of opinionare more likely to happen when agents are more sensitiveto deliberation When we include voting in the system

the grouprsquos judgment accuracy decreases due to the correctdeliberated decisions that are vetoed by a salient opinion-agent majority Strangely the effect only holds for the simplemajority decision rule Lastly the proportion of individualsthat participate in the debate and the minimal amount ofdebates have similar marginal effects on all the metrics (seeTable 4)

415 Comparing Deliberative and Pair-Wise Opinion Dynam-ics in a Nuthshell Qualitatively both models yield similaroutcomes but for the fact that deliberation never producesopinion polarization with both types of extremists (seeFigure 7) Furthermore clusters in the deliberative interac-tions model are less homogeneous in size and multiclusterconvergence occurs only when deliberation is rarely suc-cessful which happens sporadically given the calibration of

18 Complexity

Table 5 Mean difference 095 confidence intervals for metrics by model comparison

Model Metric119881119886119903(119900) 119901119903119900119901119890119909 119878ℎ 119890119888 119894119903Dyadic vs Deliberative [0191 0216] [0170 0197] [-2948 -2828] [0074 0082] [0165 0172]Mixed vs Deliberative [0142 0147] [0150 0155] [-2085 -1976] [0003 0002] [-0038 -0030]Mixed vs Dyadic [-0072 -0046] [-0045 -0018] [8332 8859] [-0079 -0071] [-0205 -0201]the model On the other hand pair-wise opinion dynamicsproduces multicluster convergence most of the time andcentral convergence only in the infrequent cases where thelatitude of acceptance is large

The comparison of these two scenarios in terms ofthe metrics is straightforward Whilst dyadic interactionsallow for more variability in opinions through pair-wiseinteractions and more mistakes in judging proposals delib-erative updates simply act as an antagonist force On aver-age the deliberative interaction model produces opiniondistributions with smaller variance and yields steady stateswith higher judgment accuracy a higher shift statistic andnaturally lower group coherence since only proposals that aredeliberated 119894119899 are eligible for voting (see Table 5)

42 Articulating Dyadic and Deliberative Social Interactionsin Opinion Dynamics So far we have considered scenarioswhere argumentation had no place and where individualsmade decisions according to opinions that were purelyconstructed from pair-wise discussions We also observedcases in which agents formed their opinions only by inte-grating deliberated proposals into their opinion updates Thescenarios we present in this section combine the two pre-ceding scenarios to account for both the effects of pair-wisediscussion and the effects of deliberation on opinions judg-ment accuracy and group coherence From the precedingobservations we conclude the following deliberative opiniondynamics reduces the variance and the proportion of extrem-ists because it shifts opinions greatly and polarizes groupsby eliminating at least one type of extremist the one whoseopinions are initially opposed to the deliberated resultsor it brings opinions close to neutrality (see Figure 7(b))This may not be desired in a group since the diversity andstability of opinions are necessary for deliberation to makesense and lead to the collective acceptance of controversialproposals Dyadic opinion dynamics on the other handtend to increase the variance of opinions and the numberof extremists while having a lesser aggregated effect on theshifts of opinion This is an immediate result of the pair-wiseinteraction dynamics inasmuch as it polarizes individualsvery quickly

To sum up in the mixed interactions model deliberationhinders the effect of voting on opinions by making someproposals that would normally be submitted for voting noteligible for voting It may also undo opinion changes dueto pair-wise interactions The mixed model can be seen asthe deliberation opinion dynamics model in which pair-wiseinteractions occur in between collective discussion and mayaffect which arguments are advanced during deliberationEquivalently it can be seen as the dyadic opinion dynamics

model in which deliberation accounts for endogenousldquoshocksrdquo on the opinion distribution

421 Qualitative Analysis of the Mixed Dynamics In Figures7(c) and 7(f) agents update their opinions through delibera-tion and pair-wise interactions Opinion trajectories visualsshow a combination of both previously described opiniontrajectories Over time as agents discuss with one anotheropinion clusters form as in the pair-wise or dyadic opiniondynamics Deliberation may disrupt the formation of theseclusters and three different situations may occur

(i) Deliberation disrupts the formation of the opinionclusters but not sufficiently to jeopardize pair-wiseopinion cluster formation (eg for high values 119905119863)Two phenomena may be responsible for it (1) thevariability of the proposal argumentrsquos support for theprinciple P and (2) too few (successive) deliberationsteps that result in opinion updates Once the opinionclusters are formed successful deliberation foreshad-ows the merging of the previously formed clustersmay bring extremists closer to moderate agents andthanks to the assimilation component of the pair-wisedynamics allow bigger and more moderate opinionclusters to form (refer to Figure 7(c))

(ii) Deliberation significantly disrupts the pair-wisedynamic formation of clusters This may happenwhen the requirements for accepting proposals areweak in the decision-making process (eg low 120572) Inthis case one has either central convergence sinceall agents follow the deliberated results in the samemanner or 3-cluster convergence with two extremeopinion foci In the latter situation the extremegroups form quickly (like in the pair-wise model) anddeliberation cannot pull extreme agents away fromthe extreme foci they are in The main reason for thisis that immediately after an extreme agent attempts toleave her foci through deliberation updates she getspulled or pushed back into it by means of a dyadicinteraction with another extreme agent

(iii) Deliberation does not disrupt the formation of theopinions cluster This is the rare case scenario forvery demanding decision-making processes in termsof inclusion (eg 119899119863) and argument acceptability (eg120572) Opinions converge as in the pair-wise opiniondynamics model (refer to Figures 7(a) and 7(d))

Last when there is voting (120572 = 0) the position ofthe moderate opinion cluster is determined by the first

Complexity 19

minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

Var(o)

sh

ec

ir

Var(o)

propex

sh

ec

ir

1

098

minus02

minus004

001

1

minus014

minus001

minus003

1

026

minus028

1

minus057 1

ＪＬＩＪ

Ｒ

Figure 8 Significant correlation estimates for themetrics of interestin the mixed opinion dyamics scenarios

accepted proposal argument andor the opinionmajority thatis formed from previous dyadic interactions (see Figure 7(f))

422 Relating the Metrics of Interest in Mixed OpinionDynamics The first thing to notice is that for all of thesescenarios all the metrics are significantly correlated (seeFigure 8) The variance of opinions and the proportion ofextremists are almost perfectly correlated To some extentthis means that a high variance of opinions known thatthe mean of opinions is statistically null equates to havingmany extremists in the population be towards one or theother side of the opinion distribution (eg extreme bipolar-ization) Similarly we find a negative and strong correlationbetween the judgment inaccuracymeasure and the coherencecoefficient (see Figure 8)mdashwhen judgment inaccuracy is high(low) coherence to deliberation is low (high) Cases likethese are telling since they may be related to sequences ofcollective decisions in which agents do not make too manymistakes when judging proposal arguments and still makedeliberated decisions that are on average in line with theirprinciples

The shift statistic interestingly is negatively correlatedwith the variance of opinions with judgment accuracy andwith the coherence coefficient In a way this means thathaving more moderate agents conduces on average to moremistakes in judging arguments and to less coherent collectivedecisions An explanation for this phenomenon could be thatas agents reach consensus in opinion important (possiblyextreme) arguments for correctly judging argument propos-als are not advanced in the deliberation arenawhich results inmore mistakes and in possibly accepting proposal argumentsthat after voting are not meant to be accepted Indeed the

negative correlation between the variance of opinions andjudgment inaccuracy points towards this direction since onaverage the more shifts there are the smaller the variance ofopinions is (see Figure 8)

423 Comparing Mixed Dynamics with Other MonolithicDynamics of Interactions Qualitatively opinion trajectoriesin the mixed model are more chaotic than in the pair-wise and deliberative interaction models (see Figure 7) Asanalyzed previously deliberation dynamics disrupt the pair-wise dynamics and create situations in which central andmulticlustered convergence (with two clusters at the extremesof the distribution) are possible at the expense of more(pair-wise dynamics) or less (deliberative dynamics) clustersIt tolerates the survival of very small isolated group ofagents that the previous models rarely tolerate in a hundredcollective decision-making steps (see Figure 7(c))

One of the most interesting observations one can makeof this family of models regards how groups change (in termsof our metrics) when dyadic social interactions are addedto deliberative interactions and vice-versa When comparingthe mixed interactions with only deliberative ones shiftsagentsrsquo aggregated change in opinions are on average lesspronounced The variance of opinions and the proportionof extremists are significantly higher for the former than forthe latter group coherence in contrast is lower with respectto the scenarios where opinions are formed only throughdeliberation Judgment accuracy follows that same trendA reason for this might be that mixed dynamics are fasterto yield semistable clusters andor opinion convergence Inconsequence the potential argument pool for deliberativeexchange shrinks faster and deliberation becomes less effec-tive over time which results in more collective mistakes Theoutcomeon coherence is predictable inasmuch as deliberativeinteractions produce group polarization If a group agrees inopinion debates are sterile because themembers of the groupwill most likely refrain from attacking the proposal argu-ments they all support and will work together to disparagethe proposal arguments they do not support (see Table 5)

In the same manner we report that the proportion ofextremists and the variance of opinions are barely lower inthe pair-wise interaction model when including deliberationin the decision-making process Shifts of opinions are alsobarely lower when there is no deliberation This can indicatethat deliberation does not undo the changes in opiniondue to pair-wise interactions nor does it amplifies them Incontrast when we consider the labeling-based metrics themixed interaction model ensures lower judgment inaccuracybecause of deliberation and lower group coherence (seeTable 5)

In sum the mixed discussion model is a compromisebetween the pair-wise interaction and deliberative opiniondynamic models in that it offers some guarantees in termsof judgment accuracy and coherence while allowing forldquoreasonablerdquo variance in the opinion pool It establishes somekind of trade-off between labeling-based metrics and thevariance of opinions that may be of interest for decision-making process design

20 Complexity

Table 6 Summary of sign or relative marginal effects for all varying parameters on dyadic deliberative and mixed interactions models inand for all experiments

Metrics Parameters119905119863 119898 119899119863 120572 119879 119880 119901119886 120574 119896 119861119890ℎ119886V119894119900119903119881119886119903(119900) XX X X O X X X X X X119901119903119900119901119890119909 XX X X O X X X X X X119878ℎ XX X X O X X X X X119890119888 XX X X XO X XOO XOO X X119894119903 XX X X O XO O X OOX XOO X XNota gray symbols correspond to dyadic dynamics light gray to deliberative dynamics and bold black to mixed dynamics Given a scenario the symbol 119883stands for constant parameter and 119874 for insignificant marginal effect and as parameters grow for positive (negative) marginal effect for the parameterFrom left to right at the parameter level the leftmost arrow corresponds to the scenario with the strongest (in absolute value) marginal effect of the parameter(with respect to to its factors) Example Let 119901 be a parameter and 119874 stands for ldquo119901 has a significant positive effect in dyadic discussion scenario negativein the deliberative one and insignificant in mixed discussion The positive effect in dyadic discussion is greater than the absolute value of the negative one inthe purely deliberative scenariordquo

424 Comments on the Differences between the Three Modelsfor OLS Regression Estimates The significance of most of theregression parameters estimated for the mixed interactionsscenarios are similar to those estimated for the pure dyadicanddeliberative scenariosNevertheless themagnitude of theeffects is different insomuch that the dynamics themselvesare different So instead of commenting the linear regressionestimates of the mixed model we compare when possiblethe standardized significant coefficients of the regressors ofthe mixed model with those of the pair-wise and deliberativeinteractionmodels (see Table 6 for a summary of the results)

In mixed interactions for instance the influence of119898 onthe opinion distribution is much more preponderant than inthe deliberative dynamics scenarios Going from 119898 = 1 to119898 = 3 increases the standard variation units of the variance ofopinions by twice as much as the one in the deliberative sce-nariosThis in particular is due to the design of the decision-making process If 119898 is higher then the number of socialinfluence steps (119905119863) in the decision process is also higher(compared to 119905119863 = 0 in the deliberative case) If we decideto consider these steps as contributing to the variance ofopinions then the effect of119898 on the variance of opinions hasto be stronger in the mixed interactions model In contrastthe number of agents participating in the debate has a weakereffect on the variance of opinions in the mixed interactionsmodel than in the deliberative dynamics specially for 119899119863 =005 (more than twice as much) The difference may comefrom the fact that in mixed discussions the part of theargument pool used in deliberation is more restrictive thanin the deliberative case We can argue that since pair-wisedynamics may get individuals closer to one another aftersome deliberation the same happens to the arguments thatindividuals will most likely use in deliberation Taking thisinto account means that deliberation will at many instancesbe unfruitful and have no effect on opinions thus failing tomoderate extreme opinions andor polarize the group In thesame direction a high number of participants in deliberationcan make the deliberation dynamics stiff In this case thedistribution of opinion is close to the uniform distributionwhich has a relatively high-to-moderate variancewith respectto the reference results

The strength of the deliberative dynamics has a positivestronger effect on the variance of opinions in themixedmodelthan in the deliberative model when 120574 = 02 and a negativeweaker one when 120574 = 01 A sound interpretation of thisresult is that when deliberation is successful and its effect onopinions does not get agents to move sufficiently away fromtheir current positions agents are pulled or pushed back totheir previously held opinions the variance of opinions doesnot changemuchOn the other hand if the effect of successfuldeliberation is strong opinions can change sufficiently tomake agents much more prone to assimilation into moderateopinion foci the variance of opinions falls

The deliberative opinion dynamics model is much moresensitive to 120572 than the mixed discussion model Differ-ences in marginal effects can double The probability ofattraction to deliberated results (119901119886) for instance has astrongmoderating effect (negative) on agentsrsquo opinions in thedeliberative opinion dynamics but a meager one (two-fold)in the mixed and dyadic dynamic models This is becauseopinion updates in the deliberative model depend only onthat parameter whereas in the mixed model more numeroustypes of interactions make opinion updates possible

In terms of judgment accuracy all parameters except thesize of the argument sacks (119896) have a significant slightlystronger effect on the deliberative dynamics than in themixedmodel In effect the difference may be attributed to theupshot of pair-wise interactions on the distribution of opin-ions and on the arguments that would be advanced duringdeliberation The result respecting parameter 119896 however iscounterintuitive Since argument sacks are static scenarioswhere shifts are higher should make 119896 more important fordeliberation to succeed and it is precisely in the scenarios ofpure deliberative dynamics that we find the biggest opinionshifts This may point to the hypothesis that in deliberativeinteractions agents change opinions substantially but do notoften change their adherence to the principle (go from anegative opinion to a positive one or vice versa) when thereis no group polarization

43 Sensitivity Analysis for the Procedural Parameters ofInterest Weperform a sensitivity analysis of the observationson the parameters of interest in themixed interactionsmodel

Complexity 21

We extend the parameter domains of the pure dyadic anddeliberative scenarios as we make it explicit in Table 2

431 Minimum Number of Debates (119898) We observe that theminimum number of debates has a significant well-observedeffect on all of our metrics excluding group coherence Forthe variance of opinions we can observe that the moredebates there are the bigger the value of the metric isalthough the higher 119899119863 and 119905119863 are the weaker the overalleffect of119898 becomes (Figures 9(d) and 9(a)) Furthermore themarginal effect of increasing minimal debates is decreasingan additional deliberation step requirement increases thevariance of opinions but less and less as it grows 119898 coupledwith 119899119863 has a shy S-shaped effect (see Figure 9(d)) on thevariance of opinions meaning that it has a stronger effectwhen 119899119863 is low which suggests a possible trade-off betweenthe number of arguments accepted in the debate and thenumber of debates required before accepting a proposal

In contrast the shifts in opinion are less and less likelyas 119898 grows and this is independent of the variations of theother parameters Again the effect is marginally decreasingand is only truly significant when 120572 = 05 An explanationto these effects may be linked to the design of the systemFirst the variance of opinions is higher when deliberation isasked for because the more deliberation steps there are thehigher the chances are that the central proposal argumentis deemed unacceptable specially when agents are focusedFurthermoremechanically speaking increasing theminimalamount of debates implies that whenever a decision is to betaken at least 119898 times 119905119863 time steps have to take place and if atany moment a proposal argument is considered undecided119905119863 time steps are added to the process So unless the debatedoes not yield 119906119899119889 labels for proposal arguments (highlyunlikely considering that 120590 = grounded semantics) the morenondeliberation steps there are in the decision process thehigher the variance of opinions is Concerning the shiftswhen 120572 = 05 either the system is too stiff to accept anyproposal argument and opinions do not changemuch or theeffects of pair-wise discussion and deliberation cancel out insuch a way that individual shifts are minimal (Figure 9(b))

On the side of labeling-based metrics the more debatesare asked for the more accurate a group is in its judgmentthe effect being smaller as 119898 grows When agents are naivethe effect is quasilinear while when they are focused thestrongest effects of acquiring more deliberation are foundwhen levels of deliberation are already low (Figure 9(f))Thiscan be explained by the fact that the more debates thereare in the decision process the closer one gets to the idealargumentation framework

432 Proportion of the Population in Deliberation Steps (119899119863)Like for 119898 119899119863 has a significant effect on the proportionof extremists and on the shifts and variance of opinions(Figure 9(e)) This may result from the fact that being ableto put more arguments in play at the same debate step mayincrease the odds of revealing cycles around the proposalargument in119860119865120576 Given that we use grounded semantics thearguments in the cycles are labeled 119906119899119889 and in consequence

debates are more often postponed when 119899119863 is higher Post-poning debates in turn increases the number of nondebatessteps in the decision process which increases the varianceof opinions and limits over time the moderating effect ofdeliberation Moreover the effect of this parameter is verydependent on the value of 120572 (Figure 9(e)) For shifts forinstance 120572 = 05 makes the effect of 119899119863 negative while120572 = 0 makes it positive but to a lesser extent Furthermorefor higher requirements of deliberation (119898) adding moreindividuals to the deliberation process has weaker effectson the variance of opinions and on the other metrics It isalso quite interesting to notice that it has the same effect onagents independently of agent behavior during deliberationThis is a surprising result as one would have expected morefocused agents in a deliberation arena to cause a raise inthe proportion of extremists as they play to knock outopposing proposal arguments and thereof reducing the effectof deliberation (lowers variance) on the whole population

Finally similar to119898 addingmore people into the deliber-ation process increases judgment accuracy (see Figure 9(c))and has no clear effect if any on group coherence (seeFigure 9(i))

433 Steps between Deliberation Steps (119905119863) In all configu-rations 119905119863 increases the proportion of extremists (variance)and decreases the shifts of opinions among agents Theshifts and the effect on the variance of opinions are presentfor 120572 = 05 insignificant otherwise (Figure 9(b)) 119905119863 ishighly linked to 119898 by construction When 119898 = 1 thecurve linking the variance of opinions and 119905119863 is convex As119898 increases the curve becomes more and more concavewhich means that 119905119863 has a more important effect on theopinion distribution as collective decisions take longer to beachieved This seems counterintuitive but in reality it reflectsthemultiplicative relationship between deliberation and pair-wise interactions and the semantics in the model If 119898 is lowand 119905119863 is high the effective number of pair-wise interactionsis on average smaller in the deliberation process Hencethe structure of the decision-making process hinders anyincrease in the variance of opinions If119898 is high the oppositeeffect is observed Additionally since the grounded semanticsyields few 119894119899 arguments with respect to other admissibility-based semantics getting closer to the ideal argumentationframeworkmaymake it difficult to obtain the expected effectsof deliberative discussion on agents Lower 119898 constrains thevariance of opinions by making deliberation more influentialon opinions

In terms of judgment accuracy and coherence we observethat 119905119863 does not explain coherence yet it tends to decreasejudgment inaccuracy This may be due to a more variedargument pool in the debate which in turn may be aconsequence of a higher variance of opinions or vice versa

434 Acceptability Voting Quota (120572) By far it is the mostinfluential parameter in our study It changes the directionand the intensity of the effect of all the other proceduralparameters and by construction heavily constrains the roadto accepting a proposal (see Figure 9)

22 Complexity

015

020

025

030

035

040

1 2 3 4 5 6

12345

Values of Ｇ

(a) 119905119863 times 119901119903119900119901119890119909 by119898

0

10

20

30

40

50

1 2 3 4 5 6

005066

Values of

(b) 119905119863 times 119878ℎ by 120572

040

045

050

055

060

1 2 3 4 5

001002003004005

Values of Ｈ＄

(c) 119898 times 119890119888 by 119899119863

015

020

025

030

035

040

1 2 3 4 5

001002003004005

Values of Ｈ＄

(d) 119898 times 119901119903119900119901119890119909 by 119899119863

0

10

20

30

40

50

001 002 003 004 005

00005066

Values of

(e) 119899119863 times 119878ℎ by 120572

040

045

050

055

060

1 2 3 4 5

BehaviorNaiveFocused

(f) 119898 times 119890119888 by 119887119890ℎ119886V119894119900119903

015

020

025

030

035

040

0 05 066Behavior

NaiveFocused

(g) 120572 times 119901119903119900119901119890119909 by 119887119890ℎ119886V119894119900119903

05

06

07

08

09

10

1 2 3 4 5 6

00005066

Values of

(h) 119905119863 times 119894119903 by 120572

08700

08725

08750

08775

08800

001 002 003 004 005

BehaviorNaiveFocused

(i) 119899119863 times 119894119903 by 119887119890ℎ119886V119894119900119903

Figure 9 Curves of mean observations for different parameters and metrics for 36000 runs Bars are 095 confidence intervals Plots withthe same observables (ordinate) except for 119894119903 are on the same scale

Complexity 23

The higher the requirement for accepting the proposal isthe higher the proportion of extremists and the variance ofopinions are and the lower the shift statistic is In few words120572 constrains the world to dyadic discussion or throws it intoa process in which deliberation is much more importantthan dyadic interactions This is why for any 120572 = 05 oneeither gives too much weight to deliberated results whichshadows pair-wise interactions or too little weight in thesense that no deliberated result is ever accepted and thusnever integrated into the opinions of the agents that havevoted for it The reason for the latter is that the latitudeacceptance is too low and thereafter pair-wise interactionsare not enough to unevenly polarize the population in such away that deliberated proposal arguments are voted favorablyby a non-negligeable majority Shifts for 120572 = 0 are stable fordifferent levels of 119905119863 which implies that deliberation keeps incheck all the shifts of opinions related to an increase in thenumber of pair-wise interactions For 120572 = 05 deliberationis less likely to be successful and shifts tend towards the shiftlevels for 120572 = 0 (see Figure 9(b))

Concerning labeling-basedmetrics120572 entirely determinesthe coherence statistic For 120572 = 05 there is no difference incoherence because of how coherence is defined it is maximalwhen 120572 = 0 and strangely maximal when 120572 = 066(see Figure 9(h)) Surprisingly 120572 has no effect on judgmentaccuracy One would have expected a higher 120572 to increasejudgment accuracy since agents would hardly collectivelyaccept a proposal argument discussed during deliberationProposal arguments have given how the argument lattice isgenerated a higher probability of being rejected than of beingaccepted in the ideal argumentation framework

435 A Word on Focused and Naive Agents Focused andnaive specifications for agent behavior are an importantparameter in the decision process since they model howagents choose arguments in the deliberation arena In the sce-narios where agents are focused agents show equal or highervariance of opinions and are therefore more extreme (seeFigure 9(g)) independently of the voting requirements Shiftsof opinion are less likely as well We can see this happeningbecause focused agents knock out proposal arguments moreoften than naive agents provided that 119899119863 is low

One could have expected naive agents to yield morecoherent decisions since they argue sincerely However thisseems not to be the case (see Figure 9(i)) Agents are unable todeliberate in away thatmakes deliberation reflect their votingintentions either because they do not have the necessaryarguments to do so or because such arguments do not exist

In terms of judgment accuracy focused agents do betterthan naive agents in finding the ideal labels for the proposalarguments This is likely because when reconstructing theframework focused agents take into account the deliberatedproposal argument and choose themost pertinent argumentsin their sacks that relate to the state of affairs at the deliber-ation arena In fine debates result in a better approximationofL119886119887120576(119868) even though one could have believed the oppositethat is focused agents are ready to sacrifice a correct collectivelabel of a proposal argument in the goal of defending theirpositions

44 Points of Discussion In this subsection we discuss theresults obtained from the simulations We attempt to makesense of them from a social sciences perspective

441 Opinion Consensus and Dissensus The deliberativeopinion dynamics corresponds to the simulated scenarios inwhich the variance of opinions is at its lowest and judgmentaccuracy at its highest Dyadic opinion dynamics are associ-ated with the scenarios in which the variance of opinions is atits highest judgment accuracy at its lowest and coherence atitsmaximumpossible levelThemixed interactions dynamicsis a combination of these aspects for all the metrics exceptfor coherence at which it is the worsemdashdeliberation leadsto some kind of loose opinion consensus that is reinforcedby pair-wise interactions Likewise the mixed discussionmodel confirms some results exposed in [16] expressingthat opinions and voting intentions in a deliberative contextldquooften changerdquo That being said and more particularly indeliberative opinion dynamics outputs align well with twointeresting results on consensus formation given by aninterpretation of Moscovici and Doisersquos [15] work Sunsteinrsquos[14] account on deliberation and group polarization andsocial impact theory [4] The first states that mild consensusis reached in deliberation as procedure disengages agentsthe second is that within discussion groups majorities tendto become larger and opinions more extrememdashnot on thebasis of competing arguments but of the initial balance ofopinions After several decision processes we find opiniondistributions that reflect these observations even though weare not able to precisely coin the set of parameters that yieldssuch distributions (see Figure 7)

Although differences between mixed and dyadic opiniondynamics in terms of the variance of opinions and shiftsare slim the former does much better in terms of judg-ment accuracy In this sense deliberation does moderateopinions but not as much when non-deliberation steps arerelatively frequent throughout the decision-making processIn terms of Moscovici and Doise theory of consensus [15]this makes sense since pair-wise discussion engages or ego-involve agents in the discussion and pushes to convergenceof opinion at the extremes of the opinion distributionHowever what remains unclear is the reason why proceduralparameters that constrain deliberation do not disengageagents from discussion and thus result in a mild consensusas also expressed by Moscovici and Doise [15] A possibleexplanation of this phenomenon may stem from the factthat deliberation in the model is always accompanied by aseries of ego-involving discussions that shadowor counter themoderating effect of deliberation Not only this deliberationcould also be responsible for crystallizing the opinions ofextremists for it favors one or the other part of the distributionof opinions as explained in [14] and to some extent in [13]Furthermore the mixed discussion model gives an exampleof polarization that results from a minority of dissenterson one side of an issue acceding to views of the majorityrsquosside [4] as opposed to the strong polarization observed indyadic interactions Situations in which the status quo orthe undecidability of the proposal argument lingers cannotpossibly yield situations of moderation In cognitive science

24 Complexity

one can be inclined to believe that the longer a deliberationprocess is the more likely individuals will take an immutablestance on the situation In view of the assumption that longdecision-making processes can be more costly for a group orsociety as a whole agents may feel forced to take a stancejust for the sake of ending the process The stance that isadopted would therefore depend on pair-wise interactionsamong individuals and not on deliberation since deliberationtakes too long to be conclusive

442 For Hypothetical Recommendations in Decision-MakingProcess Elaboration On another tune results from themodelsuggest that procedural parameters have marginally decreas-ing positive or negative effects on the metrics of interest Ina hypothetical policy analysis perspective this may indicatethat after choosing a correct number of individuals andimposing the right amount of debates one can attain rea-sonable levels of judgment accuracy coherence and varianceof opinions As accounted in [18] deliberation increasesknowledge (collective knowledge as well) and therefore itheightens the chances of making correct decisions If onebelieves that social welfare is linked to deliberation andeventually to the metrics studied previously then proceduralparameters can both be used as an instrument to attaindesired levels of social welfare and reinforce the notionof legitimacy of collective decisions It can also provide ajustification for choosing a deliberation regime rather thananother on the basis of how important (or urgent) a topic ofdiscussion is Indeed if we adopt the claim that deliberationis a forerunner of welfare then any state in which deliberativecues are at their maximum (highest number of participantsmany deliberation steps before making a decision and asfrequent as possible) has to be mapped to the highest attain-able social welfare In this case is a situation where judgmentaccuracy is at its highest yet the variance of opinions or theproportion of extremists at its lowest an ideal situation Wecannot say for sure since strong opinion consensus and longdecision-making processes in many situations may result ina loss of welfare rather than in a gain of it

Another interesting result in procedure for deliberationis the very similar estimates we find for the effects of theminimum number (119898) of debates necessary to make adecision on a proposal and the proportion (119899119863) of individualsparticipating in deliberation on judgment accuracy In thescope of social welfare this raises the question of whetheradvocating for longer decision processes by demandingmoredeliberation steps (higher 119898) rather than bigger debates(higher 119899119863) may increase welfare Depending on the objec-tives of the decider of the deliberation procedure he or shemay choose to concentrate on one or the other A recom-mendation we can assert from the mixed social interactionsmodel is that if one prefers low to high variance of opinionsandor high to low coherence the decider is better off ifshe focuses on increasing the size of the debate rather thanorganizing debates very frequently One of the reasons forthis might be that when making bigger debates all agents aretruly considered as equals in the decision-making processIn organizing many debates individuals that are chosen toparticipate in many of these will be more influential and will

somehow bias the set of arguments in the deliberation arenaThemajority rulemodeled through120572 also determines the sizeof the shifts and the distribution of opinions Because a bigger120572 is associated with higher levels of extremism and opinionvariance deliberated cues are harder to accept and thereforeit becomesmore difficult for agents to reach concensus andormoderate their opinions

On the other hand and further into the idea of legitimacythe goal of deliberative processes is the agreement amongall participants This agreement stands for the right or bestdecision from a formal or procedural perspective and it is alsothe best from a substantive point of view In other words thelegitimacy of decisions not only derives fromwell-establishedprocedural requirements (respect of the protocol) but alsobegs for the fulfillment of two essential conditions (1) satisfy-ing the procedural requirements for a correct procedure (for-mal legitimacy) and (2) the rational acceptability of the resultsof this procedure (substantive acceptability) Legitimacy canbe then obtained from the coherence statistic scrutinizedin the model insofar as it measures how well agents acceptthe results of a correct procedure (eg deliberation) So fora proposal to be legitimate one has to choose parametersthat would maximize group coherence in decision-makingprocesses

As deliberation may give rise to consensus in the delib-eration arena it may also sow the seeds of dissensus inthe group Our model clearly illustrates this phenomenonthrough the procedural parameters of deliberation and themetrics observed From another perspective in deliberativedemocracy dissensus in collective decision-making may bedesirable According to Landermore and Pagersquos accounts [36]lack of consensus can be perceived as having agents withalternative ideas that combined with other ideas can providea better approximation of the ideal status of an argumentTheauthors in [36] call this dissensus ldquopositive dissensusrdquo andit makes normative sense when solving complex collectivedecision-making problems Our model captures the ideaof having normative requirements to define a good andlegitimate collective decision requirements of high consen-sual deliberative accuracy (judgment accuracy) of legitimacy(coherence) and of diversity (variance of opinions) are alltaken into account and for such we can conclude that ourmodel is successful in describing a sound process of collectivedecision-making with interpretable outcomes

Of course the present simulation results and analysis donot prove that all opinion dynamics can be accounted forby such simple processes like the one presented here Theydo however shed light on the expressiveness of combiningdifferent paradigms to account formore interpretablemodelsand pave the way to creating original models of the sortThe simulations suggest the desirability of discovering theconsequences of relatively simple laws of communication atdifferent levels (micro meso and macro) to determine whatstill needs to be explained

5 Related Work

We see our model as a contribution to the influence andopinion dynamics field in ABM and a pragmatic application

Complexity 25

of abstract argumentation theory To our knowledge we areunaware of existing literature on agent-based modeling thatexplicitly relates collective choice and the notions of deliber-ation and opinion diffusion through abstract argumentationas we have done it

Most models in the literature of opinion diffusion areinterested in opinions because these have an influence incollective decision and in questions of social order Forinstance in [3] the authors are interested in consensus andin how a group collectively chooses among two alternativesIn other models authors are interested in the emergenceof extremism [8] and on the distribution of opinions whenextremists are introduced in the population [5 33 37] whileother authors coin the notion of opinion polarization as anemergent property of the system [8 12] They show usingmodels of ldquobounded confidencerdquo and opinion diffusion withtrust that three different kinds of steady states (unipolarbipolar and central) are possible depending on whetheragents are sufficiently uncertain about their opinions andsufficiently connected andor a certain proportion of indi-viduals are already extreme Recent articles on informationand opinion dynamics stress the importance of governanceand government intervention in the spread of emotions(opinions) in the frangibility of social consensus systemIn [10] the authors show that government intervention inthe spread of negative emotions can lead to an even fasterspread of negative emotions (single-peeked convergence) andin a faster collapse of the social consensus system Opinionas a function of trust is studied in [12] and relates to howindividuals form their opinions on the basis of how muchthey trust agents in their networks

Another stream of discrete opinion dynamics modelsthat explain the emergence of opinion can be found in theliterature Computer multiagent models of attitude changebased on Latanersquos social impact theory are presented in [4] Inthe theory an agent changes her opinion on the basis of theinformational impact she is subject to which depends on thepersuasiveness (strength) supportiveness and immediacy(group structure) of the environment and information shereceives The simulations predict two emergent groups ofphenomena the shifting of attitudes towards incompletelypolarized equilibria and the formation of coherent clusteringof subgroups with deviant attitudes Similarly but in acontinuous more argumentative fashion Mas and Flache[11] present a model of opinion diffusion using argumentsArguments are considered to be for or against a proposal andare given an agent-dependant relevance in dyadic persuasiveinteractions Arguments also determine the agentsrsquo opinionsand their dissimilarities which posits the rules of pair-wiseencounters in theirmodel as agents are assumed to form theiropinions according to the arguments they own They showthat in an argumentative opinion diffusion model the onlyobservable steady states are bipolarization and consensusand that bipolarization can emerge from interactions amongsimilar agents In other words bipolarization of opinionsis possible with homophily and without negative influenceif there is argumentation They subsequently compare theirresults to experimental data In [38] a similar approach toargumentation and opinion formation is taken but for the fact

that they introduce different types of arguments explicitlyapply social judgment theory [22] and use survey data toestablish model-to-real-world comparisons

Closer to opinion formation and abstract argumentationGabbriellini and Torroni [39] are the first to have origi-nally and soundly merged opinion diffusion and abstractargumentation They define an agentrsquos opinion as a functionof the arguments she holds and the attack relation amongthem They devised a focused peer-to-peer dialogue systemof persuasion (NetArg) inspired from Mercier and Sperberrsquosargumentative theory of reasoning [40] which used onlyabstract argumentation to study opinion polarization andopinion dynamics Moreover they considered networks andthe notions of trust and epistemic vigilance to define adynamics for knowledge revision and trust itself whichwe clearly lack in our model They showed that if a con-servative belief operator in argumentation was applied byagents when they reasoned then their dialogue protocoldid not increase polarization among agents That said theyused the model to study the effect of Groenvetterrsquos weaklink theory in the spreading of arguments and were notparticularly interested in notions like judgment accuracy orcoherence nor in tackling questions related to collectivedecision-making procedures Their system is very expressiveand helps position our work in the litterature our work is atthe frontier of pure argumentative opinion diffusion modelsopinion diffusion with arguments and continuous bounded-confidence models of opinion diffusion a la Deffuant

On another note work on collective cognitive conver-gence [41] and opinion sharing [19] shows that consensustowards a certain ldquocorrectrdquo opinion or cognitive state is alwayspossible yet dependent on noise variability and awarenessof agents In [20] the authors show that learning about anexogenous correct state of the world (represented by bits)under confidence was possible but only if the agents werenot too confident In a homogeneous population they showthat the higher the confidence the worse the learningmdashthevery confident agents do not learn the properties of the truestate of the world and disrupt the learning process of the lessconfident ones Collective cognitive convergence can be seenas a result of deliberation in truth-seeking models

When it comes to abstract argumentation theory we takean approach that wires two types of dialogues that are well-studied in the literature persuasion dialogues [42] and delib-eration dialogues [43] Another interesting line of work is theone on mechanism design [44] or the problem of devisingan argumentation protocol where strategic argumentation isnot a liability for success in debates We tackle mechanismdesign in a different way though Instead of consideringstrategy-proofness we are interested in how differences inprotocol can result in epistemologically ldquobetterrdquo collectivechoices and guarantee that opinion distributions are favorablefor deliberation For a survey on persuasion dialogue see[42]

Work on agent-based argumentation usually assumes thatthe semantic relationship between arguments is fixed Inother words if two individuals were to put two argumentsin the public arena such that one logically attacked the otherthen everyone would agree that such attack exists [45 46]

26 Complexity

Other models which do not make this restrictive assumptioncan also be found in the literature [27 47] and derive fromthe class or family of opponent models [27 48] in which twoopposing sides attempt to win a dialogue Ourmodel is in theintersection of these two but the frame combining opiniondiffusion of the kind and argumentation remains new

The idea of mixing interpersonal influence and verticalcommunication however is not original It is described andimplemented in innovation diffusion models such as [5 49]In both vertical communication is modeled as exogenoustransparent informationmdashagents are aware of the existenceof an innovation thus triggering several processes of choiceand stabilization of opinions Also in [50] an Eulerian modelis implemented to show the effect of media and exogenousinformation on opinion distributions With respect to thispoint the originality of our work is in that the informationemitted as vertical communication is endogenous It is issuedfrom a deliberation model that agents shape on the basisof their opinions arguments and behavior In the spirit of[51] where the authors control for the design of verticalcommunication (to whom the vertical communication isaddressed its timing how it influences agents opinions) wecontrol for the process generating the information for ourown set of variables length of debates majority quotas andthe frequency at which discussions take place

6 Conclusion and Perspectives

The main objective of this article was to build a bridgewhere decision-making argumentation and opinion dif-fusion could come together We proposed a model thatcombined abstract argumentation theory and a boundedconfidence opinion diffusion model and showed to whatextent it could explain variance of opinions extremismcoherence in collective decisions and judgment accuracyon arguments given an ideal state of full information Thesecond objective was to show in what way governance pro-viding agents with arenas of discussion and deliberation wasimportant in addressing the success of collective decision-making processes and the quality of their outcomes

The model revealed that allowing for more deliberationtime though in lower frequency and allowing for widerparticipation in deliberation increased the variance of opin-ions and the proportion of extremists in a group Theseobservations are consistent with the combination of resultsfound in [9] and in [13 14] which stress that deliberationmay polarize groups and may have a meager effect on theshifts of opinions and inconsistent with [16] where it isargued that opinions tend to moderate after deliberationIt happens that deliberation alone did moderate opinionyet when integrated into a more complex system in whichindividuals were allowed to interact with one another andnot all that was deliberated was accepted its influence wasshadowed by other more individualistic dynamics

Undeniably the grounded semantics played an importantrole in the weak effect of deliberation since it models askeptical way of reasoning over arguments For agentsaccepting arguments thus updating their opinions accord-ingly happened not too often Nevertheless asking for more

deliberation did increase judgment accuracy as observed in[18] yet in a marginally decreasing fashion We showed thatvoting within the deliberation protocol not only increasedthe proportion of extremists and the variance of opinions ina group but also determined how coherent deliberation andvoting procedures were with each other Lastly we showedthat agents that are focused judged arguments better hadmore stable opinions and constituted a group with higherproportion of extremists than their naive counterparts

In terms of governance the model says that there is per-haps no trade-off between extremismand judgment accuracyInstead it asserts that the higher the variance of opinions thecloser the group gets to the correct decisions This situationmay be interpreted as Landermore and Pagersquos notion of ldquoposi-tive dissensusrdquo (in complex collective decision-making tasks)[36] as opposed to consensus as a normative requirementfor correct collective decision-making If a decider has toorganize deliberation to legitimize a public policy dependingon what kind of world he or she wants different set ofparameters may be chosen to account for different levels oflegitimacy and correctness For instance one may ask for adecision to be legitimate to respect a certain level of coherenceor of judgment accuracy and to be discussed by a sundryof different-opinionated peoples If so deliberation has tobe made more often more participants have to be includedin debates or in deliberation instances and not too manytime steps should be introduced in between two deliberationinstances So the model may capture normative ideas todefine correct and legitimate collective decisions and tomake recommendations on those grounds Requirements ofhigh consensual deliberative accuracy (judgment accuracy)legitimacy (coherence) and diversity (variance of opinions)are all taken into account and for such a reason the model israther successful in describing a sound process of collectivedecision-making with interpretable outcomes

61 Extensions of the Model and Other Ideas Our model canbe extended in many ways Value-based abstract argumen-tation frameworks (VAFs) for example as an extension ofDungrsquos argumentation framework provide a formal descrip-tion of the process of decision-making in which argumentsare given values and audiences (set of agents) ignore attacksbetween arguments on the basis of their preferences over val-ues Social abstract argumentation frameworks (SAFs) alsoprovide an interesting and seemingly convenient formalismfor our model In SAFs agents vote on the acceptability ofarguments and the resulting labeling is a combination of thearguments that are voted for and labeling-based semantics Inour model we apply the same rule yet only for one argumentand given precise agent-based votingmechanisms and votingrules which are eventually parametrized

To consider our model as a specification of a mixed VAFand SAF model may be of great interest for future workThis formalism can provide solid foundations to describeexplain interpret and eventually extend the results foundhere from an argument-based perspective On those groundsfuture work will focus on creating a hybrid argumentationframeworkmdashwith its semanticsmdashthat takes heed of bothvalues or principles and voting

Complexity 27

Concerning other possible extensions of the model anatural reflex would be to argue on the validity and therobustness of the mix between the bounded confidenceopinion diffusion models and deliberation Consequentiallyadding deliberation to different bounded confidence modelsof opinion diffusion such as the Hegelsmann-Krause model[37] or to models of social impact theory like in [4] canbe of great interest for future work We acknowledge that afiner understanding of argumentation frameworks randomlattices and their respective applications for this modelare necessary Testing for different labeling-based semanticsor even trading labeling-based semantics for scoring-basedsemantics (semantics that defines acceptability on scoresgiven to arguments) such as debate semantics and ranking-based semantics (semantics that yield a preorder) may beinteresting to consider in future work

More into modeling deliberation it is not inadequateto believe that agents may also advance proposals duringdeliberation and replace to some extent the proposal forwhich the deliberation is taking place Endogenizing theprocess that produces the type of argument to be deliberatedon may be interesting to explore either by having agentsstrategically replace proposals at certain moments duringthe debate or having the central authority attempt to chooseproposals such that the probability of having them acceptedis high Agents may also be thought to be autoorganizing(independent of a central authority) and concede on the sizeof the instances of deliberation To consider differences insemantics across agents or even endogenizing the semanticson how urgent or important the proposal that is beingdiscussed is can also be an interesting direction to take

Last but not least many nontrivial modifications of ourmodel are possible and most should include better-thoughtdeliberation protocols It may be interesting to design andobserve protocols in which deliberation only affects agentsthat are actually debating for example Trust network effectsmultidimensionality of opinions new processes of argumentexchange or learning among agents are notions to furtherdevelop in order to make the model more realistic and relaxsome unreliable assumptions In sum the model is to berefined and extended with the objective of either studyingconcrete cases of deliberative polling and opinion dynamicsor implementing more intuitive thought experiments

Appendix

We present below some of the pseudocode we used toimplement the model

Algorithms

(a) Algorithm 1 implements the dyadic influence in themodel It describes how two agents interact andupdate their opinions

(b) Algorithm 2 implements the deliberative influence inthe model which describes how agents update theiropinions in sight of the accepted deliberated proposalarguments

(c) Algorithm 3 is used to obtain the grounded labelingof an argumentation framework

(d) Algorithm 4 provides an overview of a run in thesimulations119872 is the number of proposal argumentsto discuss which we set to 100 in the model

Each object in the system is indexed by a119908ℎ119900 number Hencea set of agents is annotated by a set of natural numbers Weuse this fact in the implementation of the algorithm 119874119887119895119890119888119905119894denotes the object of 119908ℎ119900 number 119894 Agentsrsquo opinions (119900) arecalibrated so that 119900 isin [minus1 1]Data Availability

The program and data used to support the findings of thisstudy are available from the corresponding author uponrequest

Disclosure

A preliminary version of this work was presented anddiscussed at the conference Bridging the Gap between FormalArgumentation and Actual Human Reasoning on October 4-5 2018 at BochumGermanyThe authors thank the audiencefor their valuable comments and suggestions

Conflicts of Interest

The authors declare that they have no conflicts of interestregarding the publication of this article

Acknowledgments

The contribution of Gabriella Pigozzi was supported by theDeutsche Forschungsgemeinschaft (DFG) and the CzechScience Foundation (GACR) as part of the joint project FromShared Evidence to Group Attitudes (RO 45486-1)

Endnotes

1 In other words the superiority of the ldquoright answerrdquo orlabeling in our case will appear as such to all

2 All agents hold the same opinion3 By sensitivity we mean how much an agent is ready to

change her opinion on the sole fact that an informationalcue has been debated on

4 The assumption is strong but this trust among agentsmay be rooted from a genuine individual motivation toreach a democratic consensus

5 We adopted a rule of thumbs that says that if an agentis neutral with respect to the principle then she willvote for any proposal that is presented to her duringdeliberation

6 If the argument is not unique she chooses one atrandom

7 If there are more than one candidate the agent will favorthe argument at minimal distance from 119868

28 Complexity

Require Vector of values for parameters (119881 = (120583 119880 119879))Ensure Successful interactions among agents119896 larr997888 0119873119890 larr997888 0

while 119896 le |119873|2 do ⊳Fill119873119890 with |119873|2 agents119896 larr997888 119896 + 1119873119890 larr997888 119877119886119899119889119900119898(|119873| + 1) ⊳Pseudo-random generator Generates integer between 0 and |119873|119873119903 larr997888 0 120575119894119895 larr997888 0 ⊳Variable to store differences in opinionfor all 119894 isin 119873119890 doRandomly pick a number 119895 isin 119873(119873119890 cup 119873119903)119873119903 larr997888 119873119903 cup 119895 120575119894119895 larr997888 |119900119894 minus 119900119895|procedure dyadic-update(119894 119895) ⊳Two agents 119894 and 119895 discuss and update opinions1199001015840 larr997888 119900119894 (120583119895 119880119895 119879119895) larr997888 (120583119894 119880119894 119879119894) larr997888 (120583 119880 119879) ⊳Save agent irsquos opinion before updating and homogenize agents

if 120575119894119895 lt 119880119894 then119900119894 larr997888 120583119894119900119895 + (1 minus 120583119894)119900119894if 120575119894119895 gt 119879119894 then119900119894 larr997888 (1 + 120583119894)119900119894 minus 120583119894119900119895if 120575119894119895 lt 119880119895 then119900119895 larr997888 1205831198951199001015840 + (1 minus 120583119895)119900119895if 120575119894119895 gt 119879119895 then119900119895 larr997888 (1 + 120583119895)119900119895 minus 1205831198951199001015840

Algorithm 1 Dyadic social influence

Require Vector of values for parameters (119881 = (120574 119901119886 119901119903))Ensure Successful interaction between proposal arguments and agents

for all 119894 isin 119873 doprocedure Probability-Update(V119886 119901119886 119901119903) ⊳Update individual probabilities of change due to deliberated cues119909119894 larr997888 119901|V119886minus119900119894 |2119886 119910119894 larr997888 1199012|V119886minus119900119894 |119903119902 larr997888 0if L119886119887(119868) = 119894119899 then

procedure deliberative-update(119894 119886) ⊳Agent i argument 119886120574119894 larr997888 120574 119902 larr997888 119906 sim U(0 1) ⊳ 119906 is a realization of a uniform random variableif 119902 lt 119909119894 then119900119894 larr997888 120574119894119900119894 + (1 minus 120574119894)V119886break

if 119902 lt 119910119894(1 minus 119909119894) then119900119894 larr997888 (1 + 120574119894)119900119894 minus 120574119894V119886Algorithm 2 Deliberation influence

8 In terms of computational complexity naive agentsmaketheirmoves in119874(|A119894| log(|A119894|)) while focused agents doat least as bad

9 Tables have memory Therefore debates are not neces-sarily trees

10 If there are less than (119899119863 times |119873|)2 agents for any of bothgroups the CA summons all of them to the deliberationtable

11 Or careful consideration of the viewpoints of otherswhich implies that citizens keep an open mind and donot reject arguments outright

12 A ldquohealthyrdquo consensus refers to the case in which manyconsensual decisions are taken but opinion-wise agentsdo not agree with each other Quotes are used on theword healthy because some political science theorists

(eg [36]) believe that in certain situations agreement inopinions for deliberation is epistemologically damaging

13 Simulations on groups of |119873| = 800 and |119873| = 400agents shows that scaling the population has an effect onall themetrics but it is confounding depending onwhichcouples of attitudes structures are considered (119880 119879) Westick to |119873| = 400 as in [9] for comparability andsimulation running time

14 By balanced wemean with as many arguments with V119886 lt0 as with V119886 gt 0

15 Experiments on the system suggest that a bigger(smaller) argument pool 119872 = 1000 (119872 = 200) isassociated with a higher (lower) variance of opinions(and extremism) and higher (lower) judgment accuracyFor the rest of the observations 119872 seem to have noeffect We choose a middle value for 119872 Similarly theargument pool chosen to be balanced or uniformly

Complexity 29

Require Argumentation framework 119860119865 = (AR)Ensure Assignment of labels to arguments from the grounded labeling of 119860119865L(A)

L0(A) larr997888 (0 0 0)119896 larr997888 0do119894119899 L119896+1(A) larr997888 119894119899 L119896(A) cup 119886 | 119886 is not labeled 119894119899 L119896(A) 119886119899119889 forall119887 isin A 119904119905 119887 R 119886 then 119887 isin 119900119906119905 L119896(A)119900119906119905 L119896+1(A) larr997888 119900119906119905 L119896(A) cup 119886 | 119886 is not labeled 119894119899 L119896(A) 119886119899119889 exist119887 isin A 119904119905 119887 R 119886 and 119887 isin 119894119899 L119896+1(A)

while L119896+1(A) = L119896(A) = 0Algorithm 3 Solve argumentation frameworks (with grounded semantics)

Require Vector of values for parameters (119881)Ensure Vector of statistics (119880)

procedure Set-parameters(119881) ⊳Initialize parameters for scenarioprocedure Init-objs(119899119898 119896 ) ⊳Create119898 arguments 119899 agents and 119860119865120576procedure Solve-Argumentation-Framework(119860119865) ⊳Assigns the epistemic labelingL120576 toA119894 larr997888 1119905 larr997888 1while 119894 le 119872 do ⊳ 119872 the number of arguments to discuss119868119894 larr997888 119866119890119899 119886119903119892(1) ⊳Generates proposal argument 119868119860 119894119905 larr997888 (119868119894 0)L119886119887(119868119894) larr997888 119906119899119889 ⊳Initialize label for 119868119895 larr997888 1 ⊳Initialize debate counter 119895while ((L119886119887(119868) = 119906119899119889 or 119898 le 119895) and 119895 le 119898) doif 119905 equiv 0(119898119900119889 119905119863) then

procedure build-argumentation-framework(119899119863 119887119890ℎ119886V119894119900119903) ⊳Build the argument graph agents deliberate(119861119894119905 119878119894119905) larr997888 (119883 119884) sub (⋃119897isin119873119899119863 A119897119894119905 ⋃119897isin119873119899119863 R119897119894119905) ⊳ 119873119899119863 sub 119873 are agents sampled by the CA119860 119894119905 larr997888 119889(119860 119894119905 (119861119894119905 119878119894119905))procedure Solve-Argumentation-Framework(A119889) ⊳FindL(A119889)L119886119887(119868119894) larr997888 L119886119887119863(119868119894)119895 larr997888 119895 + 1

else119896 larr997888 1while 119896 le 119905119863 doprocedure dyadic-social-influence(120583 119879 119880) ⊳Agents discuss one-to-one119896 larr997888 119896 + 1119905 larr997888 119905 + 1

if 120572 gt 0 andL119886119887(119868119894) = 119900119906119905 then ⊳Agents vote for the proposalif (|119897 isin 119873|119900119897 times V119868119894 ge 0| ge 120572 times 119899) then

L119886119887(119868119894) larr997888 119894119899else

L119886119887(119868119894) larr997888 119900119906119905if L119886119887(119868119894) = 119894119899 thenprocedure Deliberation-influence(120574 119901119886 119901119903) ⊳Agents are influenced by the result of the decision process119894 larr997888 119894 + 1119905 larr997888 119905 + 1119880 larr997888 119862119900119898119901119906119905119890 119878119905119886119905119894119904119905119894119888119904(119881)

return 119880 ⊳Get the statisticsAlgorithm 4 Run for one scenario

generated does not seem to play a significant explanatoryrole on any of the metrics

16 Experiments show that themaximumnumber of debatesis not an important parameter even if it is an intuitiveone when considering decision-making processes Inpractice debates end after at most 119898 + 1 deliberation

steps and even then it happens rarely However to avoidunpleasant surprises like infinite debates on a completeargument graph containing the proposal argument weadded the constraint 119898 Its value is precisely 7 becausethe biggest minimum number of debates we studyis 6

30 Complexity

17 To ensure comparability with the deliberative andmixedopinion dynamics models on judgment accuracy andcoherence agents vote on proposal arguments every 119905119863times(119898+1) 119905119863 and119898does not explain variability in judgmentaccuracy nor in coherence

18 Student t-tests for independent samples are robust tothe violation of two hypothesis on the distribution ofthe samples normality and homogeneity of variancesas long as the sample size is big enough (central limittheorem) and if the difference in size of the comparedsamples is smallrespectively

19 Studentrsquos test version forwhen variances are not assumedequal and the difference in size of the compared samplesis big

20 We do not correct for homoskedasticity nor normality oferrors in the estimates

21 Balanced as the same number of runs per scenario orsuch that no significant correlations are found betweenparameters

References

[1] R Axelrod ldquoThe dissemination of culture a model withlocal convergence and global polarizationrdquo Journal of ConflictResolution vol 41 no 2 pp 203ndash226 1997

[2] D M Rousseau Reinforcing the MicroMacro Bridge Organiza-tionalThinking and Pluralistic Vehicles SAGEPublications SageLos Angeles Calif USA 2011

[3] S Galam and SMoscovici ldquoTowards a theory of collective phe-nomena consensus and attitude changes in groupsrdquo EuropeanJournal of Social Psychology vol 21 no 1 pp 49ndash74 1991

[4] A Nowak J Szamrej and B Latane ldquoFrom private attitude topublic opinion a dynamic theory of social impactrdquo Psychologi-cal Review vol 97 no 3 pp 362ndash376 1990

[5] G Deffuant S Huet and F Amblard ldquoAn individual-basedmodel of innovation diffusion mixing social value and individ-ual benefitrdquo American Journal of Sociology vol 110 no 4 pp1041ndash1069 2005

[6] G Deffuant D Neau F Amblard and G Weisbuch ldquoMixingbeliefs among interacting agentsrdquoAdvances in Complex Systems(ACS) vol 3 no 01n04 pp 87ndash98 2000

[7] J Rouchier P Tubaro and C Emery ldquoOpinion transmission inorganizations an agent-based modeling approachrdquo Computa-tional and Mathematical OrganizationTheory vol 20 no 3 pp252ndash277 2014

[8] M Meadows and D Cliff ldquoThe relative disagreement model ofopinion dynamics where do extremists come fromrdquo in Pro-ceedings of the International Workshop on Self-Organizing Sys-tems Lecture Notes in Computer Science pp 66ndash77 Springer2013

[9] W Jager and F Amblard ldquoUniformity bipolarization and plu-riformity captured as generic stylized behavior with an agent-based simulation model of attitude changerdquo Computational ampMathematical Organization Theory vol 10 no 4 pp 295ndash3032005

[10] Y-F Zhang H-Y Duan and Z-L Geng ldquoEvolutionary mech-anism of frangibility in social consensus system based onnegative emotions spreadrdquo Complexity vol 2017 Article ID4037049 8 pages 2017

[11] A Flache andMWMacy ldquoSmall worlds and cultural polariza-tionrdquoThe Journal of Mathematical Sociology vol 35 no 1-3 pp146ndash176 2011

[12] A Tsang and K Larson ldquoOpinion dynamics of skepticalagentsrdquo in Proceedings of the 13th International Conference onAutonomous Agents and Multiagent Systems AAMAS 2014 pp277ndash284 International Foundation forAutonomousAgents andMultiagent Systems France May 2014

[13] K M Hansen Deliberative democracy and opinion formationUniversity Press of Southern Denmark Odense 2004

[14] C R Sunstein ldquoThe law of group polarizationrdquo Journal ofPolitical Philosophy vol 10 no 2 pp 175ndash195 2002 WileyOnline Library

[15] S Moscovici and W Doise Dissensions et Consensus unetheorie generale des decisions collectives Presses Universitairesde France-PUF 1992

[16] J S Fishkin and R C Luskin ldquoExperimenting with a demo-cratic ideal deliberative polling and public opinionrdquo ActaPolitica vol 40 no 3 pp 284ndash298 2005

[17] R C Luskin J S Fishkin and S Iyengar Considered Opinionson US Foreign Policy Face-To-Face versus Online Delibera-tive Polling International Communication Association NewOrleans La USA 2004

[18] J Barabas ldquoHowdeliberation affects policy opinionsrdquoAmericanPolitical Science Review vol 98 no 4 pp 687ndash701 2004

[19] O Pryymak A Rogers and N R Jennings ldquoEfficient opinionsharing in large decentralised teamsrdquo in Proceedings of the 11thInternational Conference on Autonomous Agents andMultiagentSystems vol 1 pp 543ndash550 Spain June 2012

[20] J Rouchier and E Tanimura ldquoWhen overconfident agents slowdown collective learningrdquo Simulation vol 88 no 1 pp 33ndash492012

[21] D Miller ldquoDeliberative democracy and social choicerdquo PoliticalStudies vol 40 S1 pp 54ndash67 1992

[22] M Sherif and C I Hovland Social Judgment Assimilation andContrast Effects in Communication and Attitude Change YaleUniversity Press 1961

[23] P M Dung ldquoOn the acceptability of arguments and its funda-mental role in nonmonotonic reasoning logic programmingand n-person gamesrdquo Artificial Intelligence vol 77 no 2 pp321ndash357 1995

[24] M Caminada and G Pigozzi ldquoOn judgment aggregation inabstract argumentationrdquo Autonomous Agents and Multi-AgentSystems vol 22 no 1 pp 64ndash102 2011

[25] P Baroni M Caminada andM Giacomin ldquoAn introduction toargumentation semanticsrdquoThe Knowledge Engineering Reviewvol 26 no 4 pp 365ndash410 2011

[26] S Chambers ldquoDeliberative democratic theoryrdquo Annual Reviewof Political Science vol 6 no 1 pp 307ndash326 2003

[27] E Bonzon andNMaudet ldquoOn the outcomes of multiparty per-suasionrdquo in Proceedings of the 10th International Conference onAutonomous Agents and Multiagent Systems 2011 AAMAS 2011vol 1 pp 47ndash54 International Foundation for AutonomousAgents and Multiagent Systems Taiwan May 2011

[28] J Ranciere Disagreement Politics and Philosophy University ofMinnesota Press 1999

[29] T Bench-Capon and K Atkinson ldquoAbstract argumentation andvaluesrdquo in Argumentation in Artificial Intelligence pp 45ndash64Springer 2009

[30] M Caminada ldquoOn the issue of reinstatement in argumenta-tionrdquo in Proceedings of the European Workshop on Logics in

Complexity 31

Artificial Intelligence vol 4160 of Lecture Notes in ComputerScience pp 111ndash123 Springer Berlin Heidelberg

[31] GAVreeswijk ldquoAn algorithm to computeminimally groundedand admissible defence sets in argument systemsrdquo in Pro-ceedings of the 2006 conference on Computational Models ofArgument Proceedings of COMMA 2006 vol 144 pp 109ndash120IOS Press Amsterdam The Netherlands 2006

[32] C W Sherif M Kelly H L Rodgers et al ldquoPersonal involve-ment social judgment and actionrdquo Journal of Personality andSocial Psychology vol 27 no 3 pp 311ndash328 1973

[33] G Deffuant F Amblard G Weisbuch and T Faure ldquoHow canextremism prevail A study based on the relative agreementinteraction modelrdquo Journal of Artificial Societies and SocialSimulation vol 5 no 4 p 1 2002

[34] E Awad M W Caminada G Pigozzi M Podlaszewski andI Rahwan ldquoPareto optimality and strategy-proofness in groupargument evaluationrdquo Journal of Logic and Computation vol 27no 8 pp 2581ndash2609 2017

[35] D Grossi and G Pigozzi ldquoJudgment aggregation a primerrdquoSynthesis Lectures on Artificial Intelligence and Machine Learn-ing vol 8 no 2 pp 1ndash151 2014

[36] H Landemore and S E Page ldquoDeliberation and disagreementproblem solving prediction and positive dissensusrdquo PoliticsPhilosophy and Economics vol 14 no 3 pp 229ndash254 2015

[37] R Hegselmann and U Krause ldquoOpinion dynamics andbounded confidence models analysis and simulationrdquo Journalof Artificial Societies and Social Simulation vol 5 no 3 2002

[38] A Stefanelli and R Seidl ldquoModerate and polarized opinionsUsing empirical data for an agent-based simulationrdquo inProceed-ings of the Social Simulation Conference 2014

[39] S Gabbriellini and P Torroni ldquoA new framework for ABMsbased on argumentative reasoningrdquo in Advances in SocialSimulation vol 229 pp 25ndash36 Springer Berlin Germany 2014

[40] H Mercier and D Sperber ldquoWhy do humans reason Argu-ments for an argumentative theoryrdquo Behavioral and BrainSciences vol 34 no 2 pp 57ndash74 2011

[41] H V Parunak T C Belding R Hilscher and S BruecknerldquoModeling and managing collective cognitive convergencerdquoin In Proceedings of the 7th iNternational Joint Conference onAutonomous Agents and Multiagent Systems vol 3 pp 1505ndash1508 International Foundation for Autonomous Agents andMultiagent Systems 2008

[42] H Prakken ldquoFormal systems for persuasion dialoguerdquo TheKnowledge Engineering Review vol 21 no 2 pp 163ndash188 2006

[43] K Atkinson T Bench-Capon and P Mcburney ldquoA dialoguegame protocol for multi-agent argument over proposals foractionrdquoAutonomousAgents andMulti-Agent Systems vol 11 no2 pp 153ndash171 2005

[44] M Thimm and A J Garcıa ldquoClassification and strategicalissues of argumentation games on structured argumentationframeworksrdquo in Proceedings of the 9th International Conferenceon Autonomous Agents and Multiagent Systems vol 1 pp 1247ndash1254 International Foundation for Autonomous Agents andMultiagent Systems Canada 2010

[45] I Rahwan and K Larson ldquoMechanism design for abstractargumentationrdquo in Proceedings of the 7th International JointConference on Autonomous Agents and Multiagent SystemsAAMAS 2008 pp 1031ndash1038 International Foundation forAutonomous Agents and Multiagent Systems Portugal May2008

[46] H Prakken R Riveret A Rotolo and G Sartor ldquoHeuristics inArgumentation A Game-Theoretical Investigationrdquo EuropeanUniversity Institute 2008

[47] N Hadidi Y Dimopoulos and P Moraitis ldquoTactics and con-cessions for argumentation-based negotiationrdquo in Proceedingsof the COMMA vol 245 pp 285ndash296 2012

[48] N Oren and T J Norman ldquoArguing using opponent modelsrdquoin Proceedings of the International Workshop on Argumentationin Multi-Agent Systems pp 160ndash174 Springer 2009

[49] S Delre W Jager T Bijmolt and M Janssen ldquoTargeting andtiming promotional activities An agent-based model for thetakeoff of new productsrdquo Journal of Business Research vol 60no 8 pp 826ndash835 2007

[50] A Mirtabatabaei P Jia and F Bullo ldquoEulerian opinion dynam-ics with bounded confidence and exogenous inputsrdquo SIAMJournal on Applied Dynamical Systems vol 13 no 1 pp 425ndash446 2014

[51] V Barbet N Guiraud V Laperriere and J Rouchier Hagglingon Values Towards Consensus or Trouble 2019

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18 Complexity

Table 5 Mean difference 095 confidence intervals for metrics by model comparison

Model Metric119881119886119903(119900) 119901119903119900119901119890119909 119878ℎ 119890119888 119894119903Dyadic vs Deliberative [0191 0216] [0170 0197] [-2948 -2828] [0074 0082] [0165 0172]Mixed vs Deliberative [0142 0147] [0150 0155] [-2085 -1976] [0003 0002] [-0038 -0030]Mixed vs Dyadic [-0072 -0046] [-0045 -0018] [8332 8859] [-0079 -0071] [-0205 -0201]the model On the other hand pair-wise opinion dynamicsproduces multicluster convergence most of the time andcentral convergence only in the infrequent cases where thelatitude of acceptance is large

The comparison of these two scenarios in terms ofthe metrics is straightforward Whilst dyadic interactionsallow for more variability in opinions through pair-wiseinteractions and more mistakes in judging proposals delib-erative updates simply act as an antagonist force On aver-age the deliberative interaction model produces opiniondistributions with smaller variance and yields steady stateswith higher judgment accuracy a higher shift statistic andnaturally lower group coherence since only proposals that aredeliberated 119894119899 are eligible for voting (see Table 5)

42 Articulating Dyadic and Deliberative Social Interactionsin Opinion Dynamics So far we have considered scenarioswhere argumentation had no place and where individualsmade decisions according to opinions that were purelyconstructed from pair-wise discussions We also observedcases in which agents formed their opinions only by inte-grating deliberated proposals into their opinion updates Thescenarios we present in this section combine the two pre-ceding scenarios to account for both the effects of pair-wisediscussion and the effects of deliberation on opinions judg-ment accuracy and group coherence From the precedingobservations we conclude the following deliberative opiniondynamics reduces the variance and the proportion of extrem-ists because it shifts opinions greatly and polarizes groupsby eliminating at least one type of extremist the one whoseopinions are initially opposed to the deliberated resultsor it brings opinions close to neutrality (see Figure 7(b))This may not be desired in a group since the diversity andstability of opinions are necessary for deliberation to makesense and lead to the collective acceptance of controversialproposals Dyadic opinion dynamics on the other handtend to increase the variance of opinions and the numberof extremists while having a lesser aggregated effect on theshifts of opinion This is an immediate result of the pair-wiseinteraction dynamics inasmuch as it polarizes individualsvery quickly

To sum up in the mixed interactions model deliberationhinders the effect of voting on opinions by making someproposals that would normally be submitted for voting noteligible for voting It may also undo opinion changes dueto pair-wise interactions The mixed model can be seen asthe deliberation opinion dynamics model in which pair-wiseinteractions occur in between collective discussion and mayaffect which arguments are advanced during deliberationEquivalently it can be seen as the dyadic opinion dynamics

model in which deliberation accounts for endogenousldquoshocksrdquo on the opinion distribution

421 Qualitative Analysis of the Mixed Dynamics In Figures7(c) and 7(f) agents update their opinions through delibera-tion and pair-wise interactions Opinion trajectories visualsshow a combination of both previously described opiniontrajectories Over time as agents discuss with one anotheropinion clusters form as in the pair-wise or dyadic opiniondynamics Deliberation may disrupt the formation of theseclusters and three different situations may occur

(i) Deliberation disrupts the formation of the opinionclusters but not sufficiently to jeopardize pair-wiseopinion cluster formation (eg for high values 119905119863)Two phenomena may be responsible for it (1) thevariability of the proposal argumentrsquos support for theprinciple P and (2) too few (successive) deliberationsteps that result in opinion updates Once the opinionclusters are formed successful deliberation foreshad-ows the merging of the previously formed clustersmay bring extremists closer to moderate agents andthanks to the assimilation component of the pair-wisedynamics allow bigger and more moderate opinionclusters to form (refer to Figure 7(c))

(ii) Deliberation significantly disrupts the pair-wisedynamic formation of clusters This may happenwhen the requirements for accepting proposals areweak in the decision-making process (eg low 120572) Inthis case one has either central convergence sinceall agents follow the deliberated results in the samemanner or 3-cluster convergence with two extremeopinion foci In the latter situation the extremegroups form quickly (like in the pair-wise model) anddeliberation cannot pull extreme agents away fromthe extreme foci they are in The main reason for thisis that immediately after an extreme agent attempts toleave her foci through deliberation updates she getspulled or pushed back into it by means of a dyadicinteraction with another extreme agent

(iii) Deliberation does not disrupt the formation of theopinions cluster This is the rare case scenario forvery demanding decision-making processes in termsof inclusion (eg 119899119863) and argument acceptability (eg120572) Opinions converge as in the pair-wise opiniondynamics model (refer to Figures 7(a) and 7(d))

Last when there is voting (120572 = 0) the position ofthe moderate opinion cluster is determined by the first

Complexity 19

minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

Var(o)

sh

ec

ir

Var(o)

propex

sh

ec

ir

1

098

minus02

minus004

001

1

minus014

minus001

minus003

1

026

minus028

1

minus057 1

ＪＬＩＪ

Ｒ

Figure 8 Significant correlation estimates for themetrics of interestin the mixed opinion dyamics scenarios

accepted proposal argument andor the opinionmajority thatis formed from previous dyadic interactions (see Figure 7(f))

422 Relating the Metrics of Interest in Mixed OpinionDynamics The first thing to notice is that for all of thesescenarios all the metrics are significantly correlated (seeFigure 8) The variance of opinions and the proportion ofextremists are almost perfectly correlated To some extentthis means that a high variance of opinions known thatthe mean of opinions is statistically null equates to havingmany extremists in the population be towards one or theother side of the opinion distribution (eg extreme bipolar-ization) Similarly we find a negative and strong correlationbetween the judgment inaccuracymeasure and the coherencecoefficient (see Figure 8)mdashwhen judgment inaccuracy is high(low) coherence to deliberation is low (high) Cases likethese are telling since they may be related to sequences ofcollective decisions in which agents do not make too manymistakes when judging proposal arguments and still makedeliberated decisions that are on average in line with theirprinciples

The shift statistic interestingly is negatively correlatedwith the variance of opinions with judgment accuracy andwith the coherence coefficient In a way this means thathaving more moderate agents conduces on average to moremistakes in judging arguments and to less coherent collectivedecisions An explanation for this phenomenon could be thatas agents reach consensus in opinion important (possiblyextreme) arguments for correctly judging argument propos-als are not advanced in the deliberation arenawhich results inmore mistakes and in possibly accepting proposal argumentsthat after voting are not meant to be accepted Indeed the

negative correlation between the variance of opinions andjudgment inaccuracy points towards this direction since onaverage the more shifts there are the smaller the variance ofopinions is (see Figure 8)

423 Comparing Mixed Dynamics with Other MonolithicDynamics of Interactions Qualitatively opinion trajectoriesin the mixed model are more chaotic than in the pair-wise and deliberative interaction models (see Figure 7) Asanalyzed previously deliberation dynamics disrupt the pair-wise dynamics and create situations in which central andmulticlustered convergence (with two clusters at the extremesof the distribution) are possible at the expense of more(pair-wise dynamics) or less (deliberative dynamics) clustersIt tolerates the survival of very small isolated group ofagents that the previous models rarely tolerate in a hundredcollective decision-making steps (see Figure 7(c))

One of the most interesting observations one can makeof this family of models regards how groups change (in termsof our metrics) when dyadic social interactions are addedto deliberative interactions and vice-versa When comparingthe mixed interactions with only deliberative ones shiftsagentsrsquo aggregated change in opinions are on average lesspronounced The variance of opinions and the proportionof extremists are significantly higher for the former than forthe latter group coherence in contrast is lower with respectto the scenarios where opinions are formed only throughdeliberation Judgment accuracy follows that same trendA reason for this might be that mixed dynamics are fasterto yield semistable clusters andor opinion convergence Inconsequence the potential argument pool for deliberativeexchange shrinks faster and deliberation becomes less effec-tive over time which results in more collective mistakes Theoutcomeon coherence is predictable inasmuch as deliberativeinteractions produce group polarization If a group agrees inopinion debates are sterile because themembers of the groupwill most likely refrain from attacking the proposal argu-ments they all support and will work together to disparagethe proposal arguments they do not support (see Table 5)

In the same manner we report that the proportion ofextremists and the variance of opinions are barely lower inthe pair-wise interaction model when including deliberationin the decision-making process Shifts of opinions are alsobarely lower when there is no deliberation This can indicatethat deliberation does not undo the changes in opiniondue to pair-wise interactions nor does it amplifies them Incontrast when we consider the labeling-based metrics themixed interaction model ensures lower judgment inaccuracybecause of deliberation and lower group coherence (seeTable 5)

In sum the mixed discussion model is a compromisebetween the pair-wise interaction and deliberative opiniondynamic models in that it offers some guarantees in termsof judgment accuracy and coherence while allowing forldquoreasonablerdquo variance in the opinion pool It establishes somekind of trade-off between labeling-based metrics and thevariance of opinions that may be of interest for decision-making process design

20 Complexity

Table 6 Summary of sign or relative marginal effects for all varying parameters on dyadic deliberative and mixed interactions models inand for all experiments

Metrics Parameters119905119863 119898 119899119863 120572 119879 119880 119901119886 120574 119896 119861119890ℎ119886V119894119900119903119881119886119903(119900) XX X X O X X X X X X119901119903119900119901119890119909 XX X X O X X X X X X119878ℎ XX X X O X X X X X119890119888 XX X X XO X XOO XOO X X119894119903 XX X X O XO O X OOX XOO X XNota gray symbols correspond to dyadic dynamics light gray to deliberative dynamics and bold black to mixed dynamics Given a scenario the symbol 119883stands for constant parameter and 119874 for insignificant marginal effect and as parameters grow for positive (negative) marginal effect for the parameterFrom left to right at the parameter level the leftmost arrow corresponds to the scenario with the strongest (in absolute value) marginal effect of the parameter(with respect to to its factors) Example Let 119901 be a parameter and 119874 stands for ldquo119901 has a significant positive effect in dyadic discussion scenario negativein the deliberative one and insignificant in mixed discussion The positive effect in dyadic discussion is greater than the absolute value of the negative one inthe purely deliberative scenariordquo

424 Comments on the Differences between the Three Modelsfor OLS Regression Estimates The significance of most of theregression parameters estimated for the mixed interactionsscenarios are similar to those estimated for the pure dyadicanddeliberative scenariosNevertheless themagnitude of theeffects is different insomuch that the dynamics themselvesare different So instead of commenting the linear regressionestimates of the mixed model we compare when possiblethe standardized significant coefficients of the regressors ofthe mixed model with those of the pair-wise and deliberativeinteractionmodels (see Table 6 for a summary of the results)

In mixed interactions for instance the influence of119898 onthe opinion distribution is much more preponderant than inthe deliberative dynamics scenarios Going from 119898 = 1 to119898 = 3 increases the standard variation units of the variance ofopinions by twice as much as the one in the deliberative sce-nariosThis in particular is due to the design of the decision-making process If 119898 is higher then the number of socialinfluence steps (119905119863) in the decision process is also higher(compared to 119905119863 = 0 in the deliberative case) If we decideto consider these steps as contributing to the variance ofopinions then the effect of119898 on the variance of opinions hasto be stronger in the mixed interactions model In contrastthe number of agents participating in the debate has a weakereffect on the variance of opinions in the mixed interactionsmodel than in the deliberative dynamics specially for 119899119863 =005 (more than twice as much) The difference may comefrom the fact that in mixed discussions the part of theargument pool used in deliberation is more restrictive thanin the deliberative case We can argue that since pair-wisedynamics may get individuals closer to one another aftersome deliberation the same happens to the arguments thatindividuals will most likely use in deliberation Taking thisinto account means that deliberation will at many instancesbe unfruitful and have no effect on opinions thus failing tomoderate extreme opinions andor polarize the group In thesame direction a high number of participants in deliberationcan make the deliberation dynamics stiff In this case thedistribution of opinion is close to the uniform distributionwhich has a relatively high-to-moderate variancewith respectto the reference results

The strength of the deliberative dynamics has a positivestronger effect on the variance of opinions in themixedmodelthan in the deliberative model when 120574 = 02 and a negativeweaker one when 120574 = 01 A sound interpretation of thisresult is that when deliberation is successful and its effect onopinions does not get agents to move sufficiently away fromtheir current positions agents are pulled or pushed back totheir previously held opinions the variance of opinions doesnot changemuchOn the other hand if the effect of successfuldeliberation is strong opinions can change sufficiently tomake agents much more prone to assimilation into moderateopinion foci the variance of opinions falls

The deliberative opinion dynamics model is much moresensitive to 120572 than the mixed discussion model Differ-ences in marginal effects can double The probability ofattraction to deliberated results (119901119886) for instance has astrongmoderating effect (negative) on agentsrsquo opinions in thedeliberative opinion dynamics but a meager one (two-fold)in the mixed and dyadic dynamic models This is becauseopinion updates in the deliberative model depend only onthat parameter whereas in the mixed model more numeroustypes of interactions make opinion updates possible

In terms of judgment accuracy all parameters except thesize of the argument sacks (119896) have a significant slightlystronger effect on the deliberative dynamics than in themixedmodel In effect the difference may be attributed to theupshot of pair-wise interactions on the distribution of opin-ions and on the arguments that would be advanced duringdeliberation The result respecting parameter 119896 however iscounterintuitive Since argument sacks are static scenarioswhere shifts are higher should make 119896 more important fordeliberation to succeed and it is precisely in the scenarios ofpure deliberative dynamics that we find the biggest opinionshifts This may point to the hypothesis that in deliberativeinteractions agents change opinions substantially but do notoften change their adherence to the principle (go from anegative opinion to a positive one or vice versa) when thereis no group polarization

43 Sensitivity Analysis for the Procedural Parameters ofInterest Weperform a sensitivity analysis of the observationson the parameters of interest in themixed interactionsmodel

Complexity 21

We extend the parameter domains of the pure dyadic anddeliberative scenarios as we make it explicit in Table 2

431 Minimum Number of Debates (119898) We observe that theminimum number of debates has a significant well-observedeffect on all of our metrics excluding group coherence Forthe variance of opinions we can observe that the moredebates there are the bigger the value of the metric isalthough the higher 119899119863 and 119905119863 are the weaker the overalleffect of119898 becomes (Figures 9(d) and 9(a)) Furthermore themarginal effect of increasing minimal debates is decreasingan additional deliberation step requirement increases thevariance of opinions but less and less as it grows 119898 coupledwith 119899119863 has a shy S-shaped effect (see Figure 9(d)) on thevariance of opinions meaning that it has a stronger effectwhen 119899119863 is low which suggests a possible trade-off betweenthe number of arguments accepted in the debate and thenumber of debates required before accepting a proposal

In contrast the shifts in opinion are less and less likelyas 119898 grows and this is independent of the variations of theother parameters Again the effect is marginally decreasingand is only truly significant when 120572 = 05 An explanationto these effects may be linked to the design of the systemFirst the variance of opinions is higher when deliberation isasked for because the more deliberation steps there are thehigher the chances are that the central proposal argumentis deemed unacceptable specially when agents are focusedFurthermoremechanically speaking increasing theminimalamount of debates implies that whenever a decision is to betaken at least 119898 times 119905119863 time steps have to take place and if atany moment a proposal argument is considered undecided119905119863 time steps are added to the process So unless the debatedoes not yield 119906119899119889 labels for proposal arguments (highlyunlikely considering that 120590 = grounded semantics) the morenondeliberation steps there are in the decision process thehigher the variance of opinions is Concerning the shiftswhen 120572 = 05 either the system is too stiff to accept anyproposal argument and opinions do not changemuch or theeffects of pair-wise discussion and deliberation cancel out insuch a way that individual shifts are minimal (Figure 9(b))

On the side of labeling-based metrics the more debatesare asked for the more accurate a group is in its judgmentthe effect being smaller as 119898 grows When agents are naivethe effect is quasilinear while when they are focused thestrongest effects of acquiring more deliberation are foundwhen levels of deliberation are already low (Figure 9(f))Thiscan be explained by the fact that the more debates thereare in the decision process the closer one gets to the idealargumentation framework

432 Proportion of the Population in Deliberation Steps (119899119863)Like for 119898 119899119863 has a significant effect on the proportionof extremists and on the shifts and variance of opinions(Figure 9(e)) This may result from the fact that being ableto put more arguments in play at the same debate step mayincrease the odds of revealing cycles around the proposalargument in119860119865120576 Given that we use grounded semantics thearguments in the cycles are labeled 119906119899119889 and in consequence

debates are more often postponed when 119899119863 is higher Post-poning debates in turn increases the number of nondebatessteps in the decision process which increases the varianceof opinions and limits over time the moderating effect ofdeliberation Moreover the effect of this parameter is verydependent on the value of 120572 (Figure 9(e)) For shifts forinstance 120572 = 05 makes the effect of 119899119863 negative while120572 = 0 makes it positive but to a lesser extent Furthermorefor higher requirements of deliberation (119898) adding moreindividuals to the deliberation process has weaker effectson the variance of opinions and on the other metrics It isalso quite interesting to notice that it has the same effect onagents independently of agent behavior during deliberationThis is a surprising result as one would have expected morefocused agents in a deliberation arena to cause a raise inthe proportion of extremists as they play to knock outopposing proposal arguments and thereof reducing the effectof deliberation (lowers variance) on the whole population

Finally similar to119898 addingmore people into the deliber-ation process increases judgment accuracy (see Figure 9(c))and has no clear effect if any on group coherence (seeFigure 9(i))

433 Steps between Deliberation Steps (119905119863) In all configu-rations 119905119863 increases the proportion of extremists (variance)and decreases the shifts of opinions among agents Theshifts and the effect on the variance of opinions are presentfor 120572 = 05 insignificant otherwise (Figure 9(b)) 119905119863 ishighly linked to 119898 by construction When 119898 = 1 thecurve linking the variance of opinions and 119905119863 is convex As119898 increases the curve becomes more and more concavewhich means that 119905119863 has a more important effect on theopinion distribution as collective decisions take longer to beachieved This seems counterintuitive but in reality it reflectsthemultiplicative relationship between deliberation and pair-wise interactions and the semantics in the model If 119898 is lowand 119905119863 is high the effective number of pair-wise interactionsis on average smaller in the deliberation process Hencethe structure of the decision-making process hinders anyincrease in the variance of opinions If119898 is high the oppositeeffect is observed Additionally since the grounded semanticsyields few 119894119899 arguments with respect to other admissibility-based semantics getting closer to the ideal argumentationframeworkmaymake it difficult to obtain the expected effectsof deliberative discussion on agents Lower 119898 constrains thevariance of opinions by making deliberation more influentialon opinions

In terms of judgment accuracy and coherence we observethat 119905119863 does not explain coherence yet it tends to decreasejudgment inaccuracy This may be due to a more variedargument pool in the debate which in turn may be aconsequence of a higher variance of opinions or vice versa

434 Acceptability Voting Quota (120572) By far it is the mostinfluential parameter in our study It changes the directionand the intensity of the effect of all the other proceduralparameters and by construction heavily constrains the roadto accepting a proposal (see Figure 9)

22 Complexity

015

020

025

030

035

040

1 2 3 4 5 6

12345

Values of Ｇ

(a) 119905119863 times 119901119903119900119901119890119909 by119898

0

10

20

30

40

50

1 2 3 4 5 6

005066

Values of

(b) 119905119863 times 119878ℎ by 120572

040

045

050

055

060

1 2 3 4 5

001002003004005

Values of Ｈ＄

(c) 119898 times 119890119888 by 119899119863

015

020

025

030

035

040

1 2 3 4 5

001002003004005

Values of Ｈ＄

(d) 119898 times 119901119903119900119901119890119909 by 119899119863

0

10

20

30

40

50

001 002 003 004 005

00005066

Values of

(e) 119899119863 times 119878ℎ by 120572

040

045

050

055

060

1 2 3 4 5

BehaviorNaiveFocused

(f) 119898 times 119890119888 by 119887119890ℎ119886V119894119900119903

015

020

025

030

035

040

0 05 066Behavior

NaiveFocused

(g) 120572 times 119901119903119900119901119890119909 by 119887119890ℎ119886V119894119900119903

05

06

07

08

09

10

1 2 3 4 5 6

00005066

Values of

(h) 119905119863 times 119894119903 by 120572

08700

08725

08750

08775

08800

001 002 003 004 005

BehaviorNaiveFocused

(i) 119899119863 times 119894119903 by 119887119890ℎ119886V119894119900119903

Figure 9 Curves of mean observations for different parameters and metrics for 36000 runs Bars are 095 confidence intervals Plots withthe same observables (ordinate) except for 119894119903 are on the same scale

Complexity 23

The higher the requirement for accepting the proposal isthe higher the proportion of extremists and the variance ofopinions are and the lower the shift statistic is In few words120572 constrains the world to dyadic discussion or throws it intoa process in which deliberation is much more importantthan dyadic interactions This is why for any 120572 = 05 oneeither gives too much weight to deliberated results whichshadows pair-wise interactions or too little weight in thesense that no deliberated result is ever accepted and thusnever integrated into the opinions of the agents that havevoted for it The reason for the latter is that the latitudeacceptance is too low and thereafter pair-wise interactionsare not enough to unevenly polarize the population in such away that deliberated proposal arguments are voted favorablyby a non-negligeable majority Shifts for 120572 = 0 are stable fordifferent levels of 119905119863 which implies that deliberation keeps incheck all the shifts of opinions related to an increase in thenumber of pair-wise interactions For 120572 = 05 deliberationis less likely to be successful and shifts tend towards the shiftlevels for 120572 = 0 (see Figure 9(b))

Concerning labeling-basedmetrics120572 entirely determinesthe coherence statistic For 120572 = 05 there is no difference incoherence because of how coherence is defined it is maximalwhen 120572 = 0 and strangely maximal when 120572 = 066(see Figure 9(h)) Surprisingly 120572 has no effect on judgmentaccuracy One would have expected a higher 120572 to increasejudgment accuracy since agents would hardly collectivelyaccept a proposal argument discussed during deliberationProposal arguments have given how the argument lattice isgenerated a higher probability of being rejected than of beingaccepted in the ideal argumentation framework

435 A Word on Focused and Naive Agents Focused andnaive specifications for agent behavior are an importantparameter in the decision process since they model howagents choose arguments in the deliberation arena In the sce-narios where agents are focused agents show equal or highervariance of opinions and are therefore more extreme (seeFigure 9(g)) independently of the voting requirements Shiftsof opinion are less likely as well We can see this happeningbecause focused agents knock out proposal arguments moreoften than naive agents provided that 119899119863 is low

One could have expected naive agents to yield morecoherent decisions since they argue sincerely However thisseems not to be the case (see Figure 9(i)) Agents are unable todeliberate in away thatmakes deliberation reflect their votingintentions either because they do not have the necessaryarguments to do so or because such arguments do not exist

In terms of judgment accuracy focused agents do betterthan naive agents in finding the ideal labels for the proposalarguments This is likely because when reconstructing theframework focused agents take into account the deliberatedproposal argument and choose themost pertinent argumentsin their sacks that relate to the state of affairs at the deliber-ation arena In fine debates result in a better approximationofL119886119887120576(119868) even though one could have believed the oppositethat is focused agents are ready to sacrifice a correct collectivelabel of a proposal argument in the goal of defending theirpositions

44 Points of Discussion In this subsection we discuss theresults obtained from the simulations We attempt to makesense of them from a social sciences perspective

441 Opinion Consensus and Dissensus The deliberativeopinion dynamics corresponds to the simulated scenarios inwhich the variance of opinions is at its lowest and judgmentaccuracy at its highest Dyadic opinion dynamics are associ-ated with the scenarios in which the variance of opinions is atits highest judgment accuracy at its lowest and coherence atitsmaximumpossible levelThemixed interactions dynamicsis a combination of these aspects for all the metrics exceptfor coherence at which it is the worsemdashdeliberation leadsto some kind of loose opinion consensus that is reinforcedby pair-wise interactions Likewise the mixed discussionmodel confirms some results exposed in [16] expressingthat opinions and voting intentions in a deliberative contextldquooften changerdquo That being said and more particularly indeliberative opinion dynamics outputs align well with twointeresting results on consensus formation given by aninterpretation of Moscovici and Doisersquos [15] work Sunsteinrsquos[14] account on deliberation and group polarization andsocial impact theory [4] The first states that mild consensusis reached in deliberation as procedure disengages agentsthe second is that within discussion groups majorities tendto become larger and opinions more extrememdashnot on thebasis of competing arguments but of the initial balance ofopinions After several decision processes we find opiniondistributions that reflect these observations even though weare not able to precisely coin the set of parameters that yieldssuch distributions (see Figure 7)

Although differences between mixed and dyadic opiniondynamics in terms of the variance of opinions and shiftsare slim the former does much better in terms of judg-ment accuracy In this sense deliberation does moderateopinions but not as much when non-deliberation steps arerelatively frequent throughout the decision-making processIn terms of Moscovici and Doise theory of consensus [15]this makes sense since pair-wise discussion engages or ego-involve agents in the discussion and pushes to convergenceof opinion at the extremes of the opinion distributionHowever what remains unclear is the reason why proceduralparameters that constrain deliberation do not disengageagents from discussion and thus result in a mild consensusas also expressed by Moscovici and Doise [15] A possibleexplanation of this phenomenon may stem from the factthat deliberation in the model is always accompanied by aseries of ego-involving discussions that shadowor counter themoderating effect of deliberation Not only this deliberationcould also be responsible for crystallizing the opinions ofextremists for it favors one or the other part of the distributionof opinions as explained in [14] and to some extent in [13]Furthermore the mixed discussion model gives an exampleof polarization that results from a minority of dissenterson one side of an issue acceding to views of the majorityrsquosside [4] as opposed to the strong polarization observed indyadic interactions Situations in which the status quo orthe undecidability of the proposal argument lingers cannotpossibly yield situations of moderation In cognitive science

24 Complexity

one can be inclined to believe that the longer a deliberationprocess is the more likely individuals will take an immutablestance on the situation In view of the assumption that longdecision-making processes can be more costly for a group orsociety as a whole agents may feel forced to take a stancejust for the sake of ending the process The stance that isadopted would therefore depend on pair-wise interactionsamong individuals and not on deliberation since deliberationtakes too long to be conclusive

442 For Hypothetical Recommendations in Decision-MakingProcess Elaboration On another tune results from themodelsuggest that procedural parameters have marginally decreas-ing positive or negative effects on the metrics of interest Ina hypothetical policy analysis perspective this may indicatethat after choosing a correct number of individuals andimposing the right amount of debates one can attain rea-sonable levels of judgment accuracy coherence and varianceof opinions As accounted in [18] deliberation increasesknowledge (collective knowledge as well) and therefore itheightens the chances of making correct decisions If onebelieves that social welfare is linked to deliberation andeventually to the metrics studied previously then proceduralparameters can both be used as an instrument to attaindesired levels of social welfare and reinforce the notionof legitimacy of collective decisions It can also provide ajustification for choosing a deliberation regime rather thananother on the basis of how important (or urgent) a topic ofdiscussion is Indeed if we adopt the claim that deliberationis a forerunner of welfare then any state in which deliberativecues are at their maximum (highest number of participantsmany deliberation steps before making a decision and asfrequent as possible) has to be mapped to the highest attain-able social welfare In this case is a situation where judgmentaccuracy is at its highest yet the variance of opinions or theproportion of extremists at its lowest an ideal situation Wecannot say for sure since strong opinion consensus and longdecision-making processes in many situations may result ina loss of welfare rather than in a gain of it

Another interesting result in procedure for deliberationis the very similar estimates we find for the effects of theminimum number (119898) of debates necessary to make adecision on a proposal and the proportion (119899119863) of individualsparticipating in deliberation on judgment accuracy In thescope of social welfare this raises the question of whetheradvocating for longer decision processes by demandingmoredeliberation steps (higher 119898) rather than bigger debates(higher 119899119863) may increase welfare Depending on the objec-tives of the decider of the deliberation procedure he or shemay choose to concentrate on one or the other A recom-mendation we can assert from the mixed social interactionsmodel is that if one prefers low to high variance of opinionsandor high to low coherence the decider is better off ifshe focuses on increasing the size of the debate rather thanorganizing debates very frequently One of the reasons forthis might be that when making bigger debates all agents aretruly considered as equals in the decision-making processIn organizing many debates individuals that are chosen toparticipate in many of these will be more influential and will

somehow bias the set of arguments in the deliberation arenaThemajority rulemodeled through120572 also determines the sizeof the shifts and the distribution of opinions Because a bigger120572 is associated with higher levels of extremism and opinionvariance deliberated cues are harder to accept and thereforeit becomesmore difficult for agents to reach concensus andormoderate their opinions

On the other hand and further into the idea of legitimacythe goal of deliberative processes is the agreement amongall participants This agreement stands for the right or bestdecision from a formal or procedural perspective and it is alsothe best from a substantive point of view In other words thelegitimacy of decisions not only derives fromwell-establishedprocedural requirements (respect of the protocol) but alsobegs for the fulfillment of two essential conditions (1) satisfy-ing the procedural requirements for a correct procedure (for-mal legitimacy) and (2) the rational acceptability of the resultsof this procedure (substantive acceptability) Legitimacy canbe then obtained from the coherence statistic scrutinizedin the model insofar as it measures how well agents acceptthe results of a correct procedure (eg deliberation) So fora proposal to be legitimate one has to choose parametersthat would maximize group coherence in decision-makingprocesses

As deliberation may give rise to consensus in the delib-eration arena it may also sow the seeds of dissensus inthe group Our model clearly illustrates this phenomenonthrough the procedural parameters of deliberation and themetrics observed From another perspective in deliberativedemocracy dissensus in collective decision-making may bedesirable According to Landermore and Pagersquos accounts [36]lack of consensus can be perceived as having agents withalternative ideas that combined with other ideas can providea better approximation of the ideal status of an argumentTheauthors in [36] call this dissensus ldquopositive dissensusrdquo andit makes normative sense when solving complex collectivedecision-making problems Our model captures the ideaof having normative requirements to define a good andlegitimate collective decision requirements of high consen-sual deliberative accuracy (judgment accuracy) of legitimacy(coherence) and of diversity (variance of opinions) are alltaken into account and for such we can conclude that ourmodel is successful in describing a sound process of collectivedecision-making with interpretable outcomes

Of course the present simulation results and analysis donot prove that all opinion dynamics can be accounted forby such simple processes like the one presented here Theydo however shed light on the expressiveness of combiningdifferent paradigms to account formore interpretablemodelsand pave the way to creating original models of the sortThe simulations suggest the desirability of discovering theconsequences of relatively simple laws of communication atdifferent levels (micro meso and macro) to determine whatstill needs to be explained

5 Related Work

We see our model as a contribution to the influence andopinion dynamics field in ABM and a pragmatic application

Complexity 25

of abstract argumentation theory To our knowledge we areunaware of existing literature on agent-based modeling thatexplicitly relates collective choice and the notions of deliber-ation and opinion diffusion through abstract argumentationas we have done it

Most models in the literature of opinion diffusion areinterested in opinions because these have an influence incollective decision and in questions of social order Forinstance in [3] the authors are interested in consensus andin how a group collectively chooses among two alternativesIn other models authors are interested in the emergenceof extremism [8] and on the distribution of opinions whenextremists are introduced in the population [5 33 37] whileother authors coin the notion of opinion polarization as anemergent property of the system [8 12] They show usingmodels of ldquobounded confidencerdquo and opinion diffusion withtrust that three different kinds of steady states (unipolarbipolar and central) are possible depending on whetheragents are sufficiently uncertain about their opinions andsufficiently connected andor a certain proportion of indi-viduals are already extreme Recent articles on informationand opinion dynamics stress the importance of governanceand government intervention in the spread of emotions(opinions) in the frangibility of social consensus systemIn [10] the authors show that government intervention inthe spread of negative emotions can lead to an even fasterspread of negative emotions (single-peeked convergence) andin a faster collapse of the social consensus system Opinionas a function of trust is studied in [12] and relates to howindividuals form their opinions on the basis of how muchthey trust agents in their networks

Another stream of discrete opinion dynamics modelsthat explain the emergence of opinion can be found in theliterature Computer multiagent models of attitude changebased on Latanersquos social impact theory are presented in [4] Inthe theory an agent changes her opinion on the basis of theinformational impact she is subject to which depends on thepersuasiveness (strength) supportiveness and immediacy(group structure) of the environment and information shereceives The simulations predict two emergent groups ofphenomena the shifting of attitudes towards incompletelypolarized equilibria and the formation of coherent clusteringof subgroups with deviant attitudes Similarly but in acontinuous more argumentative fashion Mas and Flache[11] present a model of opinion diffusion using argumentsArguments are considered to be for or against a proposal andare given an agent-dependant relevance in dyadic persuasiveinteractions Arguments also determine the agentsrsquo opinionsand their dissimilarities which posits the rules of pair-wiseencounters in theirmodel as agents are assumed to form theiropinions according to the arguments they own They showthat in an argumentative opinion diffusion model the onlyobservable steady states are bipolarization and consensusand that bipolarization can emerge from interactions amongsimilar agents In other words bipolarization of opinionsis possible with homophily and without negative influenceif there is argumentation They subsequently compare theirresults to experimental data In [38] a similar approach toargumentation and opinion formation is taken but for the fact

that they introduce different types of arguments explicitlyapply social judgment theory [22] and use survey data toestablish model-to-real-world comparisons

Closer to opinion formation and abstract argumentationGabbriellini and Torroni [39] are the first to have origi-nally and soundly merged opinion diffusion and abstractargumentation They define an agentrsquos opinion as a functionof the arguments she holds and the attack relation amongthem They devised a focused peer-to-peer dialogue systemof persuasion (NetArg) inspired from Mercier and Sperberrsquosargumentative theory of reasoning [40] which used onlyabstract argumentation to study opinion polarization andopinion dynamics Moreover they considered networks andthe notions of trust and epistemic vigilance to define adynamics for knowledge revision and trust itself whichwe clearly lack in our model They showed that if a con-servative belief operator in argumentation was applied byagents when they reasoned then their dialogue protocoldid not increase polarization among agents That said theyused the model to study the effect of Groenvetterrsquos weaklink theory in the spreading of arguments and were notparticularly interested in notions like judgment accuracy orcoherence nor in tackling questions related to collectivedecision-making procedures Their system is very expressiveand helps position our work in the litterature our work is atthe frontier of pure argumentative opinion diffusion modelsopinion diffusion with arguments and continuous bounded-confidence models of opinion diffusion a la Deffuant

On another note work on collective cognitive conver-gence [41] and opinion sharing [19] shows that consensustowards a certain ldquocorrectrdquo opinion or cognitive state is alwayspossible yet dependent on noise variability and awarenessof agents In [20] the authors show that learning about anexogenous correct state of the world (represented by bits)under confidence was possible but only if the agents werenot too confident In a homogeneous population they showthat the higher the confidence the worse the learningmdashthevery confident agents do not learn the properties of the truestate of the world and disrupt the learning process of the lessconfident ones Collective cognitive convergence can be seenas a result of deliberation in truth-seeking models

When it comes to abstract argumentation theory we takean approach that wires two types of dialogues that are well-studied in the literature persuasion dialogues [42] and delib-eration dialogues [43] Another interesting line of work is theone on mechanism design [44] or the problem of devisingan argumentation protocol where strategic argumentation isnot a liability for success in debates We tackle mechanismdesign in a different way though Instead of consideringstrategy-proofness we are interested in how differences inprotocol can result in epistemologically ldquobetterrdquo collectivechoices and guarantee that opinion distributions are favorablefor deliberation For a survey on persuasion dialogue see[42]

Work on agent-based argumentation usually assumes thatthe semantic relationship between arguments is fixed Inother words if two individuals were to put two argumentsin the public arena such that one logically attacked the otherthen everyone would agree that such attack exists [45 46]

26 Complexity

Other models which do not make this restrictive assumptioncan also be found in the literature [27 47] and derive fromthe class or family of opponent models [27 48] in which twoopposing sides attempt to win a dialogue Ourmodel is in theintersection of these two but the frame combining opiniondiffusion of the kind and argumentation remains new

The idea of mixing interpersonal influence and verticalcommunication however is not original It is described andimplemented in innovation diffusion models such as [5 49]In both vertical communication is modeled as exogenoustransparent informationmdashagents are aware of the existenceof an innovation thus triggering several processes of choiceand stabilization of opinions Also in [50] an Eulerian modelis implemented to show the effect of media and exogenousinformation on opinion distributions With respect to thispoint the originality of our work is in that the informationemitted as vertical communication is endogenous It is issuedfrom a deliberation model that agents shape on the basisof their opinions arguments and behavior In the spirit of[51] where the authors control for the design of verticalcommunication (to whom the vertical communication isaddressed its timing how it influences agents opinions) wecontrol for the process generating the information for ourown set of variables length of debates majority quotas andthe frequency at which discussions take place

6 Conclusion and Perspectives

The main objective of this article was to build a bridgewhere decision-making argumentation and opinion dif-fusion could come together We proposed a model thatcombined abstract argumentation theory and a boundedconfidence opinion diffusion model and showed to whatextent it could explain variance of opinions extremismcoherence in collective decisions and judgment accuracyon arguments given an ideal state of full information Thesecond objective was to show in what way governance pro-viding agents with arenas of discussion and deliberation wasimportant in addressing the success of collective decision-making processes and the quality of their outcomes

The model revealed that allowing for more deliberationtime though in lower frequency and allowing for widerparticipation in deliberation increased the variance of opin-ions and the proportion of extremists in a group Theseobservations are consistent with the combination of resultsfound in [9] and in [13 14] which stress that deliberationmay polarize groups and may have a meager effect on theshifts of opinions and inconsistent with [16] where it isargued that opinions tend to moderate after deliberationIt happens that deliberation alone did moderate opinionyet when integrated into a more complex system in whichindividuals were allowed to interact with one another andnot all that was deliberated was accepted its influence wasshadowed by other more individualistic dynamics

Undeniably the grounded semantics played an importantrole in the weak effect of deliberation since it models askeptical way of reasoning over arguments For agentsaccepting arguments thus updating their opinions accord-ingly happened not too often Nevertheless asking for more

deliberation did increase judgment accuracy as observed in[18] yet in a marginally decreasing fashion We showed thatvoting within the deliberation protocol not only increasedthe proportion of extremists and the variance of opinions ina group but also determined how coherent deliberation andvoting procedures were with each other Lastly we showedthat agents that are focused judged arguments better hadmore stable opinions and constituted a group with higherproportion of extremists than their naive counterparts

In terms of governance the model says that there is per-haps no trade-off between extremismand judgment accuracyInstead it asserts that the higher the variance of opinions thecloser the group gets to the correct decisions This situationmay be interpreted as Landermore and Pagersquos notion of ldquoposi-tive dissensusrdquo (in complex collective decision-making tasks)[36] as opposed to consensus as a normative requirementfor correct collective decision-making If a decider has toorganize deliberation to legitimize a public policy dependingon what kind of world he or she wants different set ofparameters may be chosen to account for different levels oflegitimacy and correctness For instance one may ask for adecision to be legitimate to respect a certain level of coherenceor of judgment accuracy and to be discussed by a sundryof different-opinionated peoples If so deliberation has tobe made more often more participants have to be includedin debates or in deliberation instances and not too manytime steps should be introduced in between two deliberationinstances So the model may capture normative ideas todefine correct and legitimate collective decisions and tomake recommendations on those grounds Requirements ofhigh consensual deliberative accuracy (judgment accuracy)legitimacy (coherence) and diversity (variance of opinions)are all taken into account and for such a reason the model israther successful in describing a sound process of collectivedecision-making with interpretable outcomes

61 Extensions of the Model and Other Ideas Our model canbe extended in many ways Value-based abstract argumen-tation frameworks (VAFs) for example as an extension ofDungrsquos argumentation framework provide a formal descrip-tion of the process of decision-making in which argumentsare given values and audiences (set of agents) ignore attacksbetween arguments on the basis of their preferences over val-ues Social abstract argumentation frameworks (SAFs) alsoprovide an interesting and seemingly convenient formalismfor our model In SAFs agents vote on the acceptability ofarguments and the resulting labeling is a combination of thearguments that are voted for and labeling-based semantics Inour model we apply the same rule yet only for one argumentand given precise agent-based votingmechanisms and votingrules which are eventually parametrized

To consider our model as a specification of a mixed VAFand SAF model may be of great interest for future workThis formalism can provide solid foundations to describeexplain interpret and eventually extend the results foundhere from an argument-based perspective On those groundsfuture work will focus on creating a hybrid argumentationframeworkmdashwith its semanticsmdashthat takes heed of bothvalues or principles and voting

Complexity 27

Concerning other possible extensions of the model anatural reflex would be to argue on the validity and therobustness of the mix between the bounded confidenceopinion diffusion models and deliberation Consequentiallyadding deliberation to different bounded confidence modelsof opinion diffusion such as the Hegelsmann-Krause model[37] or to models of social impact theory like in [4] canbe of great interest for future work We acknowledge that afiner understanding of argumentation frameworks randomlattices and their respective applications for this modelare necessary Testing for different labeling-based semanticsor even trading labeling-based semantics for scoring-basedsemantics (semantics that defines acceptability on scoresgiven to arguments) such as debate semantics and ranking-based semantics (semantics that yield a preorder) may beinteresting to consider in future work

More into modeling deliberation it is not inadequateto believe that agents may also advance proposals duringdeliberation and replace to some extent the proposal forwhich the deliberation is taking place Endogenizing theprocess that produces the type of argument to be deliberatedon may be interesting to explore either by having agentsstrategically replace proposals at certain moments duringthe debate or having the central authority attempt to chooseproposals such that the probability of having them acceptedis high Agents may also be thought to be autoorganizing(independent of a central authority) and concede on the sizeof the instances of deliberation To consider differences insemantics across agents or even endogenizing the semanticson how urgent or important the proposal that is beingdiscussed is can also be an interesting direction to take

Last but not least many nontrivial modifications of ourmodel are possible and most should include better-thoughtdeliberation protocols It may be interesting to design andobserve protocols in which deliberation only affects agentsthat are actually debating for example Trust network effectsmultidimensionality of opinions new processes of argumentexchange or learning among agents are notions to furtherdevelop in order to make the model more realistic and relaxsome unreliable assumptions In sum the model is to berefined and extended with the objective of either studyingconcrete cases of deliberative polling and opinion dynamicsor implementing more intuitive thought experiments

Appendix

We present below some of the pseudocode we used toimplement the model

Algorithms

(a) Algorithm 1 implements the dyadic influence in themodel It describes how two agents interact andupdate their opinions

(b) Algorithm 2 implements the deliberative influence inthe model which describes how agents update theiropinions in sight of the accepted deliberated proposalarguments

(c) Algorithm 3 is used to obtain the grounded labelingof an argumentation framework

(d) Algorithm 4 provides an overview of a run in thesimulations119872 is the number of proposal argumentsto discuss which we set to 100 in the model

Each object in the system is indexed by a119908ℎ119900 number Hencea set of agents is annotated by a set of natural numbers Weuse this fact in the implementation of the algorithm 119874119887119895119890119888119905119894denotes the object of 119908ℎ119900 number 119894 Agentsrsquo opinions (119900) arecalibrated so that 119900 isin [minus1 1]Data Availability

The program and data used to support the findings of thisstudy are available from the corresponding author uponrequest

Disclosure

A preliminary version of this work was presented anddiscussed at the conference Bridging the Gap between FormalArgumentation and Actual Human Reasoning on October 4-5 2018 at BochumGermanyThe authors thank the audiencefor their valuable comments and suggestions

Conflicts of Interest

The authors declare that they have no conflicts of interestregarding the publication of this article

Acknowledgments

The contribution of Gabriella Pigozzi was supported by theDeutsche Forschungsgemeinschaft (DFG) and the CzechScience Foundation (GACR) as part of the joint project FromShared Evidence to Group Attitudes (RO 45486-1)

Endnotes

1 In other words the superiority of the ldquoright answerrdquo orlabeling in our case will appear as such to all

2 All agents hold the same opinion3 By sensitivity we mean how much an agent is ready to

change her opinion on the sole fact that an informationalcue has been debated on

4 The assumption is strong but this trust among agentsmay be rooted from a genuine individual motivation toreach a democratic consensus

5 We adopted a rule of thumbs that says that if an agentis neutral with respect to the principle then she willvote for any proposal that is presented to her duringdeliberation

6 If the argument is not unique she chooses one atrandom

7 If there are more than one candidate the agent will favorthe argument at minimal distance from 119868

28 Complexity

Require Vector of values for parameters (119881 = (120583 119880 119879))Ensure Successful interactions among agents119896 larr997888 0119873119890 larr997888 0

while 119896 le |119873|2 do ⊳Fill119873119890 with |119873|2 agents119896 larr997888 119896 + 1119873119890 larr997888 119877119886119899119889119900119898(|119873| + 1) ⊳Pseudo-random generator Generates integer between 0 and |119873|119873119903 larr997888 0 120575119894119895 larr997888 0 ⊳Variable to store differences in opinionfor all 119894 isin 119873119890 doRandomly pick a number 119895 isin 119873(119873119890 cup 119873119903)119873119903 larr997888 119873119903 cup 119895 120575119894119895 larr997888 |119900119894 minus 119900119895|procedure dyadic-update(119894 119895) ⊳Two agents 119894 and 119895 discuss and update opinions1199001015840 larr997888 119900119894 (120583119895 119880119895 119879119895) larr997888 (120583119894 119880119894 119879119894) larr997888 (120583 119880 119879) ⊳Save agent irsquos opinion before updating and homogenize agents

if 120575119894119895 lt 119880119894 then119900119894 larr997888 120583119894119900119895 + (1 minus 120583119894)119900119894if 120575119894119895 gt 119879119894 then119900119894 larr997888 (1 + 120583119894)119900119894 minus 120583119894119900119895if 120575119894119895 lt 119880119895 then119900119895 larr997888 1205831198951199001015840 + (1 minus 120583119895)119900119895if 120575119894119895 gt 119879119895 then119900119895 larr997888 (1 + 120583119895)119900119895 minus 1205831198951199001015840

Algorithm 1 Dyadic social influence

Require Vector of values for parameters (119881 = (120574 119901119886 119901119903))Ensure Successful interaction between proposal arguments and agents

for all 119894 isin 119873 doprocedure Probability-Update(V119886 119901119886 119901119903) ⊳Update individual probabilities of change due to deliberated cues119909119894 larr997888 119901|V119886minus119900119894 |2119886 119910119894 larr997888 1199012|V119886minus119900119894 |119903119902 larr997888 0if L119886119887(119868) = 119894119899 then

procedure deliberative-update(119894 119886) ⊳Agent i argument 119886120574119894 larr997888 120574 119902 larr997888 119906 sim U(0 1) ⊳ 119906 is a realization of a uniform random variableif 119902 lt 119909119894 then119900119894 larr997888 120574119894119900119894 + (1 minus 120574119894)V119886break

if 119902 lt 119910119894(1 minus 119909119894) then119900119894 larr997888 (1 + 120574119894)119900119894 minus 120574119894V119886Algorithm 2 Deliberation influence

8 In terms of computational complexity naive agentsmaketheirmoves in119874(|A119894| log(|A119894|)) while focused agents doat least as bad

9 Tables have memory Therefore debates are not neces-sarily trees

10 If there are less than (119899119863 times |119873|)2 agents for any of bothgroups the CA summons all of them to the deliberationtable

11 Or careful consideration of the viewpoints of otherswhich implies that citizens keep an open mind and donot reject arguments outright

12 A ldquohealthyrdquo consensus refers to the case in which manyconsensual decisions are taken but opinion-wise agentsdo not agree with each other Quotes are used on theword healthy because some political science theorists

(eg [36]) believe that in certain situations agreement inopinions for deliberation is epistemologically damaging

13 Simulations on groups of |119873| = 800 and |119873| = 400agents shows that scaling the population has an effect onall themetrics but it is confounding depending onwhichcouples of attitudes structures are considered (119880 119879) Westick to |119873| = 400 as in [9] for comparability andsimulation running time

14 By balanced wemean with as many arguments with V119886 lt0 as with V119886 gt 0

15 Experiments on the system suggest that a bigger(smaller) argument pool 119872 = 1000 (119872 = 200) isassociated with a higher (lower) variance of opinions(and extremism) and higher (lower) judgment accuracyFor the rest of the observations 119872 seem to have noeffect We choose a middle value for 119872 Similarly theargument pool chosen to be balanced or uniformly

Complexity 29

Require Argumentation framework 119860119865 = (AR)Ensure Assignment of labels to arguments from the grounded labeling of 119860119865L(A)

L0(A) larr997888 (0 0 0)119896 larr997888 0do119894119899 L119896+1(A) larr997888 119894119899 L119896(A) cup 119886 | 119886 is not labeled 119894119899 L119896(A) 119886119899119889 forall119887 isin A 119904119905 119887 R 119886 then 119887 isin 119900119906119905 L119896(A)119900119906119905 L119896+1(A) larr997888 119900119906119905 L119896(A) cup 119886 | 119886 is not labeled 119894119899 L119896(A) 119886119899119889 exist119887 isin A 119904119905 119887 R 119886 and 119887 isin 119894119899 L119896+1(A)

while L119896+1(A) = L119896(A) = 0Algorithm 3 Solve argumentation frameworks (with grounded semantics)

Require Vector of values for parameters (119881)Ensure Vector of statistics (119880)

procedure Set-parameters(119881) ⊳Initialize parameters for scenarioprocedure Init-objs(119899119898 119896 ) ⊳Create119898 arguments 119899 agents and 119860119865120576procedure Solve-Argumentation-Framework(119860119865) ⊳Assigns the epistemic labelingL120576 toA119894 larr997888 1119905 larr997888 1while 119894 le 119872 do ⊳ 119872 the number of arguments to discuss119868119894 larr997888 119866119890119899 119886119903119892(1) ⊳Generates proposal argument 119868119860 119894119905 larr997888 (119868119894 0)L119886119887(119868119894) larr997888 119906119899119889 ⊳Initialize label for 119868119895 larr997888 1 ⊳Initialize debate counter 119895while ((L119886119887(119868) = 119906119899119889 or 119898 le 119895) and 119895 le 119898) doif 119905 equiv 0(119898119900119889 119905119863) then

procedure build-argumentation-framework(119899119863 119887119890ℎ119886V119894119900119903) ⊳Build the argument graph agents deliberate(119861119894119905 119878119894119905) larr997888 (119883 119884) sub (⋃119897isin119873119899119863 A119897119894119905 ⋃119897isin119873119899119863 R119897119894119905) ⊳ 119873119899119863 sub 119873 are agents sampled by the CA119860 119894119905 larr997888 119889(119860 119894119905 (119861119894119905 119878119894119905))procedure Solve-Argumentation-Framework(A119889) ⊳FindL(A119889)L119886119887(119868119894) larr997888 L119886119887119863(119868119894)119895 larr997888 119895 + 1

else119896 larr997888 1while 119896 le 119905119863 doprocedure dyadic-social-influence(120583 119879 119880) ⊳Agents discuss one-to-one119896 larr997888 119896 + 1119905 larr997888 119905 + 1

if 120572 gt 0 andL119886119887(119868119894) = 119900119906119905 then ⊳Agents vote for the proposalif (|119897 isin 119873|119900119897 times V119868119894 ge 0| ge 120572 times 119899) then

L119886119887(119868119894) larr997888 119894119899else

L119886119887(119868119894) larr997888 119900119906119905if L119886119887(119868119894) = 119894119899 thenprocedure Deliberation-influence(120574 119901119886 119901119903) ⊳Agents are influenced by the result of the decision process119894 larr997888 119894 + 1119905 larr997888 119905 + 1119880 larr997888 119862119900119898119901119906119905119890 119878119905119886119905119894119904119905119894119888119904(119881)

return 119880 ⊳Get the statisticsAlgorithm 4 Run for one scenario

generated does not seem to play a significant explanatoryrole on any of the metrics

16 Experiments show that themaximumnumber of debatesis not an important parameter even if it is an intuitiveone when considering decision-making processes Inpractice debates end after at most 119898 + 1 deliberation

steps and even then it happens rarely However to avoidunpleasant surprises like infinite debates on a completeargument graph containing the proposal argument weadded the constraint 119898 Its value is precisely 7 becausethe biggest minimum number of debates we studyis 6

30 Complexity

17 To ensure comparability with the deliberative andmixedopinion dynamics models on judgment accuracy andcoherence agents vote on proposal arguments every 119905119863times(119898+1) 119905119863 and119898does not explain variability in judgmentaccuracy nor in coherence

18 Student t-tests for independent samples are robust tothe violation of two hypothesis on the distribution ofthe samples normality and homogeneity of variancesas long as the sample size is big enough (central limittheorem) and if the difference in size of the comparedsamples is smallrespectively

19 Studentrsquos test version forwhen variances are not assumedequal and the difference in size of the compared samplesis big

20 We do not correct for homoskedasticity nor normality oferrors in the estimates

21 Balanced as the same number of runs per scenario orsuch that no significant correlations are found betweenparameters

References

[1] R Axelrod ldquoThe dissemination of culture a model withlocal convergence and global polarizationrdquo Journal of ConflictResolution vol 41 no 2 pp 203ndash226 1997

[2] D M Rousseau Reinforcing the MicroMacro Bridge Organiza-tionalThinking and Pluralistic Vehicles SAGEPublications SageLos Angeles Calif USA 2011

[3] S Galam and SMoscovici ldquoTowards a theory of collective phe-nomena consensus and attitude changes in groupsrdquo EuropeanJournal of Social Psychology vol 21 no 1 pp 49ndash74 1991

[4] A Nowak J Szamrej and B Latane ldquoFrom private attitude topublic opinion a dynamic theory of social impactrdquo Psychologi-cal Review vol 97 no 3 pp 362ndash376 1990

[5] G Deffuant S Huet and F Amblard ldquoAn individual-basedmodel of innovation diffusion mixing social value and individ-ual benefitrdquo American Journal of Sociology vol 110 no 4 pp1041ndash1069 2005

[6] G Deffuant D Neau F Amblard and G Weisbuch ldquoMixingbeliefs among interacting agentsrdquoAdvances in Complex Systems(ACS) vol 3 no 01n04 pp 87ndash98 2000

[7] J Rouchier P Tubaro and C Emery ldquoOpinion transmission inorganizations an agent-based modeling approachrdquo Computa-tional and Mathematical OrganizationTheory vol 20 no 3 pp252ndash277 2014

[8] M Meadows and D Cliff ldquoThe relative disagreement model ofopinion dynamics where do extremists come fromrdquo in Pro-ceedings of the International Workshop on Self-Organizing Sys-tems Lecture Notes in Computer Science pp 66ndash77 Springer2013

[9] W Jager and F Amblard ldquoUniformity bipolarization and plu-riformity captured as generic stylized behavior with an agent-based simulation model of attitude changerdquo Computational ampMathematical Organization Theory vol 10 no 4 pp 295ndash3032005

[10] Y-F Zhang H-Y Duan and Z-L Geng ldquoEvolutionary mech-anism of frangibility in social consensus system based onnegative emotions spreadrdquo Complexity vol 2017 Article ID4037049 8 pages 2017

[11] A Flache andMWMacy ldquoSmall worlds and cultural polariza-tionrdquoThe Journal of Mathematical Sociology vol 35 no 1-3 pp146ndash176 2011

[12] A Tsang and K Larson ldquoOpinion dynamics of skepticalagentsrdquo in Proceedings of the 13th International Conference onAutonomous Agents and Multiagent Systems AAMAS 2014 pp277ndash284 International Foundation forAutonomousAgents andMultiagent Systems France May 2014

[13] K M Hansen Deliberative democracy and opinion formationUniversity Press of Southern Denmark Odense 2004

[14] C R Sunstein ldquoThe law of group polarizationrdquo Journal ofPolitical Philosophy vol 10 no 2 pp 175ndash195 2002 WileyOnline Library

[15] S Moscovici and W Doise Dissensions et Consensus unetheorie generale des decisions collectives Presses Universitairesde France-PUF 1992

[16] J S Fishkin and R C Luskin ldquoExperimenting with a demo-cratic ideal deliberative polling and public opinionrdquo ActaPolitica vol 40 no 3 pp 284ndash298 2005

[17] R C Luskin J S Fishkin and S Iyengar Considered Opinionson US Foreign Policy Face-To-Face versus Online Delibera-tive Polling International Communication Association NewOrleans La USA 2004

[18] J Barabas ldquoHowdeliberation affects policy opinionsrdquoAmericanPolitical Science Review vol 98 no 4 pp 687ndash701 2004

[19] O Pryymak A Rogers and N R Jennings ldquoEfficient opinionsharing in large decentralised teamsrdquo in Proceedings of the 11thInternational Conference on Autonomous Agents andMultiagentSystems vol 1 pp 543ndash550 Spain June 2012

[20] J Rouchier and E Tanimura ldquoWhen overconfident agents slowdown collective learningrdquo Simulation vol 88 no 1 pp 33ndash492012

[21] D Miller ldquoDeliberative democracy and social choicerdquo PoliticalStudies vol 40 S1 pp 54ndash67 1992

[22] M Sherif and C I Hovland Social Judgment Assimilation andContrast Effects in Communication and Attitude Change YaleUniversity Press 1961

[23] P M Dung ldquoOn the acceptability of arguments and its funda-mental role in nonmonotonic reasoning logic programmingand n-person gamesrdquo Artificial Intelligence vol 77 no 2 pp321ndash357 1995

[24] M Caminada and G Pigozzi ldquoOn judgment aggregation inabstract argumentationrdquo Autonomous Agents and Multi-AgentSystems vol 22 no 1 pp 64ndash102 2011

[25] P Baroni M Caminada andM Giacomin ldquoAn introduction toargumentation semanticsrdquoThe Knowledge Engineering Reviewvol 26 no 4 pp 365ndash410 2011

[26] S Chambers ldquoDeliberative democratic theoryrdquo Annual Reviewof Political Science vol 6 no 1 pp 307ndash326 2003

[27] E Bonzon andNMaudet ldquoOn the outcomes of multiparty per-suasionrdquo in Proceedings of the 10th International Conference onAutonomous Agents and Multiagent Systems 2011 AAMAS 2011vol 1 pp 47ndash54 International Foundation for AutonomousAgents and Multiagent Systems Taiwan May 2011

[28] J Ranciere Disagreement Politics and Philosophy University ofMinnesota Press 1999

[29] T Bench-Capon and K Atkinson ldquoAbstract argumentation andvaluesrdquo in Argumentation in Artificial Intelligence pp 45ndash64Springer 2009

[30] M Caminada ldquoOn the issue of reinstatement in argumenta-tionrdquo in Proceedings of the European Workshop on Logics in

Complexity 31

Artificial Intelligence vol 4160 of Lecture Notes in ComputerScience pp 111ndash123 Springer Berlin Heidelberg

[31] GAVreeswijk ldquoAn algorithm to computeminimally groundedand admissible defence sets in argument systemsrdquo in Pro-ceedings of the 2006 conference on Computational Models ofArgument Proceedings of COMMA 2006 vol 144 pp 109ndash120IOS Press Amsterdam The Netherlands 2006

[32] C W Sherif M Kelly H L Rodgers et al ldquoPersonal involve-ment social judgment and actionrdquo Journal of Personality andSocial Psychology vol 27 no 3 pp 311ndash328 1973

[33] G Deffuant F Amblard G Weisbuch and T Faure ldquoHow canextremism prevail A study based on the relative agreementinteraction modelrdquo Journal of Artificial Societies and SocialSimulation vol 5 no 4 p 1 2002

[34] E Awad M W Caminada G Pigozzi M Podlaszewski andI Rahwan ldquoPareto optimality and strategy-proofness in groupargument evaluationrdquo Journal of Logic and Computation vol 27no 8 pp 2581ndash2609 2017

[35] D Grossi and G Pigozzi ldquoJudgment aggregation a primerrdquoSynthesis Lectures on Artificial Intelligence and Machine Learn-ing vol 8 no 2 pp 1ndash151 2014

[36] H Landemore and S E Page ldquoDeliberation and disagreementproblem solving prediction and positive dissensusrdquo PoliticsPhilosophy and Economics vol 14 no 3 pp 229ndash254 2015

[37] R Hegselmann and U Krause ldquoOpinion dynamics andbounded confidence models analysis and simulationrdquo Journalof Artificial Societies and Social Simulation vol 5 no 3 2002

[38] A Stefanelli and R Seidl ldquoModerate and polarized opinionsUsing empirical data for an agent-based simulationrdquo inProceed-ings of the Social Simulation Conference 2014

[39] S Gabbriellini and P Torroni ldquoA new framework for ABMsbased on argumentative reasoningrdquo in Advances in SocialSimulation vol 229 pp 25ndash36 Springer Berlin Germany 2014

[40] H Mercier and D Sperber ldquoWhy do humans reason Argu-ments for an argumentative theoryrdquo Behavioral and BrainSciences vol 34 no 2 pp 57ndash74 2011

[41] H V Parunak T C Belding R Hilscher and S BruecknerldquoModeling and managing collective cognitive convergencerdquoin In Proceedings of the 7th iNternational Joint Conference onAutonomous Agents and Multiagent Systems vol 3 pp 1505ndash1508 International Foundation for Autonomous Agents andMultiagent Systems 2008

[42] H Prakken ldquoFormal systems for persuasion dialoguerdquo TheKnowledge Engineering Review vol 21 no 2 pp 163ndash188 2006

[43] K Atkinson T Bench-Capon and P Mcburney ldquoA dialoguegame protocol for multi-agent argument over proposals foractionrdquoAutonomousAgents andMulti-Agent Systems vol 11 no2 pp 153ndash171 2005

[44] M Thimm and A J Garcıa ldquoClassification and strategicalissues of argumentation games on structured argumentationframeworksrdquo in Proceedings of the 9th International Conferenceon Autonomous Agents and Multiagent Systems vol 1 pp 1247ndash1254 International Foundation for Autonomous Agents andMultiagent Systems Canada 2010

[45] I Rahwan and K Larson ldquoMechanism design for abstractargumentationrdquo in Proceedings of the 7th International JointConference on Autonomous Agents and Multiagent SystemsAAMAS 2008 pp 1031ndash1038 International Foundation forAutonomous Agents and Multiagent Systems Portugal May2008

[46] H Prakken R Riveret A Rotolo and G Sartor ldquoHeuristics inArgumentation A Game-Theoretical Investigationrdquo EuropeanUniversity Institute 2008

[47] N Hadidi Y Dimopoulos and P Moraitis ldquoTactics and con-cessions for argumentation-based negotiationrdquo in Proceedingsof the COMMA vol 245 pp 285ndash296 2012

[48] N Oren and T J Norman ldquoArguing using opponent modelsrdquoin Proceedings of the International Workshop on Argumentationin Multi-Agent Systems pp 160ndash174 Springer 2009

[49] S Delre W Jager T Bijmolt and M Janssen ldquoTargeting andtiming promotional activities An agent-based model for thetakeoff of new productsrdquo Journal of Business Research vol 60no 8 pp 826ndash835 2007

[50] A Mirtabatabaei P Jia and F Bullo ldquoEulerian opinion dynam-ics with bounded confidence and exogenous inputsrdquo SIAMJournal on Applied Dynamical Systems vol 13 no 1 pp 425ndash446 2014

[51] V Barbet N Guiraud V Laperriere and J Rouchier Hagglingon Values Towards Consensus or Trouble 2019

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Complexity 19

minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1

Var(o)

sh

ec

ir

Var(o)

propex

sh

ec

ir

1

098

minus02

minus004

001

1

minus014

minus001

minus003

1

026

minus028

1

minus057 1

ＪＬＩＪ

Ｒ

Figure 8 Significant correlation estimates for themetrics of interestin the mixed opinion dyamics scenarios

accepted proposal argument andor the opinionmajority thatis formed from previous dyadic interactions (see Figure 7(f))

422 Relating the Metrics of Interest in Mixed OpinionDynamics The first thing to notice is that for all of thesescenarios all the metrics are significantly correlated (seeFigure 8) The variance of opinions and the proportion ofextremists are almost perfectly correlated To some extentthis means that a high variance of opinions known thatthe mean of opinions is statistically null equates to havingmany extremists in the population be towards one or theother side of the opinion distribution (eg extreme bipolar-ization) Similarly we find a negative and strong correlationbetween the judgment inaccuracymeasure and the coherencecoefficient (see Figure 8)mdashwhen judgment inaccuracy is high(low) coherence to deliberation is low (high) Cases likethese are telling since they may be related to sequences ofcollective decisions in which agents do not make too manymistakes when judging proposal arguments and still makedeliberated decisions that are on average in line with theirprinciples

The shift statistic interestingly is negatively correlatedwith the variance of opinions with judgment accuracy andwith the coherence coefficient In a way this means thathaving more moderate agents conduces on average to moremistakes in judging arguments and to less coherent collectivedecisions An explanation for this phenomenon could be thatas agents reach consensus in opinion important (possiblyextreme) arguments for correctly judging argument propos-als are not advanced in the deliberation arenawhich results inmore mistakes and in possibly accepting proposal argumentsthat after voting are not meant to be accepted Indeed the

negative correlation between the variance of opinions andjudgment inaccuracy points towards this direction since onaverage the more shifts there are the smaller the variance ofopinions is (see Figure 8)

423 Comparing Mixed Dynamics with Other MonolithicDynamics of Interactions Qualitatively opinion trajectoriesin the mixed model are more chaotic than in the pair-wise and deliberative interaction models (see Figure 7) Asanalyzed previously deliberation dynamics disrupt the pair-wise dynamics and create situations in which central andmulticlustered convergence (with two clusters at the extremesof the distribution) are possible at the expense of more(pair-wise dynamics) or less (deliberative dynamics) clustersIt tolerates the survival of very small isolated group ofagents that the previous models rarely tolerate in a hundredcollective decision-making steps (see Figure 7(c))

One of the most interesting observations one can makeof this family of models regards how groups change (in termsof our metrics) when dyadic social interactions are addedto deliberative interactions and vice-versa When comparingthe mixed interactions with only deliberative ones shiftsagentsrsquo aggregated change in opinions are on average lesspronounced The variance of opinions and the proportionof extremists are significantly higher for the former than forthe latter group coherence in contrast is lower with respectto the scenarios where opinions are formed only throughdeliberation Judgment accuracy follows that same trendA reason for this might be that mixed dynamics are fasterto yield semistable clusters andor opinion convergence Inconsequence the potential argument pool for deliberativeexchange shrinks faster and deliberation becomes less effec-tive over time which results in more collective mistakes Theoutcomeon coherence is predictable inasmuch as deliberativeinteractions produce group polarization If a group agrees inopinion debates are sterile because themembers of the groupwill most likely refrain from attacking the proposal argu-ments they all support and will work together to disparagethe proposal arguments they do not support (see Table 5)

In the same manner we report that the proportion ofextremists and the variance of opinions are barely lower inthe pair-wise interaction model when including deliberationin the decision-making process Shifts of opinions are alsobarely lower when there is no deliberation This can indicatethat deliberation does not undo the changes in opiniondue to pair-wise interactions nor does it amplifies them Incontrast when we consider the labeling-based metrics themixed interaction model ensures lower judgment inaccuracybecause of deliberation and lower group coherence (seeTable 5)

In sum the mixed discussion model is a compromisebetween the pair-wise interaction and deliberative opiniondynamic models in that it offers some guarantees in termsof judgment accuracy and coherence while allowing forldquoreasonablerdquo variance in the opinion pool It establishes somekind of trade-off between labeling-based metrics and thevariance of opinions that may be of interest for decision-making process design

20 Complexity

Table 6 Summary of sign or relative marginal effects for all varying parameters on dyadic deliberative and mixed interactions models inand for all experiments

Metrics Parameters119905119863 119898 119899119863 120572 119879 119880 119901119886 120574 119896 119861119890ℎ119886V119894119900119903119881119886119903(119900) XX X X O X X X X X X119901119903119900119901119890119909 XX X X O X X X X X X119878ℎ XX X X O X X X X X119890119888 XX X X XO X XOO XOO X X119894119903 XX X X O XO O X OOX XOO X XNota gray symbols correspond to dyadic dynamics light gray to deliberative dynamics and bold black to mixed dynamics Given a scenario the symbol 119883stands for constant parameter and 119874 for insignificant marginal effect and as parameters grow for positive (negative) marginal effect for the parameterFrom left to right at the parameter level the leftmost arrow corresponds to the scenario with the strongest (in absolute value) marginal effect of the parameter(with respect to to its factors) Example Let 119901 be a parameter and 119874 stands for ldquo119901 has a significant positive effect in dyadic discussion scenario negativein the deliberative one and insignificant in mixed discussion The positive effect in dyadic discussion is greater than the absolute value of the negative one inthe purely deliberative scenariordquo

424 Comments on the Differences between the Three Modelsfor OLS Regression Estimates The significance of most of theregression parameters estimated for the mixed interactionsscenarios are similar to those estimated for the pure dyadicanddeliberative scenariosNevertheless themagnitude of theeffects is different insomuch that the dynamics themselvesare different So instead of commenting the linear regressionestimates of the mixed model we compare when possiblethe standardized significant coefficients of the regressors ofthe mixed model with those of the pair-wise and deliberativeinteractionmodels (see Table 6 for a summary of the results)

In mixed interactions for instance the influence of119898 onthe opinion distribution is much more preponderant than inthe deliberative dynamics scenarios Going from 119898 = 1 to119898 = 3 increases the standard variation units of the variance ofopinions by twice as much as the one in the deliberative sce-nariosThis in particular is due to the design of the decision-making process If 119898 is higher then the number of socialinfluence steps (119905119863) in the decision process is also higher(compared to 119905119863 = 0 in the deliberative case) If we decideto consider these steps as contributing to the variance ofopinions then the effect of119898 on the variance of opinions hasto be stronger in the mixed interactions model In contrastthe number of agents participating in the debate has a weakereffect on the variance of opinions in the mixed interactionsmodel than in the deliberative dynamics specially for 119899119863 =005 (more than twice as much) The difference may comefrom the fact that in mixed discussions the part of theargument pool used in deliberation is more restrictive thanin the deliberative case We can argue that since pair-wisedynamics may get individuals closer to one another aftersome deliberation the same happens to the arguments thatindividuals will most likely use in deliberation Taking thisinto account means that deliberation will at many instancesbe unfruitful and have no effect on opinions thus failing tomoderate extreme opinions andor polarize the group In thesame direction a high number of participants in deliberationcan make the deliberation dynamics stiff In this case thedistribution of opinion is close to the uniform distributionwhich has a relatively high-to-moderate variancewith respectto the reference results

The strength of the deliberative dynamics has a positivestronger effect on the variance of opinions in themixedmodelthan in the deliberative model when 120574 = 02 and a negativeweaker one when 120574 = 01 A sound interpretation of thisresult is that when deliberation is successful and its effect onopinions does not get agents to move sufficiently away fromtheir current positions agents are pulled or pushed back totheir previously held opinions the variance of opinions doesnot changemuchOn the other hand if the effect of successfuldeliberation is strong opinions can change sufficiently tomake agents much more prone to assimilation into moderateopinion foci the variance of opinions falls

The deliberative opinion dynamics model is much moresensitive to 120572 than the mixed discussion model Differ-ences in marginal effects can double The probability ofattraction to deliberated results (119901119886) for instance has astrongmoderating effect (negative) on agentsrsquo opinions in thedeliberative opinion dynamics but a meager one (two-fold)in the mixed and dyadic dynamic models This is becauseopinion updates in the deliberative model depend only onthat parameter whereas in the mixed model more numeroustypes of interactions make opinion updates possible

In terms of judgment accuracy all parameters except thesize of the argument sacks (119896) have a significant slightlystronger effect on the deliberative dynamics than in themixedmodel In effect the difference may be attributed to theupshot of pair-wise interactions on the distribution of opin-ions and on the arguments that would be advanced duringdeliberation The result respecting parameter 119896 however iscounterintuitive Since argument sacks are static scenarioswhere shifts are higher should make 119896 more important fordeliberation to succeed and it is precisely in the scenarios ofpure deliberative dynamics that we find the biggest opinionshifts This may point to the hypothesis that in deliberativeinteractions agents change opinions substantially but do notoften change their adherence to the principle (go from anegative opinion to a positive one or vice versa) when thereis no group polarization

43 Sensitivity Analysis for the Procedural Parameters ofInterest Weperform a sensitivity analysis of the observationson the parameters of interest in themixed interactionsmodel

Complexity 21

We extend the parameter domains of the pure dyadic anddeliberative scenarios as we make it explicit in Table 2

431 Minimum Number of Debates (119898) We observe that theminimum number of debates has a significant well-observedeffect on all of our metrics excluding group coherence Forthe variance of opinions we can observe that the moredebates there are the bigger the value of the metric isalthough the higher 119899119863 and 119905119863 are the weaker the overalleffect of119898 becomes (Figures 9(d) and 9(a)) Furthermore themarginal effect of increasing minimal debates is decreasingan additional deliberation step requirement increases thevariance of opinions but less and less as it grows 119898 coupledwith 119899119863 has a shy S-shaped effect (see Figure 9(d)) on thevariance of opinions meaning that it has a stronger effectwhen 119899119863 is low which suggests a possible trade-off betweenthe number of arguments accepted in the debate and thenumber of debates required before accepting a proposal

In contrast the shifts in opinion are less and less likelyas 119898 grows and this is independent of the variations of theother parameters Again the effect is marginally decreasingand is only truly significant when 120572 = 05 An explanationto these effects may be linked to the design of the systemFirst the variance of opinions is higher when deliberation isasked for because the more deliberation steps there are thehigher the chances are that the central proposal argumentis deemed unacceptable specially when agents are focusedFurthermoremechanically speaking increasing theminimalamount of debates implies that whenever a decision is to betaken at least 119898 times 119905119863 time steps have to take place and if atany moment a proposal argument is considered undecided119905119863 time steps are added to the process So unless the debatedoes not yield 119906119899119889 labels for proposal arguments (highlyunlikely considering that 120590 = grounded semantics) the morenondeliberation steps there are in the decision process thehigher the variance of opinions is Concerning the shiftswhen 120572 = 05 either the system is too stiff to accept anyproposal argument and opinions do not changemuch or theeffects of pair-wise discussion and deliberation cancel out insuch a way that individual shifts are minimal (Figure 9(b))

On the side of labeling-based metrics the more debatesare asked for the more accurate a group is in its judgmentthe effect being smaller as 119898 grows When agents are naivethe effect is quasilinear while when they are focused thestrongest effects of acquiring more deliberation are foundwhen levels of deliberation are already low (Figure 9(f))Thiscan be explained by the fact that the more debates thereare in the decision process the closer one gets to the idealargumentation framework

432 Proportion of the Population in Deliberation Steps (119899119863)Like for 119898 119899119863 has a significant effect on the proportionof extremists and on the shifts and variance of opinions(Figure 9(e)) This may result from the fact that being ableto put more arguments in play at the same debate step mayincrease the odds of revealing cycles around the proposalargument in119860119865120576 Given that we use grounded semantics thearguments in the cycles are labeled 119906119899119889 and in consequence

debates are more often postponed when 119899119863 is higher Post-poning debates in turn increases the number of nondebatessteps in the decision process which increases the varianceof opinions and limits over time the moderating effect ofdeliberation Moreover the effect of this parameter is verydependent on the value of 120572 (Figure 9(e)) For shifts forinstance 120572 = 05 makes the effect of 119899119863 negative while120572 = 0 makes it positive but to a lesser extent Furthermorefor higher requirements of deliberation (119898) adding moreindividuals to the deliberation process has weaker effectson the variance of opinions and on the other metrics It isalso quite interesting to notice that it has the same effect onagents independently of agent behavior during deliberationThis is a surprising result as one would have expected morefocused agents in a deliberation arena to cause a raise inthe proportion of extremists as they play to knock outopposing proposal arguments and thereof reducing the effectof deliberation (lowers variance) on the whole population

Finally similar to119898 addingmore people into the deliber-ation process increases judgment accuracy (see Figure 9(c))and has no clear effect if any on group coherence (seeFigure 9(i))

433 Steps between Deliberation Steps (119905119863) In all configu-rations 119905119863 increases the proportion of extremists (variance)and decreases the shifts of opinions among agents Theshifts and the effect on the variance of opinions are presentfor 120572 = 05 insignificant otherwise (Figure 9(b)) 119905119863 ishighly linked to 119898 by construction When 119898 = 1 thecurve linking the variance of opinions and 119905119863 is convex As119898 increases the curve becomes more and more concavewhich means that 119905119863 has a more important effect on theopinion distribution as collective decisions take longer to beachieved This seems counterintuitive but in reality it reflectsthemultiplicative relationship between deliberation and pair-wise interactions and the semantics in the model If 119898 is lowand 119905119863 is high the effective number of pair-wise interactionsis on average smaller in the deliberation process Hencethe structure of the decision-making process hinders anyincrease in the variance of opinions If119898 is high the oppositeeffect is observed Additionally since the grounded semanticsyields few 119894119899 arguments with respect to other admissibility-based semantics getting closer to the ideal argumentationframeworkmaymake it difficult to obtain the expected effectsof deliberative discussion on agents Lower 119898 constrains thevariance of opinions by making deliberation more influentialon opinions

In terms of judgment accuracy and coherence we observethat 119905119863 does not explain coherence yet it tends to decreasejudgment inaccuracy This may be due to a more variedargument pool in the debate which in turn may be aconsequence of a higher variance of opinions or vice versa

434 Acceptability Voting Quota (120572) By far it is the mostinfluential parameter in our study It changes the directionand the intensity of the effect of all the other proceduralparameters and by construction heavily constrains the roadto accepting a proposal (see Figure 9)

22 Complexity

015

020

025

030

035

040

1 2 3 4 5 6

12345

Values of Ｇ

(a) 119905119863 times 119901119903119900119901119890119909 by119898

0

10

20

30

40

50

1 2 3 4 5 6

005066

Values of

(b) 119905119863 times 119878ℎ by 120572

040

045

050

055

060

1 2 3 4 5

001002003004005

Values of Ｈ＄

(c) 119898 times 119890119888 by 119899119863

015

020

025

030

035

040

1 2 3 4 5

001002003004005

Values of Ｈ＄

(d) 119898 times 119901119903119900119901119890119909 by 119899119863

0

10

20

30

40

50

001 002 003 004 005

00005066

Values of

(e) 119899119863 times 119878ℎ by 120572

040

045

050

055

060

1 2 3 4 5

BehaviorNaiveFocused

(f) 119898 times 119890119888 by 119887119890ℎ119886V119894119900119903

015

020

025

030

035

040

0 05 066Behavior

NaiveFocused

(g) 120572 times 119901119903119900119901119890119909 by 119887119890ℎ119886V119894119900119903

05

06

07

08

09

10

1 2 3 4 5 6

00005066

Values of

(h) 119905119863 times 119894119903 by 120572

08700

08725

08750

08775

08800

001 002 003 004 005

BehaviorNaiveFocused

(i) 119899119863 times 119894119903 by 119887119890ℎ119886V119894119900119903

Figure 9 Curves of mean observations for different parameters and metrics for 36000 runs Bars are 095 confidence intervals Plots withthe same observables (ordinate) except for 119894119903 are on the same scale

Complexity 23

The higher the requirement for accepting the proposal isthe higher the proportion of extremists and the variance ofopinions are and the lower the shift statistic is In few words120572 constrains the world to dyadic discussion or throws it intoa process in which deliberation is much more importantthan dyadic interactions This is why for any 120572 = 05 oneeither gives too much weight to deliberated results whichshadows pair-wise interactions or too little weight in thesense that no deliberated result is ever accepted and thusnever integrated into the opinions of the agents that havevoted for it The reason for the latter is that the latitudeacceptance is too low and thereafter pair-wise interactionsare not enough to unevenly polarize the population in such away that deliberated proposal arguments are voted favorablyby a non-negligeable majority Shifts for 120572 = 0 are stable fordifferent levels of 119905119863 which implies that deliberation keeps incheck all the shifts of opinions related to an increase in thenumber of pair-wise interactions For 120572 = 05 deliberationis less likely to be successful and shifts tend towards the shiftlevels for 120572 = 0 (see Figure 9(b))

Concerning labeling-basedmetrics120572 entirely determinesthe coherence statistic For 120572 = 05 there is no difference incoherence because of how coherence is defined it is maximalwhen 120572 = 0 and strangely maximal when 120572 = 066(see Figure 9(h)) Surprisingly 120572 has no effect on judgmentaccuracy One would have expected a higher 120572 to increasejudgment accuracy since agents would hardly collectivelyaccept a proposal argument discussed during deliberationProposal arguments have given how the argument lattice isgenerated a higher probability of being rejected than of beingaccepted in the ideal argumentation framework

435 A Word on Focused and Naive Agents Focused andnaive specifications for agent behavior are an importantparameter in the decision process since they model howagents choose arguments in the deliberation arena In the sce-narios where agents are focused agents show equal or highervariance of opinions and are therefore more extreme (seeFigure 9(g)) independently of the voting requirements Shiftsof opinion are less likely as well We can see this happeningbecause focused agents knock out proposal arguments moreoften than naive agents provided that 119899119863 is low

One could have expected naive agents to yield morecoherent decisions since they argue sincerely However thisseems not to be the case (see Figure 9(i)) Agents are unable todeliberate in away thatmakes deliberation reflect their votingintentions either because they do not have the necessaryarguments to do so or because such arguments do not exist

In terms of judgment accuracy focused agents do betterthan naive agents in finding the ideal labels for the proposalarguments This is likely because when reconstructing theframework focused agents take into account the deliberatedproposal argument and choose themost pertinent argumentsin their sacks that relate to the state of affairs at the deliber-ation arena In fine debates result in a better approximationofL119886119887120576(119868) even though one could have believed the oppositethat is focused agents are ready to sacrifice a correct collectivelabel of a proposal argument in the goal of defending theirpositions

44 Points of Discussion In this subsection we discuss theresults obtained from the simulations We attempt to makesense of them from a social sciences perspective

441 Opinion Consensus and Dissensus The deliberativeopinion dynamics corresponds to the simulated scenarios inwhich the variance of opinions is at its lowest and judgmentaccuracy at its highest Dyadic opinion dynamics are associ-ated with the scenarios in which the variance of opinions is atits highest judgment accuracy at its lowest and coherence atitsmaximumpossible levelThemixed interactions dynamicsis a combination of these aspects for all the metrics exceptfor coherence at which it is the worsemdashdeliberation leadsto some kind of loose opinion consensus that is reinforcedby pair-wise interactions Likewise the mixed discussionmodel confirms some results exposed in [16] expressingthat opinions and voting intentions in a deliberative contextldquooften changerdquo That being said and more particularly indeliberative opinion dynamics outputs align well with twointeresting results on consensus formation given by aninterpretation of Moscovici and Doisersquos [15] work Sunsteinrsquos[14] account on deliberation and group polarization andsocial impact theory [4] The first states that mild consensusis reached in deliberation as procedure disengages agentsthe second is that within discussion groups majorities tendto become larger and opinions more extrememdashnot on thebasis of competing arguments but of the initial balance ofopinions After several decision processes we find opiniondistributions that reflect these observations even though weare not able to precisely coin the set of parameters that yieldssuch distributions (see Figure 7)

Although differences between mixed and dyadic opiniondynamics in terms of the variance of opinions and shiftsare slim the former does much better in terms of judg-ment accuracy In this sense deliberation does moderateopinions but not as much when non-deliberation steps arerelatively frequent throughout the decision-making processIn terms of Moscovici and Doise theory of consensus [15]this makes sense since pair-wise discussion engages or ego-involve agents in the discussion and pushes to convergenceof opinion at the extremes of the opinion distributionHowever what remains unclear is the reason why proceduralparameters that constrain deliberation do not disengageagents from discussion and thus result in a mild consensusas also expressed by Moscovici and Doise [15] A possibleexplanation of this phenomenon may stem from the factthat deliberation in the model is always accompanied by aseries of ego-involving discussions that shadowor counter themoderating effect of deliberation Not only this deliberationcould also be responsible for crystallizing the opinions ofextremists for it favors one or the other part of the distributionof opinions as explained in [14] and to some extent in [13]Furthermore the mixed discussion model gives an exampleof polarization that results from a minority of dissenterson one side of an issue acceding to views of the majorityrsquosside [4] as opposed to the strong polarization observed indyadic interactions Situations in which the status quo orthe undecidability of the proposal argument lingers cannotpossibly yield situations of moderation In cognitive science

24 Complexity

one can be inclined to believe that the longer a deliberationprocess is the more likely individuals will take an immutablestance on the situation In view of the assumption that longdecision-making processes can be more costly for a group orsociety as a whole agents may feel forced to take a stancejust for the sake of ending the process The stance that isadopted would therefore depend on pair-wise interactionsamong individuals and not on deliberation since deliberationtakes too long to be conclusive

442 For Hypothetical Recommendations in Decision-MakingProcess Elaboration On another tune results from themodelsuggest that procedural parameters have marginally decreas-ing positive or negative effects on the metrics of interest Ina hypothetical policy analysis perspective this may indicatethat after choosing a correct number of individuals andimposing the right amount of debates one can attain rea-sonable levels of judgment accuracy coherence and varianceof opinions As accounted in [18] deliberation increasesknowledge (collective knowledge as well) and therefore itheightens the chances of making correct decisions If onebelieves that social welfare is linked to deliberation andeventually to the metrics studied previously then proceduralparameters can both be used as an instrument to attaindesired levels of social welfare and reinforce the notionof legitimacy of collective decisions It can also provide ajustification for choosing a deliberation regime rather thananother on the basis of how important (or urgent) a topic ofdiscussion is Indeed if we adopt the claim that deliberationis a forerunner of welfare then any state in which deliberativecues are at their maximum (highest number of participantsmany deliberation steps before making a decision and asfrequent as possible) has to be mapped to the highest attain-able social welfare In this case is a situation where judgmentaccuracy is at its highest yet the variance of opinions or theproportion of extremists at its lowest an ideal situation Wecannot say for sure since strong opinion consensus and longdecision-making processes in many situations may result ina loss of welfare rather than in a gain of it

Another interesting result in procedure for deliberationis the very similar estimates we find for the effects of theminimum number (119898) of debates necessary to make adecision on a proposal and the proportion (119899119863) of individualsparticipating in deliberation on judgment accuracy In thescope of social welfare this raises the question of whetheradvocating for longer decision processes by demandingmoredeliberation steps (higher 119898) rather than bigger debates(higher 119899119863) may increase welfare Depending on the objec-tives of the decider of the deliberation procedure he or shemay choose to concentrate on one or the other A recom-mendation we can assert from the mixed social interactionsmodel is that if one prefers low to high variance of opinionsandor high to low coherence the decider is better off ifshe focuses on increasing the size of the debate rather thanorganizing debates very frequently One of the reasons forthis might be that when making bigger debates all agents aretruly considered as equals in the decision-making processIn organizing many debates individuals that are chosen toparticipate in many of these will be more influential and will

somehow bias the set of arguments in the deliberation arenaThemajority rulemodeled through120572 also determines the sizeof the shifts and the distribution of opinions Because a bigger120572 is associated with higher levels of extremism and opinionvariance deliberated cues are harder to accept and thereforeit becomesmore difficult for agents to reach concensus andormoderate their opinions

On the other hand and further into the idea of legitimacythe goal of deliberative processes is the agreement amongall participants This agreement stands for the right or bestdecision from a formal or procedural perspective and it is alsothe best from a substantive point of view In other words thelegitimacy of decisions not only derives fromwell-establishedprocedural requirements (respect of the protocol) but alsobegs for the fulfillment of two essential conditions (1) satisfy-ing the procedural requirements for a correct procedure (for-mal legitimacy) and (2) the rational acceptability of the resultsof this procedure (substantive acceptability) Legitimacy canbe then obtained from the coherence statistic scrutinizedin the model insofar as it measures how well agents acceptthe results of a correct procedure (eg deliberation) So fora proposal to be legitimate one has to choose parametersthat would maximize group coherence in decision-makingprocesses

As deliberation may give rise to consensus in the delib-eration arena it may also sow the seeds of dissensus inthe group Our model clearly illustrates this phenomenonthrough the procedural parameters of deliberation and themetrics observed From another perspective in deliberativedemocracy dissensus in collective decision-making may bedesirable According to Landermore and Pagersquos accounts [36]lack of consensus can be perceived as having agents withalternative ideas that combined with other ideas can providea better approximation of the ideal status of an argumentTheauthors in [36] call this dissensus ldquopositive dissensusrdquo andit makes normative sense when solving complex collectivedecision-making problems Our model captures the ideaof having normative requirements to define a good andlegitimate collective decision requirements of high consen-sual deliberative accuracy (judgment accuracy) of legitimacy(coherence) and of diversity (variance of opinions) are alltaken into account and for such we can conclude that ourmodel is successful in describing a sound process of collectivedecision-making with interpretable outcomes

Of course the present simulation results and analysis donot prove that all opinion dynamics can be accounted forby such simple processes like the one presented here Theydo however shed light on the expressiveness of combiningdifferent paradigms to account formore interpretablemodelsand pave the way to creating original models of the sortThe simulations suggest the desirability of discovering theconsequences of relatively simple laws of communication atdifferent levels (micro meso and macro) to determine whatstill needs to be explained

5 Related Work

We see our model as a contribution to the influence andopinion dynamics field in ABM and a pragmatic application

Complexity 25

of abstract argumentation theory To our knowledge we areunaware of existing literature on agent-based modeling thatexplicitly relates collective choice and the notions of deliber-ation and opinion diffusion through abstract argumentationas we have done it

Most models in the literature of opinion diffusion areinterested in opinions because these have an influence incollective decision and in questions of social order Forinstance in [3] the authors are interested in consensus andin how a group collectively chooses among two alternativesIn other models authors are interested in the emergenceof extremism [8] and on the distribution of opinions whenextremists are introduced in the population [5 33 37] whileother authors coin the notion of opinion polarization as anemergent property of the system [8 12] They show usingmodels of ldquobounded confidencerdquo and opinion diffusion withtrust that three different kinds of steady states (unipolarbipolar and central) are possible depending on whetheragents are sufficiently uncertain about their opinions andsufficiently connected andor a certain proportion of indi-viduals are already extreme Recent articles on informationand opinion dynamics stress the importance of governanceand government intervention in the spread of emotions(opinions) in the frangibility of social consensus systemIn [10] the authors show that government intervention inthe spread of negative emotions can lead to an even fasterspread of negative emotions (single-peeked convergence) andin a faster collapse of the social consensus system Opinionas a function of trust is studied in [12] and relates to howindividuals form their opinions on the basis of how muchthey trust agents in their networks

Another stream of discrete opinion dynamics modelsthat explain the emergence of opinion can be found in theliterature Computer multiagent models of attitude changebased on Latanersquos social impact theory are presented in [4] Inthe theory an agent changes her opinion on the basis of theinformational impact she is subject to which depends on thepersuasiveness (strength) supportiveness and immediacy(group structure) of the environment and information shereceives The simulations predict two emergent groups ofphenomena the shifting of attitudes towards incompletelypolarized equilibria and the formation of coherent clusteringof subgroups with deviant attitudes Similarly but in acontinuous more argumentative fashion Mas and Flache[11] present a model of opinion diffusion using argumentsArguments are considered to be for or against a proposal andare given an agent-dependant relevance in dyadic persuasiveinteractions Arguments also determine the agentsrsquo opinionsand their dissimilarities which posits the rules of pair-wiseencounters in theirmodel as agents are assumed to form theiropinions according to the arguments they own They showthat in an argumentative opinion diffusion model the onlyobservable steady states are bipolarization and consensusand that bipolarization can emerge from interactions amongsimilar agents In other words bipolarization of opinionsis possible with homophily and without negative influenceif there is argumentation They subsequently compare theirresults to experimental data In [38] a similar approach toargumentation and opinion formation is taken but for the fact

that they introduce different types of arguments explicitlyapply social judgment theory [22] and use survey data toestablish model-to-real-world comparisons

Closer to opinion formation and abstract argumentationGabbriellini and Torroni [39] are the first to have origi-nally and soundly merged opinion diffusion and abstractargumentation They define an agentrsquos opinion as a functionof the arguments she holds and the attack relation amongthem They devised a focused peer-to-peer dialogue systemof persuasion (NetArg) inspired from Mercier and Sperberrsquosargumentative theory of reasoning [40] which used onlyabstract argumentation to study opinion polarization andopinion dynamics Moreover they considered networks andthe notions of trust and epistemic vigilance to define adynamics for knowledge revision and trust itself whichwe clearly lack in our model They showed that if a con-servative belief operator in argumentation was applied byagents when they reasoned then their dialogue protocoldid not increase polarization among agents That said theyused the model to study the effect of Groenvetterrsquos weaklink theory in the spreading of arguments and were notparticularly interested in notions like judgment accuracy orcoherence nor in tackling questions related to collectivedecision-making procedures Their system is very expressiveand helps position our work in the litterature our work is atthe frontier of pure argumentative opinion diffusion modelsopinion diffusion with arguments and continuous bounded-confidence models of opinion diffusion a la Deffuant

On another note work on collective cognitive conver-gence [41] and opinion sharing [19] shows that consensustowards a certain ldquocorrectrdquo opinion or cognitive state is alwayspossible yet dependent on noise variability and awarenessof agents In [20] the authors show that learning about anexogenous correct state of the world (represented by bits)under confidence was possible but only if the agents werenot too confident In a homogeneous population they showthat the higher the confidence the worse the learningmdashthevery confident agents do not learn the properties of the truestate of the world and disrupt the learning process of the lessconfident ones Collective cognitive convergence can be seenas a result of deliberation in truth-seeking models

When it comes to abstract argumentation theory we takean approach that wires two types of dialogues that are well-studied in the literature persuasion dialogues [42] and delib-eration dialogues [43] Another interesting line of work is theone on mechanism design [44] or the problem of devisingan argumentation protocol where strategic argumentation isnot a liability for success in debates We tackle mechanismdesign in a different way though Instead of consideringstrategy-proofness we are interested in how differences inprotocol can result in epistemologically ldquobetterrdquo collectivechoices and guarantee that opinion distributions are favorablefor deliberation For a survey on persuasion dialogue see[42]

Work on agent-based argumentation usually assumes thatthe semantic relationship between arguments is fixed Inother words if two individuals were to put two argumentsin the public arena such that one logically attacked the otherthen everyone would agree that such attack exists [45 46]

26 Complexity

Other models which do not make this restrictive assumptioncan also be found in the literature [27 47] and derive fromthe class or family of opponent models [27 48] in which twoopposing sides attempt to win a dialogue Ourmodel is in theintersection of these two but the frame combining opiniondiffusion of the kind and argumentation remains new

The idea of mixing interpersonal influence and verticalcommunication however is not original It is described andimplemented in innovation diffusion models such as [5 49]In both vertical communication is modeled as exogenoustransparent informationmdashagents are aware of the existenceof an innovation thus triggering several processes of choiceand stabilization of opinions Also in [50] an Eulerian modelis implemented to show the effect of media and exogenousinformation on opinion distributions With respect to thispoint the originality of our work is in that the informationemitted as vertical communication is endogenous It is issuedfrom a deliberation model that agents shape on the basisof their opinions arguments and behavior In the spirit of[51] where the authors control for the design of verticalcommunication (to whom the vertical communication isaddressed its timing how it influences agents opinions) wecontrol for the process generating the information for ourown set of variables length of debates majority quotas andthe frequency at which discussions take place

6 Conclusion and Perspectives

The main objective of this article was to build a bridgewhere decision-making argumentation and opinion dif-fusion could come together We proposed a model thatcombined abstract argumentation theory and a boundedconfidence opinion diffusion model and showed to whatextent it could explain variance of opinions extremismcoherence in collective decisions and judgment accuracyon arguments given an ideal state of full information Thesecond objective was to show in what way governance pro-viding agents with arenas of discussion and deliberation wasimportant in addressing the success of collective decision-making processes and the quality of their outcomes

The model revealed that allowing for more deliberationtime though in lower frequency and allowing for widerparticipation in deliberation increased the variance of opin-ions and the proportion of extremists in a group Theseobservations are consistent with the combination of resultsfound in [9] and in [13 14] which stress that deliberationmay polarize groups and may have a meager effect on theshifts of opinions and inconsistent with [16] where it isargued that opinions tend to moderate after deliberationIt happens that deliberation alone did moderate opinionyet when integrated into a more complex system in whichindividuals were allowed to interact with one another andnot all that was deliberated was accepted its influence wasshadowed by other more individualistic dynamics

Undeniably the grounded semantics played an importantrole in the weak effect of deliberation since it models askeptical way of reasoning over arguments For agentsaccepting arguments thus updating their opinions accord-ingly happened not too often Nevertheless asking for more

deliberation did increase judgment accuracy as observed in[18] yet in a marginally decreasing fashion We showed thatvoting within the deliberation protocol not only increasedthe proportion of extremists and the variance of opinions ina group but also determined how coherent deliberation andvoting procedures were with each other Lastly we showedthat agents that are focused judged arguments better hadmore stable opinions and constituted a group with higherproportion of extremists than their naive counterparts

In terms of governance the model says that there is per-haps no trade-off between extremismand judgment accuracyInstead it asserts that the higher the variance of opinions thecloser the group gets to the correct decisions This situationmay be interpreted as Landermore and Pagersquos notion of ldquoposi-tive dissensusrdquo (in complex collective decision-making tasks)[36] as opposed to consensus as a normative requirementfor correct collective decision-making If a decider has toorganize deliberation to legitimize a public policy dependingon what kind of world he or she wants different set ofparameters may be chosen to account for different levels oflegitimacy and correctness For instance one may ask for adecision to be legitimate to respect a certain level of coherenceor of judgment accuracy and to be discussed by a sundryof different-opinionated peoples If so deliberation has tobe made more often more participants have to be includedin debates or in deliberation instances and not too manytime steps should be introduced in between two deliberationinstances So the model may capture normative ideas todefine correct and legitimate collective decisions and tomake recommendations on those grounds Requirements ofhigh consensual deliberative accuracy (judgment accuracy)legitimacy (coherence) and diversity (variance of opinions)are all taken into account and for such a reason the model israther successful in describing a sound process of collectivedecision-making with interpretable outcomes

61 Extensions of the Model and Other Ideas Our model canbe extended in many ways Value-based abstract argumen-tation frameworks (VAFs) for example as an extension ofDungrsquos argumentation framework provide a formal descrip-tion of the process of decision-making in which argumentsare given values and audiences (set of agents) ignore attacksbetween arguments on the basis of their preferences over val-ues Social abstract argumentation frameworks (SAFs) alsoprovide an interesting and seemingly convenient formalismfor our model In SAFs agents vote on the acceptability ofarguments and the resulting labeling is a combination of thearguments that are voted for and labeling-based semantics Inour model we apply the same rule yet only for one argumentand given precise agent-based votingmechanisms and votingrules which are eventually parametrized

To consider our model as a specification of a mixed VAFand SAF model may be of great interest for future workThis formalism can provide solid foundations to describeexplain interpret and eventually extend the results foundhere from an argument-based perspective On those groundsfuture work will focus on creating a hybrid argumentationframeworkmdashwith its semanticsmdashthat takes heed of bothvalues or principles and voting

Complexity 27

Concerning other possible extensions of the model anatural reflex would be to argue on the validity and therobustness of the mix between the bounded confidenceopinion diffusion models and deliberation Consequentiallyadding deliberation to different bounded confidence modelsof opinion diffusion such as the Hegelsmann-Krause model[37] or to models of social impact theory like in [4] canbe of great interest for future work We acknowledge that afiner understanding of argumentation frameworks randomlattices and their respective applications for this modelare necessary Testing for different labeling-based semanticsor even trading labeling-based semantics for scoring-basedsemantics (semantics that defines acceptability on scoresgiven to arguments) such as debate semantics and ranking-based semantics (semantics that yield a preorder) may beinteresting to consider in future work

More into modeling deliberation it is not inadequateto believe that agents may also advance proposals duringdeliberation and replace to some extent the proposal forwhich the deliberation is taking place Endogenizing theprocess that produces the type of argument to be deliberatedon may be interesting to explore either by having agentsstrategically replace proposals at certain moments duringthe debate or having the central authority attempt to chooseproposals such that the probability of having them acceptedis high Agents may also be thought to be autoorganizing(independent of a central authority) and concede on the sizeof the instances of deliberation To consider differences insemantics across agents or even endogenizing the semanticson how urgent or important the proposal that is beingdiscussed is can also be an interesting direction to take

Last but not least many nontrivial modifications of ourmodel are possible and most should include better-thoughtdeliberation protocols It may be interesting to design andobserve protocols in which deliberation only affects agentsthat are actually debating for example Trust network effectsmultidimensionality of opinions new processes of argumentexchange or learning among agents are notions to furtherdevelop in order to make the model more realistic and relaxsome unreliable assumptions In sum the model is to berefined and extended with the objective of either studyingconcrete cases of deliberative polling and opinion dynamicsor implementing more intuitive thought experiments

Appendix

We present below some of the pseudocode we used toimplement the model

Algorithms

(a) Algorithm 1 implements the dyadic influence in themodel It describes how two agents interact andupdate their opinions

(b) Algorithm 2 implements the deliberative influence inthe model which describes how agents update theiropinions in sight of the accepted deliberated proposalarguments

(c) Algorithm 3 is used to obtain the grounded labelingof an argumentation framework

(d) Algorithm 4 provides an overview of a run in thesimulations119872 is the number of proposal argumentsto discuss which we set to 100 in the model

Each object in the system is indexed by a119908ℎ119900 number Hencea set of agents is annotated by a set of natural numbers Weuse this fact in the implementation of the algorithm 119874119887119895119890119888119905119894denotes the object of 119908ℎ119900 number 119894 Agentsrsquo opinions (119900) arecalibrated so that 119900 isin [minus1 1]Data Availability

The program and data used to support the findings of thisstudy are available from the corresponding author uponrequest

Disclosure

A preliminary version of this work was presented anddiscussed at the conference Bridging the Gap between FormalArgumentation and Actual Human Reasoning on October 4-5 2018 at BochumGermanyThe authors thank the audiencefor their valuable comments and suggestions

Conflicts of Interest

The authors declare that they have no conflicts of interestregarding the publication of this article

Acknowledgments

The contribution of Gabriella Pigozzi was supported by theDeutsche Forschungsgemeinschaft (DFG) and the CzechScience Foundation (GACR) as part of the joint project FromShared Evidence to Group Attitudes (RO 45486-1)

Endnotes

1 In other words the superiority of the ldquoright answerrdquo orlabeling in our case will appear as such to all

2 All agents hold the same opinion3 By sensitivity we mean how much an agent is ready to

change her opinion on the sole fact that an informationalcue has been debated on

4 The assumption is strong but this trust among agentsmay be rooted from a genuine individual motivation toreach a democratic consensus

5 We adopted a rule of thumbs that says that if an agentis neutral with respect to the principle then she willvote for any proposal that is presented to her duringdeliberation

6 If the argument is not unique she chooses one atrandom

7 If there are more than one candidate the agent will favorthe argument at minimal distance from 119868

28 Complexity

Require Vector of values for parameters (119881 = (120583 119880 119879))Ensure Successful interactions among agents119896 larr997888 0119873119890 larr997888 0

while 119896 le |119873|2 do ⊳Fill119873119890 with |119873|2 agents119896 larr997888 119896 + 1119873119890 larr997888 119877119886119899119889119900119898(|119873| + 1) ⊳Pseudo-random generator Generates integer between 0 and |119873|119873119903 larr997888 0 120575119894119895 larr997888 0 ⊳Variable to store differences in opinionfor all 119894 isin 119873119890 doRandomly pick a number 119895 isin 119873(119873119890 cup 119873119903)119873119903 larr997888 119873119903 cup 119895 120575119894119895 larr997888 |119900119894 minus 119900119895|procedure dyadic-update(119894 119895) ⊳Two agents 119894 and 119895 discuss and update opinions1199001015840 larr997888 119900119894 (120583119895 119880119895 119879119895) larr997888 (120583119894 119880119894 119879119894) larr997888 (120583 119880 119879) ⊳Save agent irsquos opinion before updating and homogenize agents

if 120575119894119895 lt 119880119894 then119900119894 larr997888 120583119894119900119895 + (1 minus 120583119894)119900119894if 120575119894119895 gt 119879119894 then119900119894 larr997888 (1 + 120583119894)119900119894 minus 120583119894119900119895if 120575119894119895 lt 119880119895 then119900119895 larr997888 1205831198951199001015840 + (1 minus 120583119895)119900119895if 120575119894119895 gt 119879119895 then119900119895 larr997888 (1 + 120583119895)119900119895 minus 1205831198951199001015840

Algorithm 1 Dyadic social influence

Require Vector of values for parameters (119881 = (120574 119901119886 119901119903))Ensure Successful interaction between proposal arguments and agents

for all 119894 isin 119873 doprocedure Probability-Update(V119886 119901119886 119901119903) ⊳Update individual probabilities of change due to deliberated cues119909119894 larr997888 119901|V119886minus119900119894 |2119886 119910119894 larr997888 1199012|V119886minus119900119894 |119903119902 larr997888 0if L119886119887(119868) = 119894119899 then

procedure deliberative-update(119894 119886) ⊳Agent i argument 119886120574119894 larr997888 120574 119902 larr997888 119906 sim U(0 1) ⊳ 119906 is a realization of a uniform random variableif 119902 lt 119909119894 then119900119894 larr997888 120574119894119900119894 + (1 minus 120574119894)V119886break

if 119902 lt 119910119894(1 minus 119909119894) then119900119894 larr997888 (1 + 120574119894)119900119894 minus 120574119894V119886Algorithm 2 Deliberation influence

8 In terms of computational complexity naive agentsmaketheirmoves in119874(|A119894| log(|A119894|)) while focused agents doat least as bad

9 Tables have memory Therefore debates are not neces-sarily trees

10 If there are less than (119899119863 times |119873|)2 agents for any of bothgroups the CA summons all of them to the deliberationtable

11 Or careful consideration of the viewpoints of otherswhich implies that citizens keep an open mind and donot reject arguments outright

12 A ldquohealthyrdquo consensus refers to the case in which manyconsensual decisions are taken but opinion-wise agentsdo not agree with each other Quotes are used on theword healthy because some political science theorists

(eg [36]) believe that in certain situations agreement inopinions for deliberation is epistemologically damaging

13 Simulations on groups of |119873| = 800 and |119873| = 400agents shows that scaling the population has an effect onall themetrics but it is confounding depending onwhichcouples of attitudes structures are considered (119880 119879) Westick to |119873| = 400 as in [9] for comparability andsimulation running time

14 By balanced wemean with as many arguments with V119886 lt0 as with V119886 gt 0

15 Experiments on the system suggest that a bigger(smaller) argument pool 119872 = 1000 (119872 = 200) isassociated with a higher (lower) variance of opinions(and extremism) and higher (lower) judgment accuracyFor the rest of the observations 119872 seem to have noeffect We choose a middle value for 119872 Similarly theargument pool chosen to be balanced or uniformly

Complexity 29

Require Argumentation framework 119860119865 = (AR)Ensure Assignment of labels to arguments from the grounded labeling of 119860119865L(A)

L0(A) larr997888 (0 0 0)119896 larr997888 0do119894119899 L119896+1(A) larr997888 119894119899 L119896(A) cup 119886 | 119886 is not labeled 119894119899 L119896(A) 119886119899119889 forall119887 isin A 119904119905 119887 R 119886 then 119887 isin 119900119906119905 L119896(A)119900119906119905 L119896+1(A) larr997888 119900119906119905 L119896(A) cup 119886 | 119886 is not labeled 119894119899 L119896(A) 119886119899119889 exist119887 isin A 119904119905 119887 R 119886 and 119887 isin 119894119899 L119896+1(A)

while L119896+1(A) = L119896(A) = 0Algorithm 3 Solve argumentation frameworks (with grounded semantics)

Require Vector of values for parameters (119881)Ensure Vector of statistics (119880)

procedure Set-parameters(119881) ⊳Initialize parameters for scenarioprocedure Init-objs(119899119898 119896 ) ⊳Create119898 arguments 119899 agents and 119860119865120576procedure Solve-Argumentation-Framework(119860119865) ⊳Assigns the epistemic labelingL120576 toA119894 larr997888 1119905 larr997888 1while 119894 le 119872 do ⊳ 119872 the number of arguments to discuss119868119894 larr997888 119866119890119899 119886119903119892(1) ⊳Generates proposal argument 119868119860 119894119905 larr997888 (119868119894 0)L119886119887(119868119894) larr997888 119906119899119889 ⊳Initialize label for 119868119895 larr997888 1 ⊳Initialize debate counter 119895while ((L119886119887(119868) = 119906119899119889 or 119898 le 119895) and 119895 le 119898) doif 119905 equiv 0(119898119900119889 119905119863) then

procedure build-argumentation-framework(119899119863 119887119890ℎ119886V119894119900119903) ⊳Build the argument graph agents deliberate(119861119894119905 119878119894119905) larr997888 (119883 119884) sub (⋃119897isin119873119899119863 A119897119894119905 ⋃119897isin119873119899119863 R119897119894119905) ⊳ 119873119899119863 sub 119873 are agents sampled by the CA119860 119894119905 larr997888 119889(119860 119894119905 (119861119894119905 119878119894119905))procedure Solve-Argumentation-Framework(A119889) ⊳FindL(A119889)L119886119887(119868119894) larr997888 L119886119887119863(119868119894)119895 larr997888 119895 + 1

else119896 larr997888 1while 119896 le 119905119863 doprocedure dyadic-social-influence(120583 119879 119880) ⊳Agents discuss one-to-one119896 larr997888 119896 + 1119905 larr997888 119905 + 1

if 120572 gt 0 andL119886119887(119868119894) = 119900119906119905 then ⊳Agents vote for the proposalif (|119897 isin 119873|119900119897 times V119868119894 ge 0| ge 120572 times 119899) then

L119886119887(119868119894) larr997888 119894119899else

L119886119887(119868119894) larr997888 119900119906119905if L119886119887(119868119894) = 119894119899 thenprocedure Deliberation-influence(120574 119901119886 119901119903) ⊳Agents are influenced by the result of the decision process119894 larr997888 119894 + 1119905 larr997888 119905 + 1119880 larr997888 119862119900119898119901119906119905119890 119878119905119886119905119894119904119905119894119888119904(119881)

return 119880 ⊳Get the statisticsAlgorithm 4 Run for one scenario

generated does not seem to play a significant explanatoryrole on any of the metrics

16 Experiments show that themaximumnumber of debatesis not an important parameter even if it is an intuitiveone when considering decision-making processes Inpractice debates end after at most 119898 + 1 deliberation

steps and even then it happens rarely However to avoidunpleasant surprises like infinite debates on a completeargument graph containing the proposal argument weadded the constraint 119898 Its value is precisely 7 becausethe biggest minimum number of debates we studyis 6

30 Complexity

17 To ensure comparability with the deliberative andmixedopinion dynamics models on judgment accuracy andcoherence agents vote on proposal arguments every 119905119863times(119898+1) 119905119863 and119898does not explain variability in judgmentaccuracy nor in coherence

18 Student t-tests for independent samples are robust tothe violation of two hypothesis on the distribution ofthe samples normality and homogeneity of variancesas long as the sample size is big enough (central limittheorem) and if the difference in size of the comparedsamples is smallrespectively

19 Studentrsquos test version forwhen variances are not assumedequal and the difference in size of the compared samplesis big

20 We do not correct for homoskedasticity nor normality oferrors in the estimates

21 Balanced as the same number of runs per scenario orsuch that no significant correlations are found betweenparameters

References

[1] R Axelrod ldquoThe dissemination of culture a model withlocal convergence and global polarizationrdquo Journal of ConflictResolution vol 41 no 2 pp 203ndash226 1997

[2] D M Rousseau Reinforcing the MicroMacro Bridge Organiza-tionalThinking and Pluralistic Vehicles SAGEPublications SageLos Angeles Calif USA 2011

[3] S Galam and SMoscovici ldquoTowards a theory of collective phe-nomena consensus and attitude changes in groupsrdquo EuropeanJournal of Social Psychology vol 21 no 1 pp 49ndash74 1991

[4] A Nowak J Szamrej and B Latane ldquoFrom private attitude topublic opinion a dynamic theory of social impactrdquo Psychologi-cal Review vol 97 no 3 pp 362ndash376 1990

[5] G Deffuant S Huet and F Amblard ldquoAn individual-basedmodel of innovation diffusion mixing social value and individ-ual benefitrdquo American Journal of Sociology vol 110 no 4 pp1041ndash1069 2005

[6] G Deffuant D Neau F Amblard and G Weisbuch ldquoMixingbeliefs among interacting agentsrdquoAdvances in Complex Systems(ACS) vol 3 no 01n04 pp 87ndash98 2000

[7] J Rouchier P Tubaro and C Emery ldquoOpinion transmission inorganizations an agent-based modeling approachrdquo Computa-tional and Mathematical OrganizationTheory vol 20 no 3 pp252ndash277 2014

[8] M Meadows and D Cliff ldquoThe relative disagreement model ofopinion dynamics where do extremists come fromrdquo in Pro-ceedings of the International Workshop on Self-Organizing Sys-tems Lecture Notes in Computer Science pp 66ndash77 Springer2013

[9] W Jager and F Amblard ldquoUniformity bipolarization and plu-riformity captured as generic stylized behavior with an agent-based simulation model of attitude changerdquo Computational ampMathematical Organization Theory vol 10 no 4 pp 295ndash3032005

[10] Y-F Zhang H-Y Duan and Z-L Geng ldquoEvolutionary mech-anism of frangibility in social consensus system based onnegative emotions spreadrdquo Complexity vol 2017 Article ID4037049 8 pages 2017

[11] A Flache andMWMacy ldquoSmall worlds and cultural polariza-tionrdquoThe Journal of Mathematical Sociology vol 35 no 1-3 pp146ndash176 2011

[12] A Tsang and K Larson ldquoOpinion dynamics of skepticalagentsrdquo in Proceedings of the 13th International Conference onAutonomous Agents and Multiagent Systems AAMAS 2014 pp277ndash284 International Foundation forAutonomousAgents andMultiagent Systems France May 2014

[13] K M Hansen Deliberative democracy and opinion formationUniversity Press of Southern Denmark Odense 2004

[14] C R Sunstein ldquoThe law of group polarizationrdquo Journal ofPolitical Philosophy vol 10 no 2 pp 175ndash195 2002 WileyOnline Library

[15] S Moscovici and W Doise Dissensions et Consensus unetheorie generale des decisions collectives Presses Universitairesde France-PUF 1992

[16] J S Fishkin and R C Luskin ldquoExperimenting with a demo-cratic ideal deliberative polling and public opinionrdquo ActaPolitica vol 40 no 3 pp 284ndash298 2005

[17] R C Luskin J S Fishkin and S Iyengar Considered Opinionson US Foreign Policy Face-To-Face versus Online Delibera-tive Polling International Communication Association NewOrleans La USA 2004

[18] J Barabas ldquoHowdeliberation affects policy opinionsrdquoAmericanPolitical Science Review vol 98 no 4 pp 687ndash701 2004

[19] O Pryymak A Rogers and N R Jennings ldquoEfficient opinionsharing in large decentralised teamsrdquo in Proceedings of the 11thInternational Conference on Autonomous Agents andMultiagentSystems vol 1 pp 543ndash550 Spain June 2012

[20] J Rouchier and E Tanimura ldquoWhen overconfident agents slowdown collective learningrdquo Simulation vol 88 no 1 pp 33ndash492012

[21] D Miller ldquoDeliberative democracy and social choicerdquo PoliticalStudies vol 40 S1 pp 54ndash67 1992

[22] M Sherif and C I Hovland Social Judgment Assimilation andContrast Effects in Communication and Attitude Change YaleUniversity Press 1961

[23] P M Dung ldquoOn the acceptability of arguments and its funda-mental role in nonmonotonic reasoning logic programmingand n-person gamesrdquo Artificial Intelligence vol 77 no 2 pp321ndash357 1995

[24] M Caminada and G Pigozzi ldquoOn judgment aggregation inabstract argumentationrdquo Autonomous Agents and Multi-AgentSystems vol 22 no 1 pp 64ndash102 2011

[25] P Baroni M Caminada andM Giacomin ldquoAn introduction toargumentation semanticsrdquoThe Knowledge Engineering Reviewvol 26 no 4 pp 365ndash410 2011

[26] S Chambers ldquoDeliberative democratic theoryrdquo Annual Reviewof Political Science vol 6 no 1 pp 307ndash326 2003

[27] E Bonzon andNMaudet ldquoOn the outcomes of multiparty per-suasionrdquo in Proceedings of the 10th International Conference onAutonomous Agents and Multiagent Systems 2011 AAMAS 2011vol 1 pp 47ndash54 International Foundation for AutonomousAgents and Multiagent Systems Taiwan May 2011

[28] J Ranciere Disagreement Politics and Philosophy University ofMinnesota Press 1999

[29] T Bench-Capon and K Atkinson ldquoAbstract argumentation andvaluesrdquo in Argumentation in Artificial Intelligence pp 45ndash64Springer 2009

[30] M Caminada ldquoOn the issue of reinstatement in argumenta-tionrdquo in Proceedings of the European Workshop on Logics in

Complexity 31

Artificial Intelligence vol 4160 of Lecture Notes in ComputerScience pp 111ndash123 Springer Berlin Heidelberg

[31] GAVreeswijk ldquoAn algorithm to computeminimally groundedand admissible defence sets in argument systemsrdquo in Pro-ceedings of the 2006 conference on Computational Models ofArgument Proceedings of COMMA 2006 vol 144 pp 109ndash120IOS Press Amsterdam The Netherlands 2006

[32] C W Sherif M Kelly H L Rodgers et al ldquoPersonal involve-ment social judgment and actionrdquo Journal of Personality andSocial Psychology vol 27 no 3 pp 311ndash328 1973

[33] G Deffuant F Amblard G Weisbuch and T Faure ldquoHow canextremism prevail A study based on the relative agreementinteraction modelrdquo Journal of Artificial Societies and SocialSimulation vol 5 no 4 p 1 2002

[34] E Awad M W Caminada G Pigozzi M Podlaszewski andI Rahwan ldquoPareto optimality and strategy-proofness in groupargument evaluationrdquo Journal of Logic and Computation vol 27no 8 pp 2581ndash2609 2017

[35] D Grossi and G Pigozzi ldquoJudgment aggregation a primerrdquoSynthesis Lectures on Artificial Intelligence and Machine Learn-ing vol 8 no 2 pp 1ndash151 2014

[36] H Landemore and S E Page ldquoDeliberation and disagreementproblem solving prediction and positive dissensusrdquo PoliticsPhilosophy and Economics vol 14 no 3 pp 229ndash254 2015

[37] R Hegselmann and U Krause ldquoOpinion dynamics andbounded confidence models analysis and simulationrdquo Journalof Artificial Societies and Social Simulation vol 5 no 3 2002

[38] A Stefanelli and R Seidl ldquoModerate and polarized opinionsUsing empirical data for an agent-based simulationrdquo inProceed-ings of the Social Simulation Conference 2014

[39] S Gabbriellini and P Torroni ldquoA new framework for ABMsbased on argumentative reasoningrdquo in Advances in SocialSimulation vol 229 pp 25ndash36 Springer Berlin Germany 2014

[40] H Mercier and D Sperber ldquoWhy do humans reason Argu-ments for an argumentative theoryrdquo Behavioral and BrainSciences vol 34 no 2 pp 57ndash74 2011

[41] H V Parunak T C Belding R Hilscher and S BruecknerldquoModeling and managing collective cognitive convergencerdquoin In Proceedings of the 7th iNternational Joint Conference onAutonomous Agents and Multiagent Systems vol 3 pp 1505ndash1508 International Foundation for Autonomous Agents andMultiagent Systems 2008

[42] H Prakken ldquoFormal systems for persuasion dialoguerdquo TheKnowledge Engineering Review vol 21 no 2 pp 163ndash188 2006

[43] K Atkinson T Bench-Capon and P Mcburney ldquoA dialoguegame protocol for multi-agent argument over proposals foractionrdquoAutonomousAgents andMulti-Agent Systems vol 11 no2 pp 153ndash171 2005

[44] M Thimm and A J Garcıa ldquoClassification and strategicalissues of argumentation games on structured argumentationframeworksrdquo in Proceedings of the 9th International Conferenceon Autonomous Agents and Multiagent Systems vol 1 pp 1247ndash1254 International Foundation for Autonomous Agents andMultiagent Systems Canada 2010

[45] I Rahwan and K Larson ldquoMechanism design for abstractargumentationrdquo in Proceedings of the 7th International JointConference on Autonomous Agents and Multiagent SystemsAAMAS 2008 pp 1031ndash1038 International Foundation forAutonomous Agents and Multiagent Systems Portugal May2008

[46] H Prakken R Riveret A Rotolo and G Sartor ldquoHeuristics inArgumentation A Game-Theoretical Investigationrdquo EuropeanUniversity Institute 2008

[47] N Hadidi Y Dimopoulos and P Moraitis ldquoTactics and con-cessions for argumentation-based negotiationrdquo in Proceedingsof the COMMA vol 245 pp 285ndash296 2012

[48] N Oren and T J Norman ldquoArguing using opponent modelsrdquoin Proceedings of the International Workshop on Argumentationin Multi-Agent Systems pp 160ndash174 Springer 2009

[49] S Delre W Jager T Bijmolt and M Janssen ldquoTargeting andtiming promotional activities An agent-based model for thetakeoff of new productsrdquo Journal of Business Research vol 60no 8 pp 826ndash835 2007

[50] A Mirtabatabaei P Jia and F Bullo ldquoEulerian opinion dynam-ics with bounded confidence and exogenous inputsrdquo SIAMJournal on Applied Dynamical Systems vol 13 no 1 pp 425ndash446 2014

[51] V Barbet N Guiraud V Laperriere and J Rouchier Hagglingon Values Towards Consensus or Trouble 2019

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20 Complexity

Table 6 Summary of sign or relative marginal effects for all varying parameters on dyadic deliberative and mixed interactions models inand for all experiments

Metrics Parameters119905119863 119898 119899119863 120572 119879 119880 119901119886 120574 119896 119861119890ℎ119886V119894119900119903119881119886119903(119900) XX X X O X X X X X X119901119903119900119901119890119909 XX X X O X X X X X X119878ℎ XX X X O X X X X X119890119888 XX X X XO X XOO XOO X X119894119903 XX X X O XO O X OOX XOO X XNota gray symbols correspond to dyadic dynamics light gray to deliberative dynamics and bold black to mixed dynamics Given a scenario the symbol 119883stands for constant parameter and 119874 for insignificant marginal effect and as parameters grow for positive (negative) marginal effect for the parameterFrom left to right at the parameter level the leftmost arrow corresponds to the scenario with the strongest (in absolute value) marginal effect of the parameter(with respect to to its factors) Example Let 119901 be a parameter and 119874 stands for ldquo119901 has a significant positive effect in dyadic discussion scenario negativein the deliberative one and insignificant in mixed discussion The positive effect in dyadic discussion is greater than the absolute value of the negative one inthe purely deliberative scenariordquo

424 Comments on the Differences between the Three Modelsfor OLS Regression Estimates The significance of most of theregression parameters estimated for the mixed interactionsscenarios are similar to those estimated for the pure dyadicanddeliberative scenariosNevertheless themagnitude of theeffects is different insomuch that the dynamics themselvesare different So instead of commenting the linear regressionestimates of the mixed model we compare when possiblethe standardized significant coefficients of the regressors ofthe mixed model with those of the pair-wise and deliberativeinteractionmodels (see Table 6 for a summary of the results)

In mixed interactions for instance the influence of119898 onthe opinion distribution is much more preponderant than inthe deliberative dynamics scenarios Going from 119898 = 1 to119898 = 3 increases the standard variation units of the variance ofopinions by twice as much as the one in the deliberative sce-nariosThis in particular is due to the design of the decision-making process If 119898 is higher then the number of socialinfluence steps (119905119863) in the decision process is also higher(compared to 119905119863 = 0 in the deliberative case) If we decideto consider these steps as contributing to the variance ofopinions then the effect of119898 on the variance of opinions hasto be stronger in the mixed interactions model In contrastthe number of agents participating in the debate has a weakereffect on the variance of opinions in the mixed interactionsmodel than in the deliberative dynamics specially for 119899119863 =005 (more than twice as much) The difference may comefrom the fact that in mixed discussions the part of theargument pool used in deliberation is more restrictive thanin the deliberative case We can argue that since pair-wisedynamics may get individuals closer to one another aftersome deliberation the same happens to the arguments thatindividuals will most likely use in deliberation Taking thisinto account means that deliberation will at many instancesbe unfruitful and have no effect on opinions thus failing tomoderate extreme opinions andor polarize the group In thesame direction a high number of participants in deliberationcan make the deliberation dynamics stiff In this case thedistribution of opinion is close to the uniform distributionwhich has a relatively high-to-moderate variancewith respectto the reference results

The strength of the deliberative dynamics has a positivestronger effect on the variance of opinions in themixedmodelthan in the deliberative model when 120574 = 02 and a negativeweaker one when 120574 = 01 A sound interpretation of thisresult is that when deliberation is successful and its effect onopinions does not get agents to move sufficiently away fromtheir current positions agents are pulled or pushed back totheir previously held opinions the variance of opinions doesnot changemuchOn the other hand if the effect of successfuldeliberation is strong opinions can change sufficiently tomake agents much more prone to assimilation into moderateopinion foci the variance of opinions falls

The deliberative opinion dynamics model is much moresensitive to 120572 than the mixed discussion model Differ-ences in marginal effects can double The probability ofattraction to deliberated results (119901119886) for instance has astrongmoderating effect (negative) on agentsrsquo opinions in thedeliberative opinion dynamics but a meager one (two-fold)in the mixed and dyadic dynamic models This is becauseopinion updates in the deliberative model depend only onthat parameter whereas in the mixed model more numeroustypes of interactions make opinion updates possible

In terms of judgment accuracy all parameters except thesize of the argument sacks (119896) have a significant slightlystronger effect on the deliberative dynamics than in themixedmodel In effect the difference may be attributed to theupshot of pair-wise interactions on the distribution of opin-ions and on the arguments that would be advanced duringdeliberation The result respecting parameter 119896 however iscounterintuitive Since argument sacks are static scenarioswhere shifts are higher should make 119896 more important fordeliberation to succeed and it is precisely in the scenarios ofpure deliberative dynamics that we find the biggest opinionshifts This may point to the hypothesis that in deliberativeinteractions agents change opinions substantially but do notoften change their adherence to the principle (go from anegative opinion to a positive one or vice versa) when thereis no group polarization

43 Sensitivity Analysis for the Procedural Parameters ofInterest Weperform a sensitivity analysis of the observationson the parameters of interest in themixed interactionsmodel

Complexity 21

We extend the parameter domains of the pure dyadic anddeliberative scenarios as we make it explicit in Table 2

431 Minimum Number of Debates (119898) We observe that theminimum number of debates has a significant well-observedeffect on all of our metrics excluding group coherence Forthe variance of opinions we can observe that the moredebates there are the bigger the value of the metric isalthough the higher 119899119863 and 119905119863 are the weaker the overalleffect of119898 becomes (Figures 9(d) and 9(a)) Furthermore themarginal effect of increasing minimal debates is decreasingan additional deliberation step requirement increases thevariance of opinions but less and less as it grows 119898 coupledwith 119899119863 has a shy S-shaped effect (see Figure 9(d)) on thevariance of opinions meaning that it has a stronger effectwhen 119899119863 is low which suggests a possible trade-off betweenthe number of arguments accepted in the debate and thenumber of debates required before accepting a proposal

In contrast the shifts in opinion are less and less likelyas 119898 grows and this is independent of the variations of theother parameters Again the effect is marginally decreasingand is only truly significant when 120572 = 05 An explanationto these effects may be linked to the design of the systemFirst the variance of opinions is higher when deliberation isasked for because the more deliberation steps there are thehigher the chances are that the central proposal argumentis deemed unacceptable specially when agents are focusedFurthermoremechanically speaking increasing theminimalamount of debates implies that whenever a decision is to betaken at least 119898 times 119905119863 time steps have to take place and if atany moment a proposal argument is considered undecided119905119863 time steps are added to the process So unless the debatedoes not yield 119906119899119889 labels for proposal arguments (highlyunlikely considering that 120590 = grounded semantics) the morenondeliberation steps there are in the decision process thehigher the variance of opinions is Concerning the shiftswhen 120572 = 05 either the system is too stiff to accept anyproposal argument and opinions do not changemuch or theeffects of pair-wise discussion and deliberation cancel out insuch a way that individual shifts are minimal (Figure 9(b))

On the side of labeling-based metrics the more debatesare asked for the more accurate a group is in its judgmentthe effect being smaller as 119898 grows When agents are naivethe effect is quasilinear while when they are focused thestrongest effects of acquiring more deliberation are foundwhen levels of deliberation are already low (Figure 9(f))Thiscan be explained by the fact that the more debates thereare in the decision process the closer one gets to the idealargumentation framework

432 Proportion of the Population in Deliberation Steps (119899119863)Like for 119898 119899119863 has a significant effect on the proportionof extremists and on the shifts and variance of opinions(Figure 9(e)) This may result from the fact that being ableto put more arguments in play at the same debate step mayincrease the odds of revealing cycles around the proposalargument in119860119865120576 Given that we use grounded semantics thearguments in the cycles are labeled 119906119899119889 and in consequence

debates are more often postponed when 119899119863 is higher Post-poning debates in turn increases the number of nondebatessteps in the decision process which increases the varianceof opinions and limits over time the moderating effect ofdeliberation Moreover the effect of this parameter is verydependent on the value of 120572 (Figure 9(e)) For shifts forinstance 120572 = 05 makes the effect of 119899119863 negative while120572 = 0 makes it positive but to a lesser extent Furthermorefor higher requirements of deliberation (119898) adding moreindividuals to the deliberation process has weaker effectson the variance of opinions and on the other metrics It isalso quite interesting to notice that it has the same effect onagents independently of agent behavior during deliberationThis is a surprising result as one would have expected morefocused agents in a deliberation arena to cause a raise inthe proportion of extremists as they play to knock outopposing proposal arguments and thereof reducing the effectof deliberation (lowers variance) on the whole population

Finally similar to119898 addingmore people into the deliber-ation process increases judgment accuracy (see Figure 9(c))and has no clear effect if any on group coherence (seeFigure 9(i))

433 Steps between Deliberation Steps (119905119863) In all configu-rations 119905119863 increases the proportion of extremists (variance)and decreases the shifts of opinions among agents Theshifts and the effect on the variance of opinions are presentfor 120572 = 05 insignificant otherwise (Figure 9(b)) 119905119863 ishighly linked to 119898 by construction When 119898 = 1 thecurve linking the variance of opinions and 119905119863 is convex As119898 increases the curve becomes more and more concavewhich means that 119905119863 has a more important effect on theopinion distribution as collective decisions take longer to beachieved This seems counterintuitive but in reality it reflectsthemultiplicative relationship between deliberation and pair-wise interactions and the semantics in the model If 119898 is lowand 119905119863 is high the effective number of pair-wise interactionsis on average smaller in the deliberation process Hencethe structure of the decision-making process hinders anyincrease in the variance of opinions If119898 is high the oppositeeffect is observed Additionally since the grounded semanticsyields few 119894119899 arguments with respect to other admissibility-based semantics getting closer to the ideal argumentationframeworkmaymake it difficult to obtain the expected effectsof deliberative discussion on agents Lower 119898 constrains thevariance of opinions by making deliberation more influentialon opinions

In terms of judgment accuracy and coherence we observethat 119905119863 does not explain coherence yet it tends to decreasejudgment inaccuracy This may be due to a more variedargument pool in the debate which in turn may be aconsequence of a higher variance of opinions or vice versa

434 Acceptability Voting Quota (120572) By far it is the mostinfluential parameter in our study It changes the directionand the intensity of the effect of all the other proceduralparameters and by construction heavily constrains the roadto accepting a proposal (see Figure 9)

22 Complexity

015

020

025

030

035

040

1 2 3 4 5 6

12345

Values of Ｇ

(a) 119905119863 times 119901119903119900119901119890119909 by119898

0

10

20

30

40

50

1 2 3 4 5 6

005066

Values of

(b) 119905119863 times 119878ℎ by 120572

040

045

050

055

060

1 2 3 4 5

001002003004005

Values of Ｈ＄

(c) 119898 times 119890119888 by 119899119863

015

020

025

030

035

040

1 2 3 4 5

001002003004005

Values of Ｈ＄

(d) 119898 times 119901119903119900119901119890119909 by 119899119863

0

10

20

30

40

50

001 002 003 004 005

00005066

Values of

(e) 119899119863 times 119878ℎ by 120572

040

045

050

055

060

1 2 3 4 5

BehaviorNaiveFocused

(f) 119898 times 119890119888 by 119887119890ℎ119886V119894119900119903

015

020

025

030

035

040

0 05 066Behavior

NaiveFocused

(g) 120572 times 119901119903119900119901119890119909 by 119887119890ℎ119886V119894119900119903

05

06

07

08

09

10

1 2 3 4 5 6

00005066

Values of

(h) 119905119863 times 119894119903 by 120572

08700

08725

08750

08775

08800

001 002 003 004 005

BehaviorNaiveFocused

(i) 119899119863 times 119894119903 by 119887119890ℎ119886V119894119900119903

Figure 9 Curves of mean observations for different parameters and metrics for 36000 runs Bars are 095 confidence intervals Plots withthe same observables (ordinate) except for 119894119903 are on the same scale

Complexity 23

The higher the requirement for accepting the proposal isthe higher the proportion of extremists and the variance ofopinions are and the lower the shift statistic is In few words120572 constrains the world to dyadic discussion or throws it intoa process in which deliberation is much more importantthan dyadic interactions This is why for any 120572 = 05 oneeither gives too much weight to deliberated results whichshadows pair-wise interactions or too little weight in thesense that no deliberated result is ever accepted and thusnever integrated into the opinions of the agents that havevoted for it The reason for the latter is that the latitudeacceptance is too low and thereafter pair-wise interactionsare not enough to unevenly polarize the population in such away that deliberated proposal arguments are voted favorablyby a non-negligeable majority Shifts for 120572 = 0 are stable fordifferent levels of 119905119863 which implies that deliberation keeps incheck all the shifts of opinions related to an increase in thenumber of pair-wise interactions For 120572 = 05 deliberationis less likely to be successful and shifts tend towards the shiftlevels for 120572 = 0 (see Figure 9(b))

Concerning labeling-basedmetrics120572 entirely determinesthe coherence statistic For 120572 = 05 there is no difference incoherence because of how coherence is defined it is maximalwhen 120572 = 0 and strangely maximal when 120572 = 066(see Figure 9(h)) Surprisingly 120572 has no effect on judgmentaccuracy One would have expected a higher 120572 to increasejudgment accuracy since agents would hardly collectivelyaccept a proposal argument discussed during deliberationProposal arguments have given how the argument lattice isgenerated a higher probability of being rejected than of beingaccepted in the ideal argumentation framework

435 A Word on Focused and Naive Agents Focused andnaive specifications for agent behavior are an importantparameter in the decision process since they model howagents choose arguments in the deliberation arena In the sce-narios where agents are focused agents show equal or highervariance of opinions and are therefore more extreme (seeFigure 9(g)) independently of the voting requirements Shiftsof opinion are less likely as well We can see this happeningbecause focused agents knock out proposal arguments moreoften than naive agents provided that 119899119863 is low

One could have expected naive agents to yield morecoherent decisions since they argue sincerely However thisseems not to be the case (see Figure 9(i)) Agents are unable todeliberate in away thatmakes deliberation reflect their votingintentions either because they do not have the necessaryarguments to do so or because such arguments do not exist

In terms of judgment accuracy focused agents do betterthan naive agents in finding the ideal labels for the proposalarguments This is likely because when reconstructing theframework focused agents take into account the deliberatedproposal argument and choose themost pertinent argumentsin their sacks that relate to the state of affairs at the deliber-ation arena In fine debates result in a better approximationofL119886119887120576(119868) even though one could have believed the oppositethat is focused agents are ready to sacrifice a correct collectivelabel of a proposal argument in the goal of defending theirpositions

44 Points of Discussion In this subsection we discuss theresults obtained from the simulations We attempt to makesense of them from a social sciences perspective

441 Opinion Consensus and Dissensus The deliberativeopinion dynamics corresponds to the simulated scenarios inwhich the variance of opinions is at its lowest and judgmentaccuracy at its highest Dyadic opinion dynamics are associ-ated with the scenarios in which the variance of opinions is atits highest judgment accuracy at its lowest and coherence atitsmaximumpossible levelThemixed interactions dynamicsis a combination of these aspects for all the metrics exceptfor coherence at which it is the worsemdashdeliberation leadsto some kind of loose opinion consensus that is reinforcedby pair-wise interactions Likewise the mixed discussionmodel confirms some results exposed in [16] expressingthat opinions and voting intentions in a deliberative contextldquooften changerdquo That being said and more particularly indeliberative opinion dynamics outputs align well with twointeresting results on consensus formation given by aninterpretation of Moscovici and Doisersquos [15] work Sunsteinrsquos[14] account on deliberation and group polarization andsocial impact theory [4] The first states that mild consensusis reached in deliberation as procedure disengages agentsthe second is that within discussion groups majorities tendto become larger and opinions more extrememdashnot on thebasis of competing arguments but of the initial balance ofopinions After several decision processes we find opiniondistributions that reflect these observations even though weare not able to precisely coin the set of parameters that yieldssuch distributions (see Figure 7)

Although differences between mixed and dyadic opiniondynamics in terms of the variance of opinions and shiftsare slim the former does much better in terms of judg-ment accuracy In this sense deliberation does moderateopinions but not as much when non-deliberation steps arerelatively frequent throughout the decision-making processIn terms of Moscovici and Doise theory of consensus [15]this makes sense since pair-wise discussion engages or ego-involve agents in the discussion and pushes to convergenceof opinion at the extremes of the opinion distributionHowever what remains unclear is the reason why proceduralparameters that constrain deliberation do not disengageagents from discussion and thus result in a mild consensusas also expressed by Moscovici and Doise [15] A possibleexplanation of this phenomenon may stem from the factthat deliberation in the model is always accompanied by aseries of ego-involving discussions that shadowor counter themoderating effect of deliberation Not only this deliberationcould also be responsible for crystallizing the opinions ofextremists for it favors one or the other part of the distributionof opinions as explained in [14] and to some extent in [13]Furthermore the mixed discussion model gives an exampleof polarization that results from a minority of dissenterson one side of an issue acceding to views of the majorityrsquosside [4] as opposed to the strong polarization observed indyadic interactions Situations in which the status quo orthe undecidability of the proposal argument lingers cannotpossibly yield situations of moderation In cognitive science

24 Complexity

one can be inclined to believe that the longer a deliberationprocess is the more likely individuals will take an immutablestance on the situation In view of the assumption that longdecision-making processes can be more costly for a group orsociety as a whole agents may feel forced to take a stancejust for the sake of ending the process The stance that isadopted would therefore depend on pair-wise interactionsamong individuals and not on deliberation since deliberationtakes too long to be conclusive

442 For Hypothetical Recommendations in Decision-MakingProcess Elaboration On another tune results from themodelsuggest that procedural parameters have marginally decreas-ing positive or negative effects on the metrics of interest Ina hypothetical policy analysis perspective this may indicatethat after choosing a correct number of individuals andimposing the right amount of debates one can attain rea-sonable levels of judgment accuracy coherence and varianceof opinions As accounted in [18] deliberation increasesknowledge (collective knowledge as well) and therefore itheightens the chances of making correct decisions If onebelieves that social welfare is linked to deliberation andeventually to the metrics studied previously then proceduralparameters can both be used as an instrument to attaindesired levels of social welfare and reinforce the notionof legitimacy of collective decisions It can also provide ajustification for choosing a deliberation regime rather thananother on the basis of how important (or urgent) a topic ofdiscussion is Indeed if we adopt the claim that deliberationis a forerunner of welfare then any state in which deliberativecues are at their maximum (highest number of participantsmany deliberation steps before making a decision and asfrequent as possible) has to be mapped to the highest attain-able social welfare In this case is a situation where judgmentaccuracy is at its highest yet the variance of opinions or theproportion of extremists at its lowest an ideal situation Wecannot say for sure since strong opinion consensus and longdecision-making processes in many situations may result ina loss of welfare rather than in a gain of it

Another interesting result in procedure for deliberationis the very similar estimates we find for the effects of theminimum number (119898) of debates necessary to make adecision on a proposal and the proportion (119899119863) of individualsparticipating in deliberation on judgment accuracy In thescope of social welfare this raises the question of whetheradvocating for longer decision processes by demandingmoredeliberation steps (higher 119898) rather than bigger debates(higher 119899119863) may increase welfare Depending on the objec-tives of the decider of the deliberation procedure he or shemay choose to concentrate on one or the other A recom-mendation we can assert from the mixed social interactionsmodel is that if one prefers low to high variance of opinionsandor high to low coherence the decider is better off ifshe focuses on increasing the size of the debate rather thanorganizing debates very frequently One of the reasons forthis might be that when making bigger debates all agents aretruly considered as equals in the decision-making processIn organizing many debates individuals that are chosen toparticipate in many of these will be more influential and will

somehow bias the set of arguments in the deliberation arenaThemajority rulemodeled through120572 also determines the sizeof the shifts and the distribution of opinions Because a bigger120572 is associated with higher levels of extremism and opinionvariance deliberated cues are harder to accept and thereforeit becomesmore difficult for agents to reach concensus andormoderate their opinions

On the other hand and further into the idea of legitimacythe goal of deliberative processes is the agreement amongall participants This agreement stands for the right or bestdecision from a formal or procedural perspective and it is alsothe best from a substantive point of view In other words thelegitimacy of decisions not only derives fromwell-establishedprocedural requirements (respect of the protocol) but alsobegs for the fulfillment of two essential conditions (1) satisfy-ing the procedural requirements for a correct procedure (for-mal legitimacy) and (2) the rational acceptability of the resultsof this procedure (substantive acceptability) Legitimacy canbe then obtained from the coherence statistic scrutinizedin the model insofar as it measures how well agents acceptthe results of a correct procedure (eg deliberation) So fora proposal to be legitimate one has to choose parametersthat would maximize group coherence in decision-makingprocesses

As deliberation may give rise to consensus in the delib-eration arena it may also sow the seeds of dissensus inthe group Our model clearly illustrates this phenomenonthrough the procedural parameters of deliberation and themetrics observed From another perspective in deliberativedemocracy dissensus in collective decision-making may bedesirable According to Landermore and Pagersquos accounts [36]lack of consensus can be perceived as having agents withalternative ideas that combined with other ideas can providea better approximation of the ideal status of an argumentTheauthors in [36] call this dissensus ldquopositive dissensusrdquo andit makes normative sense when solving complex collectivedecision-making problems Our model captures the ideaof having normative requirements to define a good andlegitimate collective decision requirements of high consen-sual deliberative accuracy (judgment accuracy) of legitimacy(coherence) and of diversity (variance of opinions) are alltaken into account and for such we can conclude that ourmodel is successful in describing a sound process of collectivedecision-making with interpretable outcomes

Of course the present simulation results and analysis donot prove that all opinion dynamics can be accounted forby such simple processes like the one presented here Theydo however shed light on the expressiveness of combiningdifferent paradigms to account formore interpretablemodelsand pave the way to creating original models of the sortThe simulations suggest the desirability of discovering theconsequences of relatively simple laws of communication atdifferent levels (micro meso and macro) to determine whatstill needs to be explained

5 Related Work

We see our model as a contribution to the influence andopinion dynamics field in ABM and a pragmatic application

Complexity 25

of abstract argumentation theory To our knowledge we areunaware of existing literature on agent-based modeling thatexplicitly relates collective choice and the notions of deliber-ation and opinion diffusion through abstract argumentationas we have done it

Most models in the literature of opinion diffusion areinterested in opinions because these have an influence incollective decision and in questions of social order Forinstance in [3] the authors are interested in consensus andin how a group collectively chooses among two alternativesIn other models authors are interested in the emergenceof extremism [8] and on the distribution of opinions whenextremists are introduced in the population [5 33 37] whileother authors coin the notion of opinion polarization as anemergent property of the system [8 12] They show usingmodels of ldquobounded confidencerdquo and opinion diffusion withtrust that three different kinds of steady states (unipolarbipolar and central) are possible depending on whetheragents are sufficiently uncertain about their opinions andsufficiently connected andor a certain proportion of indi-viduals are already extreme Recent articles on informationand opinion dynamics stress the importance of governanceand government intervention in the spread of emotions(opinions) in the frangibility of social consensus systemIn [10] the authors show that government intervention inthe spread of negative emotions can lead to an even fasterspread of negative emotions (single-peeked convergence) andin a faster collapse of the social consensus system Opinionas a function of trust is studied in [12] and relates to howindividuals form their opinions on the basis of how muchthey trust agents in their networks

Another stream of discrete opinion dynamics modelsthat explain the emergence of opinion can be found in theliterature Computer multiagent models of attitude changebased on Latanersquos social impact theory are presented in [4] Inthe theory an agent changes her opinion on the basis of theinformational impact she is subject to which depends on thepersuasiveness (strength) supportiveness and immediacy(group structure) of the environment and information shereceives The simulations predict two emergent groups ofphenomena the shifting of attitudes towards incompletelypolarized equilibria and the formation of coherent clusteringof subgroups with deviant attitudes Similarly but in acontinuous more argumentative fashion Mas and Flache[11] present a model of opinion diffusion using argumentsArguments are considered to be for or against a proposal andare given an agent-dependant relevance in dyadic persuasiveinteractions Arguments also determine the agentsrsquo opinionsand their dissimilarities which posits the rules of pair-wiseencounters in theirmodel as agents are assumed to form theiropinions according to the arguments they own They showthat in an argumentative opinion diffusion model the onlyobservable steady states are bipolarization and consensusand that bipolarization can emerge from interactions amongsimilar agents In other words bipolarization of opinionsis possible with homophily and without negative influenceif there is argumentation They subsequently compare theirresults to experimental data In [38] a similar approach toargumentation and opinion formation is taken but for the fact

that they introduce different types of arguments explicitlyapply social judgment theory [22] and use survey data toestablish model-to-real-world comparisons

Closer to opinion formation and abstract argumentationGabbriellini and Torroni [39] are the first to have origi-nally and soundly merged opinion diffusion and abstractargumentation They define an agentrsquos opinion as a functionof the arguments she holds and the attack relation amongthem They devised a focused peer-to-peer dialogue systemof persuasion (NetArg) inspired from Mercier and Sperberrsquosargumentative theory of reasoning [40] which used onlyabstract argumentation to study opinion polarization andopinion dynamics Moreover they considered networks andthe notions of trust and epistemic vigilance to define adynamics for knowledge revision and trust itself whichwe clearly lack in our model They showed that if a con-servative belief operator in argumentation was applied byagents when they reasoned then their dialogue protocoldid not increase polarization among agents That said theyused the model to study the effect of Groenvetterrsquos weaklink theory in the spreading of arguments and were notparticularly interested in notions like judgment accuracy orcoherence nor in tackling questions related to collectivedecision-making procedures Their system is very expressiveand helps position our work in the litterature our work is atthe frontier of pure argumentative opinion diffusion modelsopinion diffusion with arguments and continuous bounded-confidence models of opinion diffusion a la Deffuant

On another note work on collective cognitive conver-gence [41] and opinion sharing [19] shows that consensustowards a certain ldquocorrectrdquo opinion or cognitive state is alwayspossible yet dependent on noise variability and awarenessof agents In [20] the authors show that learning about anexogenous correct state of the world (represented by bits)under confidence was possible but only if the agents werenot too confident In a homogeneous population they showthat the higher the confidence the worse the learningmdashthevery confident agents do not learn the properties of the truestate of the world and disrupt the learning process of the lessconfident ones Collective cognitive convergence can be seenas a result of deliberation in truth-seeking models

When it comes to abstract argumentation theory we takean approach that wires two types of dialogues that are well-studied in the literature persuasion dialogues [42] and delib-eration dialogues [43] Another interesting line of work is theone on mechanism design [44] or the problem of devisingan argumentation protocol where strategic argumentation isnot a liability for success in debates We tackle mechanismdesign in a different way though Instead of consideringstrategy-proofness we are interested in how differences inprotocol can result in epistemologically ldquobetterrdquo collectivechoices and guarantee that opinion distributions are favorablefor deliberation For a survey on persuasion dialogue see[42]

Work on agent-based argumentation usually assumes thatthe semantic relationship between arguments is fixed Inother words if two individuals were to put two argumentsin the public arena such that one logically attacked the otherthen everyone would agree that such attack exists [45 46]

26 Complexity

Other models which do not make this restrictive assumptioncan also be found in the literature [27 47] and derive fromthe class or family of opponent models [27 48] in which twoopposing sides attempt to win a dialogue Ourmodel is in theintersection of these two but the frame combining opiniondiffusion of the kind and argumentation remains new

The idea of mixing interpersonal influence and verticalcommunication however is not original It is described andimplemented in innovation diffusion models such as [5 49]In both vertical communication is modeled as exogenoustransparent informationmdashagents are aware of the existenceof an innovation thus triggering several processes of choiceand stabilization of opinions Also in [50] an Eulerian modelis implemented to show the effect of media and exogenousinformation on opinion distributions With respect to thispoint the originality of our work is in that the informationemitted as vertical communication is endogenous It is issuedfrom a deliberation model that agents shape on the basisof their opinions arguments and behavior In the spirit of[51] where the authors control for the design of verticalcommunication (to whom the vertical communication isaddressed its timing how it influences agents opinions) wecontrol for the process generating the information for ourown set of variables length of debates majority quotas andthe frequency at which discussions take place

6 Conclusion and Perspectives

The main objective of this article was to build a bridgewhere decision-making argumentation and opinion dif-fusion could come together We proposed a model thatcombined abstract argumentation theory and a boundedconfidence opinion diffusion model and showed to whatextent it could explain variance of opinions extremismcoherence in collective decisions and judgment accuracyon arguments given an ideal state of full information Thesecond objective was to show in what way governance pro-viding agents with arenas of discussion and deliberation wasimportant in addressing the success of collective decision-making processes and the quality of their outcomes

The model revealed that allowing for more deliberationtime though in lower frequency and allowing for widerparticipation in deliberation increased the variance of opin-ions and the proportion of extremists in a group Theseobservations are consistent with the combination of resultsfound in [9] and in [13 14] which stress that deliberationmay polarize groups and may have a meager effect on theshifts of opinions and inconsistent with [16] where it isargued that opinions tend to moderate after deliberationIt happens that deliberation alone did moderate opinionyet when integrated into a more complex system in whichindividuals were allowed to interact with one another andnot all that was deliberated was accepted its influence wasshadowed by other more individualistic dynamics

Undeniably the grounded semantics played an importantrole in the weak effect of deliberation since it models askeptical way of reasoning over arguments For agentsaccepting arguments thus updating their opinions accord-ingly happened not too often Nevertheless asking for more

deliberation did increase judgment accuracy as observed in[18] yet in a marginally decreasing fashion We showed thatvoting within the deliberation protocol not only increasedthe proportion of extremists and the variance of opinions ina group but also determined how coherent deliberation andvoting procedures were with each other Lastly we showedthat agents that are focused judged arguments better hadmore stable opinions and constituted a group with higherproportion of extremists than their naive counterparts

In terms of governance the model says that there is per-haps no trade-off between extremismand judgment accuracyInstead it asserts that the higher the variance of opinions thecloser the group gets to the correct decisions This situationmay be interpreted as Landermore and Pagersquos notion of ldquoposi-tive dissensusrdquo (in complex collective decision-making tasks)[36] as opposed to consensus as a normative requirementfor correct collective decision-making If a decider has toorganize deliberation to legitimize a public policy dependingon what kind of world he or she wants different set ofparameters may be chosen to account for different levels oflegitimacy and correctness For instance one may ask for adecision to be legitimate to respect a certain level of coherenceor of judgment accuracy and to be discussed by a sundryof different-opinionated peoples If so deliberation has tobe made more often more participants have to be includedin debates or in deliberation instances and not too manytime steps should be introduced in between two deliberationinstances So the model may capture normative ideas todefine correct and legitimate collective decisions and tomake recommendations on those grounds Requirements ofhigh consensual deliberative accuracy (judgment accuracy)legitimacy (coherence) and diversity (variance of opinions)are all taken into account and for such a reason the model israther successful in describing a sound process of collectivedecision-making with interpretable outcomes

61 Extensions of the Model and Other Ideas Our model canbe extended in many ways Value-based abstract argumen-tation frameworks (VAFs) for example as an extension ofDungrsquos argumentation framework provide a formal descrip-tion of the process of decision-making in which argumentsare given values and audiences (set of agents) ignore attacksbetween arguments on the basis of their preferences over val-ues Social abstract argumentation frameworks (SAFs) alsoprovide an interesting and seemingly convenient formalismfor our model In SAFs agents vote on the acceptability ofarguments and the resulting labeling is a combination of thearguments that are voted for and labeling-based semantics Inour model we apply the same rule yet only for one argumentand given precise agent-based votingmechanisms and votingrules which are eventually parametrized

To consider our model as a specification of a mixed VAFand SAF model may be of great interest for future workThis formalism can provide solid foundations to describeexplain interpret and eventually extend the results foundhere from an argument-based perspective On those groundsfuture work will focus on creating a hybrid argumentationframeworkmdashwith its semanticsmdashthat takes heed of bothvalues or principles and voting

Complexity 27

Concerning other possible extensions of the model anatural reflex would be to argue on the validity and therobustness of the mix between the bounded confidenceopinion diffusion models and deliberation Consequentiallyadding deliberation to different bounded confidence modelsof opinion diffusion such as the Hegelsmann-Krause model[37] or to models of social impact theory like in [4] canbe of great interest for future work We acknowledge that afiner understanding of argumentation frameworks randomlattices and their respective applications for this modelare necessary Testing for different labeling-based semanticsor even trading labeling-based semantics for scoring-basedsemantics (semantics that defines acceptability on scoresgiven to arguments) such as debate semantics and ranking-based semantics (semantics that yield a preorder) may beinteresting to consider in future work

More into modeling deliberation it is not inadequateto believe that agents may also advance proposals duringdeliberation and replace to some extent the proposal forwhich the deliberation is taking place Endogenizing theprocess that produces the type of argument to be deliberatedon may be interesting to explore either by having agentsstrategically replace proposals at certain moments duringthe debate or having the central authority attempt to chooseproposals such that the probability of having them acceptedis high Agents may also be thought to be autoorganizing(independent of a central authority) and concede on the sizeof the instances of deliberation To consider differences insemantics across agents or even endogenizing the semanticson how urgent or important the proposal that is beingdiscussed is can also be an interesting direction to take

Last but not least many nontrivial modifications of ourmodel are possible and most should include better-thoughtdeliberation protocols It may be interesting to design andobserve protocols in which deliberation only affects agentsthat are actually debating for example Trust network effectsmultidimensionality of opinions new processes of argumentexchange or learning among agents are notions to furtherdevelop in order to make the model more realistic and relaxsome unreliable assumptions In sum the model is to berefined and extended with the objective of either studyingconcrete cases of deliberative polling and opinion dynamicsor implementing more intuitive thought experiments

Appendix

We present below some of the pseudocode we used toimplement the model

Algorithms

(a) Algorithm 1 implements the dyadic influence in themodel It describes how two agents interact andupdate their opinions

(b) Algorithm 2 implements the deliberative influence inthe model which describes how agents update theiropinions in sight of the accepted deliberated proposalarguments

(c) Algorithm 3 is used to obtain the grounded labelingof an argumentation framework

(d) Algorithm 4 provides an overview of a run in thesimulations119872 is the number of proposal argumentsto discuss which we set to 100 in the model

Each object in the system is indexed by a119908ℎ119900 number Hencea set of agents is annotated by a set of natural numbers Weuse this fact in the implementation of the algorithm 119874119887119895119890119888119905119894denotes the object of 119908ℎ119900 number 119894 Agentsrsquo opinions (119900) arecalibrated so that 119900 isin [minus1 1]Data Availability

The program and data used to support the findings of thisstudy are available from the corresponding author uponrequest

Disclosure

A preliminary version of this work was presented anddiscussed at the conference Bridging the Gap between FormalArgumentation and Actual Human Reasoning on October 4-5 2018 at BochumGermanyThe authors thank the audiencefor their valuable comments and suggestions

Conflicts of Interest

The authors declare that they have no conflicts of interestregarding the publication of this article

Acknowledgments

The contribution of Gabriella Pigozzi was supported by theDeutsche Forschungsgemeinschaft (DFG) and the CzechScience Foundation (GACR) as part of the joint project FromShared Evidence to Group Attitudes (RO 45486-1)

Endnotes

1 In other words the superiority of the ldquoright answerrdquo orlabeling in our case will appear as such to all

2 All agents hold the same opinion3 By sensitivity we mean how much an agent is ready to

change her opinion on the sole fact that an informationalcue has been debated on

4 The assumption is strong but this trust among agentsmay be rooted from a genuine individual motivation toreach a democratic consensus

5 We adopted a rule of thumbs that says that if an agentis neutral with respect to the principle then she willvote for any proposal that is presented to her duringdeliberation

6 If the argument is not unique she chooses one atrandom

7 If there are more than one candidate the agent will favorthe argument at minimal distance from 119868

28 Complexity

Require Vector of values for parameters (119881 = (120583 119880 119879))Ensure Successful interactions among agents119896 larr997888 0119873119890 larr997888 0

while 119896 le |119873|2 do ⊳Fill119873119890 with |119873|2 agents119896 larr997888 119896 + 1119873119890 larr997888 119877119886119899119889119900119898(|119873| + 1) ⊳Pseudo-random generator Generates integer between 0 and |119873|119873119903 larr997888 0 120575119894119895 larr997888 0 ⊳Variable to store differences in opinionfor all 119894 isin 119873119890 doRandomly pick a number 119895 isin 119873(119873119890 cup 119873119903)119873119903 larr997888 119873119903 cup 119895 120575119894119895 larr997888 |119900119894 minus 119900119895|procedure dyadic-update(119894 119895) ⊳Two agents 119894 and 119895 discuss and update opinions1199001015840 larr997888 119900119894 (120583119895 119880119895 119879119895) larr997888 (120583119894 119880119894 119879119894) larr997888 (120583 119880 119879) ⊳Save agent irsquos opinion before updating and homogenize agents

if 120575119894119895 lt 119880119894 then119900119894 larr997888 120583119894119900119895 + (1 minus 120583119894)119900119894if 120575119894119895 gt 119879119894 then119900119894 larr997888 (1 + 120583119894)119900119894 minus 120583119894119900119895if 120575119894119895 lt 119880119895 then119900119895 larr997888 1205831198951199001015840 + (1 minus 120583119895)119900119895if 120575119894119895 gt 119879119895 then119900119895 larr997888 (1 + 120583119895)119900119895 minus 1205831198951199001015840

Algorithm 1 Dyadic social influence

Require Vector of values for parameters (119881 = (120574 119901119886 119901119903))Ensure Successful interaction between proposal arguments and agents

for all 119894 isin 119873 doprocedure Probability-Update(V119886 119901119886 119901119903) ⊳Update individual probabilities of change due to deliberated cues119909119894 larr997888 119901|V119886minus119900119894 |2119886 119910119894 larr997888 1199012|V119886minus119900119894 |119903119902 larr997888 0if L119886119887(119868) = 119894119899 then

procedure deliberative-update(119894 119886) ⊳Agent i argument 119886120574119894 larr997888 120574 119902 larr997888 119906 sim U(0 1) ⊳ 119906 is a realization of a uniform random variableif 119902 lt 119909119894 then119900119894 larr997888 120574119894119900119894 + (1 minus 120574119894)V119886break

if 119902 lt 119910119894(1 minus 119909119894) then119900119894 larr997888 (1 + 120574119894)119900119894 minus 120574119894V119886Algorithm 2 Deliberation influence

8 In terms of computational complexity naive agentsmaketheirmoves in119874(|A119894| log(|A119894|)) while focused agents doat least as bad

9 Tables have memory Therefore debates are not neces-sarily trees

10 If there are less than (119899119863 times |119873|)2 agents for any of bothgroups the CA summons all of them to the deliberationtable

11 Or careful consideration of the viewpoints of otherswhich implies that citizens keep an open mind and donot reject arguments outright

12 A ldquohealthyrdquo consensus refers to the case in which manyconsensual decisions are taken but opinion-wise agentsdo not agree with each other Quotes are used on theword healthy because some political science theorists

(eg [36]) believe that in certain situations agreement inopinions for deliberation is epistemologically damaging

13 Simulations on groups of |119873| = 800 and |119873| = 400agents shows that scaling the population has an effect onall themetrics but it is confounding depending onwhichcouples of attitudes structures are considered (119880 119879) Westick to |119873| = 400 as in [9] for comparability andsimulation running time

14 By balanced wemean with as many arguments with V119886 lt0 as with V119886 gt 0

15 Experiments on the system suggest that a bigger(smaller) argument pool 119872 = 1000 (119872 = 200) isassociated with a higher (lower) variance of opinions(and extremism) and higher (lower) judgment accuracyFor the rest of the observations 119872 seem to have noeffect We choose a middle value for 119872 Similarly theargument pool chosen to be balanced or uniformly

Complexity 29

Require Argumentation framework 119860119865 = (AR)Ensure Assignment of labels to arguments from the grounded labeling of 119860119865L(A)

L0(A) larr997888 (0 0 0)119896 larr997888 0do119894119899 L119896+1(A) larr997888 119894119899 L119896(A) cup 119886 | 119886 is not labeled 119894119899 L119896(A) 119886119899119889 forall119887 isin A 119904119905 119887 R 119886 then 119887 isin 119900119906119905 L119896(A)119900119906119905 L119896+1(A) larr997888 119900119906119905 L119896(A) cup 119886 | 119886 is not labeled 119894119899 L119896(A) 119886119899119889 exist119887 isin A 119904119905 119887 R 119886 and 119887 isin 119894119899 L119896+1(A)

while L119896+1(A) = L119896(A) = 0Algorithm 3 Solve argumentation frameworks (with grounded semantics)

Require Vector of values for parameters (119881)Ensure Vector of statistics (119880)

procedure Set-parameters(119881) ⊳Initialize parameters for scenarioprocedure Init-objs(119899119898 119896 ) ⊳Create119898 arguments 119899 agents and 119860119865120576procedure Solve-Argumentation-Framework(119860119865) ⊳Assigns the epistemic labelingL120576 toA119894 larr997888 1119905 larr997888 1while 119894 le 119872 do ⊳ 119872 the number of arguments to discuss119868119894 larr997888 119866119890119899 119886119903119892(1) ⊳Generates proposal argument 119868119860 119894119905 larr997888 (119868119894 0)L119886119887(119868119894) larr997888 119906119899119889 ⊳Initialize label for 119868119895 larr997888 1 ⊳Initialize debate counter 119895while ((L119886119887(119868) = 119906119899119889 or 119898 le 119895) and 119895 le 119898) doif 119905 equiv 0(119898119900119889 119905119863) then

procedure build-argumentation-framework(119899119863 119887119890ℎ119886V119894119900119903) ⊳Build the argument graph agents deliberate(119861119894119905 119878119894119905) larr997888 (119883 119884) sub (⋃119897isin119873119899119863 A119897119894119905 ⋃119897isin119873119899119863 R119897119894119905) ⊳ 119873119899119863 sub 119873 are agents sampled by the CA119860 119894119905 larr997888 119889(119860 119894119905 (119861119894119905 119878119894119905))procedure Solve-Argumentation-Framework(A119889) ⊳FindL(A119889)L119886119887(119868119894) larr997888 L119886119887119863(119868119894)119895 larr997888 119895 + 1

else119896 larr997888 1while 119896 le 119905119863 doprocedure dyadic-social-influence(120583 119879 119880) ⊳Agents discuss one-to-one119896 larr997888 119896 + 1119905 larr997888 119905 + 1

if 120572 gt 0 andL119886119887(119868119894) = 119900119906119905 then ⊳Agents vote for the proposalif (|119897 isin 119873|119900119897 times V119868119894 ge 0| ge 120572 times 119899) then

L119886119887(119868119894) larr997888 119894119899else

L119886119887(119868119894) larr997888 119900119906119905if L119886119887(119868119894) = 119894119899 thenprocedure Deliberation-influence(120574 119901119886 119901119903) ⊳Agents are influenced by the result of the decision process119894 larr997888 119894 + 1119905 larr997888 119905 + 1119880 larr997888 119862119900119898119901119906119905119890 119878119905119886119905119894119904119905119894119888119904(119881)

return 119880 ⊳Get the statisticsAlgorithm 4 Run for one scenario

generated does not seem to play a significant explanatoryrole on any of the metrics

16 Experiments show that themaximumnumber of debatesis not an important parameter even if it is an intuitiveone when considering decision-making processes Inpractice debates end after at most 119898 + 1 deliberation

steps and even then it happens rarely However to avoidunpleasant surprises like infinite debates on a completeargument graph containing the proposal argument weadded the constraint 119898 Its value is precisely 7 becausethe biggest minimum number of debates we studyis 6

30 Complexity

17 To ensure comparability with the deliberative andmixedopinion dynamics models on judgment accuracy andcoherence agents vote on proposal arguments every 119905119863times(119898+1) 119905119863 and119898does not explain variability in judgmentaccuracy nor in coherence

18 Student t-tests for independent samples are robust tothe violation of two hypothesis on the distribution ofthe samples normality and homogeneity of variancesas long as the sample size is big enough (central limittheorem) and if the difference in size of the comparedsamples is smallrespectively

19 Studentrsquos test version forwhen variances are not assumedequal and the difference in size of the compared samplesis big

20 We do not correct for homoskedasticity nor normality oferrors in the estimates

21 Balanced as the same number of runs per scenario orsuch that no significant correlations are found betweenparameters

References

[1] R Axelrod ldquoThe dissemination of culture a model withlocal convergence and global polarizationrdquo Journal of ConflictResolution vol 41 no 2 pp 203ndash226 1997

[2] D M Rousseau Reinforcing the MicroMacro Bridge Organiza-tionalThinking and Pluralistic Vehicles SAGEPublications SageLos Angeles Calif USA 2011

[3] S Galam and SMoscovici ldquoTowards a theory of collective phe-nomena consensus and attitude changes in groupsrdquo EuropeanJournal of Social Psychology vol 21 no 1 pp 49ndash74 1991

[4] A Nowak J Szamrej and B Latane ldquoFrom private attitude topublic opinion a dynamic theory of social impactrdquo Psychologi-cal Review vol 97 no 3 pp 362ndash376 1990

[5] G Deffuant S Huet and F Amblard ldquoAn individual-basedmodel of innovation diffusion mixing social value and individ-ual benefitrdquo American Journal of Sociology vol 110 no 4 pp1041ndash1069 2005

[6] G Deffuant D Neau F Amblard and G Weisbuch ldquoMixingbeliefs among interacting agentsrdquoAdvances in Complex Systems(ACS) vol 3 no 01n04 pp 87ndash98 2000

[7] J Rouchier P Tubaro and C Emery ldquoOpinion transmission inorganizations an agent-based modeling approachrdquo Computa-tional and Mathematical OrganizationTheory vol 20 no 3 pp252ndash277 2014

[8] M Meadows and D Cliff ldquoThe relative disagreement model ofopinion dynamics where do extremists come fromrdquo in Pro-ceedings of the International Workshop on Self-Organizing Sys-tems Lecture Notes in Computer Science pp 66ndash77 Springer2013

[9] W Jager and F Amblard ldquoUniformity bipolarization and plu-riformity captured as generic stylized behavior with an agent-based simulation model of attitude changerdquo Computational ampMathematical Organization Theory vol 10 no 4 pp 295ndash3032005

[10] Y-F Zhang H-Y Duan and Z-L Geng ldquoEvolutionary mech-anism of frangibility in social consensus system based onnegative emotions spreadrdquo Complexity vol 2017 Article ID4037049 8 pages 2017

[11] A Flache andMWMacy ldquoSmall worlds and cultural polariza-tionrdquoThe Journal of Mathematical Sociology vol 35 no 1-3 pp146ndash176 2011

[12] A Tsang and K Larson ldquoOpinion dynamics of skepticalagentsrdquo in Proceedings of the 13th International Conference onAutonomous Agents and Multiagent Systems AAMAS 2014 pp277ndash284 International Foundation forAutonomousAgents andMultiagent Systems France May 2014

[13] K M Hansen Deliberative democracy and opinion formationUniversity Press of Southern Denmark Odense 2004

[14] C R Sunstein ldquoThe law of group polarizationrdquo Journal ofPolitical Philosophy vol 10 no 2 pp 175ndash195 2002 WileyOnline Library

[15] S Moscovici and W Doise Dissensions et Consensus unetheorie generale des decisions collectives Presses Universitairesde France-PUF 1992

[16] J S Fishkin and R C Luskin ldquoExperimenting with a demo-cratic ideal deliberative polling and public opinionrdquo ActaPolitica vol 40 no 3 pp 284ndash298 2005

[17] R C Luskin J S Fishkin and S Iyengar Considered Opinionson US Foreign Policy Face-To-Face versus Online Delibera-tive Polling International Communication Association NewOrleans La USA 2004

[18] J Barabas ldquoHowdeliberation affects policy opinionsrdquoAmericanPolitical Science Review vol 98 no 4 pp 687ndash701 2004

[19] O Pryymak A Rogers and N R Jennings ldquoEfficient opinionsharing in large decentralised teamsrdquo in Proceedings of the 11thInternational Conference on Autonomous Agents andMultiagentSystems vol 1 pp 543ndash550 Spain June 2012

[20] J Rouchier and E Tanimura ldquoWhen overconfident agents slowdown collective learningrdquo Simulation vol 88 no 1 pp 33ndash492012

[21] D Miller ldquoDeliberative democracy and social choicerdquo PoliticalStudies vol 40 S1 pp 54ndash67 1992

[22] M Sherif and C I Hovland Social Judgment Assimilation andContrast Effects in Communication and Attitude Change YaleUniversity Press 1961

[23] P M Dung ldquoOn the acceptability of arguments and its funda-mental role in nonmonotonic reasoning logic programmingand n-person gamesrdquo Artificial Intelligence vol 77 no 2 pp321ndash357 1995

[24] M Caminada and G Pigozzi ldquoOn judgment aggregation inabstract argumentationrdquo Autonomous Agents and Multi-AgentSystems vol 22 no 1 pp 64ndash102 2011

[25] P Baroni M Caminada andM Giacomin ldquoAn introduction toargumentation semanticsrdquoThe Knowledge Engineering Reviewvol 26 no 4 pp 365ndash410 2011

[26] S Chambers ldquoDeliberative democratic theoryrdquo Annual Reviewof Political Science vol 6 no 1 pp 307ndash326 2003

[27] E Bonzon andNMaudet ldquoOn the outcomes of multiparty per-suasionrdquo in Proceedings of the 10th International Conference onAutonomous Agents and Multiagent Systems 2011 AAMAS 2011vol 1 pp 47ndash54 International Foundation for AutonomousAgents and Multiagent Systems Taiwan May 2011

[28] J Ranciere Disagreement Politics and Philosophy University ofMinnesota Press 1999

[29] T Bench-Capon and K Atkinson ldquoAbstract argumentation andvaluesrdquo in Argumentation in Artificial Intelligence pp 45ndash64Springer 2009

[30] M Caminada ldquoOn the issue of reinstatement in argumenta-tionrdquo in Proceedings of the European Workshop on Logics in

Complexity 31

Artificial Intelligence vol 4160 of Lecture Notes in ComputerScience pp 111ndash123 Springer Berlin Heidelberg

[31] GAVreeswijk ldquoAn algorithm to computeminimally groundedand admissible defence sets in argument systemsrdquo in Pro-ceedings of the 2006 conference on Computational Models ofArgument Proceedings of COMMA 2006 vol 144 pp 109ndash120IOS Press Amsterdam The Netherlands 2006

[32] C W Sherif M Kelly H L Rodgers et al ldquoPersonal involve-ment social judgment and actionrdquo Journal of Personality andSocial Psychology vol 27 no 3 pp 311ndash328 1973

[33] G Deffuant F Amblard G Weisbuch and T Faure ldquoHow canextremism prevail A study based on the relative agreementinteraction modelrdquo Journal of Artificial Societies and SocialSimulation vol 5 no 4 p 1 2002

[34] E Awad M W Caminada G Pigozzi M Podlaszewski andI Rahwan ldquoPareto optimality and strategy-proofness in groupargument evaluationrdquo Journal of Logic and Computation vol 27no 8 pp 2581ndash2609 2017

[35] D Grossi and G Pigozzi ldquoJudgment aggregation a primerrdquoSynthesis Lectures on Artificial Intelligence and Machine Learn-ing vol 8 no 2 pp 1ndash151 2014

[36] H Landemore and S E Page ldquoDeliberation and disagreementproblem solving prediction and positive dissensusrdquo PoliticsPhilosophy and Economics vol 14 no 3 pp 229ndash254 2015

[37] R Hegselmann and U Krause ldquoOpinion dynamics andbounded confidence models analysis and simulationrdquo Journalof Artificial Societies and Social Simulation vol 5 no 3 2002

[38] A Stefanelli and R Seidl ldquoModerate and polarized opinionsUsing empirical data for an agent-based simulationrdquo inProceed-ings of the Social Simulation Conference 2014

[39] S Gabbriellini and P Torroni ldquoA new framework for ABMsbased on argumentative reasoningrdquo in Advances in SocialSimulation vol 229 pp 25ndash36 Springer Berlin Germany 2014

[40] H Mercier and D Sperber ldquoWhy do humans reason Argu-ments for an argumentative theoryrdquo Behavioral and BrainSciences vol 34 no 2 pp 57ndash74 2011

[41] H V Parunak T C Belding R Hilscher and S BruecknerldquoModeling and managing collective cognitive convergencerdquoin In Proceedings of the 7th iNternational Joint Conference onAutonomous Agents and Multiagent Systems vol 3 pp 1505ndash1508 International Foundation for Autonomous Agents andMultiagent Systems 2008

[42] H Prakken ldquoFormal systems for persuasion dialoguerdquo TheKnowledge Engineering Review vol 21 no 2 pp 163ndash188 2006

[43] K Atkinson T Bench-Capon and P Mcburney ldquoA dialoguegame protocol for multi-agent argument over proposals foractionrdquoAutonomousAgents andMulti-Agent Systems vol 11 no2 pp 153ndash171 2005

[44] M Thimm and A J Garcıa ldquoClassification and strategicalissues of argumentation games on structured argumentationframeworksrdquo in Proceedings of the 9th International Conferenceon Autonomous Agents and Multiagent Systems vol 1 pp 1247ndash1254 International Foundation for Autonomous Agents andMultiagent Systems Canada 2010

[45] I Rahwan and K Larson ldquoMechanism design for abstractargumentationrdquo in Proceedings of the 7th International JointConference on Autonomous Agents and Multiagent SystemsAAMAS 2008 pp 1031ndash1038 International Foundation forAutonomous Agents and Multiagent Systems Portugal May2008

[46] H Prakken R Riveret A Rotolo and G Sartor ldquoHeuristics inArgumentation A Game-Theoretical Investigationrdquo EuropeanUniversity Institute 2008

[47] N Hadidi Y Dimopoulos and P Moraitis ldquoTactics and con-cessions for argumentation-based negotiationrdquo in Proceedingsof the COMMA vol 245 pp 285ndash296 2012

[48] N Oren and T J Norman ldquoArguing using opponent modelsrdquoin Proceedings of the International Workshop on Argumentationin Multi-Agent Systems pp 160ndash174 Springer 2009

[49] S Delre W Jager T Bijmolt and M Janssen ldquoTargeting andtiming promotional activities An agent-based model for thetakeoff of new productsrdquo Journal of Business Research vol 60no 8 pp 826ndash835 2007

[50] A Mirtabatabaei P Jia and F Bullo ldquoEulerian opinion dynam-ics with bounded confidence and exogenous inputsrdquo SIAMJournal on Applied Dynamical Systems vol 13 no 1 pp 425ndash446 2014

[51] V Barbet N Guiraud V Laperriere and J Rouchier Hagglingon Values Towards Consensus or Trouble 2019

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Complexity 21

We extend the parameter domains of the pure dyadic anddeliberative scenarios as we make it explicit in Table 2

431 Minimum Number of Debates (119898) We observe that theminimum number of debates has a significant well-observedeffect on all of our metrics excluding group coherence Forthe variance of opinions we can observe that the moredebates there are the bigger the value of the metric isalthough the higher 119899119863 and 119905119863 are the weaker the overalleffect of119898 becomes (Figures 9(d) and 9(a)) Furthermore themarginal effect of increasing minimal debates is decreasingan additional deliberation step requirement increases thevariance of opinions but less and less as it grows 119898 coupledwith 119899119863 has a shy S-shaped effect (see Figure 9(d)) on thevariance of opinions meaning that it has a stronger effectwhen 119899119863 is low which suggests a possible trade-off betweenthe number of arguments accepted in the debate and thenumber of debates required before accepting a proposal

In contrast the shifts in opinion are less and less likelyas 119898 grows and this is independent of the variations of theother parameters Again the effect is marginally decreasingand is only truly significant when 120572 = 05 An explanationto these effects may be linked to the design of the systemFirst the variance of opinions is higher when deliberation isasked for because the more deliberation steps there are thehigher the chances are that the central proposal argumentis deemed unacceptable specially when agents are focusedFurthermoremechanically speaking increasing theminimalamount of debates implies that whenever a decision is to betaken at least 119898 times 119905119863 time steps have to take place and if atany moment a proposal argument is considered undecided119905119863 time steps are added to the process So unless the debatedoes not yield 119906119899119889 labels for proposal arguments (highlyunlikely considering that 120590 = grounded semantics) the morenondeliberation steps there are in the decision process thehigher the variance of opinions is Concerning the shiftswhen 120572 = 05 either the system is too stiff to accept anyproposal argument and opinions do not changemuch or theeffects of pair-wise discussion and deliberation cancel out insuch a way that individual shifts are minimal (Figure 9(b))

On the side of labeling-based metrics the more debatesare asked for the more accurate a group is in its judgmentthe effect being smaller as 119898 grows When agents are naivethe effect is quasilinear while when they are focused thestrongest effects of acquiring more deliberation are foundwhen levels of deliberation are already low (Figure 9(f))Thiscan be explained by the fact that the more debates thereare in the decision process the closer one gets to the idealargumentation framework

432 Proportion of the Population in Deliberation Steps (119899119863)Like for 119898 119899119863 has a significant effect on the proportionof extremists and on the shifts and variance of opinions(Figure 9(e)) This may result from the fact that being ableto put more arguments in play at the same debate step mayincrease the odds of revealing cycles around the proposalargument in119860119865120576 Given that we use grounded semantics thearguments in the cycles are labeled 119906119899119889 and in consequence

debates are more often postponed when 119899119863 is higher Post-poning debates in turn increases the number of nondebatessteps in the decision process which increases the varianceof opinions and limits over time the moderating effect ofdeliberation Moreover the effect of this parameter is verydependent on the value of 120572 (Figure 9(e)) For shifts forinstance 120572 = 05 makes the effect of 119899119863 negative while120572 = 0 makes it positive but to a lesser extent Furthermorefor higher requirements of deliberation (119898) adding moreindividuals to the deliberation process has weaker effectson the variance of opinions and on the other metrics It isalso quite interesting to notice that it has the same effect onagents independently of agent behavior during deliberationThis is a surprising result as one would have expected morefocused agents in a deliberation arena to cause a raise inthe proportion of extremists as they play to knock outopposing proposal arguments and thereof reducing the effectof deliberation (lowers variance) on the whole population

Finally similar to119898 addingmore people into the deliber-ation process increases judgment accuracy (see Figure 9(c))and has no clear effect if any on group coherence (seeFigure 9(i))

433 Steps between Deliberation Steps (119905119863) In all configu-rations 119905119863 increases the proportion of extremists (variance)and decreases the shifts of opinions among agents Theshifts and the effect on the variance of opinions are presentfor 120572 = 05 insignificant otherwise (Figure 9(b)) 119905119863 ishighly linked to 119898 by construction When 119898 = 1 thecurve linking the variance of opinions and 119905119863 is convex As119898 increases the curve becomes more and more concavewhich means that 119905119863 has a more important effect on theopinion distribution as collective decisions take longer to beachieved This seems counterintuitive but in reality it reflectsthemultiplicative relationship between deliberation and pair-wise interactions and the semantics in the model If 119898 is lowand 119905119863 is high the effective number of pair-wise interactionsis on average smaller in the deliberation process Hencethe structure of the decision-making process hinders anyincrease in the variance of opinions If119898 is high the oppositeeffect is observed Additionally since the grounded semanticsyields few 119894119899 arguments with respect to other admissibility-based semantics getting closer to the ideal argumentationframeworkmaymake it difficult to obtain the expected effectsof deliberative discussion on agents Lower 119898 constrains thevariance of opinions by making deliberation more influentialon opinions

In terms of judgment accuracy and coherence we observethat 119905119863 does not explain coherence yet it tends to decreasejudgment inaccuracy This may be due to a more variedargument pool in the debate which in turn may be aconsequence of a higher variance of opinions or vice versa

434 Acceptability Voting Quota (120572) By far it is the mostinfluential parameter in our study It changes the directionand the intensity of the effect of all the other proceduralparameters and by construction heavily constrains the roadto accepting a proposal (see Figure 9)

22 Complexity

015

020

025

030

035

040

1 2 3 4 5 6

12345

Values of Ｇ

(a) 119905119863 times 119901119903119900119901119890119909 by119898

0

10

20

30

40

50

1 2 3 4 5 6

005066

Values of

(b) 119905119863 times 119878ℎ by 120572

040

045

050

055

060

1 2 3 4 5

001002003004005

Values of Ｈ＄

(c) 119898 times 119890119888 by 119899119863

015

020

025

030

035

040

1 2 3 4 5

001002003004005

Values of Ｈ＄

(d) 119898 times 119901119903119900119901119890119909 by 119899119863

0

10

20

30

40

50

001 002 003 004 005

00005066

Values of

(e) 119899119863 times 119878ℎ by 120572

040

045

050

055

060

1 2 3 4 5

BehaviorNaiveFocused

(f) 119898 times 119890119888 by 119887119890ℎ119886V119894119900119903

015

020

025

030

035

040

0 05 066Behavior

NaiveFocused

(g) 120572 times 119901119903119900119901119890119909 by 119887119890ℎ119886V119894119900119903

05

06

07

08

09

10

1 2 3 4 5 6

00005066

Values of

(h) 119905119863 times 119894119903 by 120572

08700

08725

08750

08775

08800

001 002 003 004 005

BehaviorNaiveFocused

(i) 119899119863 times 119894119903 by 119887119890ℎ119886V119894119900119903

Figure 9 Curves of mean observations for different parameters and metrics for 36000 runs Bars are 095 confidence intervals Plots withthe same observables (ordinate) except for 119894119903 are on the same scale

Complexity 23

The higher the requirement for accepting the proposal isthe higher the proportion of extremists and the variance ofopinions are and the lower the shift statistic is In few words120572 constrains the world to dyadic discussion or throws it intoa process in which deliberation is much more importantthan dyadic interactions This is why for any 120572 = 05 oneeither gives too much weight to deliberated results whichshadows pair-wise interactions or too little weight in thesense that no deliberated result is ever accepted and thusnever integrated into the opinions of the agents that havevoted for it The reason for the latter is that the latitudeacceptance is too low and thereafter pair-wise interactionsare not enough to unevenly polarize the population in such away that deliberated proposal arguments are voted favorablyby a non-negligeable majority Shifts for 120572 = 0 are stable fordifferent levels of 119905119863 which implies that deliberation keeps incheck all the shifts of opinions related to an increase in thenumber of pair-wise interactions For 120572 = 05 deliberationis less likely to be successful and shifts tend towards the shiftlevels for 120572 = 0 (see Figure 9(b))

Concerning labeling-basedmetrics120572 entirely determinesthe coherence statistic For 120572 = 05 there is no difference incoherence because of how coherence is defined it is maximalwhen 120572 = 0 and strangely maximal when 120572 = 066(see Figure 9(h)) Surprisingly 120572 has no effect on judgmentaccuracy One would have expected a higher 120572 to increasejudgment accuracy since agents would hardly collectivelyaccept a proposal argument discussed during deliberationProposal arguments have given how the argument lattice isgenerated a higher probability of being rejected than of beingaccepted in the ideal argumentation framework

435 A Word on Focused and Naive Agents Focused andnaive specifications for agent behavior are an importantparameter in the decision process since they model howagents choose arguments in the deliberation arena In the sce-narios where agents are focused agents show equal or highervariance of opinions and are therefore more extreme (seeFigure 9(g)) independently of the voting requirements Shiftsof opinion are less likely as well We can see this happeningbecause focused agents knock out proposal arguments moreoften than naive agents provided that 119899119863 is low

One could have expected naive agents to yield morecoherent decisions since they argue sincerely However thisseems not to be the case (see Figure 9(i)) Agents are unable todeliberate in away thatmakes deliberation reflect their votingintentions either because they do not have the necessaryarguments to do so or because such arguments do not exist

In terms of judgment accuracy focused agents do betterthan naive agents in finding the ideal labels for the proposalarguments This is likely because when reconstructing theframework focused agents take into account the deliberatedproposal argument and choose themost pertinent argumentsin their sacks that relate to the state of affairs at the deliber-ation arena In fine debates result in a better approximationofL119886119887120576(119868) even though one could have believed the oppositethat is focused agents are ready to sacrifice a correct collectivelabel of a proposal argument in the goal of defending theirpositions

44 Points of Discussion In this subsection we discuss theresults obtained from the simulations We attempt to makesense of them from a social sciences perspective

441 Opinion Consensus and Dissensus The deliberativeopinion dynamics corresponds to the simulated scenarios inwhich the variance of opinions is at its lowest and judgmentaccuracy at its highest Dyadic opinion dynamics are associ-ated with the scenarios in which the variance of opinions is atits highest judgment accuracy at its lowest and coherence atitsmaximumpossible levelThemixed interactions dynamicsis a combination of these aspects for all the metrics exceptfor coherence at which it is the worsemdashdeliberation leadsto some kind of loose opinion consensus that is reinforcedby pair-wise interactions Likewise the mixed discussionmodel confirms some results exposed in [16] expressingthat opinions and voting intentions in a deliberative contextldquooften changerdquo That being said and more particularly indeliberative opinion dynamics outputs align well with twointeresting results on consensus formation given by aninterpretation of Moscovici and Doisersquos [15] work Sunsteinrsquos[14] account on deliberation and group polarization andsocial impact theory [4] The first states that mild consensusis reached in deliberation as procedure disengages agentsthe second is that within discussion groups majorities tendto become larger and opinions more extrememdashnot on thebasis of competing arguments but of the initial balance ofopinions After several decision processes we find opiniondistributions that reflect these observations even though weare not able to precisely coin the set of parameters that yieldssuch distributions (see Figure 7)

Although differences between mixed and dyadic opiniondynamics in terms of the variance of opinions and shiftsare slim the former does much better in terms of judg-ment accuracy In this sense deliberation does moderateopinions but not as much when non-deliberation steps arerelatively frequent throughout the decision-making processIn terms of Moscovici and Doise theory of consensus [15]this makes sense since pair-wise discussion engages or ego-involve agents in the discussion and pushes to convergenceof opinion at the extremes of the opinion distributionHowever what remains unclear is the reason why proceduralparameters that constrain deliberation do not disengageagents from discussion and thus result in a mild consensusas also expressed by Moscovici and Doise [15] A possibleexplanation of this phenomenon may stem from the factthat deliberation in the model is always accompanied by aseries of ego-involving discussions that shadowor counter themoderating effect of deliberation Not only this deliberationcould also be responsible for crystallizing the opinions ofextremists for it favors one or the other part of the distributionof opinions as explained in [14] and to some extent in [13]Furthermore the mixed discussion model gives an exampleof polarization that results from a minority of dissenterson one side of an issue acceding to views of the majorityrsquosside [4] as opposed to the strong polarization observed indyadic interactions Situations in which the status quo orthe undecidability of the proposal argument lingers cannotpossibly yield situations of moderation In cognitive science

24 Complexity

one can be inclined to believe that the longer a deliberationprocess is the more likely individuals will take an immutablestance on the situation In view of the assumption that longdecision-making processes can be more costly for a group orsociety as a whole agents may feel forced to take a stancejust for the sake of ending the process The stance that isadopted would therefore depend on pair-wise interactionsamong individuals and not on deliberation since deliberationtakes too long to be conclusive

442 For Hypothetical Recommendations in Decision-MakingProcess Elaboration On another tune results from themodelsuggest that procedural parameters have marginally decreas-ing positive or negative effects on the metrics of interest Ina hypothetical policy analysis perspective this may indicatethat after choosing a correct number of individuals andimposing the right amount of debates one can attain rea-sonable levels of judgment accuracy coherence and varianceof opinions As accounted in [18] deliberation increasesknowledge (collective knowledge as well) and therefore itheightens the chances of making correct decisions If onebelieves that social welfare is linked to deliberation andeventually to the metrics studied previously then proceduralparameters can both be used as an instrument to attaindesired levels of social welfare and reinforce the notionof legitimacy of collective decisions It can also provide ajustification for choosing a deliberation regime rather thananother on the basis of how important (or urgent) a topic ofdiscussion is Indeed if we adopt the claim that deliberationis a forerunner of welfare then any state in which deliberativecues are at their maximum (highest number of participantsmany deliberation steps before making a decision and asfrequent as possible) has to be mapped to the highest attain-able social welfare In this case is a situation where judgmentaccuracy is at its highest yet the variance of opinions or theproportion of extremists at its lowest an ideal situation Wecannot say for sure since strong opinion consensus and longdecision-making processes in many situations may result ina loss of welfare rather than in a gain of it

Another interesting result in procedure for deliberationis the very similar estimates we find for the effects of theminimum number (119898) of debates necessary to make adecision on a proposal and the proportion (119899119863) of individualsparticipating in deliberation on judgment accuracy In thescope of social welfare this raises the question of whetheradvocating for longer decision processes by demandingmoredeliberation steps (higher 119898) rather than bigger debates(higher 119899119863) may increase welfare Depending on the objec-tives of the decider of the deliberation procedure he or shemay choose to concentrate on one or the other A recom-mendation we can assert from the mixed social interactionsmodel is that if one prefers low to high variance of opinionsandor high to low coherence the decider is better off ifshe focuses on increasing the size of the debate rather thanorganizing debates very frequently One of the reasons forthis might be that when making bigger debates all agents aretruly considered as equals in the decision-making processIn organizing many debates individuals that are chosen toparticipate in many of these will be more influential and will

somehow bias the set of arguments in the deliberation arenaThemajority rulemodeled through120572 also determines the sizeof the shifts and the distribution of opinions Because a bigger120572 is associated with higher levels of extremism and opinionvariance deliberated cues are harder to accept and thereforeit becomesmore difficult for agents to reach concensus andormoderate their opinions

On the other hand and further into the idea of legitimacythe goal of deliberative processes is the agreement amongall participants This agreement stands for the right or bestdecision from a formal or procedural perspective and it is alsothe best from a substantive point of view In other words thelegitimacy of decisions not only derives fromwell-establishedprocedural requirements (respect of the protocol) but alsobegs for the fulfillment of two essential conditions (1) satisfy-ing the procedural requirements for a correct procedure (for-mal legitimacy) and (2) the rational acceptability of the resultsof this procedure (substantive acceptability) Legitimacy canbe then obtained from the coherence statistic scrutinizedin the model insofar as it measures how well agents acceptthe results of a correct procedure (eg deliberation) So fora proposal to be legitimate one has to choose parametersthat would maximize group coherence in decision-makingprocesses

As deliberation may give rise to consensus in the delib-eration arena it may also sow the seeds of dissensus inthe group Our model clearly illustrates this phenomenonthrough the procedural parameters of deliberation and themetrics observed From another perspective in deliberativedemocracy dissensus in collective decision-making may bedesirable According to Landermore and Pagersquos accounts [36]lack of consensus can be perceived as having agents withalternative ideas that combined with other ideas can providea better approximation of the ideal status of an argumentTheauthors in [36] call this dissensus ldquopositive dissensusrdquo andit makes normative sense when solving complex collectivedecision-making problems Our model captures the ideaof having normative requirements to define a good andlegitimate collective decision requirements of high consen-sual deliberative accuracy (judgment accuracy) of legitimacy(coherence) and of diversity (variance of opinions) are alltaken into account and for such we can conclude that ourmodel is successful in describing a sound process of collectivedecision-making with interpretable outcomes

Of course the present simulation results and analysis donot prove that all opinion dynamics can be accounted forby such simple processes like the one presented here Theydo however shed light on the expressiveness of combiningdifferent paradigms to account formore interpretablemodelsand pave the way to creating original models of the sortThe simulations suggest the desirability of discovering theconsequences of relatively simple laws of communication atdifferent levels (micro meso and macro) to determine whatstill needs to be explained

5 Related Work

We see our model as a contribution to the influence andopinion dynamics field in ABM and a pragmatic application

Complexity 25

of abstract argumentation theory To our knowledge we areunaware of existing literature on agent-based modeling thatexplicitly relates collective choice and the notions of deliber-ation and opinion diffusion through abstract argumentationas we have done it

Most models in the literature of opinion diffusion areinterested in opinions because these have an influence incollective decision and in questions of social order Forinstance in [3] the authors are interested in consensus andin how a group collectively chooses among two alternativesIn other models authors are interested in the emergenceof extremism [8] and on the distribution of opinions whenextremists are introduced in the population [5 33 37] whileother authors coin the notion of opinion polarization as anemergent property of the system [8 12] They show usingmodels of ldquobounded confidencerdquo and opinion diffusion withtrust that three different kinds of steady states (unipolarbipolar and central) are possible depending on whetheragents are sufficiently uncertain about their opinions andsufficiently connected andor a certain proportion of indi-viduals are already extreme Recent articles on informationand opinion dynamics stress the importance of governanceand government intervention in the spread of emotions(opinions) in the frangibility of social consensus systemIn [10] the authors show that government intervention inthe spread of negative emotions can lead to an even fasterspread of negative emotions (single-peeked convergence) andin a faster collapse of the social consensus system Opinionas a function of trust is studied in [12] and relates to howindividuals form their opinions on the basis of how muchthey trust agents in their networks

Another stream of discrete opinion dynamics modelsthat explain the emergence of opinion can be found in theliterature Computer multiagent models of attitude changebased on Latanersquos social impact theory are presented in [4] Inthe theory an agent changes her opinion on the basis of theinformational impact she is subject to which depends on thepersuasiveness (strength) supportiveness and immediacy(group structure) of the environment and information shereceives The simulations predict two emergent groups ofphenomena the shifting of attitudes towards incompletelypolarized equilibria and the formation of coherent clusteringof subgroups with deviant attitudes Similarly but in acontinuous more argumentative fashion Mas and Flache[11] present a model of opinion diffusion using argumentsArguments are considered to be for or against a proposal andare given an agent-dependant relevance in dyadic persuasiveinteractions Arguments also determine the agentsrsquo opinionsand their dissimilarities which posits the rules of pair-wiseencounters in theirmodel as agents are assumed to form theiropinions according to the arguments they own They showthat in an argumentative opinion diffusion model the onlyobservable steady states are bipolarization and consensusand that bipolarization can emerge from interactions amongsimilar agents In other words bipolarization of opinionsis possible with homophily and without negative influenceif there is argumentation They subsequently compare theirresults to experimental data In [38] a similar approach toargumentation and opinion formation is taken but for the fact

that they introduce different types of arguments explicitlyapply social judgment theory [22] and use survey data toestablish model-to-real-world comparisons

Closer to opinion formation and abstract argumentationGabbriellini and Torroni [39] are the first to have origi-nally and soundly merged opinion diffusion and abstractargumentation They define an agentrsquos opinion as a functionof the arguments she holds and the attack relation amongthem They devised a focused peer-to-peer dialogue systemof persuasion (NetArg) inspired from Mercier and Sperberrsquosargumentative theory of reasoning [40] which used onlyabstract argumentation to study opinion polarization andopinion dynamics Moreover they considered networks andthe notions of trust and epistemic vigilance to define adynamics for knowledge revision and trust itself whichwe clearly lack in our model They showed that if a con-servative belief operator in argumentation was applied byagents when they reasoned then their dialogue protocoldid not increase polarization among agents That said theyused the model to study the effect of Groenvetterrsquos weaklink theory in the spreading of arguments and were notparticularly interested in notions like judgment accuracy orcoherence nor in tackling questions related to collectivedecision-making procedures Their system is very expressiveand helps position our work in the litterature our work is atthe frontier of pure argumentative opinion diffusion modelsopinion diffusion with arguments and continuous bounded-confidence models of opinion diffusion a la Deffuant

On another note work on collective cognitive conver-gence [41] and opinion sharing [19] shows that consensustowards a certain ldquocorrectrdquo opinion or cognitive state is alwayspossible yet dependent on noise variability and awarenessof agents In [20] the authors show that learning about anexogenous correct state of the world (represented by bits)under confidence was possible but only if the agents werenot too confident In a homogeneous population they showthat the higher the confidence the worse the learningmdashthevery confident agents do not learn the properties of the truestate of the world and disrupt the learning process of the lessconfident ones Collective cognitive convergence can be seenas a result of deliberation in truth-seeking models

When it comes to abstract argumentation theory we takean approach that wires two types of dialogues that are well-studied in the literature persuasion dialogues [42] and delib-eration dialogues [43] Another interesting line of work is theone on mechanism design [44] or the problem of devisingan argumentation protocol where strategic argumentation isnot a liability for success in debates We tackle mechanismdesign in a different way though Instead of consideringstrategy-proofness we are interested in how differences inprotocol can result in epistemologically ldquobetterrdquo collectivechoices and guarantee that opinion distributions are favorablefor deliberation For a survey on persuasion dialogue see[42]

Work on agent-based argumentation usually assumes thatthe semantic relationship between arguments is fixed Inother words if two individuals were to put two argumentsin the public arena such that one logically attacked the otherthen everyone would agree that such attack exists [45 46]

26 Complexity

Other models which do not make this restrictive assumptioncan also be found in the literature [27 47] and derive fromthe class or family of opponent models [27 48] in which twoopposing sides attempt to win a dialogue Ourmodel is in theintersection of these two but the frame combining opiniondiffusion of the kind and argumentation remains new

The idea of mixing interpersonal influence and verticalcommunication however is not original It is described andimplemented in innovation diffusion models such as [5 49]In both vertical communication is modeled as exogenoustransparent informationmdashagents are aware of the existenceof an innovation thus triggering several processes of choiceand stabilization of opinions Also in [50] an Eulerian modelis implemented to show the effect of media and exogenousinformation on opinion distributions With respect to thispoint the originality of our work is in that the informationemitted as vertical communication is endogenous It is issuedfrom a deliberation model that agents shape on the basisof their opinions arguments and behavior In the spirit of[51] where the authors control for the design of verticalcommunication (to whom the vertical communication isaddressed its timing how it influences agents opinions) wecontrol for the process generating the information for ourown set of variables length of debates majority quotas andthe frequency at which discussions take place

6 Conclusion and Perspectives

The main objective of this article was to build a bridgewhere decision-making argumentation and opinion dif-fusion could come together We proposed a model thatcombined abstract argumentation theory and a boundedconfidence opinion diffusion model and showed to whatextent it could explain variance of opinions extremismcoherence in collective decisions and judgment accuracyon arguments given an ideal state of full information Thesecond objective was to show in what way governance pro-viding agents with arenas of discussion and deliberation wasimportant in addressing the success of collective decision-making processes and the quality of their outcomes

The model revealed that allowing for more deliberationtime though in lower frequency and allowing for widerparticipation in deliberation increased the variance of opin-ions and the proportion of extremists in a group Theseobservations are consistent with the combination of resultsfound in [9] and in [13 14] which stress that deliberationmay polarize groups and may have a meager effect on theshifts of opinions and inconsistent with [16] where it isargued that opinions tend to moderate after deliberationIt happens that deliberation alone did moderate opinionyet when integrated into a more complex system in whichindividuals were allowed to interact with one another andnot all that was deliberated was accepted its influence wasshadowed by other more individualistic dynamics

Undeniably the grounded semantics played an importantrole in the weak effect of deliberation since it models askeptical way of reasoning over arguments For agentsaccepting arguments thus updating their opinions accord-ingly happened not too often Nevertheless asking for more

deliberation did increase judgment accuracy as observed in[18] yet in a marginally decreasing fashion We showed thatvoting within the deliberation protocol not only increasedthe proportion of extremists and the variance of opinions ina group but also determined how coherent deliberation andvoting procedures were with each other Lastly we showedthat agents that are focused judged arguments better hadmore stable opinions and constituted a group with higherproportion of extremists than their naive counterparts

In terms of governance the model says that there is per-haps no trade-off between extremismand judgment accuracyInstead it asserts that the higher the variance of opinions thecloser the group gets to the correct decisions This situationmay be interpreted as Landermore and Pagersquos notion of ldquoposi-tive dissensusrdquo (in complex collective decision-making tasks)[36] as opposed to consensus as a normative requirementfor correct collective decision-making If a decider has toorganize deliberation to legitimize a public policy dependingon what kind of world he or she wants different set ofparameters may be chosen to account for different levels oflegitimacy and correctness For instance one may ask for adecision to be legitimate to respect a certain level of coherenceor of judgment accuracy and to be discussed by a sundryof different-opinionated peoples If so deliberation has tobe made more often more participants have to be includedin debates or in deliberation instances and not too manytime steps should be introduced in between two deliberationinstances So the model may capture normative ideas todefine correct and legitimate collective decisions and tomake recommendations on those grounds Requirements ofhigh consensual deliberative accuracy (judgment accuracy)legitimacy (coherence) and diversity (variance of opinions)are all taken into account and for such a reason the model israther successful in describing a sound process of collectivedecision-making with interpretable outcomes

61 Extensions of the Model and Other Ideas Our model canbe extended in many ways Value-based abstract argumen-tation frameworks (VAFs) for example as an extension ofDungrsquos argumentation framework provide a formal descrip-tion of the process of decision-making in which argumentsare given values and audiences (set of agents) ignore attacksbetween arguments on the basis of their preferences over val-ues Social abstract argumentation frameworks (SAFs) alsoprovide an interesting and seemingly convenient formalismfor our model In SAFs agents vote on the acceptability ofarguments and the resulting labeling is a combination of thearguments that are voted for and labeling-based semantics Inour model we apply the same rule yet only for one argumentand given precise agent-based votingmechanisms and votingrules which are eventually parametrized

To consider our model as a specification of a mixed VAFand SAF model may be of great interest for future workThis formalism can provide solid foundations to describeexplain interpret and eventually extend the results foundhere from an argument-based perspective On those groundsfuture work will focus on creating a hybrid argumentationframeworkmdashwith its semanticsmdashthat takes heed of bothvalues or principles and voting

Complexity 27

Concerning other possible extensions of the model anatural reflex would be to argue on the validity and therobustness of the mix between the bounded confidenceopinion diffusion models and deliberation Consequentiallyadding deliberation to different bounded confidence modelsof opinion diffusion such as the Hegelsmann-Krause model[37] or to models of social impact theory like in [4] canbe of great interest for future work We acknowledge that afiner understanding of argumentation frameworks randomlattices and their respective applications for this modelare necessary Testing for different labeling-based semanticsor even trading labeling-based semantics for scoring-basedsemantics (semantics that defines acceptability on scoresgiven to arguments) such as debate semantics and ranking-based semantics (semantics that yield a preorder) may beinteresting to consider in future work

More into modeling deliberation it is not inadequateto believe that agents may also advance proposals duringdeliberation and replace to some extent the proposal forwhich the deliberation is taking place Endogenizing theprocess that produces the type of argument to be deliberatedon may be interesting to explore either by having agentsstrategically replace proposals at certain moments duringthe debate or having the central authority attempt to chooseproposals such that the probability of having them acceptedis high Agents may also be thought to be autoorganizing(independent of a central authority) and concede on the sizeof the instances of deliberation To consider differences insemantics across agents or even endogenizing the semanticson how urgent or important the proposal that is beingdiscussed is can also be an interesting direction to take

Last but not least many nontrivial modifications of ourmodel are possible and most should include better-thoughtdeliberation protocols It may be interesting to design andobserve protocols in which deliberation only affects agentsthat are actually debating for example Trust network effectsmultidimensionality of opinions new processes of argumentexchange or learning among agents are notions to furtherdevelop in order to make the model more realistic and relaxsome unreliable assumptions In sum the model is to berefined and extended with the objective of either studyingconcrete cases of deliberative polling and opinion dynamicsor implementing more intuitive thought experiments

Appendix

We present below some of the pseudocode we used toimplement the model

Algorithms

(a) Algorithm 1 implements the dyadic influence in themodel It describes how two agents interact andupdate their opinions

(b) Algorithm 2 implements the deliberative influence inthe model which describes how agents update theiropinions in sight of the accepted deliberated proposalarguments

(c) Algorithm 3 is used to obtain the grounded labelingof an argumentation framework

(d) Algorithm 4 provides an overview of a run in thesimulations119872 is the number of proposal argumentsto discuss which we set to 100 in the model

Each object in the system is indexed by a119908ℎ119900 number Hencea set of agents is annotated by a set of natural numbers Weuse this fact in the implementation of the algorithm 119874119887119895119890119888119905119894denotes the object of 119908ℎ119900 number 119894 Agentsrsquo opinions (119900) arecalibrated so that 119900 isin [minus1 1]Data Availability

The program and data used to support the findings of thisstudy are available from the corresponding author uponrequest

Disclosure

A preliminary version of this work was presented anddiscussed at the conference Bridging the Gap between FormalArgumentation and Actual Human Reasoning on October 4-5 2018 at BochumGermanyThe authors thank the audiencefor their valuable comments and suggestions

Conflicts of Interest

The authors declare that they have no conflicts of interestregarding the publication of this article

Acknowledgments

The contribution of Gabriella Pigozzi was supported by theDeutsche Forschungsgemeinschaft (DFG) and the CzechScience Foundation (GACR) as part of the joint project FromShared Evidence to Group Attitudes (RO 45486-1)

Endnotes

1 In other words the superiority of the ldquoright answerrdquo orlabeling in our case will appear as such to all

2 All agents hold the same opinion3 By sensitivity we mean how much an agent is ready to

change her opinion on the sole fact that an informationalcue has been debated on

4 The assumption is strong but this trust among agentsmay be rooted from a genuine individual motivation toreach a democratic consensus

5 We adopted a rule of thumbs that says that if an agentis neutral with respect to the principle then she willvote for any proposal that is presented to her duringdeliberation

6 If the argument is not unique she chooses one atrandom

7 If there are more than one candidate the agent will favorthe argument at minimal distance from 119868

28 Complexity

Require Vector of values for parameters (119881 = (120583 119880 119879))Ensure Successful interactions among agents119896 larr997888 0119873119890 larr997888 0

while 119896 le |119873|2 do ⊳Fill119873119890 with |119873|2 agents119896 larr997888 119896 + 1119873119890 larr997888 119877119886119899119889119900119898(|119873| + 1) ⊳Pseudo-random generator Generates integer between 0 and |119873|119873119903 larr997888 0 120575119894119895 larr997888 0 ⊳Variable to store differences in opinionfor all 119894 isin 119873119890 doRandomly pick a number 119895 isin 119873(119873119890 cup 119873119903)119873119903 larr997888 119873119903 cup 119895 120575119894119895 larr997888 |119900119894 minus 119900119895|procedure dyadic-update(119894 119895) ⊳Two agents 119894 and 119895 discuss and update opinions1199001015840 larr997888 119900119894 (120583119895 119880119895 119879119895) larr997888 (120583119894 119880119894 119879119894) larr997888 (120583 119880 119879) ⊳Save agent irsquos opinion before updating and homogenize agents

if 120575119894119895 lt 119880119894 then119900119894 larr997888 120583119894119900119895 + (1 minus 120583119894)119900119894if 120575119894119895 gt 119879119894 then119900119894 larr997888 (1 + 120583119894)119900119894 minus 120583119894119900119895if 120575119894119895 lt 119880119895 then119900119895 larr997888 1205831198951199001015840 + (1 minus 120583119895)119900119895if 120575119894119895 gt 119879119895 then119900119895 larr997888 (1 + 120583119895)119900119895 minus 1205831198951199001015840

Algorithm 1 Dyadic social influence

Require Vector of values for parameters (119881 = (120574 119901119886 119901119903))Ensure Successful interaction between proposal arguments and agents

for all 119894 isin 119873 doprocedure Probability-Update(V119886 119901119886 119901119903) ⊳Update individual probabilities of change due to deliberated cues119909119894 larr997888 119901|V119886minus119900119894 |2119886 119910119894 larr997888 1199012|V119886minus119900119894 |119903119902 larr997888 0if L119886119887(119868) = 119894119899 then

procedure deliberative-update(119894 119886) ⊳Agent i argument 119886120574119894 larr997888 120574 119902 larr997888 119906 sim U(0 1) ⊳ 119906 is a realization of a uniform random variableif 119902 lt 119909119894 then119900119894 larr997888 120574119894119900119894 + (1 minus 120574119894)V119886break

if 119902 lt 119910119894(1 minus 119909119894) then119900119894 larr997888 (1 + 120574119894)119900119894 minus 120574119894V119886Algorithm 2 Deliberation influence

8 In terms of computational complexity naive agentsmaketheirmoves in119874(|A119894| log(|A119894|)) while focused agents doat least as bad

9 Tables have memory Therefore debates are not neces-sarily trees

10 If there are less than (119899119863 times |119873|)2 agents for any of bothgroups the CA summons all of them to the deliberationtable

11 Or careful consideration of the viewpoints of otherswhich implies that citizens keep an open mind and donot reject arguments outright

12 A ldquohealthyrdquo consensus refers to the case in which manyconsensual decisions are taken but opinion-wise agentsdo not agree with each other Quotes are used on theword healthy because some political science theorists

(eg [36]) believe that in certain situations agreement inopinions for deliberation is epistemologically damaging

13 Simulations on groups of |119873| = 800 and |119873| = 400agents shows that scaling the population has an effect onall themetrics but it is confounding depending onwhichcouples of attitudes structures are considered (119880 119879) Westick to |119873| = 400 as in [9] for comparability andsimulation running time

14 By balanced wemean with as many arguments with V119886 lt0 as with V119886 gt 0

15 Experiments on the system suggest that a bigger(smaller) argument pool 119872 = 1000 (119872 = 200) isassociated with a higher (lower) variance of opinions(and extremism) and higher (lower) judgment accuracyFor the rest of the observations 119872 seem to have noeffect We choose a middle value for 119872 Similarly theargument pool chosen to be balanced or uniformly

Complexity 29

Require Argumentation framework 119860119865 = (AR)Ensure Assignment of labels to arguments from the grounded labeling of 119860119865L(A)

L0(A) larr997888 (0 0 0)119896 larr997888 0do119894119899 L119896+1(A) larr997888 119894119899 L119896(A) cup 119886 | 119886 is not labeled 119894119899 L119896(A) 119886119899119889 forall119887 isin A 119904119905 119887 R 119886 then 119887 isin 119900119906119905 L119896(A)119900119906119905 L119896+1(A) larr997888 119900119906119905 L119896(A) cup 119886 | 119886 is not labeled 119894119899 L119896(A) 119886119899119889 exist119887 isin A 119904119905 119887 R 119886 and 119887 isin 119894119899 L119896+1(A)

while L119896+1(A) = L119896(A) = 0Algorithm 3 Solve argumentation frameworks (with grounded semantics)

Require Vector of values for parameters (119881)Ensure Vector of statistics (119880)

procedure Set-parameters(119881) ⊳Initialize parameters for scenarioprocedure Init-objs(119899119898 119896 ) ⊳Create119898 arguments 119899 agents and 119860119865120576procedure Solve-Argumentation-Framework(119860119865) ⊳Assigns the epistemic labelingL120576 toA119894 larr997888 1119905 larr997888 1while 119894 le 119872 do ⊳ 119872 the number of arguments to discuss119868119894 larr997888 119866119890119899 119886119903119892(1) ⊳Generates proposal argument 119868119860 119894119905 larr997888 (119868119894 0)L119886119887(119868119894) larr997888 119906119899119889 ⊳Initialize label for 119868119895 larr997888 1 ⊳Initialize debate counter 119895while ((L119886119887(119868) = 119906119899119889 or 119898 le 119895) and 119895 le 119898) doif 119905 equiv 0(119898119900119889 119905119863) then

procedure build-argumentation-framework(119899119863 119887119890ℎ119886V119894119900119903) ⊳Build the argument graph agents deliberate(119861119894119905 119878119894119905) larr997888 (119883 119884) sub (⋃119897isin119873119899119863 A119897119894119905 ⋃119897isin119873119899119863 R119897119894119905) ⊳ 119873119899119863 sub 119873 are agents sampled by the CA119860 119894119905 larr997888 119889(119860 119894119905 (119861119894119905 119878119894119905))procedure Solve-Argumentation-Framework(A119889) ⊳FindL(A119889)L119886119887(119868119894) larr997888 L119886119887119863(119868119894)119895 larr997888 119895 + 1

else119896 larr997888 1while 119896 le 119905119863 doprocedure dyadic-social-influence(120583 119879 119880) ⊳Agents discuss one-to-one119896 larr997888 119896 + 1119905 larr997888 119905 + 1

if 120572 gt 0 andL119886119887(119868119894) = 119900119906119905 then ⊳Agents vote for the proposalif (|119897 isin 119873|119900119897 times V119868119894 ge 0| ge 120572 times 119899) then

L119886119887(119868119894) larr997888 119894119899else

L119886119887(119868119894) larr997888 119900119906119905if L119886119887(119868119894) = 119894119899 thenprocedure Deliberation-influence(120574 119901119886 119901119903) ⊳Agents are influenced by the result of the decision process119894 larr997888 119894 + 1119905 larr997888 119905 + 1119880 larr997888 119862119900119898119901119906119905119890 119878119905119886119905119894119904119905119894119888119904(119881)

return 119880 ⊳Get the statisticsAlgorithm 4 Run for one scenario

generated does not seem to play a significant explanatoryrole on any of the metrics

16 Experiments show that themaximumnumber of debatesis not an important parameter even if it is an intuitiveone when considering decision-making processes Inpractice debates end after at most 119898 + 1 deliberation

steps and even then it happens rarely However to avoidunpleasant surprises like infinite debates on a completeargument graph containing the proposal argument weadded the constraint 119898 Its value is precisely 7 becausethe biggest minimum number of debates we studyis 6

30 Complexity

17 To ensure comparability with the deliberative andmixedopinion dynamics models on judgment accuracy andcoherence agents vote on proposal arguments every 119905119863times(119898+1) 119905119863 and119898does not explain variability in judgmentaccuracy nor in coherence

18 Student t-tests for independent samples are robust tothe violation of two hypothesis on the distribution ofthe samples normality and homogeneity of variancesas long as the sample size is big enough (central limittheorem) and if the difference in size of the comparedsamples is smallrespectively

19 Studentrsquos test version forwhen variances are not assumedequal and the difference in size of the compared samplesis big

20 We do not correct for homoskedasticity nor normality oferrors in the estimates

21 Balanced as the same number of runs per scenario orsuch that no significant correlations are found betweenparameters

References

[1] R Axelrod ldquoThe dissemination of culture a model withlocal convergence and global polarizationrdquo Journal of ConflictResolution vol 41 no 2 pp 203ndash226 1997

[2] D M Rousseau Reinforcing the MicroMacro Bridge Organiza-tionalThinking and Pluralistic Vehicles SAGEPublications SageLos Angeles Calif USA 2011

[3] S Galam and SMoscovici ldquoTowards a theory of collective phe-nomena consensus and attitude changes in groupsrdquo EuropeanJournal of Social Psychology vol 21 no 1 pp 49ndash74 1991

[4] A Nowak J Szamrej and B Latane ldquoFrom private attitude topublic opinion a dynamic theory of social impactrdquo Psychologi-cal Review vol 97 no 3 pp 362ndash376 1990

[5] G Deffuant S Huet and F Amblard ldquoAn individual-basedmodel of innovation diffusion mixing social value and individ-ual benefitrdquo American Journal of Sociology vol 110 no 4 pp1041ndash1069 2005

[6] G Deffuant D Neau F Amblard and G Weisbuch ldquoMixingbeliefs among interacting agentsrdquoAdvances in Complex Systems(ACS) vol 3 no 01n04 pp 87ndash98 2000

[7] J Rouchier P Tubaro and C Emery ldquoOpinion transmission inorganizations an agent-based modeling approachrdquo Computa-tional and Mathematical OrganizationTheory vol 20 no 3 pp252ndash277 2014

[8] M Meadows and D Cliff ldquoThe relative disagreement model ofopinion dynamics where do extremists come fromrdquo in Pro-ceedings of the International Workshop on Self-Organizing Sys-tems Lecture Notes in Computer Science pp 66ndash77 Springer2013

[9] W Jager and F Amblard ldquoUniformity bipolarization and plu-riformity captured as generic stylized behavior with an agent-based simulation model of attitude changerdquo Computational ampMathematical Organization Theory vol 10 no 4 pp 295ndash3032005

[10] Y-F Zhang H-Y Duan and Z-L Geng ldquoEvolutionary mech-anism of frangibility in social consensus system based onnegative emotions spreadrdquo Complexity vol 2017 Article ID4037049 8 pages 2017

[11] A Flache andMWMacy ldquoSmall worlds and cultural polariza-tionrdquoThe Journal of Mathematical Sociology vol 35 no 1-3 pp146ndash176 2011

[12] A Tsang and K Larson ldquoOpinion dynamics of skepticalagentsrdquo in Proceedings of the 13th International Conference onAutonomous Agents and Multiagent Systems AAMAS 2014 pp277ndash284 International Foundation forAutonomousAgents andMultiagent Systems France May 2014

[13] K M Hansen Deliberative democracy and opinion formationUniversity Press of Southern Denmark Odense 2004

[14] C R Sunstein ldquoThe law of group polarizationrdquo Journal ofPolitical Philosophy vol 10 no 2 pp 175ndash195 2002 WileyOnline Library

[15] S Moscovici and W Doise Dissensions et Consensus unetheorie generale des decisions collectives Presses Universitairesde France-PUF 1992

[16] J S Fishkin and R C Luskin ldquoExperimenting with a demo-cratic ideal deliberative polling and public opinionrdquo ActaPolitica vol 40 no 3 pp 284ndash298 2005

[17] R C Luskin J S Fishkin and S Iyengar Considered Opinionson US Foreign Policy Face-To-Face versus Online Delibera-tive Polling International Communication Association NewOrleans La USA 2004

[18] J Barabas ldquoHowdeliberation affects policy opinionsrdquoAmericanPolitical Science Review vol 98 no 4 pp 687ndash701 2004

[19] O Pryymak A Rogers and N R Jennings ldquoEfficient opinionsharing in large decentralised teamsrdquo in Proceedings of the 11thInternational Conference on Autonomous Agents andMultiagentSystems vol 1 pp 543ndash550 Spain June 2012

[20] J Rouchier and E Tanimura ldquoWhen overconfident agents slowdown collective learningrdquo Simulation vol 88 no 1 pp 33ndash492012

[21] D Miller ldquoDeliberative democracy and social choicerdquo PoliticalStudies vol 40 S1 pp 54ndash67 1992

[22] M Sherif and C I Hovland Social Judgment Assimilation andContrast Effects in Communication and Attitude Change YaleUniversity Press 1961

[23] P M Dung ldquoOn the acceptability of arguments and its funda-mental role in nonmonotonic reasoning logic programmingand n-person gamesrdquo Artificial Intelligence vol 77 no 2 pp321ndash357 1995

[24] M Caminada and G Pigozzi ldquoOn judgment aggregation inabstract argumentationrdquo Autonomous Agents and Multi-AgentSystems vol 22 no 1 pp 64ndash102 2011

[25] P Baroni M Caminada andM Giacomin ldquoAn introduction toargumentation semanticsrdquoThe Knowledge Engineering Reviewvol 26 no 4 pp 365ndash410 2011

[26] S Chambers ldquoDeliberative democratic theoryrdquo Annual Reviewof Political Science vol 6 no 1 pp 307ndash326 2003

[27] E Bonzon andNMaudet ldquoOn the outcomes of multiparty per-suasionrdquo in Proceedings of the 10th International Conference onAutonomous Agents and Multiagent Systems 2011 AAMAS 2011vol 1 pp 47ndash54 International Foundation for AutonomousAgents and Multiagent Systems Taiwan May 2011

[28] J Ranciere Disagreement Politics and Philosophy University ofMinnesota Press 1999

[29] T Bench-Capon and K Atkinson ldquoAbstract argumentation andvaluesrdquo in Argumentation in Artificial Intelligence pp 45ndash64Springer 2009

[30] M Caminada ldquoOn the issue of reinstatement in argumenta-tionrdquo in Proceedings of the European Workshop on Logics in

Complexity 31

Artificial Intelligence vol 4160 of Lecture Notes in ComputerScience pp 111ndash123 Springer Berlin Heidelberg

[31] GAVreeswijk ldquoAn algorithm to computeminimally groundedand admissible defence sets in argument systemsrdquo in Pro-ceedings of the 2006 conference on Computational Models ofArgument Proceedings of COMMA 2006 vol 144 pp 109ndash120IOS Press Amsterdam The Netherlands 2006

[32] C W Sherif M Kelly H L Rodgers et al ldquoPersonal involve-ment social judgment and actionrdquo Journal of Personality andSocial Psychology vol 27 no 3 pp 311ndash328 1973

[33] G Deffuant F Amblard G Weisbuch and T Faure ldquoHow canextremism prevail A study based on the relative agreementinteraction modelrdquo Journal of Artificial Societies and SocialSimulation vol 5 no 4 p 1 2002

[34] E Awad M W Caminada G Pigozzi M Podlaszewski andI Rahwan ldquoPareto optimality and strategy-proofness in groupargument evaluationrdquo Journal of Logic and Computation vol 27no 8 pp 2581ndash2609 2017

[35] D Grossi and G Pigozzi ldquoJudgment aggregation a primerrdquoSynthesis Lectures on Artificial Intelligence and Machine Learn-ing vol 8 no 2 pp 1ndash151 2014

[36] H Landemore and S E Page ldquoDeliberation and disagreementproblem solving prediction and positive dissensusrdquo PoliticsPhilosophy and Economics vol 14 no 3 pp 229ndash254 2015

[37] R Hegselmann and U Krause ldquoOpinion dynamics andbounded confidence models analysis and simulationrdquo Journalof Artificial Societies and Social Simulation vol 5 no 3 2002

[38] A Stefanelli and R Seidl ldquoModerate and polarized opinionsUsing empirical data for an agent-based simulationrdquo inProceed-ings of the Social Simulation Conference 2014

[39] S Gabbriellini and P Torroni ldquoA new framework for ABMsbased on argumentative reasoningrdquo in Advances in SocialSimulation vol 229 pp 25ndash36 Springer Berlin Germany 2014

[40] H Mercier and D Sperber ldquoWhy do humans reason Argu-ments for an argumentative theoryrdquo Behavioral and BrainSciences vol 34 no 2 pp 57ndash74 2011

[41] H V Parunak T C Belding R Hilscher and S BruecknerldquoModeling and managing collective cognitive convergencerdquoin In Proceedings of the 7th iNternational Joint Conference onAutonomous Agents and Multiagent Systems vol 3 pp 1505ndash1508 International Foundation for Autonomous Agents andMultiagent Systems 2008

[42] H Prakken ldquoFormal systems for persuasion dialoguerdquo TheKnowledge Engineering Review vol 21 no 2 pp 163ndash188 2006

[43] K Atkinson T Bench-Capon and P Mcburney ldquoA dialoguegame protocol for multi-agent argument over proposals foractionrdquoAutonomousAgents andMulti-Agent Systems vol 11 no2 pp 153ndash171 2005

[44] M Thimm and A J Garcıa ldquoClassification and strategicalissues of argumentation games on structured argumentationframeworksrdquo in Proceedings of the 9th International Conferenceon Autonomous Agents and Multiagent Systems vol 1 pp 1247ndash1254 International Foundation for Autonomous Agents andMultiagent Systems Canada 2010

[45] I Rahwan and K Larson ldquoMechanism design for abstractargumentationrdquo in Proceedings of the 7th International JointConference on Autonomous Agents and Multiagent SystemsAAMAS 2008 pp 1031ndash1038 International Foundation forAutonomous Agents and Multiagent Systems Portugal May2008

[46] H Prakken R Riveret A Rotolo and G Sartor ldquoHeuristics inArgumentation A Game-Theoretical Investigationrdquo EuropeanUniversity Institute 2008

[47] N Hadidi Y Dimopoulos and P Moraitis ldquoTactics and con-cessions for argumentation-based negotiationrdquo in Proceedingsof the COMMA vol 245 pp 285ndash296 2012

[48] N Oren and T J Norman ldquoArguing using opponent modelsrdquoin Proceedings of the International Workshop on Argumentationin Multi-Agent Systems pp 160ndash174 Springer 2009

[49] S Delre W Jager T Bijmolt and M Janssen ldquoTargeting andtiming promotional activities An agent-based model for thetakeoff of new productsrdquo Journal of Business Research vol 60no 8 pp 826ndash835 2007

[50] A Mirtabatabaei P Jia and F Bullo ldquoEulerian opinion dynam-ics with bounded confidence and exogenous inputsrdquo SIAMJournal on Applied Dynamical Systems vol 13 no 1 pp 425ndash446 2014

[51] V Barbet N Guiraud V Laperriere and J Rouchier Hagglingon Values Towards Consensus or Trouble 2019

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

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Hindawiwwwhindawicom Volume 2018

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22 Complexity

015

020

025

030

035

040

1 2 3 4 5 6

12345

Values of Ｇ

(a) 119905119863 times 119901119903119900119901119890119909 by119898

0

10

20

30

40

50

1 2 3 4 5 6

005066

Values of

(b) 119905119863 times 119878ℎ by 120572

040

045

050

055

060

1 2 3 4 5

001002003004005

Values of Ｈ＄

(c) 119898 times 119890119888 by 119899119863

015

020

025

030

035

040

1 2 3 4 5

001002003004005

Values of Ｈ＄

(d) 119898 times 119901119903119900119901119890119909 by 119899119863

0

10

20

30

40

50

001 002 003 004 005

00005066

Values of

(e) 119899119863 times 119878ℎ by 120572

040

045

050

055

060

1 2 3 4 5

BehaviorNaiveFocused

(f) 119898 times 119890119888 by 119887119890ℎ119886V119894119900119903

015

020

025

030

035

040

0 05 066Behavior

NaiveFocused

(g) 120572 times 119901119903119900119901119890119909 by 119887119890ℎ119886V119894119900119903

05

06

07

08

09

10

1 2 3 4 5 6

00005066

Values of

(h) 119905119863 times 119894119903 by 120572

08700

08725

08750

08775

08800

001 002 003 004 005

BehaviorNaiveFocused

(i) 119899119863 times 119894119903 by 119887119890ℎ119886V119894119900119903

Figure 9 Curves of mean observations for different parameters and metrics for 36000 runs Bars are 095 confidence intervals Plots withthe same observables (ordinate) except for 119894119903 are on the same scale

Complexity 23

The higher the requirement for accepting the proposal isthe higher the proportion of extremists and the variance ofopinions are and the lower the shift statistic is In few words120572 constrains the world to dyadic discussion or throws it intoa process in which deliberation is much more importantthan dyadic interactions This is why for any 120572 = 05 oneeither gives too much weight to deliberated results whichshadows pair-wise interactions or too little weight in thesense that no deliberated result is ever accepted and thusnever integrated into the opinions of the agents that havevoted for it The reason for the latter is that the latitudeacceptance is too low and thereafter pair-wise interactionsare not enough to unevenly polarize the population in such away that deliberated proposal arguments are voted favorablyby a non-negligeable majority Shifts for 120572 = 0 are stable fordifferent levels of 119905119863 which implies that deliberation keeps incheck all the shifts of opinions related to an increase in thenumber of pair-wise interactions For 120572 = 05 deliberationis less likely to be successful and shifts tend towards the shiftlevels for 120572 = 0 (see Figure 9(b))

Concerning labeling-basedmetrics120572 entirely determinesthe coherence statistic For 120572 = 05 there is no difference incoherence because of how coherence is defined it is maximalwhen 120572 = 0 and strangely maximal when 120572 = 066(see Figure 9(h)) Surprisingly 120572 has no effect on judgmentaccuracy One would have expected a higher 120572 to increasejudgment accuracy since agents would hardly collectivelyaccept a proposal argument discussed during deliberationProposal arguments have given how the argument lattice isgenerated a higher probability of being rejected than of beingaccepted in the ideal argumentation framework

435 A Word on Focused and Naive Agents Focused andnaive specifications for agent behavior are an importantparameter in the decision process since they model howagents choose arguments in the deliberation arena In the sce-narios where agents are focused agents show equal or highervariance of opinions and are therefore more extreme (seeFigure 9(g)) independently of the voting requirements Shiftsof opinion are less likely as well We can see this happeningbecause focused agents knock out proposal arguments moreoften than naive agents provided that 119899119863 is low

One could have expected naive agents to yield morecoherent decisions since they argue sincerely However thisseems not to be the case (see Figure 9(i)) Agents are unable todeliberate in away thatmakes deliberation reflect their votingintentions either because they do not have the necessaryarguments to do so or because such arguments do not exist

In terms of judgment accuracy focused agents do betterthan naive agents in finding the ideal labels for the proposalarguments This is likely because when reconstructing theframework focused agents take into account the deliberatedproposal argument and choose themost pertinent argumentsin their sacks that relate to the state of affairs at the deliber-ation arena In fine debates result in a better approximationofL119886119887120576(119868) even though one could have believed the oppositethat is focused agents are ready to sacrifice a correct collectivelabel of a proposal argument in the goal of defending theirpositions

44 Points of Discussion In this subsection we discuss theresults obtained from the simulations We attempt to makesense of them from a social sciences perspective

441 Opinion Consensus and Dissensus The deliberativeopinion dynamics corresponds to the simulated scenarios inwhich the variance of opinions is at its lowest and judgmentaccuracy at its highest Dyadic opinion dynamics are associ-ated with the scenarios in which the variance of opinions is atits highest judgment accuracy at its lowest and coherence atitsmaximumpossible levelThemixed interactions dynamicsis a combination of these aspects for all the metrics exceptfor coherence at which it is the worsemdashdeliberation leadsto some kind of loose opinion consensus that is reinforcedby pair-wise interactions Likewise the mixed discussionmodel confirms some results exposed in [16] expressingthat opinions and voting intentions in a deliberative contextldquooften changerdquo That being said and more particularly indeliberative opinion dynamics outputs align well with twointeresting results on consensus formation given by aninterpretation of Moscovici and Doisersquos [15] work Sunsteinrsquos[14] account on deliberation and group polarization andsocial impact theory [4] The first states that mild consensusis reached in deliberation as procedure disengages agentsthe second is that within discussion groups majorities tendto become larger and opinions more extrememdashnot on thebasis of competing arguments but of the initial balance ofopinions After several decision processes we find opiniondistributions that reflect these observations even though weare not able to precisely coin the set of parameters that yieldssuch distributions (see Figure 7)

Although differences between mixed and dyadic opiniondynamics in terms of the variance of opinions and shiftsare slim the former does much better in terms of judg-ment accuracy In this sense deliberation does moderateopinions but not as much when non-deliberation steps arerelatively frequent throughout the decision-making processIn terms of Moscovici and Doise theory of consensus [15]this makes sense since pair-wise discussion engages or ego-involve agents in the discussion and pushes to convergenceof opinion at the extremes of the opinion distributionHowever what remains unclear is the reason why proceduralparameters that constrain deliberation do not disengageagents from discussion and thus result in a mild consensusas also expressed by Moscovici and Doise [15] A possibleexplanation of this phenomenon may stem from the factthat deliberation in the model is always accompanied by aseries of ego-involving discussions that shadowor counter themoderating effect of deliberation Not only this deliberationcould also be responsible for crystallizing the opinions ofextremists for it favors one or the other part of the distributionof opinions as explained in [14] and to some extent in [13]Furthermore the mixed discussion model gives an exampleof polarization that results from a minority of dissenterson one side of an issue acceding to views of the majorityrsquosside [4] as opposed to the strong polarization observed indyadic interactions Situations in which the status quo orthe undecidability of the proposal argument lingers cannotpossibly yield situations of moderation In cognitive science

24 Complexity

one can be inclined to believe that the longer a deliberationprocess is the more likely individuals will take an immutablestance on the situation In view of the assumption that longdecision-making processes can be more costly for a group orsociety as a whole agents may feel forced to take a stancejust for the sake of ending the process The stance that isadopted would therefore depend on pair-wise interactionsamong individuals and not on deliberation since deliberationtakes too long to be conclusive

442 For Hypothetical Recommendations in Decision-MakingProcess Elaboration On another tune results from themodelsuggest that procedural parameters have marginally decreas-ing positive or negative effects on the metrics of interest Ina hypothetical policy analysis perspective this may indicatethat after choosing a correct number of individuals andimposing the right amount of debates one can attain rea-sonable levels of judgment accuracy coherence and varianceof opinions As accounted in [18] deliberation increasesknowledge (collective knowledge as well) and therefore itheightens the chances of making correct decisions If onebelieves that social welfare is linked to deliberation andeventually to the metrics studied previously then proceduralparameters can both be used as an instrument to attaindesired levels of social welfare and reinforce the notionof legitimacy of collective decisions It can also provide ajustification for choosing a deliberation regime rather thananother on the basis of how important (or urgent) a topic ofdiscussion is Indeed if we adopt the claim that deliberationis a forerunner of welfare then any state in which deliberativecues are at their maximum (highest number of participantsmany deliberation steps before making a decision and asfrequent as possible) has to be mapped to the highest attain-able social welfare In this case is a situation where judgmentaccuracy is at its highest yet the variance of opinions or theproportion of extremists at its lowest an ideal situation Wecannot say for sure since strong opinion consensus and longdecision-making processes in many situations may result ina loss of welfare rather than in a gain of it

Another interesting result in procedure for deliberationis the very similar estimates we find for the effects of theminimum number (119898) of debates necessary to make adecision on a proposal and the proportion (119899119863) of individualsparticipating in deliberation on judgment accuracy In thescope of social welfare this raises the question of whetheradvocating for longer decision processes by demandingmoredeliberation steps (higher 119898) rather than bigger debates(higher 119899119863) may increase welfare Depending on the objec-tives of the decider of the deliberation procedure he or shemay choose to concentrate on one or the other A recom-mendation we can assert from the mixed social interactionsmodel is that if one prefers low to high variance of opinionsandor high to low coherence the decider is better off ifshe focuses on increasing the size of the debate rather thanorganizing debates very frequently One of the reasons forthis might be that when making bigger debates all agents aretruly considered as equals in the decision-making processIn organizing many debates individuals that are chosen toparticipate in many of these will be more influential and will

somehow bias the set of arguments in the deliberation arenaThemajority rulemodeled through120572 also determines the sizeof the shifts and the distribution of opinions Because a bigger120572 is associated with higher levels of extremism and opinionvariance deliberated cues are harder to accept and thereforeit becomesmore difficult for agents to reach concensus andormoderate their opinions

On the other hand and further into the idea of legitimacythe goal of deliberative processes is the agreement amongall participants This agreement stands for the right or bestdecision from a formal or procedural perspective and it is alsothe best from a substantive point of view In other words thelegitimacy of decisions not only derives fromwell-establishedprocedural requirements (respect of the protocol) but alsobegs for the fulfillment of two essential conditions (1) satisfy-ing the procedural requirements for a correct procedure (for-mal legitimacy) and (2) the rational acceptability of the resultsof this procedure (substantive acceptability) Legitimacy canbe then obtained from the coherence statistic scrutinizedin the model insofar as it measures how well agents acceptthe results of a correct procedure (eg deliberation) So fora proposal to be legitimate one has to choose parametersthat would maximize group coherence in decision-makingprocesses

As deliberation may give rise to consensus in the delib-eration arena it may also sow the seeds of dissensus inthe group Our model clearly illustrates this phenomenonthrough the procedural parameters of deliberation and themetrics observed From another perspective in deliberativedemocracy dissensus in collective decision-making may bedesirable According to Landermore and Pagersquos accounts [36]lack of consensus can be perceived as having agents withalternative ideas that combined with other ideas can providea better approximation of the ideal status of an argumentTheauthors in [36] call this dissensus ldquopositive dissensusrdquo andit makes normative sense when solving complex collectivedecision-making problems Our model captures the ideaof having normative requirements to define a good andlegitimate collective decision requirements of high consen-sual deliberative accuracy (judgment accuracy) of legitimacy(coherence) and of diversity (variance of opinions) are alltaken into account and for such we can conclude that ourmodel is successful in describing a sound process of collectivedecision-making with interpretable outcomes

Of course the present simulation results and analysis donot prove that all opinion dynamics can be accounted forby such simple processes like the one presented here Theydo however shed light on the expressiveness of combiningdifferent paradigms to account formore interpretablemodelsand pave the way to creating original models of the sortThe simulations suggest the desirability of discovering theconsequences of relatively simple laws of communication atdifferent levels (micro meso and macro) to determine whatstill needs to be explained

5 Related Work

We see our model as a contribution to the influence andopinion dynamics field in ABM and a pragmatic application

Complexity 25

of abstract argumentation theory To our knowledge we areunaware of existing literature on agent-based modeling thatexplicitly relates collective choice and the notions of deliber-ation and opinion diffusion through abstract argumentationas we have done it

Most models in the literature of opinion diffusion areinterested in opinions because these have an influence incollective decision and in questions of social order Forinstance in [3] the authors are interested in consensus andin how a group collectively chooses among two alternativesIn other models authors are interested in the emergenceof extremism [8] and on the distribution of opinions whenextremists are introduced in the population [5 33 37] whileother authors coin the notion of opinion polarization as anemergent property of the system [8 12] They show usingmodels of ldquobounded confidencerdquo and opinion diffusion withtrust that three different kinds of steady states (unipolarbipolar and central) are possible depending on whetheragents are sufficiently uncertain about their opinions andsufficiently connected andor a certain proportion of indi-viduals are already extreme Recent articles on informationand opinion dynamics stress the importance of governanceand government intervention in the spread of emotions(opinions) in the frangibility of social consensus systemIn [10] the authors show that government intervention inthe spread of negative emotions can lead to an even fasterspread of negative emotions (single-peeked convergence) andin a faster collapse of the social consensus system Opinionas a function of trust is studied in [12] and relates to howindividuals form their opinions on the basis of how muchthey trust agents in their networks

Another stream of discrete opinion dynamics modelsthat explain the emergence of opinion can be found in theliterature Computer multiagent models of attitude changebased on Latanersquos social impact theory are presented in [4] Inthe theory an agent changes her opinion on the basis of theinformational impact she is subject to which depends on thepersuasiveness (strength) supportiveness and immediacy(group structure) of the environment and information shereceives The simulations predict two emergent groups ofphenomena the shifting of attitudes towards incompletelypolarized equilibria and the formation of coherent clusteringof subgroups with deviant attitudes Similarly but in acontinuous more argumentative fashion Mas and Flache[11] present a model of opinion diffusion using argumentsArguments are considered to be for or against a proposal andare given an agent-dependant relevance in dyadic persuasiveinteractions Arguments also determine the agentsrsquo opinionsand their dissimilarities which posits the rules of pair-wiseencounters in theirmodel as agents are assumed to form theiropinions according to the arguments they own They showthat in an argumentative opinion diffusion model the onlyobservable steady states are bipolarization and consensusand that bipolarization can emerge from interactions amongsimilar agents In other words bipolarization of opinionsis possible with homophily and without negative influenceif there is argumentation They subsequently compare theirresults to experimental data In [38] a similar approach toargumentation and opinion formation is taken but for the fact

that they introduce different types of arguments explicitlyapply social judgment theory [22] and use survey data toestablish model-to-real-world comparisons

Closer to opinion formation and abstract argumentationGabbriellini and Torroni [39] are the first to have origi-nally and soundly merged opinion diffusion and abstractargumentation They define an agentrsquos opinion as a functionof the arguments she holds and the attack relation amongthem They devised a focused peer-to-peer dialogue systemof persuasion (NetArg) inspired from Mercier and Sperberrsquosargumentative theory of reasoning [40] which used onlyabstract argumentation to study opinion polarization andopinion dynamics Moreover they considered networks andthe notions of trust and epistemic vigilance to define adynamics for knowledge revision and trust itself whichwe clearly lack in our model They showed that if a con-servative belief operator in argumentation was applied byagents when they reasoned then their dialogue protocoldid not increase polarization among agents That said theyused the model to study the effect of Groenvetterrsquos weaklink theory in the spreading of arguments and were notparticularly interested in notions like judgment accuracy orcoherence nor in tackling questions related to collectivedecision-making procedures Their system is very expressiveand helps position our work in the litterature our work is atthe frontier of pure argumentative opinion diffusion modelsopinion diffusion with arguments and continuous bounded-confidence models of opinion diffusion a la Deffuant

On another note work on collective cognitive conver-gence [41] and opinion sharing [19] shows that consensustowards a certain ldquocorrectrdquo opinion or cognitive state is alwayspossible yet dependent on noise variability and awarenessof agents In [20] the authors show that learning about anexogenous correct state of the world (represented by bits)under confidence was possible but only if the agents werenot too confident In a homogeneous population they showthat the higher the confidence the worse the learningmdashthevery confident agents do not learn the properties of the truestate of the world and disrupt the learning process of the lessconfident ones Collective cognitive convergence can be seenas a result of deliberation in truth-seeking models

When it comes to abstract argumentation theory we takean approach that wires two types of dialogues that are well-studied in the literature persuasion dialogues [42] and delib-eration dialogues [43] Another interesting line of work is theone on mechanism design [44] or the problem of devisingan argumentation protocol where strategic argumentation isnot a liability for success in debates We tackle mechanismdesign in a different way though Instead of consideringstrategy-proofness we are interested in how differences inprotocol can result in epistemologically ldquobetterrdquo collectivechoices and guarantee that opinion distributions are favorablefor deliberation For a survey on persuasion dialogue see[42]

Work on agent-based argumentation usually assumes thatthe semantic relationship between arguments is fixed Inother words if two individuals were to put two argumentsin the public arena such that one logically attacked the otherthen everyone would agree that such attack exists [45 46]

26 Complexity

Other models which do not make this restrictive assumptioncan also be found in the literature [27 47] and derive fromthe class or family of opponent models [27 48] in which twoopposing sides attempt to win a dialogue Ourmodel is in theintersection of these two but the frame combining opiniondiffusion of the kind and argumentation remains new

The idea of mixing interpersonal influence and verticalcommunication however is not original It is described andimplemented in innovation diffusion models such as [5 49]In both vertical communication is modeled as exogenoustransparent informationmdashagents are aware of the existenceof an innovation thus triggering several processes of choiceand stabilization of opinions Also in [50] an Eulerian modelis implemented to show the effect of media and exogenousinformation on opinion distributions With respect to thispoint the originality of our work is in that the informationemitted as vertical communication is endogenous It is issuedfrom a deliberation model that agents shape on the basisof their opinions arguments and behavior In the spirit of[51] where the authors control for the design of verticalcommunication (to whom the vertical communication isaddressed its timing how it influences agents opinions) wecontrol for the process generating the information for ourown set of variables length of debates majority quotas andthe frequency at which discussions take place

6 Conclusion and Perspectives

The main objective of this article was to build a bridgewhere decision-making argumentation and opinion dif-fusion could come together We proposed a model thatcombined abstract argumentation theory and a boundedconfidence opinion diffusion model and showed to whatextent it could explain variance of opinions extremismcoherence in collective decisions and judgment accuracyon arguments given an ideal state of full information Thesecond objective was to show in what way governance pro-viding agents with arenas of discussion and deliberation wasimportant in addressing the success of collective decision-making processes and the quality of their outcomes

The model revealed that allowing for more deliberationtime though in lower frequency and allowing for widerparticipation in deliberation increased the variance of opin-ions and the proportion of extremists in a group Theseobservations are consistent with the combination of resultsfound in [9] and in [13 14] which stress that deliberationmay polarize groups and may have a meager effect on theshifts of opinions and inconsistent with [16] where it isargued that opinions tend to moderate after deliberationIt happens that deliberation alone did moderate opinionyet when integrated into a more complex system in whichindividuals were allowed to interact with one another andnot all that was deliberated was accepted its influence wasshadowed by other more individualistic dynamics

Undeniably the grounded semantics played an importantrole in the weak effect of deliberation since it models askeptical way of reasoning over arguments For agentsaccepting arguments thus updating their opinions accord-ingly happened not too often Nevertheless asking for more

deliberation did increase judgment accuracy as observed in[18] yet in a marginally decreasing fashion We showed thatvoting within the deliberation protocol not only increasedthe proportion of extremists and the variance of opinions ina group but also determined how coherent deliberation andvoting procedures were with each other Lastly we showedthat agents that are focused judged arguments better hadmore stable opinions and constituted a group with higherproportion of extremists than their naive counterparts

In terms of governance the model says that there is per-haps no trade-off between extremismand judgment accuracyInstead it asserts that the higher the variance of opinions thecloser the group gets to the correct decisions This situationmay be interpreted as Landermore and Pagersquos notion of ldquoposi-tive dissensusrdquo (in complex collective decision-making tasks)[36] as opposed to consensus as a normative requirementfor correct collective decision-making If a decider has toorganize deliberation to legitimize a public policy dependingon what kind of world he or she wants different set ofparameters may be chosen to account for different levels oflegitimacy and correctness For instance one may ask for adecision to be legitimate to respect a certain level of coherenceor of judgment accuracy and to be discussed by a sundryof different-opinionated peoples If so deliberation has tobe made more often more participants have to be includedin debates or in deliberation instances and not too manytime steps should be introduced in between two deliberationinstances So the model may capture normative ideas todefine correct and legitimate collective decisions and tomake recommendations on those grounds Requirements ofhigh consensual deliberative accuracy (judgment accuracy)legitimacy (coherence) and diversity (variance of opinions)are all taken into account and for such a reason the model israther successful in describing a sound process of collectivedecision-making with interpretable outcomes

61 Extensions of the Model and Other Ideas Our model canbe extended in many ways Value-based abstract argumen-tation frameworks (VAFs) for example as an extension ofDungrsquos argumentation framework provide a formal descrip-tion of the process of decision-making in which argumentsare given values and audiences (set of agents) ignore attacksbetween arguments on the basis of their preferences over val-ues Social abstract argumentation frameworks (SAFs) alsoprovide an interesting and seemingly convenient formalismfor our model In SAFs agents vote on the acceptability ofarguments and the resulting labeling is a combination of thearguments that are voted for and labeling-based semantics Inour model we apply the same rule yet only for one argumentand given precise agent-based votingmechanisms and votingrules which are eventually parametrized

To consider our model as a specification of a mixed VAFand SAF model may be of great interest for future workThis formalism can provide solid foundations to describeexplain interpret and eventually extend the results foundhere from an argument-based perspective On those groundsfuture work will focus on creating a hybrid argumentationframeworkmdashwith its semanticsmdashthat takes heed of bothvalues or principles and voting

Complexity 27

Concerning other possible extensions of the model anatural reflex would be to argue on the validity and therobustness of the mix between the bounded confidenceopinion diffusion models and deliberation Consequentiallyadding deliberation to different bounded confidence modelsof opinion diffusion such as the Hegelsmann-Krause model[37] or to models of social impact theory like in [4] canbe of great interest for future work We acknowledge that afiner understanding of argumentation frameworks randomlattices and their respective applications for this modelare necessary Testing for different labeling-based semanticsor even trading labeling-based semantics for scoring-basedsemantics (semantics that defines acceptability on scoresgiven to arguments) such as debate semantics and ranking-based semantics (semantics that yield a preorder) may beinteresting to consider in future work

More into modeling deliberation it is not inadequateto believe that agents may also advance proposals duringdeliberation and replace to some extent the proposal forwhich the deliberation is taking place Endogenizing theprocess that produces the type of argument to be deliberatedon may be interesting to explore either by having agentsstrategically replace proposals at certain moments duringthe debate or having the central authority attempt to chooseproposals such that the probability of having them acceptedis high Agents may also be thought to be autoorganizing(independent of a central authority) and concede on the sizeof the instances of deliberation To consider differences insemantics across agents or even endogenizing the semanticson how urgent or important the proposal that is beingdiscussed is can also be an interesting direction to take

Last but not least many nontrivial modifications of ourmodel are possible and most should include better-thoughtdeliberation protocols It may be interesting to design andobserve protocols in which deliberation only affects agentsthat are actually debating for example Trust network effectsmultidimensionality of opinions new processes of argumentexchange or learning among agents are notions to furtherdevelop in order to make the model more realistic and relaxsome unreliable assumptions In sum the model is to berefined and extended with the objective of either studyingconcrete cases of deliberative polling and opinion dynamicsor implementing more intuitive thought experiments

Appendix

We present below some of the pseudocode we used toimplement the model

Algorithms

(a) Algorithm 1 implements the dyadic influence in themodel It describes how two agents interact andupdate their opinions

(b) Algorithm 2 implements the deliberative influence inthe model which describes how agents update theiropinions in sight of the accepted deliberated proposalarguments

(c) Algorithm 3 is used to obtain the grounded labelingof an argumentation framework

(d) Algorithm 4 provides an overview of a run in thesimulations119872 is the number of proposal argumentsto discuss which we set to 100 in the model

Each object in the system is indexed by a119908ℎ119900 number Hencea set of agents is annotated by a set of natural numbers Weuse this fact in the implementation of the algorithm 119874119887119895119890119888119905119894denotes the object of 119908ℎ119900 number 119894 Agentsrsquo opinions (119900) arecalibrated so that 119900 isin [minus1 1]Data Availability

The program and data used to support the findings of thisstudy are available from the corresponding author uponrequest

Disclosure

A preliminary version of this work was presented anddiscussed at the conference Bridging the Gap between FormalArgumentation and Actual Human Reasoning on October 4-5 2018 at BochumGermanyThe authors thank the audiencefor their valuable comments and suggestions

Conflicts of Interest

The authors declare that they have no conflicts of interestregarding the publication of this article

Acknowledgments

The contribution of Gabriella Pigozzi was supported by theDeutsche Forschungsgemeinschaft (DFG) and the CzechScience Foundation (GACR) as part of the joint project FromShared Evidence to Group Attitudes (RO 45486-1)

Endnotes

1 In other words the superiority of the ldquoright answerrdquo orlabeling in our case will appear as such to all

2 All agents hold the same opinion3 By sensitivity we mean how much an agent is ready to

change her opinion on the sole fact that an informationalcue has been debated on

4 The assumption is strong but this trust among agentsmay be rooted from a genuine individual motivation toreach a democratic consensus

5 We adopted a rule of thumbs that says that if an agentis neutral with respect to the principle then she willvote for any proposal that is presented to her duringdeliberation

6 If the argument is not unique she chooses one atrandom

7 If there are more than one candidate the agent will favorthe argument at minimal distance from 119868

28 Complexity

Require Vector of values for parameters (119881 = (120583 119880 119879))Ensure Successful interactions among agents119896 larr997888 0119873119890 larr997888 0

while 119896 le |119873|2 do ⊳Fill119873119890 with |119873|2 agents119896 larr997888 119896 + 1119873119890 larr997888 119877119886119899119889119900119898(|119873| + 1) ⊳Pseudo-random generator Generates integer between 0 and |119873|119873119903 larr997888 0 120575119894119895 larr997888 0 ⊳Variable to store differences in opinionfor all 119894 isin 119873119890 doRandomly pick a number 119895 isin 119873(119873119890 cup 119873119903)119873119903 larr997888 119873119903 cup 119895 120575119894119895 larr997888 |119900119894 minus 119900119895|procedure dyadic-update(119894 119895) ⊳Two agents 119894 and 119895 discuss and update opinions1199001015840 larr997888 119900119894 (120583119895 119880119895 119879119895) larr997888 (120583119894 119880119894 119879119894) larr997888 (120583 119880 119879) ⊳Save agent irsquos opinion before updating and homogenize agents

if 120575119894119895 lt 119880119894 then119900119894 larr997888 120583119894119900119895 + (1 minus 120583119894)119900119894if 120575119894119895 gt 119879119894 then119900119894 larr997888 (1 + 120583119894)119900119894 minus 120583119894119900119895if 120575119894119895 lt 119880119895 then119900119895 larr997888 1205831198951199001015840 + (1 minus 120583119895)119900119895if 120575119894119895 gt 119879119895 then119900119895 larr997888 (1 + 120583119895)119900119895 minus 1205831198951199001015840

Algorithm 1 Dyadic social influence

Require Vector of values for parameters (119881 = (120574 119901119886 119901119903))Ensure Successful interaction between proposal arguments and agents

for all 119894 isin 119873 doprocedure Probability-Update(V119886 119901119886 119901119903) ⊳Update individual probabilities of change due to deliberated cues119909119894 larr997888 119901|V119886minus119900119894 |2119886 119910119894 larr997888 1199012|V119886minus119900119894 |119903119902 larr997888 0if L119886119887(119868) = 119894119899 then

procedure deliberative-update(119894 119886) ⊳Agent i argument 119886120574119894 larr997888 120574 119902 larr997888 119906 sim U(0 1) ⊳ 119906 is a realization of a uniform random variableif 119902 lt 119909119894 then119900119894 larr997888 120574119894119900119894 + (1 minus 120574119894)V119886break

if 119902 lt 119910119894(1 minus 119909119894) then119900119894 larr997888 (1 + 120574119894)119900119894 minus 120574119894V119886Algorithm 2 Deliberation influence

8 In terms of computational complexity naive agentsmaketheirmoves in119874(|A119894| log(|A119894|)) while focused agents doat least as bad

9 Tables have memory Therefore debates are not neces-sarily trees

10 If there are less than (119899119863 times |119873|)2 agents for any of bothgroups the CA summons all of them to the deliberationtable

11 Or careful consideration of the viewpoints of otherswhich implies that citizens keep an open mind and donot reject arguments outright

12 A ldquohealthyrdquo consensus refers to the case in which manyconsensual decisions are taken but opinion-wise agentsdo not agree with each other Quotes are used on theword healthy because some political science theorists

(eg [36]) believe that in certain situations agreement inopinions for deliberation is epistemologically damaging

13 Simulations on groups of |119873| = 800 and |119873| = 400agents shows that scaling the population has an effect onall themetrics but it is confounding depending onwhichcouples of attitudes structures are considered (119880 119879) Westick to |119873| = 400 as in [9] for comparability andsimulation running time

14 By balanced wemean with as many arguments with V119886 lt0 as with V119886 gt 0

15 Experiments on the system suggest that a bigger(smaller) argument pool 119872 = 1000 (119872 = 200) isassociated with a higher (lower) variance of opinions(and extremism) and higher (lower) judgment accuracyFor the rest of the observations 119872 seem to have noeffect We choose a middle value for 119872 Similarly theargument pool chosen to be balanced or uniformly

Complexity 29

Require Argumentation framework 119860119865 = (AR)Ensure Assignment of labels to arguments from the grounded labeling of 119860119865L(A)

L0(A) larr997888 (0 0 0)119896 larr997888 0do119894119899 L119896+1(A) larr997888 119894119899 L119896(A) cup 119886 | 119886 is not labeled 119894119899 L119896(A) 119886119899119889 forall119887 isin A 119904119905 119887 R 119886 then 119887 isin 119900119906119905 L119896(A)119900119906119905 L119896+1(A) larr997888 119900119906119905 L119896(A) cup 119886 | 119886 is not labeled 119894119899 L119896(A) 119886119899119889 exist119887 isin A 119904119905 119887 R 119886 and 119887 isin 119894119899 L119896+1(A)

while L119896+1(A) = L119896(A) = 0Algorithm 3 Solve argumentation frameworks (with grounded semantics)

Require Vector of values for parameters (119881)Ensure Vector of statistics (119880)

procedure Set-parameters(119881) ⊳Initialize parameters for scenarioprocedure Init-objs(119899119898 119896 ) ⊳Create119898 arguments 119899 agents and 119860119865120576procedure Solve-Argumentation-Framework(119860119865) ⊳Assigns the epistemic labelingL120576 toA119894 larr997888 1119905 larr997888 1while 119894 le 119872 do ⊳ 119872 the number of arguments to discuss119868119894 larr997888 119866119890119899 119886119903119892(1) ⊳Generates proposal argument 119868119860 119894119905 larr997888 (119868119894 0)L119886119887(119868119894) larr997888 119906119899119889 ⊳Initialize label for 119868119895 larr997888 1 ⊳Initialize debate counter 119895while ((L119886119887(119868) = 119906119899119889 or 119898 le 119895) and 119895 le 119898) doif 119905 equiv 0(119898119900119889 119905119863) then

procedure build-argumentation-framework(119899119863 119887119890ℎ119886V119894119900119903) ⊳Build the argument graph agents deliberate(119861119894119905 119878119894119905) larr997888 (119883 119884) sub (⋃119897isin119873119899119863 A119897119894119905 ⋃119897isin119873119899119863 R119897119894119905) ⊳ 119873119899119863 sub 119873 are agents sampled by the CA119860 119894119905 larr997888 119889(119860 119894119905 (119861119894119905 119878119894119905))procedure Solve-Argumentation-Framework(A119889) ⊳FindL(A119889)L119886119887(119868119894) larr997888 L119886119887119863(119868119894)119895 larr997888 119895 + 1

else119896 larr997888 1while 119896 le 119905119863 doprocedure dyadic-social-influence(120583 119879 119880) ⊳Agents discuss one-to-one119896 larr997888 119896 + 1119905 larr997888 119905 + 1

if 120572 gt 0 andL119886119887(119868119894) = 119900119906119905 then ⊳Agents vote for the proposalif (|119897 isin 119873|119900119897 times V119868119894 ge 0| ge 120572 times 119899) then

L119886119887(119868119894) larr997888 119894119899else

L119886119887(119868119894) larr997888 119900119906119905if L119886119887(119868119894) = 119894119899 thenprocedure Deliberation-influence(120574 119901119886 119901119903) ⊳Agents are influenced by the result of the decision process119894 larr997888 119894 + 1119905 larr997888 119905 + 1119880 larr997888 119862119900119898119901119906119905119890 119878119905119886119905119894119904119905119894119888119904(119881)

return 119880 ⊳Get the statisticsAlgorithm 4 Run for one scenario

generated does not seem to play a significant explanatoryrole on any of the metrics

16 Experiments show that themaximumnumber of debatesis not an important parameter even if it is an intuitiveone when considering decision-making processes Inpractice debates end after at most 119898 + 1 deliberation

steps and even then it happens rarely However to avoidunpleasant surprises like infinite debates on a completeargument graph containing the proposal argument weadded the constraint 119898 Its value is precisely 7 becausethe biggest minimum number of debates we studyis 6

30 Complexity

17 To ensure comparability with the deliberative andmixedopinion dynamics models on judgment accuracy andcoherence agents vote on proposal arguments every 119905119863times(119898+1) 119905119863 and119898does not explain variability in judgmentaccuracy nor in coherence

18 Student t-tests for independent samples are robust tothe violation of two hypothesis on the distribution ofthe samples normality and homogeneity of variancesas long as the sample size is big enough (central limittheorem) and if the difference in size of the comparedsamples is smallrespectively

19 Studentrsquos test version forwhen variances are not assumedequal and the difference in size of the compared samplesis big

20 We do not correct for homoskedasticity nor normality oferrors in the estimates

21 Balanced as the same number of runs per scenario orsuch that no significant correlations are found betweenparameters

References

[1] R Axelrod ldquoThe dissemination of culture a model withlocal convergence and global polarizationrdquo Journal of ConflictResolution vol 41 no 2 pp 203ndash226 1997

[2] D M Rousseau Reinforcing the MicroMacro Bridge Organiza-tionalThinking and Pluralistic Vehicles SAGEPublications SageLos Angeles Calif USA 2011

[3] S Galam and SMoscovici ldquoTowards a theory of collective phe-nomena consensus and attitude changes in groupsrdquo EuropeanJournal of Social Psychology vol 21 no 1 pp 49ndash74 1991

[4] A Nowak J Szamrej and B Latane ldquoFrom private attitude topublic opinion a dynamic theory of social impactrdquo Psychologi-cal Review vol 97 no 3 pp 362ndash376 1990

[5] G Deffuant S Huet and F Amblard ldquoAn individual-basedmodel of innovation diffusion mixing social value and individ-ual benefitrdquo American Journal of Sociology vol 110 no 4 pp1041ndash1069 2005

[6] G Deffuant D Neau F Amblard and G Weisbuch ldquoMixingbeliefs among interacting agentsrdquoAdvances in Complex Systems(ACS) vol 3 no 01n04 pp 87ndash98 2000

[7] J Rouchier P Tubaro and C Emery ldquoOpinion transmission inorganizations an agent-based modeling approachrdquo Computa-tional and Mathematical OrganizationTheory vol 20 no 3 pp252ndash277 2014

[8] M Meadows and D Cliff ldquoThe relative disagreement model ofopinion dynamics where do extremists come fromrdquo in Pro-ceedings of the International Workshop on Self-Organizing Sys-tems Lecture Notes in Computer Science pp 66ndash77 Springer2013

[9] W Jager and F Amblard ldquoUniformity bipolarization and plu-riformity captured as generic stylized behavior with an agent-based simulation model of attitude changerdquo Computational ampMathematical Organization Theory vol 10 no 4 pp 295ndash3032005

[10] Y-F Zhang H-Y Duan and Z-L Geng ldquoEvolutionary mech-anism of frangibility in social consensus system based onnegative emotions spreadrdquo Complexity vol 2017 Article ID4037049 8 pages 2017

[11] A Flache andMWMacy ldquoSmall worlds and cultural polariza-tionrdquoThe Journal of Mathematical Sociology vol 35 no 1-3 pp146ndash176 2011

[12] A Tsang and K Larson ldquoOpinion dynamics of skepticalagentsrdquo in Proceedings of the 13th International Conference onAutonomous Agents and Multiagent Systems AAMAS 2014 pp277ndash284 International Foundation forAutonomousAgents andMultiagent Systems France May 2014

[13] K M Hansen Deliberative democracy and opinion formationUniversity Press of Southern Denmark Odense 2004

[14] C R Sunstein ldquoThe law of group polarizationrdquo Journal ofPolitical Philosophy vol 10 no 2 pp 175ndash195 2002 WileyOnline Library

[15] S Moscovici and W Doise Dissensions et Consensus unetheorie generale des decisions collectives Presses Universitairesde France-PUF 1992

[16] J S Fishkin and R C Luskin ldquoExperimenting with a demo-cratic ideal deliberative polling and public opinionrdquo ActaPolitica vol 40 no 3 pp 284ndash298 2005

[17] R C Luskin J S Fishkin and S Iyengar Considered Opinionson US Foreign Policy Face-To-Face versus Online Delibera-tive Polling International Communication Association NewOrleans La USA 2004

[18] J Barabas ldquoHowdeliberation affects policy opinionsrdquoAmericanPolitical Science Review vol 98 no 4 pp 687ndash701 2004

[19] O Pryymak A Rogers and N R Jennings ldquoEfficient opinionsharing in large decentralised teamsrdquo in Proceedings of the 11thInternational Conference on Autonomous Agents andMultiagentSystems vol 1 pp 543ndash550 Spain June 2012

[20] J Rouchier and E Tanimura ldquoWhen overconfident agents slowdown collective learningrdquo Simulation vol 88 no 1 pp 33ndash492012

[21] D Miller ldquoDeliberative democracy and social choicerdquo PoliticalStudies vol 40 S1 pp 54ndash67 1992

[22] M Sherif and C I Hovland Social Judgment Assimilation andContrast Effects in Communication and Attitude Change YaleUniversity Press 1961

[23] P M Dung ldquoOn the acceptability of arguments and its funda-mental role in nonmonotonic reasoning logic programmingand n-person gamesrdquo Artificial Intelligence vol 77 no 2 pp321ndash357 1995

[24] M Caminada and G Pigozzi ldquoOn judgment aggregation inabstract argumentationrdquo Autonomous Agents and Multi-AgentSystems vol 22 no 1 pp 64ndash102 2011

[25] P Baroni M Caminada andM Giacomin ldquoAn introduction toargumentation semanticsrdquoThe Knowledge Engineering Reviewvol 26 no 4 pp 365ndash410 2011

[26] S Chambers ldquoDeliberative democratic theoryrdquo Annual Reviewof Political Science vol 6 no 1 pp 307ndash326 2003

[27] E Bonzon andNMaudet ldquoOn the outcomes of multiparty per-suasionrdquo in Proceedings of the 10th International Conference onAutonomous Agents and Multiagent Systems 2011 AAMAS 2011vol 1 pp 47ndash54 International Foundation for AutonomousAgents and Multiagent Systems Taiwan May 2011

[28] J Ranciere Disagreement Politics and Philosophy University ofMinnesota Press 1999

[29] T Bench-Capon and K Atkinson ldquoAbstract argumentation andvaluesrdquo in Argumentation in Artificial Intelligence pp 45ndash64Springer 2009

[30] M Caminada ldquoOn the issue of reinstatement in argumenta-tionrdquo in Proceedings of the European Workshop on Logics in

Complexity 31

Artificial Intelligence vol 4160 of Lecture Notes in ComputerScience pp 111ndash123 Springer Berlin Heidelberg

[31] GAVreeswijk ldquoAn algorithm to computeminimally groundedand admissible defence sets in argument systemsrdquo in Pro-ceedings of the 2006 conference on Computational Models ofArgument Proceedings of COMMA 2006 vol 144 pp 109ndash120IOS Press Amsterdam The Netherlands 2006

[32] C W Sherif M Kelly H L Rodgers et al ldquoPersonal involve-ment social judgment and actionrdquo Journal of Personality andSocial Psychology vol 27 no 3 pp 311ndash328 1973

[33] G Deffuant F Amblard G Weisbuch and T Faure ldquoHow canextremism prevail A study based on the relative agreementinteraction modelrdquo Journal of Artificial Societies and SocialSimulation vol 5 no 4 p 1 2002

[34] E Awad M W Caminada G Pigozzi M Podlaszewski andI Rahwan ldquoPareto optimality and strategy-proofness in groupargument evaluationrdquo Journal of Logic and Computation vol 27no 8 pp 2581ndash2609 2017

[35] D Grossi and G Pigozzi ldquoJudgment aggregation a primerrdquoSynthesis Lectures on Artificial Intelligence and Machine Learn-ing vol 8 no 2 pp 1ndash151 2014

[36] H Landemore and S E Page ldquoDeliberation and disagreementproblem solving prediction and positive dissensusrdquo PoliticsPhilosophy and Economics vol 14 no 3 pp 229ndash254 2015

[37] R Hegselmann and U Krause ldquoOpinion dynamics andbounded confidence models analysis and simulationrdquo Journalof Artificial Societies and Social Simulation vol 5 no 3 2002

[38] A Stefanelli and R Seidl ldquoModerate and polarized opinionsUsing empirical data for an agent-based simulationrdquo inProceed-ings of the Social Simulation Conference 2014

[39] S Gabbriellini and P Torroni ldquoA new framework for ABMsbased on argumentative reasoningrdquo in Advances in SocialSimulation vol 229 pp 25ndash36 Springer Berlin Germany 2014

[40] H Mercier and D Sperber ldquoWhy do humans reason Argu-ments for an argumentative theoryrdquo Behavioral and BrainSciences vol 34 no 2 pp 57ndash74 2011

[41] H V Parunak T C Belding R Hilscher and S BruecknerldquoModeling and managing collective cognitive convergencerdquoin In Proceedings of the 7th iNternational Joint Conference onAutonomous Agents and Multiagent Systems vol 3 pp 1505ndash1508 International Foundation for Autonomous Agents andMultiagent Systems 2008

[42] H Prakken ldquoFormal systems for persuasion dialoguerdquo TheKnowledge Engineering Review vol 21 no 2 pp 163ndash188 2006

[43] K Atkinson T Bench-Capon and P Mcburney ldquoA dialoguegame protocol for multi-agent argument over proposals foractionrdquoAutonomousAgents andMulti-Agent Systems vol 11 no2 pp 153ndash171 2005

[44] M Thimm and A J Garcıa ldquoClassification and strategicalissues of argumentation games on structured argumentationframeworksrdquo in Proceedings of the 9th International Conferenceon Autonomous Agents and Multiagent Systems vol 1 pp 1247ndash1254 International Foundation for Autonomous Agents andMultiagent Systems Canada 2010

[45] I Rahwan and K Larson ldquoMechanism design for abstractargumentationrdquo in Proceedings of the 7th International JointConference on Autonomous Agents and Multiagent SystemsAAMAS 2008 pp 1031ndash1038 International Foundation forAutonomous Agents and Multiagent Systems Portugal May2008

[46] H Prakken R Riveret A Rotolo and G Sartor ldquoHeuristics inArgumentation A Game-Theoretical Investigationrdquo EuropeanUniversity Institute 2008

[47] N Hadidi Y Dimopoulos and P Moraitis ldquoTactics and con-cessions for argumentation-based negotiationrdquo in Proceedingsof the COMMA vol 245 pp 285ndash296 2012

[48] N Oren and T J Norman ldquoArguing using opponent modelsrdquoin Proceedings of the International Workshop on Argumentationin Multi-Agent Systems pp 160ndash174 Springer 2009

[49] S Delre W Jager T Bijmolt and M Janssen ldquoTargeting andtiming promotional activities An agent-based model for thetakeoff of new productsrdquo Journal of Business Research vol 60no 8 pp 826ndash835 2007

[50] A Mirtabatabaei P Jia and F Bullo ldquoEulerian opinion dynam-ics with bounded confidence and exogenous inputsrdquo SIAMJournal on Applied Dynamical Systems vol 13 no 1 pp 425ndash446 2014

[51] V Barbet N Guiraud V Laperriere and J Rouchier Hagglingon Values Towards Consensus or Trouble 2019

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Complexity 23

The higher the requirement for accepting the proposal isthe higher the proportion of extremists and the variance ofopinions are and the lower the shift statistic is In few words120572 constrains the world to dyadic discussion or throws it intoa process in which deliberation is much more importantthan dyadic interactions This is why for any 120572 = 05 oneeither gives too much weight to deliberated results whichshadows pair-wise interactions or too little weight in thesense that no deliberated result is ever accepted and thusnever integrated into the opinions of the agents that havevoted for it The reason for the latter is that the latitudeacceptance is too low and thereafter pair-wise interactionsare not enough to unevenly polarize the population in such away that deliberated proposal arguments are voted favorablyby a non-negligeable majority Shifts for 120572 = 0 are stable fordifferent levels of 119905119863 which implies that deliberation keeps incheck all the shifts of opinions related to an increase in thenumber of pair-wise interactions For 120572 = 05 deliberationis less likely to be successful and shifts tend towards the shiftlevels for 120572 = 0 (see Figure 9(b))

Concerning labeling-basedmetrics120572 entirely determinesthe coherence statistic For 120572 = 05 there is no difference incoherence because of how coherence is defined it is maximalwhen 120572 = 0 and strangely maximal when 120572 = 066(see Figure 9(h)) Surprisingly 120572 has no effect on judgmentaccuracy One would have expected a higher 120572 to increasejudgment accuracy since agents would hardly collectivelyaccept a proposal argument discussed during deliberationProposal arguments have given how the argument lattice isgenerated a higher probability of being rejected than of beingaccepted in the ideal argumentation framework

435 A Word on Focused and Naive Agents Focused andnaive specifications for agent behavior are an importantparameter in the decision process since they model howagents choose arguments in the deliberation arena In the sce-narios where agents are focused agents show equal or highervariance of opinions and are therefore more extreme (seeFigure 9(g)) independently of the voting requirements Shiftsof opinion are less likely as well We can see this happeningbecause focused agents knock out proposal arguments moreoften than naive agents provided that 119899119863 is low

One could have expected naive agents to yield morecoherent decisions since they argue sincerely However thisseems not to be the case (see Figure 9(i)) Agents are unable todeliberate in away thatmakes deliberation reflect their votingintentions either because they do not have the necessaryarguments to do so or because such arguments do not exist

In terms of judgment accuracy focused agents do betterthan naive agents in finding the ideal labels for the proposalarguments This is likely because when reconstructing theframework focused agents take into account the deliberatedproposal argument and choose themost pertinent argumentsin their sacks that relate to the state of affairs at the deliber-ation arena In fine debates result in a better approximationofL119886119887120576(119868) even though one could have believed the oppositethat is focused agents are ready to sacrifice a correct collectivelabel of a proposal argument in the goal of defending theirpositions

44 Points of Discussion In this subsection we discuss theresults obtained from the simulations We attempt to makesense of them from a social sciences perspective

441 Opinion Consensus and Dissensus The deliberativeopinion dynamics corresponds to the simulated scenarios inwhich the variance of opinions is at its lowest and judgmentaccuracy at its highest Dyadic opinion dynamics are associ-ated with the scenarios in which the variance of opinions is atits highest judgment accuracy at its lowest and coherence atitsmaximumpossible levelThemixed interactions dynamicsis a combination of these aspects for all the metrics exceptfor coherence at which it is the worsemdashdeliberation leadsto some kind of loose opinion consensus that is reinforcedby pair-wise interactions Likewise the mixed discussionmodel confirms some results exposed in [16] expressingthat opinions and voting intentions in a deliberative contextldquooften changerdquo That being said and more particularly indeliberative opinion dynamics outputs align well with twointeresting results on consensus formation given by aninterpretation of Moscovici and Doisersquos [15] work Sunsteinrsquos[14] account on deliberation and group polarization andsocial impact theory [4] The first states that mild consensusis reached in deliberation as procedure disengages agentsthe second is that within discussion groups majorities tendto become larger and opinions more extrememdashnot on thebasis of competing arguments but of the initial balance ofopinions After several decision processes we find opiniondistributions that reflect these observations even though weare not able to precisely coin the set of parameters that yieldssuch distributions (see Figure 7)

Although differences between mixed and dyadic opiniondynamics in terms of the variance of opinions and shiftsare slim the former does much better in terms of judg-ment accuracy In this sense deliberation does moderateopinions but not as much when non-deliberation steps arerelatively frequent throughout the decision-making processIn terms of Moscovici and Doise theory of consensus [15]this makes sense since pair-wise discussion engages or ego-involve agents in the discussion and pushes to convergenceof opinion at the extremes of the opinion distributionHowever what remains unclear is the reason why proceduralparameters that constrain deliberation do not disengageagents from discussion and thus result in a mild consensusas also expressed by Moscovici and Doise [15] A possibleexplanation of this phenomenon may stem from the factthat deliberation in the model is always accompanied by aseries of ego-involving discussions that shadowor counter themoderating effect of deliberation Not only this deliberationcould also be responsible for crystallizing the opinions ofextremists for it favors one or the other part of the distributionof opinions as explained in [14] and to some extent in [13]Furthermore the mixed discussion model gives an exampleof polarization that results from a minority of dissenterson one side of an issue acceding to views of the majorityrsquosside [4] as opposed to the strong polarization observed indyadic interactions Situations in which the status quo orthe undecidability of the proposal argument lingers cannotpossibly yield situations of moderation In cognitive science

24 Complexity

one can be inclined to believe that the longer a deliberationprocess is the more likely individuals will take an immutablestance on the situation In view of the assumption that longdecision-making processes can be more costly for a group orsociety as a whole agents may feel forced to take a stancejust for the sake of ending the process The stance that isadopted would therefore depend on pair-wise interactionsamong individuals and not on deliberation since deliberationtakes too long to be conclusive

442 For Hypothetical Recommendations in Decision-MakingProcess Elaboration On another tune results from themodelsuggest that procedural parameters have marginally decreas-ing positive or negative effects on the metrics of interest Ina hypothetical policy analysis perspective this may indicatethat after choosing a correct number of individuals andimposing the right amount of debates one can attain rea-sonable levels of judgment accuracy coherence and varianceof opinions As accounted in [18] deliberation increasesknowledge (collective knowledge as well) and therefore itheightens the chances of making correct decisions If onebelieves that social welfare is linked to deliberation andeventually to the metrics studied previously then proceduralparameters can both be used as an instrument to attaindesired levels of social welfare and reinforce the notionof legitimacy of collective decisions It can also provide ajustification for choosing a deliberation regime rather thananother on the basis of how important (or urgent) a topic ofdiscussion is Indeed if we adopt the claim that deliberationis a forerunner of welfare then any state in which deliberativecues are at their maximum (highest number of participantsmany deliberation steps before making a decision and asfrequent as possible) has to be mapped to the highest attain-able social welfare In this case is a situation where judgmentaccuracy is at its highest yet the variance of opinions or theproportion of extremists at its lowest an ideal situation Wecannot say for sure since strong opinion consensus and longdecision-making processes in many situations may result ina loss of welfare rather than in a gain of it

Another interesting result in procedure for deliberationis the very similar estimates we find for the effects of theminimum number (119898) of debates necessary to make adecision on a proposal and the proportion (119899119863) of individualsparticipating in deliberation on judgment accuracy In thescope of social welfare this raises the question of whetheradvocating for longer decision processes by demandingmoredeliberation steps (higher 119898) rather than bigger debates(higher 119899119863) may increase welfare Depending on the objec-tives of the decider of the deliberation procedure he or shemay choose to concentrate on one or the other A recom-mendation we can assert from the mixed social interactionsmodel is that if one prefers low to high variance of opinionsandor high to low coherence the decider is better off ifshe focuses on increasing the size of the debate rather thanorganizing debates very frequently One of the reasons forthis might be that when making bigger debates all agents aretruly considered as equals in the decision-making processIn organizing many debates individuals that are chosen toparticipate in many of these will be more influential and will

somehow bias the set of arguments in the deliberation arenaThemajority rulemodeled through120572 also determines the sizeof the shifts and the distribution of opinions Because a bigger120572 is associated with higher levels of extremism and opinionvariance deliberated cues are harder to accept and thereforeit becomesmore difficult for agents to reach concensus andormoderate their opinions

On the other hand and further into the idea of legitimacythe goal of deliberative processes is the agreement amongall participants This agreement stands for the right or bestdecision from a formal or procedural perspective and it is alsothe best from a substantive point of view In other words thelegitimacy of decisions not only derives fromwell-establishedprocedural requirements (respect of the protocol) but alsobegs for the fulfillment of two essential conditions (1) satisfy-ing the procedural requirements for a correct procedure (for-mal legitimacy) and (2) the rational acceptability of the resultsof this procedure (substantive acceptability) Legitimacy canbe then obtained from the coherence statistic scrutinizedin the model insofar as it measures how well agents acceptthe results of a correct procedure (eg deliberation) So fora proposal to be legitimate one has to choose parametersthat would maximize group coherence in decision-makingprocesses

As deliberation may give rise to consensus in the delib-eration arena it may also sow the seeds of dissensus inthe group Our model clearly illustrates this phenomenonthrough the procedural parameters of deliberation and themetrics observed From another perspective in deliberativedemocracy dissensus in collective decision-making may bedesirable According to Landermore and Pagersquos accounts [36]lack of consensus can be perceived as having agents withalternative ideas that combined with other ideas can providea better approximation of the ideal status of an argumentTheauthors in [36] call this dissensus ldquopositive dissensusrdquo andit makes normative sense when solving complex collectivedecision-making problems Our model captures the ideaof having normative requirements to define a good andlegitimate collective decision requirements of high consen-sual deliberative accuracy (judgment accuracy) of legitimacy(coherence) and of diversity (variance of opinions) are alltaken into account and for such we can conclude that ourmodel is successful in describing a sound process of collectivedecision-making with interpretable outcomes

Of course the present simulation results and analysis donot prove that all opinion dynamics can be accounted forby such simple processes like the one presented here Theydo however shed light on the expressiveness of combiningdifferent paradigms to account formore interpretablemodelsand pave the way to creating original models of the sortThe simulations suggest the desirability of discovering theconsequences of relatively simple laws of communication atdifferent levels (micro meso and macro) to determine whatstill needs to be explained

5 Related Work

We see our model as a contribution to the influence andopinion dynamics field in ABM and a pragmatic application

Complexity 25

of abstract argumentation theory To our knowledge we areunaware of existing literature on agent-based modeling thatexplicitly relates collective choice and the notions of deliber-ation and opinion diffusion through abstract argumentationas we have done it

Most models in the literature of opinion diffusion areinterested in opinions because these have an influence incollective decision and in questions of social order Forinstance in [3] the authors are interested in consensus andin how a group collectively chooses among two alternativesIn other models authors are interested in the emergenceof extremism [8] and on the distribution of opinions whenextremists are introduced in the population [5 33 37] whileother authors coin the notion of opinion polarization as anemergent property of the system [8 12] They show usingmodels of ldquobounded confidencerdquo and opinion diffusion withtrust that three different kinds of steady states (unipolarbipolar and central) are possible depending on whetheragents are sufficiently uncertain about their opinions andsufficiently connected andor a certain proportion of indi-viduals are already extreme Recent articles on informationand opinion dynamics stress the importance of governanceand government intervention in the spread of emotions(opinions) in the frangibility of social consensus systemIn [10] the authors show that government intervention inthe spread of negative emotions can lead to an even fasterspread of negative emotions (single-peeked convergence) andin a faster collapse of the social consensus system Opinionas a function of trust is studied in [12] and relates to howindividuals form their opinions on the basis of how muchthey trust agents in their networks

Another stream of discrete opinion dynamics modelsthat explain the emergence of opinion can be found in theliterature Computer multiagent models of attitude changebased on Latanersquos social impact theory are presented in [4] Inthe theory an agent changes her opinion on the basis of theinformational impact she is subject to which depends on thepersuasiveness (strength) supportiveness and immediacy(group structure) of the environment and information shereceives The simulations predict two emergent groups ofphenomena the shifting of attitudes towards incompletelypolarized equilibria and the formation of coherent clusteringof subgroups with deviant attitudes Similarly but in acontinuous more argumentative fashion Mas and Flache[11] present a model of opinion diffusion using argumentsArguments are considered to be for or against a proposal andare given an agent-dependant relevance in dyadic persuasiveinteractions Arguments also determine the agentsrsquo opinionsand their dissimilarities which posits the rules of pair-wiseencounters in theirmodel as agents are assumed to form theiropinions according to the arguments they own They showthat in an argumentative opinion diffusion model the onlyobservable steady states are bipolarization and consensusand that bipolarization can emerge from interactions amongsimilar agents In other words bipolarization of opinionsis possible with homophily and without negative influenceif there is argumentation They subsequently compare theirresults to experimental data In [38] a similar approach toargumentation and opinion formation is taken but for the fact

that they introduce different types of arguments explicitlyapply social judgment theory [22] and use survey data toestablish model-to-real-world comparisons

Closer to opinion formation and abstract argumentationGabbriellini and Torroni [39] are the first to have origi-nally and soundly merged opinion diffusion and abstractargumentation They define an agentrsquos opinion as a functionof the arguments she holds and the attack relation amongthem They devised a focused peer-to-peer dialogue systemof persuasion (NetArg) inspired from Mercier and Sperberrsquosargumentative theory of reasoning [40] which used onlyabstract argumentation to study opinion polarization andopinion dynamics Moreover they considered networks andthe notions of trust and epistemic vigilance to define adynamics for knowledge revision and trust itself whichwe clearly lack in our model They showed that if a con-servative belief operator in argumentation was applied byagents when they reasoned then their dialogue protocoldid not increase polarization among agents That said theyused the model to study the effect of Groenvetterrsquos weaklink theory in the spreading of arguments and were notparticularly interested in notions like judgment accuracy orcoherence nor in tackling questions related to collectivedecision-making procedures Their system is very expressiveand helps position our work in the litterature our work is atthe frontier of pure argumentative opinion diffusion modelsopinion diffusion with arguments and continuous bounded-confidence models of opinion diffusion a la Deffuant

On another note work on collective cognitive conver-gence [41] and opinion sharing [19] shows that consensustowards a certain ldquocorrectrdquo opinion or cognitive state is alwayspossible yet dependent on noise variability and awarenessof agents In [20] the authors show that learning about anexogenous correct state of the world (represented by bits)under confidence was possible but only if the agents werenot too confident In a homogeneous population they showthat the higher the confidence the worse the learningmdashthevery confident agents do not learn the properties of the truestate of the world and disrupt the learning process of the lessconfident ones Collective cognitive convergence can be seenas a result of deliberation in truth-seeking models

When it comes to abstract argumentation theory we takean approach that wires two types of dialogues that are well-studied in the literature persuasion dialogues [42] and delib-eration dialogues [43] Another interesting line of work is theone on mechanism design [44] or the problem of devisingan argumentation protocol where strategic argumentation isnot a liability for success in debates We tackle mechanismdesign in a different way though Instead of consideringstrategy-proofness we are interested in how differences inprotocol can result in epistemologically ldquobetterrdquo collectivechoices and guarantee that opinion distributions are favorablefor deliberation For a survey on persuasion dialogue see[42]

Work on agent-based argumentation usually assumes thatthe semantic relationship between arguments is fixed Inother words if two individuals were to put two argumentsin the public arena such that one logically attacked the otherthen everyone would agree that such attack exists [45 46]

26 Complexity

Other models which do not make this restrictive assumptioncan also be found in the literature [27 47] and derive fromthe class or family of opponent models [27 48] in which twoopposing sides attempt to win a dialogue Ourmodel is in theintersection of these two but the frame combining opiniondiffusion of the kind and argumentation remains new

The idea of mixing interpersonal influence and verticalcommunication however is not original It is described andimplemented in innovation diffusion models such as [5 49]In both vertical communication is modeled as exogenoustransparent informationmdashagents are aware of the existenceof an innovation thus triggering several processes of choiceand stabilization of opinions Also in [50] an Eulerian modelis implemented to show the effect of media and exogenousinformation on opinion distributions With respect to thispoint the originality of our work is in that the informationemitted as vertical communication is endogenous It is issuedfrom a deliberation model that agents shape on the basisof their opinions arguments and behavior In the spirit of[51] where the authors control for the design of verticalcommunication (to whom the vertical communication isaddressed its timing how it influences agents opinions) wecontrol for the process generating the information for ourown set of variables length of debates majority quotas andthe frequency at which discussions take place

6 Conclusion and Perspectives

The main objective of this article was to build a bridgewhere decision-making argumentation and opinion dif-fusion could come together We proposed a model thatcombined abstract argumentation theory and a boundedconfidence opinion diffusion model and showed to whatextent it could explain variance of opinions extremismcoherence in collective decisions and judgment accuracyon arguments given an ideal state of full information Thesecond objective was to show in what way governance pro-viding agents with arenas of discussion and deliberation wasimportant in addressing the success of collective decision-making processes and the quality of their outcomes

The model revealed that allowing for more deliberationtime though in lower frequency and allowing for widerparticipation in deliberation increased the variance of opin-ions and the proportion of extremists in a group Theseobservations are consistent with the combination of resultsfound in [9] and in [13 14] which stress that deliberationmay polarize groups and may have a meager effect on theshifts of opinions and inconsistent with [16] where it isargued that opinions tend to moderate after deliberationIt happens that deliberation alone did moderate opinionyet when integrated into a more complex system in whichindividuals were allowed to interact with one another andnot all that was deliberated was accepted its influence wasshadowed by other more individualistic dynamics

Undeniably the grounded semantics played an importantrole in the weak effect of deliberation since it models askeptical way of reasoning over arguments For agentsaccepting arguments thus updating their opinions accord-ingly happened not too often Nevertheless asking for more

deliberation did increase judgment accuracy as observed in[18] yet in a marginally decreasing fashion We showed thatvoting within the deliberation protocol not only increasedthe proportion of extremists and the variance of opinions ina group but also determined how coherent deliberation andvoting procedures were with each other Lastly we showedthat agents that are focused judged arguments better hadmore stable opinions and constituted a group with higherproportion of extremists than their naive counterparts

In terms of governance the model says that there is per-haps no trade-off between extremismand judgment accuracyInstead it asserts that the higher the variance of opinions thecloser the group gets to the correct decisions This situationmay be interpreted as Landermore and Pagersquos notion of ldquoposi-tive dissensusrdquo (in complex collective decision-making tasks)[36] as opposed to consensus as a normative requirementfor correct collective decision-making If a decider has toorganize deliberation to legitimize a public policy dependingon what kind of world he or she wants different set ofparameters may be chosen to account for different levels oflegitimacy and correctness For instance one may ask for adecision to be legitimate to respect a certain level of coherenceor of judgment accuracy and to be discussed by a sundryof different-opinionated peoples If so deliberation has tobe made more often more participants have to be includedin debates or in deliberation instances and not too manytime steps should be introduced in between two deliberationinstances So the model may capture normative ideas todefine correct and legitimate collective decisions and tomake recommendations on those grounds Requirements ofhigh consensual deliberative accuracy (judgment accuracy)legitimacy (coherence) and diversity (variance of opinions)are all taken into account and for such a reason the model israther successful in describing a sound process of collectivedecision-making with interpretable outcomes

61 Extensions of the Model and Other Ideas Our model canbe extended in many ways Value-based abstract argumen-tation frameworks (VAFs) for example as an extension ofDungrsquos argumentation framework provide a formal descrip-tion of the process of decision-making in which argumentsare given values and audiences (set of agents) ignore attacksbetween arguments on the basis of their preferences over val-ues Social abstract argumentation frameworks (SAFs) alsoprovide an interesting and seemingly convenient formalismfor our model In SAFs agents vote on the acceptability ofarguments and the resulting labeling is a combination of thearguments that are voted for and labeling-based semantics Inour model we apply the same rule yet only for one argumentand given precise agent-based votingmechanisms and votingrules which are eventually parametrized

To consider our model as a specification of a mixed VAFand SAF model may be of great interest for future workThis formalism can provide solid foundations to describeexplain interpret and eventually extend the results foundhere from an argument-based perspective On those groundsfuture work will focus on creating a hybrid argumentationframeworkmdashwith its semanticsmdashthat takes heed of bothvalues or principles and voting

Complexity 27

Concerning other possible extensions of the model anatural reflex would be to argue on the validity and therobustness of the mix between the bounded confidenceopinion diffusion models and deliberation Consequentiallyadding deliberation to different bounded confidence modelsof opinion diffusion such as the Hegelsmann-Krause model[37] or to models of social impact theory like in [4] canbe of great interest for future work We acknowledge that afiner understanding of argumentation frameworks randomlattices and their respective applications for this modelare necessary Testing for different labeling-based semanticsor even trading labeling-based semantics for scoring-basedsemantics (semantics that defines acceptability on scoresgiven to arguments) such as debate semantics and ranking-based semantics (semantics that yield a preorder) may beinteresting to consider in future work

More into modeling deliberation it is not inadequateto believe that agents may also advance proposals duringdeliberation and replace to some extent the proposal forwhich the deliberation is taking place Endogenizing theprocess that produces the type of argument to be deliberatedon may be interesting to explore either by having agentsstrategically replace proposals at certain moments duringthe debate or having the central authority attempt to chooseproposals such that the probability of having them acceptedis high Agents may also be thought to be autoorganizing(independent of a central authority) and concede on the sizeof the instances of deliberation To consider differences insemantics across agents or even endogenizing the semanticson how urgent or important the proposal that is beingdiscussed is can also be an interesting direction to take

Last but not least many nontrivial modifications of ourmodel are possible and most should include better-thoughtdeliberation protocols It may be interesting to design andobserve protocols in which deliberation only affects agentsthat are actually debating for example Trust network effectsmultidimensionality of opinions new processes of argumentexchange or learning among agents are notions to furtherdevelop in order to make the model more realistic and relaxsome unreliable assumptions In sum the model is to berefined and extended with the objective of either studyingconcrete cases of deliberative polling and opinion dynamicsor implementing more intuitive thought experiments

Appendix

We present below some of the pseudocode we used toimplement the model

Algorithms

(a) Algorithm 1 implements the dyadic influence in themodel It describes how two agents interact andupdate their opinions

(b) Algorithm 2 implements the deliberative influence inthe model which describes how agents update theiropinions in sight of the accepted deliberated proposalarguments

(c) Algorithm 3 is used to obtain the grounded labelingof an argumentation framework

(d) Algorithm 4 provides an overview of a run in thesimulations119872 is the number of proposal argumentsto discuss which we set to 100 in the model

Each object in the system is indexed by a119908ℎ119900 number Hencea set of agents is annotated by a set of natural numbers Weuse this fact in the implementation of the algorithm 119874119887119895119890119888119905119894denotes the object of 119908ℎ119900 number 119894 Agentsrsquo opinions (119900) arecalibrated so that 119900 isin [minus1 1]Data Availability

The program and data used to support the findings of thisstudy are available from the corresponding author uponrequest

Disclosure

A preliminary version of this work was presented anddiscussed at the conference Bridging the Gap between FormalArgumentation and Actual Human Reasoning on October 4-5 2018 at BochumGermanyThe authors thank the audiencefor their valuable comments and suggestions

Conflicts of Interest

The authors declare that they have no conflicts of interestregarding the publication of this article

Acknowledgments

The contribution of Gabriella Pigozzi was supported by theDeutsche Forschungsgemeinschaft (DFG) and the CzechScience Foundation (GACR) as part of the joint project FromShared Evidence to Group Attitudes (RO 45486-1)

Endnotes

1 In other words the superiority of the ldquoright answerrdquo orlabeling in our case will appear as such to all

2 All agents hold the same opinion3 By sensitivity we mean how much an agent is ready to

change her opinion on the sole fact that an informationalcue has been debated on

4 The assumption is strong but this trust among agentsmay be rooted from a genuine individual motivation toreach a democratic consensus

5 We adopted a rule of thumbs that says that if an agentis neutral with respect to the principle then she willvote for any proposal that is presented to her duringdeliberation

6 If the argument is not unique she chooses one atrandom

7 If there are more than one candidate the agent will favorthe argument at minimal distance from 119868

28 Complexity

Require Vector of values for parameters (119881 = (120583 119880 119879))Ensure Successful interactions among agents119896 larr997888 0119873119890 larr997888 0

while 119896 le |119873|2 do ⊳Fill119873119890 with |119873|2 agents119896 larr997888 119896 + 1119873119890 larr997888 119877119886119899119889119900119898(|119873| + 1) ⊳Pseudo-random generator Generates integer between 0 and |119873|119873119903 larr997888 0 120575119894119895 larr997888 0 ⊳Variable to store differences in opinionfor all 119894 isin 119873119890 doRandomly pick a number 119895 isin 119873(119873119890 cup 119873119903)119873119903 larr997888 119873119903 cup 119895 120575119894119895 larr997888 |119900119894 minus 119900119895|procedure dyadic-update(119894 119895) ⊳Two agents 119894 and 119895 discuss and update opinions1199001015840 larr997888 119900119894 (120583119895 119880119895 119879119895) larr997888 (120583119894 119880119894 119879119894) larr997888 (120583 119880 119879) ⊳Save agent irsquos opinion before updating and homogenize agents

if 120575119894119895 lt 119880119894 then119900119894 larr997888 120583119894119900119895 + (1 minus 120583119894)119900119894if 120575119894119895 gt 119879119894 then119900119894 larr997888 (1 + 120583119894)119900119894 minus 120583119894119900119895if 120575119894119895 lt 119880119895 then119900119895 larr997888 1205831198951199001015840 + (1 minus 120583119895)119900119895if 120575119894119895 gt 119879119895 then119900119895 larr997888 (1 + 120583119895)119900119895 minus 1205831198951199001015840

Algorithm 1 Dyadic social influence

Require Vector of values for parameters (119881 = (120574 119901119886 119901119903))Ensure Successful interaction between proposal arguments and agents

for all 119894 isin 119873 doprocedure Probability-Update(V119886 119901119886 119901119903) ⊳Update individual probabilities of change due to deliberated cues119909119894 larr997888 119901|V119886minus119900119894 |2119886 119910119894 larr997888 1199012|V119886minus119900119894 |119903119902 larr997888 0if L119886119887(119868) = 119894119899 then

procedure deliberative-update(119894 119886) ⊳Agent i argument 119886120574119894 larr997888 120574 119902 larr997888 119906 sim U(0 1) ⊳ 119906 is a realization of a uniform random variableif 119902 lt 119909119894 then119900119894 larr997888 120574119894119900119894 + (1 minus 120574119894)V119886break

if 119902 lt 119910119894(1 minus 119909119894) then119900119894 larr997888 (1 + 120574119894)119900119894 minus 120574119894V119886Algorithm 2 Deliberation influence

8 In terms of computational complexity naive agentsmaketheirmoves in119874(|A119894| log(|A119894|)) while focused agents doat least as bad

9 Tables have memory Therefore debates are not neces-sarily trees

10 If there are less than (119899119863 times |119873|)2 agents for any of bothgroups the CA summons all of them to the deliberationtable

11 Or careful consideration of the viewpoints of otherswhich implies that citizens keep an open mind and donot reject arguments outright

12 A ldquohealthyrdquo consensus refers to the case in which manyconsensual decisions are taken but opinion-wise agentsdo not agree with each other Quotes are used on theword healthy because some political science theorists

(eg [36]) believe that in certain situations agreement inopinions for deliberation is epistemologically damaging

13 Simulations on groups of |119873| = 800 and |119873| = 400agents shows that scaling the population has an effect onall themetrics but it is confounding depending onwhichcouples of attitudes structures are considered (119880 119879) Westick to |119873| = 400 as in [9] for comparability andsimulation running time

14 By balanced wemean with as many arguments with V119886 lt0 as with V119886 gt 0

15 Experiments on the system suggest that a bigger(smaller) argument pool 119872 = 1000 (119872 = 200) isassociated with a higher (lower) variance of opinions(and extremism) and higher (lower) judgment accuracyFor the rest of the observations 119872 seem to have noeffect We choose a middle value for 119872 Similarly theargument pool chosen to be balanced or uniformly

Complexity 29

Require Argumentation framework 119860119865 = (AR)Ensure Assignment of labels to arguments from the grounded labeling of 119860119865L(A)

L0(A) larr997888 (0 0 0)119896 larr997888 0do119894119899 L119896+1(A) larr997888 119894119899 L119896(A) cup 119886 | 119886 is not labeled 119894119899 L119896(A) 119886119899119889 forall119887 isin A 119904119905 119887 R 119886 then 119887 isin 119900119906119905 L119896(A)119900119906119905 L119896+1(A) larr997888 119900119906119905 L119896(A) cup 119886 | 119886 is not labeled 119894119899 L119896(A) 119886119899119889 exist119887 isin A 119904119905 119887 R 119886 and 119887 isin 119894119899 L119896+1(A)

while L119896+1(A) = L119896(A) = 0Algorithm 3 Solve argumentation frameworks (with grounded semantics)

Require Vector of values for parameters (119881)Ensure Vector of statistics (119880)

procedure Set-parameters(119881) ⊳Initialize parameters for scenarioprocedure Init-objs(119899119898 119896 ) ⊳Create119898 arguments 119899 agents and 119860119865120576procedure Solve-Argumentation-Framework(119860119865) ⊳Assigns the epistemic labelingL120576 toA119894 larr997888 1119905 larr997888 1while 119894 le 119872 do ⊳ 119872 the number of arguments to discuss119868119894 larr997888 119866119890119899 119886119903119892(1) ⊳Generates proposal argument 119868119860 119894119905 larr997888 (119868119894 0)L119886119887(119868119894) larr997888 119906119899119889 ⊳Initialize label for 119868119895 larr997888 1 ⊳Initialize debate counter 119895while ((L119886119887(119868) = 119906119899119889 or 119898 le 119895) and 119895 le 119898) doif 119905 equiv 0(119898119900119889 119905119863) then

procedure build-argumentation-framework(119899119863 119887119890ℎ119886V119894119900119903) ⊳Build the argument graph agents deliberate(119861119894119905 119878119894119905) larr997888 (119883 119884) sub (⋃119897isin119873119899119863 A119897119894119905 ⋃119897isin119873119899119863 R119897119894119905) ⊳ 119873119899119863 sub 119873 are agents sampled by the CA119860 119894119905 larr997888 119889(119860 119894119905 (119861119894119905 119878119894119905))procedure Solve-Argumentation-Framework(A119889) ⊳FindL(A119889)L119886119887(119868119894) larr997888 L119886119887119863(119868119894)119895 larr997888 119895 + 1

else119896 larr997888 1while 119896 le 119905119863 doprocedure dyadic-social-influence(120583 119879 119880) ⊳Agents discuss one-to-one119896 larr997888 119896 + 1119905 larr997888 119905 + 1

if 120572 gt 0 andL119886119887(119868119894) = 119900119906119905 then ⊳Agents vote for the proposalif (|119897 isin 119873|119900119897 times V119868119894 ge 0| ge 120572 times 119899) then

L119886119887(119868119894) larr997888 119894119899else

L119886119887(119868119894) larr997888 119900119906119905if L119886119887(119868119894) = 119894119899 thenprocedure Deliberation-influence(120574 119901119886 119901119903) ⊳Agents are influenced by the result of the decision process119894 larr997888 119894 + 1119905 larr997888 119905 + 1119880 larr997888 119862119900119898119901119906119905119890 119878119905119886119905119894119904119905119894119888119904(119881)

return 119880 ⊳Get the statisticsAlgorithm 4 Run for one scenario

generated does not seem to play a significant explanatoryrole on any of the metrics

16 Experiments show that themaximumnumber of debatesis not an important parameter even if it is an intuitiveone when considering decision-making processes Inpractice debates end after at most 119898 + 1 deliberation

steps and even then it happens rarely However to avoidunpleasant surprises like infinite debates on a completeargument graph containing the proposal argument weadded the constraint 119898 Its value is precisely 7 becausethe biggest minimum number of debates we studyis 6

30 Complexity

17 To ensure comparability with the deliberative andmixedopinion dynamics models on judgment accuracy andcoherence agents vote on proposal arguments every 119905119863times(119898+1) 119905119863 and119898does not explain variability in judgmentaccuracy nor in coherence

18 Student t-tests for independent samples are robust tothe violation of two hypothesis on the distribution ofthe samples normality and homogeneity of variancesas long as the sample size is big enough (central limittheorem) and if the difference in size of the comparedsamples is smallrespectively

19 Studentrsquos test version forwhen variances are not assumedequal and the difference in size of the compared samplesis big

20 We do not correct for homoskedasticity nor normality oferrors in the estimates

21 Balanced as the same number of runs per scenario orsuch that no significant correlations are found betweenparameters

References

[1] R Axelrod ldquoThe dissemination of culture a model withlocal convergence and global polarizationrdquo Journal of ConflictResolution vol 41 no 2 pp 203ndash226 1997

[2] D M Rousseau Reinforcing the MicroMacro Bridge Organiza-tionalThinking and Pluralistic Vehicles SAGEPublications SageLos Angeles Calif USA 2011

[3] S Galam and SMoscovici ldquoTowards a theory of collective phe-nomena consensus and attitude changes in groupsrdquo EuropeanJournal of Social Psychology vol 21 no 1 pp 49ndash74 1991

[4] A Nowak J Szamrej and B Latane ldquoFrom private attitude topublic opinion a dynamic theory of social impactrdquo Psychologi-cal Review vol 97 no 3 pp 362ndash376 1990

[5] G Deffuant S Huet and F Amblard ldquoAn individual-basedmodel of innovation diffusion mixing social value and individ-ual benefitrdquo American Journal of Sociology vol 110 no 4 pp1041ndash1069 2005

[6] G Deffuant D Neau F Amblard and G Weisbuch ldquoMixingbeliefs among interacting agentsrdquoAdvances in Complex Systems(ACS) vol 3 no 01n04 pp 87ndash98 2000

[7] J Rouchier P Tubaro and C Emery ldquoOpinion transmission inorganizations an agent-based modeling approachrdquo Computa-tional and Mathematical OrganizationTheory vol 20 no 3 pp252ndash277 2014

[8] M Meadows and D Cliff ldquoThe relative disagreement model ofopinion dynamics where do extremists come fromrdquo in Pro-ceedings of the International Workshop on Self-Organizing Sys-tems Lecture Notes in Computer Science pp 66ndash77 Springer2013

[9] W Jager and F Amblard ldquoUniformity bipolarization and plu-riformity captured as generic stylized behavior with an agent-based simulation model of attitude changerdquo Computational ampMathematical Organization Theory vol 10 no 4 pp 295ndash3032005

[10] Y-F Zhang H-Y Duan and Z-L Geng ldquoEvolutionary mech-anism of frangibility in social consensus system based onnegative emotions spreadrdquo Complexity vol 2017 Article ID4037049 8 pages 2017

[11] A Flache andMWMacy ldquoSmall worlds and cultural polariza-tionrdquoThe Journal of Mathematical Sociology vol 35 no 1-3 pp146ndash176 2011

[12] A Tsang and K Larson ldquoOpinion dynamics of skepticalagentsrdquo in Proceedings of the 13th International Conference onAutonomous Agents and Multiagent Systems AAMAS 2014 pp277ndash284 International Foundation forAutonomousAgents andMultiagent Systems France May 2014

[13] K M Hansen Deliberative democracy and opinion formationUniversity Press of Southern Denmark Odense 2004

[14] C R Sunstein ldquoThe law of group polarizationrdquo Journal ofPolitical Philosophy vol 10 no 2 pp 175ndash195 2002 WileyOnline Library

[15] S Moscovici and W Doise Dissensions et Consensus unetheorie generale des decisions collectives Presses Universitairesde France-PUF 1992

[16] J S Fishkin and R C Luskin ldquoExperimenting with a demo-cratic ideal deliberative polling and public opinionrdquo ActaPolitica vol 40 no 3 pp 284ndash298 2005

[17] R C Luskin J S Fishkin and S Iyengar Considered Opinionson US Foreign Policy Face-To-Face versus Online Delibera-tive Polling International Communication Association NewOrleans La USA 2004

[18] J Barabas ldquoHowdeliberation affects policy opinionsrdquoAmericanPolitical Science Review vol 98 no 4 pp 687ndash701 2004

[19] O Pryymak A Rogers and N R Jennings ldquoEfficient opinionsharing in large decentralised teamsrdquo in Proceedings of the 11thInternational Conference on Autonomous Agents andMultiagentSystems vol 1 pp 543ndash550 Spain June 2012

[20] J Rouchier and E Tanimura ldquoWhen overconfident agents slowdown collective learningrdquo Simulation vol 88 no 1 pp 33ndash492012

[21] D Miller ldquoDeliberative democracy and social choicerdquo PoliticalStudies vol 40 S1 pp 54ndash67 1992

[22] M Sherif and C I Hovland Social Judgment Assimilation andContrast Effects in Communication and Attitude Change YaleUniversity Press 1961

[23] P M Dung ldquoOn the acceptability of arguments and its funda-mental role in nonmonotonic reasoning logic programmingand n-person gamesrdquo Artificial Intelligence vol 77 no 2 pp321ndash357 1995

[24] M Caminada and G Pigozzi ldquoOn judgment aggregation inabstract argumentationrdquo Autonomous Agents and Multi-AgentSystems vol 22 no 1 pp 64ndash102 2011

[25] P Baroni M Caminada andM Giacomin ldquoAn introduction toargumentation semanticsrdquoThe Knowledge Engineering Reviewvol 26 no 4 pp 365ndash410 2011

[26] S Chambers ldquoDeliberative democratic theoryrdquo Annual Reviewof Political Science vol 6 no 1 pp 307ndash326 2003

[27] E Bonzon andNMaudet ldquoOn the outcomes of multiparty per-suasionrdquo in Proceedings of the 10th International Conference onAutonomous Agents and Multiagent Systems 2011 AAMAS 2011vol 1 pp 47ndash54 International Foundation for AutonomousAgents and Multiagent Systems Taiwan May 2011

[28] J Ranciere Disagreement Politics and Philosophy University ofMinnesota Press 1999

[29] T Bench-Capon and K Atkinson ldquoAbstract argumentation andvaluesrdquo in Argumentation in Artificial Intelligence pp 45ndash64Springer 2009

[30] M Caminada ldquoOn the issue of reinstatement in argumenta-tionrdquo in Proceedings of the European Workshop on Logics in

Complexity 31

Artificial Intelligence vol 4160 of Lecture Notes in ComputerScience pp 111ndash123 Springer Berlin Heidelberg

[31] GAVreeswijk ldquoAn algorithm to computeminimally groundedand admissible defence sets in argument systemsrdquo in Pro-ceedings of the 2006 conference on Computational Models ofArgument Proceedings of COMMA 2006 vol 144 pp 109ndash120IOS Press Amsterdam The Netherlands 2006

[32] C W Sherif M Kelly H L Rodgers et al ldquoPersonal involve-ment social judgment and actionrdquo Journal of Personality andSocial Psychology vol 27 no 3 pp 311ndash328 1973

[33] G Deffuant F Amblard G Weisbuch and T Faure ldquoHow canextremism prevail A study based on the relative agreementinteraction modelrdquo Journal of Artificial Societies and SocialSimulation vol 5 no 4 p 1 2002

[34] E Awad M W Caminada G Pigozzi M Podlaszewski andI Rahwan ldquoPareto optimality and strategy-proofness in groupargument evaluationrdquo Journal of Logic and Computation vol 27no 8 pp 2581ndash2609 2017

[35] D Grossi and G Pigozzi ldquoJudgment aggregation a primerrdquoSynthesis Lectures on Artificial Intelligence and Machine Learn-ing vol 8 no 2 pp 1ndash151 2014

[36] H Landemore and S E Page ldquoDeliberation and disagreementproblem solving prediction and positive dissensusrdquo PoliticsPhilosophy and Economics vol 14 no 3 pp 229ndash254 2015

[37] R Hegselmann and U Krause ldquoOpinion dynamics andbounded confidence models analysis and simulationrdquo Journalof Artificial Societies and Social Simulation vol 5 no 3 2002

[38] A Stefanelli and R Seidl ldquoModerate and polarized opinionsUsing empirical data for an agent-based simulationrdquo inProceed-ings of the Social Simulation Conference 2014

[39] S Gabbriellini and P Torroni ldquoA new framework for ABMsbased on argumentative reasoningrdquo in Advances in SocialSimulation vol 229 pp 25ndash36 Springer Berlin Germany 2014

[40] H Mercier and D Sperber ldquoWhy do humans reason Argu-ments for an argumentative theoryrdquo Behavioral and BrainSciences vol 34 no 2 pp 57ndash74 2011

[41] H V Parunak T C Belding R Hilscher and S BruecknerldquoModeling and managing collective cognitive convergencerdquoin In Proceedings of the 7th iNternational Joint Conference onAutonomous Agents and Multiagent Systems vol 3 pp 1505ndash1508 International Foundation for Autonomous Agents andMultiagent Systems 2008

[42] H Prakken ldquoFormal systems for persuasion dialoguerdquo TheKnowledge Engineering Review vol 21 no 2 pp 163ndash188 2006

[43] K Atkinson T Bench-Capon and P Mcburney ldquoA dialoguegame protocol for multi-agent argument over proposals foractionrdquoAutonomousAgents andMulti-Agent Systems vol 11 no2 pp 153ndash171 2005

[44] M Thimm and A J Garcıa ldquoClassification and strategicalissues of argumentation games on structured argumentationframeworksrdquo in Proceedings of the 9th International Conferenceon Autonomous Agents and Multiagent Systems vol 1 pp 1247ndash1254 International Foundation for Autonomous Agents andMultiagent Systems Canada 2010

[45] I Rahwan and K Larson ldquoMechanism design for abstractargumentationrdquo in Proceedings of the 7th International JointConference on Autonomous Agents and Multiagent SystemsAAMAS 2008 pp 1031ndash1038 International Foundation forAutonomous Agents and Multiagent Systems Portugal May2008

[46] H Prakken R Riveret A Rotolo and G Sartor ldquoHeuristics inArgumentation A Game-Theoretical Investigationrdquo EuropeanUniversity Institute 2008

[47] N Hadidi Y Dimopoulos and P Moraitis ldquoTactics and con-cessions for argumentation-based negotiationrdquo in Proceedingsof the COMMA vol 245 pp 285ndash296 2012

[48] N Oren and T J Norman ldquoArguing using opponent modelsrdquoin Proceedings of the International Workshop on Argumentationin Multi-Agent Systems pp 160ndash174 Springer 2009

[49] S Delre W Jager T Bijmolt and M Janssen ldquoTargeting andtiming promotional activities An agent-based model for thetakeoff of new productsrdquo Journal of Business Research vol 60no 8 pp 826ndash835 2007

[50] A Mirtabatabaei P Jia and F Bullo ldquoEulerian opinion dynam-ics with bounded confidence and exogenous inputsrdquo SIAMJournal on Applied Dynamical Systems vol 13 no 1 pp 425ndash446 2014

[51] V Barbet N Guiraud V Laperriere and J Rouchier Hagglingon Values Towards Consensus or Trouble 2019

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24 Complexity

one can be inclined to believe that the longer a deliberationprocess is the more likely individuals will take an immutablestance on the situation In view of the assumption that longdecision-making processes can be more costly for a group orsociety as a whole agents may feel forced to take a stancejust for the sake of ending the process The stance that isadopted would therefore depend on pair-wise interactionsamong individuals and not on deliberation since deliberationtakes too long to be conclusive

442 For Hypothetical Recommendations in Decision-MakingProcess Elaboration On another tune results from themodelsuggest that procedural parameters have marginally decreas-ing positive or negative effects on the metrics of interest Ina hypothetical policy analysis perspective this may indicatethat after choosing a correct number of individuals andimposing the right amount of debates one can attain rea-sonable levels of judgment accuracy coherence and varianceof opinions As accounted in [18] deliberation increasesknowledge (collective knowledge as well) and therefore itheightens the chances of making correct decisions If onebelieves that social welfare is linked to deliberation andeventually to the metrics studied previously then proceduralparameters can both be used as an instrument to attaindesired levels of social welfare and reinforce the notionof legitimacy of collective decisions It can also provide ajustification for choosing a deliberation regime rather thananother on the basis of how important (or urgent) a topic ofdiscussion is Indeed if we adopt the claim that deliberationis a forerunner of welfare then any state in which deliberativecues are at their maximum (highest number of participantsmany deliberation steps before making a decision and asfrequent as possible) has to be mapped to the highest attain-able social welfare In this case is a situation where judgmentaccuracy is at its highest yet the variance of opinions or theproportion of extremists at its lowest an ideal situation Wecannot say for sure since strong opinion consensus and longdecision-making processes in many situations may result ina loss of welfare rather than in a gain of it

Another interesting result in procedure for deliberationis the very similar estimates we find for the effects of theminimum number (119898) of debates necessary to make adecision on a proposal and the proportion (119899119863) of individualsparticipating in deliberation on judgment accuracy In thescope of social welfare this raises the question of whetheradvocating for longer decision processes by demandingmoredeliberation steps (higher 119898) rather than bigger debates(higher 119899119863) may increase welfare Depending on the objec-tives of the decider of the deliberation procedure he or shemay choose to concentrate on one or the other A recom-mendation we can assert from the mixed social interactionsmodel is that if one prefers low to high variance of opinionsandor high to low coherence the decider is better off ifshe focuses on increasing the size of the debate rather thanorganizing debates very frequently One of the reasons forthis might be that when making bigger debates all agents aretruly considered as equals in the decision-making processIn organizing many debates individuals that are chosen toparticipate in many of these will be more influential and will

somehow bias the set of arguments in the deliberation arenaThemajority rulemodeled through120572 also determines the sizeof the shifts and the distribution of opinions Because a bigger120572 is associated with higher levels of extremism and opinionvariance deliberated cues are harder to accept and thereforeit becomesmore difficult for agents to reach concensus andormoderate their opinions

On the other hand and further into the idea of legitimacythe goal of deliberative processes is the agreement amongall participants This agreement stands for the right or bestdecision from a formal or procedural perspective and it is alsothe best from a substantive point of view In other words thelegitimacy of decisions not only derives fromwell-establishedprocedural requirements (respect of the protocol) but alsobegs for the fulfillment of two essential conditions (1) satisfy-ing the procedural requirements for a correct procedure (for-mal legitimacy) and (2) the rational acceptability of the resultsof this procedure (substantive acceptability) Legitimacy canbe then obtained from the coherence statistic scrutinizedin the model insofar as it measures how well agents acceptthe results of a correct procedure (eg deliberation) So fora proposal to be legitimate one has to choose parametersthat would maximize group coherence in decision-makingprocesses

As deliberation may give rise to consensus in the delib-eration arena it may also sow the seeds of dissensus inthe group Our model clearly illustrates this phenomenonthrough the procedural parameters of deliberation and themetrics observed From another perspective in deliberativedemocracy dissensus in collective decision-making may bedesirable According to Landermore and Pagersquos accounts [36]lack of consensus can be perceived as having agents withalternative ideas that combined with other ideas can providea better approximation of the ideal status of an argumentTheauthors in [36] call this dissensus ldquopositive dissensusrdquo andit makes normative sense when solving complex collectivedecision-making problems Our model captures the ideaof having normative requirements to define a good andlegitimate collective decision requirements of high consen-sual deliberative accuracy (judgment accuracy) of legitimacy(coherence) and of diversity (variance of opinions) are alltaken into account and for such we can conclude that ourmodel is successful in describing a sound process of collectivedecision-making with interpretable outcomes

Of course the present simulation results and analysis donot prove that all opinion dynamics can be accounted forby such simple processes like the one presented here Theydo however shed light on the expressiveness of combiningdifferent paradigms to account formore interpretablemodelsand pave the way to creating original models of the sortThe simulations suggest the desirability of discovering theconsequences of relatively simple laws of communication atdifferent levels (micro meso and macro) to determine whatstill needs to be explained

5 Related Work

We see our model as a contribution to the influence andopinion dynamics field in ABM and a pragmatic application

Complexity 25

of abstract argumentation theory To our knowledge we areunaware of existing literature on agent-based modeling thatexplicitly relates collective choice and the notions of deliber-ation and opinion diffusion through abstract argumentationas we have done it

Most models in the literature of opinion diffusion areinterested in opinions because these have an influence incollective decision and in questions of social order Forinstance in [3] the authors are interested in consensus andin how a group collectively chooses among two alternativesIn other models authors are interested in the emergenceof extremism [8] and on the distribution of opinions whenextremists are introduced in the population [5 33 37] whileother authors coin the notion of opinion polarization as anemergent property of the system [8 12] They show usingmodels of ldquobounded confidencerdquo and opinion diffusion withtrust that three different kinds of steady states (unipolarbipolar and central) are possible depending on whetheragents are sufficiently uncertain about their opinions andsufficiently connected andor a certain proportion of indi-viduals are already extreme Recent articles on informationand opinion dynamics stress the importance of governanceand government intervention in the spread of emotions(opinions) in the frangibility of social consensus systemIn [10] the authors show that government intervention inthe spread of negative emotions can lead to an even fasterspread of negative emotions (single-peeked convergence) andin a faster collapse of the social consensus system Opinionas a function of trust is studied in [12] and relates to howindividuals form their opinions on the basis of how muchthey trust agents in their networks

Another stream of discrete opinion dynamics modelsthat explain the emergence of opinion can be found in theliterature Computer multiagent models of attitude changebased on Latanersquos social impact theory are presented in [4] Inthe theory an agent changes her opinion on the basis of theinformational impact she is subject to which depends on thepersuasiveness (strength) supportiveness and immediacy(group structure) of the environment and information shereceives The simulations predict two emergent groups ofphenomena the shifting of attitudes towards incompletelypolarized equilibria and the formation of coherent clusteringof subgroups with deviant attitudes Similarly but in acontinuous more argumentative fashion Mas and Flache[11] present a model of opinion diffusion using argumentsArguments are considered to be for or against a proposal andare given an agent-dependant relevance in dyadic persuasiveinteractions Arguments also determine the agentsrsquo opinionsand their dissimilarities which posits the rules of pair-wiseencounters in theirmodel as agents are assumed to form theiropinions according to the arguments they own They showthat in an argumentative opinion diffusion model the onlyobservable steady states are bipolarization and consensusand that bipolarization can emerge from interactions amongsimilar agents In other words bipolarization of opinionsis possible with homophily and without negative influenceif there is argumentation They subsequently compare theirresults to experimental data In [38] a similar approach toargumentation and opinion formation is taken but for the fact

that they introduce different types of arguments explicitlyapply social judgment theory [22] and use survey data toestablish model-to-real-world comparisons

Closer to opinion formation and abstract argumentationGabbriellini and Torroni [39] are the first to have origi-nally and soundly merged opinion diffusion and abstractargumentation They define an agentrsquos opinion as a functionof the arguments she holds and the attack relation amongthem They devised a focused peer-to-peer dialogue systemof persuasion (NetArg) inspired from Mercier and Sperberrsquosargumentative theory of reasoning [40] which used onlyabstract argumentation to study opinion polarization andopinion dynamics Moreover they considered networks andthe notions of trust and epistemic vigilance to define adynamics for knowledge revision and trust itself whichwe clearly lack in our model They showed that if a con-servative belief operator in argumentation was applied byagents when they reasoned then their dialogue protocoldid not increase polarization among agents That said theyused the model to study the effect of Groenvetterrsquos weaklink theory in the spreading of arguments and were notparticularly interested in notions like judgment accuracy orcoherence nor in tackling questions related to collectivedecision-making procedures Their system is very expressiveand helps position our work in the litterature our work is atthe frontier of pure argumentative opinion diffusion modelsopinion diffusion with arguments and continuous bounded-confidence models of opinion diffusion a la Deffuant

On another note work on collective cognitive conver-gence [41] and opinion sharing [19] shows that consensustowards a certain ldquocorrectrdquo opinion or cognitive state is alwayspossible yet dependent on noise variability and awarenessof agents In [20] the authors show that learning about anexogenous correct state of the world (represented by bits)under confidence was possible but only if the agents werenot too confident In a homogeneous population they showthat the higher the confidence the worse the learningmdashthevery confident agents do not learn the properties of the truestate of the world and disrupt the learning process of the lessconfident ones Collective cognitive convergence can be seenas a result of deliberation in truth-seeking models

When it comes to abstract argumentation theory we takean approach that wires two types of dialogues that are well-studied in the literature persuasion dialogues [42] and delib-eration dialogues [43] Another interesting line of work is theone on mechanism design [44] or the problem of devisingan argumentation protocol where strategic argumentation isnot a liability for success in debates We tackle mechanismdesign in a different way though Instead of consideringstrategy-proofness we are interested in how differences inprotocol can result in epistemologically ldquobetterrdquo collectivechoices and guarantee that opinion distributions are favorablefor deliberation For a survey on persuasion dialogue see[42]

Work on agent-based argumentation usually assumes thatthe semantic relationship between arguments is fixed Inother words if two individuals were to put two argumentsin the public arena such that one logically attacked the otherthen everyone would agree that such attack exists [45 46]

26 Complexity

Other models which do not make this restrictive assumptioncan also be found in the literature [27 47] and derive fromthe class or family of opponent models [27 48] in which twoopposing sides attempt to win a dialogue Ourmodel is in theintersection of these two but the frame combining opiniondiffusion of the kind and argumentation remains new

The idea of mixing interpersonal influence and verticalcommunication however is not original It is described andimplemented in innovation diffusion models such as [5 49]In both vertical communication is modeled as exogenoustransparent informationmdashagents are aware of the existenceof an innovation thus triggering several processes of choiceand stabilization of opinions Also in [50] an Eulerian modelis implemented to show the effect of media and exogenousinformation on opinion distributions With respect to thispoint the originality of our work is in that the informationemitted as vertical communication is endogenous It is issuedfrom a deliberation model that agents shape on the basisof their opinions arguments and behavior In the spirit of[51] where the authors control for the design of verticalcommunication (to whom the vertical communication isaddressed its timing how it influences agents opinions) wecontrol for the process generating the information for ourown set of variables length of debates majority quotas andthe frequency at which discussions take place

6 Conclusion and Perspectives

The main objective of this article was to build a bridgewhere decision-making argumentation and opinion dif-fusion could come together We proposed a model thatcombined abstract argumentation theory and a boundedconfidence opinion diffusion model and showed to whatextent it could explain variance of opinions extremismcoherence in collective decisions and judgment accuracyon arguments given an ideal state of full information Thesecond objective was to show in what way governance pro-viding agents with arenas of discussion and deliberation wasimportant in addressing the success of collective decision-making processes and the quality of their outcomes

The model revealed that allowing for more deliberationtime though in lower frequency and allowing for widerparticipation in deliberation increased the variance of opin-ions and the proportion of extremists in a group Theseobservations are consistent with the combination of resultsfound in [9] and in [13 14] which stress that deliberationmay polarize groups and may have a meager effect on theshifts of opinions and inconsistent with [16] where it isargued that opinions tend to moderate after deliberationIt happens that deliberation alone did moderate opinionyet when integrated into a more complex system in whichindividuals were allowed to interact with one another andnot all that was deliberated was accepted its influence wasshadowed by other more individualistic dynamics

Undeniably the grounded semantics played an importantrole in the weak effect of deliberation since it models askeptical way of reasoning over arguments For agentsaccepting arguments thus updating their opinions accord-ingly happened not too often Nevertheless asking for more

deliberation did increase judgment accuracy as observed in[18] yet in a marginally decreasing fashion We showed thatvoting within the deliberation protocol not only increasedthe proportion of extremists and the variance of opinions ina group but also determined how coherent deliberation andvoting procedures were with each other Lastly we showedthat agents that are focused judged arguments better hadmore stable opinions and constituted a group with higherproportion of extremists than their naive counterparts

In terms of governance the model says that there is per-haps no trade-off between extremismand judgment accuracyInstead it asserts that the higher the variance of opinions thecloser the group gets to the correct decisions This situationmay be interpreted as Landermore and Pagersquos notion of ldquoposi-tive dissensusrdquo (in complex collective decision-making tasks)[36] as opposed to consensus as a normative requirementfor correct collective decision-making If a decider has toorganize deliberation to legitimize a public policy dependingon what kind of world he or she wants different set ofparameters may be chosen to account for different levels oflegitimacy and correctness For instance one may ask for adecision to be legitimate to respect a certain level of coherenceor of judgment accuracy and to be discussed by a sundryof different-opinionated peoples If so deliberation has tobe made more often more participants have to be includedin debates or in deliberation instances and not too manytime steps should be introduced in between two deliberationinstances So the model may capture normative ideas todefine correct and legitimate collective decisions and tomake recommendations on those grounds Requirements ofhigh consensual deliberative accuracy (judgment accuracy)legitimacy (coherence) and diversity (variance of opinions)are all taken into account and for such a reason the model israther successful in describing a sound process of collectivedecision-making with interpretable outcomes

61 Extensions of the Model and Other Ideas Our model canbe extended in many ways Value-based abstract argumen-tation frameworks (VAFs) for example as an extension ofDungrsquos argumentation framework provide a formal descrip-tion of the process of decision-making in which argumentsare given values and audiences (set of agents) ignore attacksbetween arguments on the basis of their preferences over val-ues Social abstract argumentation frameworks (SAFs) alsoprovide an interesting and seemingly convenient formalismfor our model In SAFs agents vote on the acceptability ofarguments and the resulting labeling is a combination of thearguments that are voted for and labeling-based semantics Inour model we apply the same rule yet only for one argumentand given precise agent-based votingmechanisms and votingrules which are eventually parametrized

To consider our model as a specification of a mixed VAFand SAF model may be of great interest for future workThis formalism can provide solid foundations to describeexplain interpret and eventually extend the results foundhere from an argument-based perspective On those groundsfuture work will focus on creating a hybrid argumentationframeworkmdashwith its semanticsmdashthat takes heed of bothvalues or principles and voting

Complexity 27

Concerning other possible extensions of the model anatural reflex would be to argue on the validity and therobustness of the mix between the bounded confidenceopinion diffusion models and deliberation Consequentiallyadding deliberation to different bounded confidence modelsof opinion diffusion such as the Hegelsmann-Krause model[37] or to models of social impact theory like in [4] canbe of great interest for future work We acknowledge that afiner understanding of argumentation frameworks randomlattices and their respective applications for this modelare necessary Testing for different labeling-based semanticsor even trading labeling-based semantics for scoring-basedsemantics (semantics that defines acceptability on scoresgiven to arguments) such as debate semantics and ranking-based semantics (semantics that yield a preorder) may beinteresting to consider in future work

More into modeling deliberation it is not inadequateto believe that agents may also advance proposals duringdeliberation and replace to some extent the proposal forwhich the deliberation is taking place Endogenizing theprocess that produces the type of argument to be deliberatedon may be interesting to explore either by having agentsstrategically replace proposals at certain moments duringthe debate or having the central authority attempt to chooseproposals such that the probability of having them acceptedis high Agents may also be thought to be autoorganizing(independent of a central authority) and concede on the sizeof the instances of deliberation To consider differences insemantics across agents or even endogenizing the semanticson how urgent or important the proposal that is beingdiscussed is can also be an interesting direction to take

Last but not least many nontrivial modifications of ourmodel are possible and most should include better-thoughtdeliberation protocols It may be interesting to design andobserve protocols in which deliberation only affects agentsthat are actually debating for example Trust network effectsmultidimensionality of opinions new processes of argumentexchange or learning among agents are notions to furtherdevelop in order to make the model more realistic and relaxsome unreliable assumptions In sum the model is to berefined and extended with the objective of either studyingconcrete cases of deliberative polling and opinion dynamicsor implementing more intuitive thought experiments

Appendix

We present below some of the pseudocode we used toimplement the model

Algorithms

(a) Algorithm 1 implements the dyadic influence in themodel It describes how two agents interact andupdate their opinions

(b) Algorithm 2 implements the deliberative influence inthe model which describes how agents update theiropinions in sight of the accepted deliberated proposalarguments

(c) Algorithm 3 is used to obtain the grounded labelingof an argumentation framework

(d) Algorithm 4 provides an overview of a run in thesimulations119872 is the number of proposal argumentsto discuss which we set to 100 in the model

Each object in the system is indexed by a119908ℎ119900 number Hencea set of agents is annotated by a set of natural numbers Weuse this fact in the implementation of the algorithm 119874119887119895119890119888119905119894denotes the object of 119908ℎ119900 number 119894 Agentsrsquo opinions (119900) arecalibrated so that 119900 isin [minus1 1]Data Availability

The program and data used to support the findings of thisstudy are available from the corresponding author uponrequest

Disclosure

A preliminary version of this work was presented anddiscussed at the conference Bridging the Gap between FormalArgumentation and Actual Human Reasoning on October 4-5 2018 at BochumGermanyThe authors thank the audiencefor their valuable comments and suggestions

Conflicts of Interest

The authors declare that they have no conflicts of interestregarding the publication of this article

Acknowledgments

The contribution of Gabriella Pigozzi was supported by theDeutsche Forschungsgemeinschaft (DFG) and the CzechScience Foundation (GACR) as part of the joint project FromShared Evidence to Group Attitudes (RO 45486-1)

Endnotes

1 In other words the superiority of the ldquoright answerrdquo orlabeling in our case will appear as such to all

2 All agents hold the same opinion3 By sensitivity we mean how much an agent is ready to

change her opinion on the sole fact that an informationalcue has been debated on

4 The assumption is strong but this trust among agentsmay be rooted from a genuine individual motivation toreach a democratic consensus

5 We adopted a rule of thumbs that says that if an agentis neutral with respect to the principle then she willvote for any proposal that is presented to her duringdeliberation

6 If the argument is not unique she chooses one atrandom

7 If there are more than one candidate the agent will favorthe argument at minimal distance from 119868

28 Complexity

Require Vector of values for parameters (119881 = (120583 119880 119879))Ensure Successful interactions among agents119896 larr997888 0119873119890 larr997888 0

while 119896 le |119873|2 do ⊳Fill119873119890 with |119873|2 agents119896 larr997888 119896 + 1119873119890 larr997888 119877119886119899119889119900119898(|119873| + 1) ⊳Pseudo-random generator Generates integer between 0 and |119873|119873119903 larr997888 0 120575119894119895 larr997888 0 ⊳Variable to store differences in opinionfor all 119894 isin 119873119890 doRandomly pick a number 119895 isin 119873(119873119890 cup 119873119903)119873119903 larr997888 119873119903 cup 119895 120575119894119895 larr997888 |119900119894 minus 119900119895|procedure dyadic-update(119894 119895) ⊳Two agents 119894 and 119895 discuss and update opinions1199001015840 larr997888 119900119894 (120583119895 119880119895 119879119895) larr997888 (120583119894 119880119894 119879119894) larr997888 (120583 119880 119879) ⊳Save agent irsquos opinion before updating and homogenize agents

if 120575119894119895 lt 119880119894 then119900119894 larr997888 120583119894119900119895 + (1 minus 120583119894)119900119894if 120575119894119895 gt 119879119894 then119900119894 larr997888 (1 + 120583119894)119900119894 minus 120583119894119900119895if 120575119894119895 lt 119880119895 then119900119895 larr997888 1205831198951199001015840 + (1 minus 120583119895)119900119895if 120575119894119895 gt 119879119895 then119900119895 larr997888 (1 + 120583119895)119900119895 minus 1205831198951199001015840

Algorithm 1 Dyadic social influence

Require Vector of values for parameters (119881 = (120574 119901119886 119901119903))Ensure Successful interaction between proposal arguments and agents

for all 119894 isin 119873 doprocedure Probability-Update(V119886 119901119886 119901119903) ⊳Update individual probabilities of change due to deliberated cues119909119894 larr997888 119901|V119886minus119900119894 |2119886 119910119894 larr997888 1199012|V119886minus119900119894 |119903119902 larr997888 0if L119886119887(119868) = 119894119899 then

procedure deliberative-update(119894 119886) ⊳Agent i argument 119886120574119894 larr997888 120574 119902 larr997888 119906 sim U(0 1) ⊳ 119906 is a realization of a uniform random variableif 119902 lt 119909119894 then119900119894 larr997888 120574119894119900119894 + (1 minus 120574119894)V119886break

if 119902 lt 119910119894(1 minus 119909119894) then119900119894 larr997888 (1 + 120574119894)119900119894 minus 120574119894V119886Algorithm 2 Deliberation influence

8 In terms of computational complexity naive agentsmaketheirmoves in119874(|A119894| log(|A119894|)) while focused agents doat least as bad

9 Tables have memory Therefore debates are not neces-sarily trees

10 If there are less than (119899119863 times |119873|)2 agents for any of bothgroups the CA summons all of them to the deliberationtable

11 Or careful consideration of the viewpoints of otherswhich implies that citizens keep an open mind and donot reject arguments outright

12 A ldquohealthyrdquo consensus refers to the case in which manyconsensual decisions are taken but opinion-wise agentsdo not agree with each other Quotes are used on theword healthy because some political science theorists

(eg [36]) believe that in certain situations agreement inopinions for deliberation is epistemologically damaging

13 Simulations on groups of |119873| = 800 and |119873| = 400agents shows that scaling the population has an effect onall themetrics but it is confounding depending onwhichcouples of attitudes structures are considered (119880 119879) Westick to |119873| = 400 as in [9] for comparability andsimulation running time

14 By balanced wemean with as many arguments with V119886 lt0 as with V119886 gt 0

15 Experiments on the system suggest that a bigger(smaller) argument pool 119872 = 1000 (119872 = 200) isassociated with a higher (lower) variance of opinions(and extremism) and higher (lower) judgment accuracyFor the rest of the observations 119872 seem to have noeffect We choose a middle value for 119872 Similarly theargument pool chosen to be balanced or uniformly

Complexity 29

Require Argumentation framework 119860119865 = (AR)Ensure Assignment of labels to arguments from the grounded labeling of 119860119865L(A)

L0(A) larr997888 (0 0 0)119896 larr997888 0do119894119899 L119896+1(A) larr997888 119894119899 L119896(A) cup 119886 | 119886 is not labeled 119894119899 L119896(A) 119886119899119889 forall119887 isin A 119904119905 119887 R 119886 then 119887 isin 119900119906119905 L119896(A)119900119906119905 L119896+1(A) larr997888 119900119906119905 L119896(A) cup 119886 | 119886 is not labeled 119894119899 L119896(A) 119886119899119889 exist119887 isin A 119904119905 119887 R 119886 and 119887 isin 119894119899 L119896+1(A)

while L119896+1(A) = L119896(A) = 0Algorithm 3 Solve argumentation frameworks (with grounded semantics)

Require Vector of values for parameters (119881)Ensure Vector of statistics (119880)

procedure Set-parameters(119881) ⊳Initialize parameters for scenarioprocedure Init-objs(119899119898 119896 ) ⊳Create119898 arguments 119899 agents and 119860119865120576procedure Solve-Argumentation-Framework(119860119865) ⊳Assigns the epistemic labelingL120576 toA119894 larr997888 1119905 larr997888 1while 119894 le 119872 do ⊳ 119872 the number of arguments to discuss119868119894 larr997888 119866119890119899 119886119903119892(1) ⊳Generates proposal argument 119868119860 119894119905 larr997888 (119868119894 0)L119886119887(119868119894) larr997888 119906119899119889 ⊳Initialize label for 119868119895 larr997888 1 ⊳Initialize debate counter 119895while ((L119886119887(119868) = 119906119899119889 or 119898 le 119895) and 119895 le 119898) doif 119905 equiv 0(119898119900119889 119905119863) then

procedure build-argumentation-framework(119899119863 119887119890ℎ119886V119894119900119903) ⊳Build the argument graph agents deliberate(119861119894119905 119878119894119905) larr997888 (119883 119884) sub (⋃119897isin119873119899119863 A119897119894119905 ⋃119897isin119873119899119863 R119897119894119905) ⊳ 119873119899119863 sub 119873 are agents sampled by the CA119860 119894119905 larr997888 119889(119860 119894119905 (119861119894119905 119878119894119905))procedure Solve-Argumentation-Framework(A119889) ⊳FindL(A119889)L119886119887(119868119894) larr997888 L119886119887119863(119868119894)119895 larr997888 119895 + 1

else119896 larr997888 1while 119896 le 119905119863 doprocedure dyadic-social-influence(120583 119879 119880) ⊳Agents discuss one-to-one119896 larr997888 119896 + 1119905 larr997888 119905 + 1

if 120572 gt 0 andL119886119887(119868119894) = 119900119906119905 then ⊳Agents vote for the proposalif (|119897 isin 119873|119900119897 times V119868119894 ge 0| ge 120572 times 119899) then

L119886119887(119868119894) larr997888 119894119899else

L119886119887(119868119894) larr997888 119900119906119905if L119886119887(119868119894) = 119894119899 thenprocedure Deliberation-influence(120574 119901119886 119901119903) ⊳Agents are influenced by the result of the decision process119894 larr997888 119894 + 1119905 larr997888 119905 + 1119880 larr997888 119862119900119898119901119906119905119890 119878119905119886119905119894119904119905119894119888119904(119881)

return 119880 ⊳Get the statisticsAlgorithm 4 Run for one scenario

generated does not seem to play a significant explanatoryrole on any of the metrics

16 Experiments show that themaximumnumber of debatesis not an important parameter even if it is an intuitiveone when considering decision-making processes Inpractice debates end after at most 119898 + 1 deliberation

steps and even then it happens rarely However to avoidunpleasant surprises like infinite debates on a completeargument graph containing the proposal argument weadded the constraint 119898 Its value is precisely 7 becausethe biggest minimum number of debates we studyis 6

30 Complexity

17 To ensure comparability with the deliberative andmixedopinion dynamics models on judgment accuracy andcoherence agents vote on proposal arguments every 119905119863times(119898+1) 119905119863 and119898does not explain variability in judgmentaccuracy nor in coherence

18 Student t-tests for independent samples are robust tothe violation of two hypothesis on the distribution ofthe samples normality and homogeneity of variancesas long as the sample size is big enough (central limittheorem) and if the difference in size of the comparedsamples is smallrespectively

19 Studentrsquos test version forwhen variances are not assumedequal and the difference in size of the compared samplesis big

20 We do not correct for homoskedasticity nor normality oferrors in the estimates

21 Balanced as the same number of runs per scenario orsuch that no significant correlations are found betweenparameters

References

[1] R Axelrod ldquoThe dissemination of culture a model withlocal convergence and global polarizationrdquo Journal of ConflictResolution vol 41 no 2 pp 203ndash226 1997

[2] D M Rousseau Reinforcing the MicroMacro Bridge Organiza-tionalThinking and Pluralistic Vehicles SAGEPublications SageLos Angeles Calif USA 2011

[3] S Galam and SMoscovici ldquoTowards a theory of collective phe-nomena consensus and attitude changes in groupsrdquo EuropeanJournal of Social Psychology vol 21 no 1 pp 49ndash74 1991

[4] A Nowak J Szamrej and B Latane ldquoFrom private attitude topublic opinion a dynamic theory of social impactrdquo Psychologi-cal Review vol 97 no 3 pp 362ndash376 1990

[5] G Deffuant S Huet and F Amblard ldquoAn individual-basedmodel of innovation diffusion mixing social value and individ-ual benefitrdquo American Journal of Sociology vol 110 no 4 pp1041ndash1069 2005

[6] G Deffuant D Neau F Amblard and G Weisbuch ldquoMixingbeliefs among interacting agentsrdquoAdvances in Complex Systems(ACS) vol 3 no 01n04 pp 87ndash98 2000

[7] J Rouchier P Tubaro and C Emery ldquoOpinion transmission inorganizations an agent-based modeling approachrdquo Computa-tional and Mathematical OrganizationTheory vol 20 no 3 pp252ndash277 2014

[8] M Meadows and D Cliff ldquoThe relative disagreement model ofopinion dynamics where do extremists come fromrdquo in Pro-ceedings of the International Workshop on Self-Organizing Sys-tems Lecture Notes in Computer Science pp 66ndash77 Springer2013

[9] W Jager and F Amblard ldquoUniformity bipolarization and plu-riformity captured as generic stylized behavior with an agent-based simulation model of attitude changerdquo Computational ampMathematical Organization Theory vol 10 no 4 pp 295ndash3032005

[10] Y-F Zhang H-Y Duan and Z-L Geng ldquoEvolutionary mech-anism of frangibility in social consensus system based onnegative emotions spreadrdquo Complexity vol 2017 Article ID4037049 8 pages 2017

[11] A Flache andMWMacy ldquoSmall worlds and cultural polariza-tionrdquoThe Journal of Mathematical Sociology vol 35 no 1-3 pp146ndash176 2011

[12] A Tsang and K Larson ldquoOpinion dynamics of skepticalagentsrdquo in Proceedings of the 13th International Conference onAutonomous Agents and Multiagent Systems AAMAS 2014 pp277ndash284 International Foundation forAutonomousAgents andMultiagent Systems France May 2014

[13] K M Hansen Deliberative democracy and opinion formationUniversity Press of Southern Denmark Odense 2004

[14] C R Sunstein ldquoThe law of group polarizationrdquo Journal ofPolitical Philosophy vol 10 no 2 pp 175ndash195 2002 WileyOnline Library

[15] S Moscovici and W Doise Dissensions et Consensus unetheorie generale des decisions collectives Presses Universitairesde France-PUF 1992

[16] J S Fishkin and R C Luskin ldquoExperimenting with a demo-cratic ideal deliberative polling and public opinionrdquo ActaPolitica vol 40 no 3 pp 284ndash298 2005

[17] R C Luskin J S Fishkin and S Iyengar Considered Opinionson US Foreign Policy Face-To-Face versus Online Delibera-tive Polling International Communication Association NewOrleans La USA 2004

[18] J Barabas ldquoHowdeliberation affects policy opinionsrdquoAmericanPolitical Science Review vol 98 no 4 pp 687ndash701 2004

[19] O Pryymak A Rogers and N R Jennings ldquoEfficient opinionsharing in large decentralised teamsrdquo in Proceedings of the 11thInternational Conference on Autonomous Agents andMultiagentSystems vol 1 pp 543ndash550 Spain June 2012

[20] J Rouchier and E Tanimura ldquoWhen overconfident agents slowdown collective learningrdquo Simulation vol 88 no 1 pp 33ndash492012

[21] D Miller ldquoDeliberative democracy and social choicerdquo PoliticalStudies vol 40 S1 pp 54ndash67 1992

[22] M Sherif and C I Hovland Social Judgment Assimilation andContrast Effects in Communication and Attitude Change YaleUniversity Press 1961

[23] P M Dung ldquoOn the acceptability of arguments and its funda-mental role in nonmonotonic reasoning logic programmingand n-person gamesrdquo Artificial Intelligence vol 77 no 2 pp321ndash357 1995

[24] M Caminada and G Pigozzi ldquoOn judgment aggregation inabstract argumentationrdquo Autonomous Agents and Multi-AgentSystems vol 22 no 1 pp 64ndash102 2011

[25] P Baroni M Caminada andM Giacomin ldquoAn introduction toargumentation semanticsrdquoThe Knowledge Engineering Reviewvol 26 no 4 pp 365ndash410 2011

[26] S Chambers ldquoDeliberative democratic theoryrdquo Annual Reviewof Political Science vol 6 no 1 pp 307ndash326 2003

[27] E Bonzon andNMaudet ldquoOn the outcomes of multiparty per-suasionrdquo in Proceedings of the 10th International Conference onAutonomous Agents and Multiagent Systems 2011 AAMAS 2011vol 1 pp 47ndash54 International Foundation for AutonomousAgents and Multiagent Systems Taiwan May 2011

[28] J Ranciere Disagreement Politics and Philosophy University ofMinnesota Press 1999

[29] T Bench-Capon and K Atkinson ldquoAbstract argumentation andvaluesrdquo in Argumentation in Artificial Intelligence pp 45ndash64Springer 2009

[30] M Caminada ldquoOn the issue of reinstatement in argumenta-tionrdquo in Proceedings of the European Workshop on Logics in

Complexity 31

Artificial Intelligence vol 4160 of Lecture Notes in ComputerScience pp 111ndash123 Springer Berlin Heidelberg

[31] GAVreeswijk ldquoAn algorithm to computeminimally groundedand admissible defence sets in argument systemsrdquo in Pro-ceedings of the 2006 conference on Computational Models ofArgument Proceedings of COMMA 2006 vol 144 pp 109ndash120IOS Press Amsterdam The Netherlands 2006

[32] C W Sherif M Kelly H L Rodgers et al ldquoPersonal involve-ment social judgment and actionrdquo Journal of Personality andSocial Psychology vol 27 no 3 pp 311ndash328 1973

[33] G Deffuant F Amblard G Weisbuch and T Faure ldquoHow canextremism prevail A study based on the relative agreementinteraction modelrdquo Journal of Artificial Societies and SocialSimulation vol 5 no 4 p 1 2002

[34] E Awad M W Caminada G Pigozzi M Podlaszewski andI Rahwan ldquoPareto optimality and strategy-proofness in groupargument evaluationrdquo Journal of Logic and Computation vol 27no 8 pp 2581ndash2609 2017

[35] D Grossi and G Pigozzi ldquoJudgment aggregation a primerrdquoSynthesis Lectures on Artificial Intelligence and Machine Learn-ing vol 8 no 2 pp 1ndash151 2014

[36] H Landemore and S E Page ldquoDeliberation and disagreementproblem solving prediction and positive dissensusrdquo PoliticsPhilosophy and Economics vol 14 no 3 pp 229ndash254 2015

[37] R Hegselmann and U Krause ldquoOpinion dynamics andbounded confidence models analysis and simulationrdquo Journalof Artificial Societies and Social Simulation vol 5 no 3 2002

[38] A Stefanelli and R Seidl ldquoModerate and polarized opinionsUsing empirical data for an agent-based simulationrdquo inProceed-ings of the Social Simulation Conference 2014

[39] S Gabbriellini and P Torroni ldquoA new framework for ABMsbased on argumentative reasoningrdquo in Advances in SocialSimulation vol 229 pp 25ndash36 Springer Berlin Germany 2014

[40] H Mercier and D Sperber ldquoWhy do humans reason Argu-ments for an argumentative theoryrdquo Behavioral and BrainSciences vol 34 no 2 pp 57ndash74 2011

[41] H V Parunak T C Belding R Hilscher and S BruecknerldquoModeling and managing collective cognitive convergencerdquoin In Proceedings of the 7th iNternational Joint Conference onAutonomous Agents and Multiagent Systems vol 3 pp 1505ndash1508 International Foundation for Autonomous Agents andMultiagent Systems 2008

[42] H Prakken ldquoFormal systems for persuasion dialoguerdquo TheKnowledge Engineering Review vol 21 no 2 pp 163ndash188 2006

[43] K Atkinson T Bench-Capon and P Mcburney ldquoA dialoguegame protocol for multi-agent argument over proposals foractionrdquoAutonomousAgents andMulti-Agent Systems vol 11 no2 pp 153ndash171 2005

[44] M Thimm and A J Garcıa ldquoClassification and strategicalissues of argumentation games on structured argumentationframeworksrdquo in Proceedings of the 9th International Conferenceon Autonomous Agents and Multiagent Systems vol 1 pp 1247ndash1254 International Foundation for Autonomous Agents andMultiagent Systems Canada 2010

[45] I Rahwan and K Larson ldquoMechanism design for abstractargumentationrdquo in Proceedings of the 7th International JointConference on Autonomous Agents and Multiagent SystemsAAMAS 2008 pp 1031ndash1038 International Foundation forAutonomous Agents and Multiagent Systems Portugal May2008

[46] H Prakken R Riveret A Rotolo and G Sartor ldquoHeuristics inArgumentation A Game-Theoretical Investigationrdquo EuropeanUniversity Institute 2008

[47] N Hadidi Y Dimopoulos and P Moraitis ldquoTactics and con-cessions for argumentation-based negotiationrdquo in Proceedingsof the COMMA vol 245 pp 285ndash296 2012

[48] N Oren and T J Norman ldquoArguing using opponent modelsrdquoin Proceedings of the International Workshop on Argumentationin Multi-Agent Systems pp 160ndash174 Springer 2009

[49] S Delre W Jager T Bijmolt and M Janssen ldquoTargeting andtiming promotional activities An agent-based model for thetakeoff of new productsrdquo Journal of Business Research vol 60no 8 pp 826ndash835 2007

[50] A Mirtabatabaei P Jia and F Bullo ldquoEulerian opinion dynam-ics with bounded confidence and exogenous inputsrdquo SIAMJournal on Applied Dynamical Systems vol 13 no 1 pp 425ndash446 2014

[51] V Barbet N Guiraud V Laperriere and J Rouchier Hagglingon Values Towards Consensus or Trouble 2019

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Complexity 25

of abstract argumentation theory To our knowledge we areunaware of existing literature on agent-based modeling thatexplicitly relates collective choice and the notions of deliber-ation and opinion diffusion through abstract argumentationas we have done it

Most models in the literature of opinion diffusion areinterested in opinions because these have an influence incollective decision and in questions of social order Forinstance in [3] the authors are interested in consensus andin how a group collectively chooses among two alternativesIn other models authors are interested in the emergenceof extremism [8] and on the distribution of opinions whenextremists are introduced in the population [5 33 37] whileother authors coin the notion of opinion polarization as anemergent property of the system [8 12] They show usingmodels of ldquobounded confidencerdquo and opinion diffusion withtrust that three different kinds of steady states (unipolarbipolar and central) are possible depending on whetheragents are sufficiently uncertain about their opinions andsufficiently connected andor a certain proportion of indi-viduals are already extreme Recent articles on informationand opinion dynamics stress the importance of governanceand government intervention in the spread of emotions(opinions) in the frangibility of social consensus systemIn [10] the authors show that government intervention inthe spread of negative emotions can lead to an even fasterspread of negative emotions (single-peeked convergence) andin a faster collapse of the social consensus system Opinionas a function of trust is studied in [12] and relates to howindividuals form their opinions on the basis of how muchthey trust agents in their networks

Another stream of discrete opinion dynamics modelsthat explain the emergence of opinion can be found in theliterature Computer multiagent models of attitude changebased on Latanersquos social impact theory are presented in [4] Inthe theory an agent changes her opinion on the basis of theinformational impact she is subject to which depends on thepersuasiveness (strength) supportiveness and immediacy(group structure) of the environment and information shereceives The simulations predict two emergent groups ofphenomena the shifting of attitudes towards incompletelypolarized equilibria and the formation of coherent clusteringof subgroups with deviant attitudes Similarly but in acontinuous more argumentative fashion Mas and Flache[11] present a model of opinion diffusion using argumentsArguments are considered to be for or against a proposal andare given an agent-dependant relevance in dyadic persuasiveinteractions Arguments also determine the agentsrsquo opinionsand their dissimilarities which posits the rules of pair-wiseencounters in theirmodel as agents are assumed to form theiropinions according to the arguments they own They showthat in an argumentative opinion diffusion model the onlyobservable steady states are bipolarization and consensusand that bipolarization can emerge from interactions amongsimilar agents In other words bipolarization of opinionsis possible with homophily and without negative influenceif there is argumentation They subsequently compare theirresults to experimental data In [38] a similar approach toargumentation and opinion formation is taken but for the fact

that they introduce different types of arguments explicitlyapply social judgment theory [22] and use survey data toestablish model-to-real-world comparisons

Closer to opinion formation and abstract argumentationGabbriellini and Torroni [39] are the first to have origi-nally and soundly merged opinion diffusion and abstractargumentation They define an agentrsquos opinion as a functionof the arguments she holds and the attack relation amongthem They devised a focused peer-to-peer dialogue systemof persuasion (NetArg) inspired from Mercier and Sperberrsquosargumentative theory of reasoning [40] which used onlyabstract argumentation to study opinion polarization andopinion dynamics Moreover they considered networks andthe notions of trust and epistemic vigilance to define adynamics for knowledge revision and trust itself whichwe clearly lack in our model They showed that if a con-servative belief operator in argumentation was applied byagents when they reasoned then their dialogue protocoldid not increase polarization among agents That said theyused the model to study the effect of Groenvetterrsquos weaklink theory in the spreading of arguments and were notparticularly interested in notions like judgment accuracy orcoherence nor in tackling questions related to collectivedecision-making procedures Their system is very expressiveand helps position our work in the litterature our work is atthe frontier of pure argumentative opinion diffusion modelsopinion diffusion with arguments and continuous bounded-confidence models of opinion diffusion a la Deffuant

On another note work on collective cognitive conver-gence [41] and opinion sharing [19] shows that consensustowards a certain ldquocorrectrdquo opinion or cognitive state is alwayspossible yet dependent on noise variability and awarenessof agents In [20] the authors show that learning about anexogenous correct state of the world (represented by bits)under confidence was possible but only if the agents werenot too confident In a homogeneous population they showthat the higher the confidence the worse the learningmdashthevery confident agents do not learn the properties of the truestate of the world and disrupt the learning process of the lessconfident ones Collective cognitive convergence can be seenas a result of deliberation in truth-seeking models

When it comes to abstract argumentation theory we takean approach that wires two types of dialogues that are well-studied in the literature persuasion dialogues [42] and delib-eration dialogues [43] Another interesting line of work is theone on mechanism design [44] or the problem of devisingan argumentation protocol where strategic argumentation isnot a liability for success in debates We tackle mechanismdesign in a different way though Instead of consideringstrategy-proofness we are interested in how differences inprotocol can result in epistemologically ldquobetterrdquo collectivechoices and guarantee that opinion distributions are favorablefor deliberation For a survey on persuasion dialogue see[42]

Work on agent-based argumentation usually assumes thatthe semantic relationship between arguments is fixed Inother words if two individuals were to put two argumentsin the public arena such that one logically attacked the otherthen everyone would agree that such attack exists [45 46]

26 Complexity

Other models which do not make this restrictive assumptioncan also be found in the literature [27 47] and derive fromthe class or family of opponent models [27 48] in which twoopposing sides attempt to win a dialogue Ourmodel is in theintersection of these two but the frame combining opiniondiffusion of the kind and argumentation remains new

The idea of mixing interpersonal influence and verticalcommunication however is not original It is described andimplemented in innovation diffusion models such as [5 49]In both vertical communication is modeled as exogenoustransparent informationmdashagents are aware of the existenceof an innovation thus triggering several processes of choiceand stabilization of opinions Also in [50] an Eulerian modelis implemented to show the effect of media and exogenousinformation on opinion distributions With respect to thispoint the originality of our work is in that the informationemitted as vertical communication is endogenous It is issuedfrom a deliberation model that agents shape on the basisof their opinions arguments and behavior In the spirit of[51] where the authors control for the design of verticalcommunication (to whom the vertical communication isaddressed its timing how it influences agents opinions) wecontrol for the process generating the information for ourown set of variables length of debates majority quotas andthe frequency at which discussions take place

6 Conclusion and Perspectives

The main objective of this article was to build a bridgewhere decision-making argumentation and opinion dif-fusion could come together We proposed a model thatcombined abstract argumentation theory and a boundedconfidence opinion diffusion model and showed to whatextent it could explain variance of opinions extremismcoherence in collective decisions and judgment accuracyon arguments given an ideal state of full information Thesecond objective was to show in what way governance pro-viding agents with arenas of discussion and deliberation wasimportant in addressing the success of collective decision-making processes and the quality of their outcomes

The model revealed that allowing for more deliberationtime though in lower frequency and allowing for widerparticipation in deliberation increased the variance of opin-ions and the proportion of extremists in a group Theseobservations are consistent with the combination of resultsfound in [9] and in [13 14] which stress that deliberationmay polarize groups and may have a meager effect on theshifts of opinions and inconsistent with [16] where it isargued that opinions tend to moderate after deliberationIt happens that deliberation alone did moderate opinionyet when integrated into a more complex system in whichindividuals were allowed to interact with one another andnot all that was deliberated was accepted its influence wasshadowed by other more individualistic dynamics

Undeniably the grounded semantics played an importantrole in the weak effect of deliberation since it models askeptical way of reasoning over arguments For agentsaccepting arguments thus updating their opinions accord-ingly happened not too often Nevertheless asking for more

deliberation did increase judgment accuracy as observed in[18] yet in a marginally decreasing fashion We showed thatvoting within the deliberation protocol not only increasedthe proportion of extremists and the variance of opinions ina group but also determined how coherent deliberation andvoting procedures were with each other Lastly we showedthat agents that are focused judged arguments better hadmore stable opinions and constituted a group with higherproportion of extremists than their naive counterparts

In terms of governance the model says that there is per-haps no trade-off between extremismand judgment accuracyInstead it asserts that the higher the variance of opinions thecloser the group gets to the correct decisions This situationmay be interpreted as Landermore and Pagersquos notion of ldquoposi-tive dissensusrdquo (in complex collective decision-making tasks)[36] as opposed to consensus as a normative requirementfor correct collective decision-making If a decider has toorganize deliberation to legitimize a public policy dependingon what kind of world he or she wants different set ofparameters may be chosen to account for different levels oflegitimacy and correctness For instance one may ask for adecision to be legitimate to respect a certain level of coherenceor of judgment accuracy and to be discussed by a sundryof different-opinionated peoples If so deliberation has tobe made more often more participants have to be includedin debates or in deliberation instances and not too manytime steps should be introduced in between two deliberationinstances So the model may capture normative ideas todefine correct and legitimate collective decisions and tomake recommendations on those grounds Requirements ofhigh consensual deliberative accuracy (judgment accuracy)legitimacy (coherence) and diversity (variance of opinions)are all taken into account and for such a reason the model israther successful in describing a sound process of collectivedecision-making with interpretable outcomes

61 Extensions of the Model and Other Ideas Our model canbe extended in many ways Value-based abstract argumen-tation frameworks (VAFs) for example as an extension ofDungrsquos argumentation framework provide a formal descrip-tion of the process of decision-making in which argumentsare given values and audiences (set of agents) ignore attacksbetween arguments on the basis of their preferences over val-ues Social abstract argumentation frameworks (SAFs) alsoprovide an interesting and seemingly convenient formalismfor our model In SAFs agents vote on the acceptability ofarguments and the resulting labeling is a combination of thearguments that are voted for and labeling-based semantics Inour model we apply the same rule yet only for one argumentand given precise agent-based votingmechanisms and votingrules which are eventually parametrized

To consider our model as a specification of a mixed VAFand SAF model may be of great interest for future workThis formalism can provide solid foundations to describeexplain interpret and eventually extend the results foundhere from an argument-based perspective On those groundsfuture work will focus on creating a hybrid argumentationframeworkmdashwith its semanticsmdashthat takes heed of bothvalues or principles and voting

Complexity 27

Concerning other possible extensions of the model anatural reflex would be to argue on the validity and therobustness of the mix between the bounded confidenceopinion diffusion models and deliberation Consequentiallyadding deliberation to different bounded confidence modelsof opinion diffusion such as the Hegelsmann-Krause model[37] or to models of social impact theory like in [4] canbe of great interest for future work We acknowledge that afiner understanding of argumentation frameworks randomlattices and their respective applications for this modelare necessary Testing for different labeling-based semanticsor even trading labeling-based semantics for scoring-basedsemantics (semantics that defines acceptability on scoresgiven to arguments) such as debate semantics and ranking-based semantics (semantics that yield a preorder) may beinteresting to consider in future work

More into modeling deliberation it is not inadequateto believe that agents may also advance proposals duringdeliberation and replace to some extent the proposal forwhich the deliberation is taking place Endogenizing theprocess that produces the type of argument to be deliberatedon may be interesting to explore either by having agentsstrategically replace proposals at certain moments duringthe debate or having the central authority attempt to chooseproposals such that the probability of having them acceptedis high Agents may also be thought to be autoorganizing(independent of a central authority) and concede on the sizeof the instances of deliberation To consider differences insemantics across agents or even endogenizing the semanticson how urgent or important the proposal that is beingdiscussed is can also be an interesting direction to take

Last but not least many nontrivial modifications of ourmodel are possible and most should include better-thoughtdeliberation protocols It may be interesting to design andobserve protocols in which deliberation only affects agentsthat are actually debating for example Trust network effectsmultidimensionality of opinions new processes of argumentexchange or learning among agents are notions to furtherdevelop in order to make the model more realistic and relaxsome unreliable assumptions In sum the model is to berefined and extended with the objective of either studyingconcrete cases of deliberative polling and opinion dynamicsor implementing more intuitive thought experiments

Appendix

We present below some of the pseudocode we used toimplement the model

Algorithms

(a) Algorithm 1 implements the dyadic influence in themodel It describes how two agents interact andupdate their opinions

(b) Algorithm 2 implements the deliberative influence inthe model which describes how agents update theiropinions in sight of the accepted deliberated proposalarguments

(c) Algorithm 3 is used to obtain the grounded labelingof an argumentation framework

(d) Algorithm 4 provides an overview of a run in thesimulations119872 is the number of proposal argumentsto discuss which we set to 100 in the model

Each object in the system is indexed by a119908ℎ119900 number Hencea set of agents is annotated by a set of natural numbers Weuse this fact in the implementation of the algorithm 119874119887119895119890119888119905119894denotes the object of 119908ℎ119900 number 119894 Agentsrsquo opinions (119900) arecalibrated so that 119900 isin [minus1 1]Data Availability

The program and data used to support the findings of thisstudy are available from the corresponding author uponrequest

Disclosure

A preliminary version of this work was presented anddiscussed at the conference Bridging the Gap between FormalArgumentation and Actual Human Reasoning on October 4-5 2018 at BochumGermanyThe authors thank the audiencefor their valuable comments and suggestions

Conflicts of Interest

The authors declare that they have no conflicts of interestregarding the publication of this article

Acknowledgments

The contribution of Gabriella Pigozzi was supported by theDeutsche Forschungsgemeinschaft (DFG) and the CzechScience Foundation (GACR) as part of the joint project FromShared Evidence to Group Attitudes (RO 45486-1)

Endnotes

1 In other words the superiority of the ldquoright answerrdquo orlabeling in our case will appear as such to all

2 All agents hold the same opinion3 By sensitivity we mean how much an agent is ready to

change her opinion on the sole fact that an informationalcue has been debated on

4 The assumption is strong but this trust among agentsmay be rooted from a genuine individual motivation toreach a democratic consensus

5 We adopted a rule of thumbs that says that if an agentis neutral with respect to the principle then she willvote for any proposal that is presented to her duringdeliberation

6 If the argument is not unique she chooses one atrandom

7 If there are more than one candidate the agent will favorthe argument at minimal distance from 119868

28 Complexity

Require Vector of values for parameters (119881 = (120583 119880 119879))Ensure Successful interactions among agents119896 larr997888 0119873119890 larr997888 0

while 119896 le |119873|2 do ⊳Fill119873119890 with |119873|2 agents119896 larr997888 119896 + 1119873119890 larr997888 119877119886119899119889119900119898(|119873| + 1) ⊳Pseudo-random generator Generates integer between 0 and |119873|119873119903 larr997888 0 120575119894119895 larr997888 0 ⊳Variable to store differences in opinionfor all 119894 isin 119873119890 doRandomly pick a number 119895 isin 119873(119873119890 cup 119873119903)119873119903 larr997888 119873119903 cup 119895 120575119894119895 larr997888 |119900119894 minus 119900119895|procedure dyadic-update(119894 119895) ⊳Two agents 119894 and 119895 discuss and update opinions1199001015840 larr997888 119900119894 (120583119895 119880119895 119879119895) larr997888 (120583119894 119880119894 119879119894) larr997888 (120583 119880 119879) ⊳Save agent irsquos opinion before updating and homogenize agents

if 120575119894119895 lt 119880119894 then119900119894 larr997888 120583119894119900119895 + (1 minus 120583119894)119900119894if 120575119894119895 gt 119879119894 then119900119894 larr997888 (1 + 120583119894)119900119894 minus 120583119894119900119895if 120575119894119895 lt 119880119895 then119900119895 larr997888 1205831198951199001015840 + (1 minus 120583119895)119900119895if 120575119894119895 gt 119879119895 then119900119895 larr997888 (1 + 120583119895)119900119895 minus 1205831198951199001015840

Algorithm 1 Dyadic social influence

Require Vector of values for parameters (119881 = (120574 119901119886 119901119903))Ensure Successful interaction between proposal arguments and agents

for all 119894 isin 119873 doprocedure Probability-Update(V119886 119901119886 119901119903) ⊳Update individual probabilities of change due to deliberated cues119909119894 larr997888 119901|V119886minus119900119894 |2119886 119910119894 larr997888 1199012|V119886minus119900119894 |119903119902 larr997888 0if L119886119887(119868) = 119894119899 then

procedure deliberative-update(119894 119886) ⊳Agent i argument 119886120574119894 larr997888 120574 119902 larr997888 119906 sim U(0 1) ⊳ 119906 is a realization of a uniform random variableif 119902 lt 119909119894 then119900119894 larr997888 120574119894119900119894 + (1 minus 120574119894)V119886break

if 119902 lt 119910119894(1 minus 119909119894) then119900119894 larr997888 (1 + 120574119894)119900119894 minus 120574119894V119886Algorithm 2 Deliberation influence

8 In terms of computational complexity naive agentsmaketheirmoves in119874(|A119894| log(|A119894|)) while focused agents doat least as bad

9 Tables have memory Therefore debates are not neces-sarily trees

10 If there are less than (119899119863 times |119873|)2 agents for any of bothgroups the CA summons all of them to the deliberationtable

11 Or careful consideration of the viewpoints of otherswhich implies that citizens keep an open mind and donot reject arguments outright

12 A ldquohealthyrdquo consensus refers to the case in which manyconsensual decisions are taken but opinion-wise agentsdo not agree with each other Quotes are used on theword healthy because some political science theorists

(eg [36]) believe that in certain situations agreement inopinions for deliberation is epistemologically damaging

13 Simulations on groups of |119873| = 800 and |119873| = 400agents shows that scaling the population has an effect onall themetrics but it is confounding depending onwhichcouples of attitudes structures are considered (119880 119879) Westick to |119873| = 400 as in [9] for comparability andsimulation running time

14 By balanced wemean with as many arguments with V119886 lt0 as with V119886 gt 0

15 Experiments on the system suggest that a bigger(smaller) argument pool 119872 = 1000 (119872 = 200) isassociated with a higher (lower) variance of opinions(and extremism) and higher (lower) judgment accuracyFor the rest of the observations 119872 seem to have noeffect We choose a middle value for 119872 Similarly theargument pool chosen to be balanced or uniformly

Complexity 29

Require Argumentation framework 119860119865 = (AR)Ensure Assignment of labels to arguments from the grounded labeling of 119860119865L(A)

L0(A) larr997888 (0 0 0)119896 larr997888 0do119894119899 L119896+1(A) larr997888 119894119899 L119896(A) cup 119886 | 119886 is not labeled 119894119899 L119896(A) 119886119899119889 forall119887 isin A 119904119905 119887 R 119886 then 119887 isin 119900119906119905 L119896(A)119900119906119905 L119896+1(A) larr997888 119900119906119905 L119896(A) cup 119886 | 119886 is not labeled 119894119899 L119896(A) 119886119899119889 exist119887 isin A 119904119905 119887 R 119886 and 119887 isin 119894119899 L119896+1(A)

while L119896+1(A) = L119896(A) = 0Algorithm 3 Solve argumentation frameworks (with grounded semantics)

Require Vector of values for parameters (119881)Ensure Vector of statistics (119880)

procedure Set-parameters(119881) ⊳Initialize parameters for scenarioprocedure Init-objs(119899119898 119896 ) ⊳Create119898 arguments 119899 agents and 119860119865120576procedure Solve-Argumentation-Framework(119860119865) ⊳Assigns the epistemic labelingL120576 toA119894 larr997888 1119905 larr997888 1while 119894 le 119872 do ⊳ 119872 the number of arguments to discuss119868119894 larr997888 119866119890119899 119886119903119892(1) ⊳Generates proposal argument 119868119860 119894119905 larr997888 (119868119894 0)L119886119887(119868119894) larr997888 119906119899119889 ⊳Initialize label for 119868119895 larr997888 1 ⊳Initialize debate counter 119895while ((L119886119887(119868) = 119906119899119889 or 119898 le 119895) and 119895 le 119898) doif 119905 equiv 0(119898119900119889 119905119863) then

procedure build-argumentation-framework(119899119863 119887119890ℎ119886V119894119900119903) ⊳Build the argument graph agents deliberate(119861119894119905 119878119894119905) larr997888 (119883 119884) sub (⋃119897isin119873119899119863 A119897119894119905 ⋃119897isin119873119899119863 R119897119894119905) ⊳ 119873119899119863 sub 119873 are agents sampled by the CA119860 119894119905 larr997888 119889(119860 119894119905 (119861119894119905 119878119894119905))procedure Solve-Argumentation-Framework(A119889) ⊳FindL(A119889)L119886119887(119868119894) larr997888 L119886119887119863(119868119894)119895 larr997888 119895 + 1

else119896 larr997888 1while 119896 le 119905119863 doprocedure dyadic-social-influence(120583 119879 119880) ⊳Agents discuss one-to-one119896 larr997888 119896 + 1119905 larr997888 119905 + 1

if 120572 gt 0 andL119886119887(119868119894) = 119900119906119905 then ⊳Agents vote for the proposalif (|119897 isin 119873|119900119897 times V119868119894 ge 0| ge 120572 times 119899) then

L119886119887(119868119894) larr997888 119894119899else

L119886119887(119868119894) larr997888 119900119906119905if L119886119887(119868119894) = 119894119899 thenprocedure Deliberation-influence(120574 119901119886 119901119903) ⊳Agents are influenced by the result of the decision process119894 larr997888 119894 + 1119905 larr997888 119905 + 1119880 larr997888 119862119900119898119901119906119905119890 119878119905119886119905119894119904119905119894119888119904(119881)

return 119880 ⊳Get the statisticsAlgorithm 4 Run for one scenario

generated does not seem to play a significant explanatoryrole on any of the metrics

16 Experiments show that themaximumnumber of debatesis not an important parameter even if it is an intuitiveone when considering decision-making processes Inpractice debates end after at most 119898 + 1 deliberation

steps and even then it happens rarely However to avoidunpleasant surprises like infinite debates on a completeargument graph containing the proposal argument weadded the constraint 119898 Its value is precisely 7 becausethe biggest minimum number of debates we studyis 6

30 Complexity

17 To ensure comparability with the deliberative andmixedopinion dynamics models on judgment accuracy andcoherence agents vote on proposal arguments every 119905119863times(119898+1) 119905119863 and119898does not explain variability in judgmentaccuracy nor in coherence

18 Student t-tests for independent samples are robust tothe violation of two hypothesis on the distribution ofthe samples normality and homogeneity of variancesas long as the sample size is big enough (central limittheorem) and if the difference in size of the comparedsamples is smallrespectively

19 Studentrsquos test version forwhen variances are not assumedequal and the difference in size of the compared samplesis big

20 We do not correct for homoskedasticity nor normality oferrors in the estimates

21 Balanced as the same number of runs per scenario orsuch that no significant correlations are found betweenparameters

References

[1] R Axelrod ldquoThe dissemination of culture a model withlocal convergence and global polarizationrdquo Journal of ConflictResolution vol 41 no 2 pp 203ndash226 1997

[2] D M Rousseau Reinforcing the MicroMacro Bridge Organiza-tionalThinking and Pluralistic Vehicles SAGEPublications SageLos Angeles Calif USA 2011

[3] S Galam and SMoscovici ldquoTowards a theory of collective phe-nomena consensus and attitude changes in groupsrdquo EuropeanJournal of Social Psychology vol 21 no 1 pp 49ndash74 1991

[4] A Nowak J Szamrej and B Latane ldquoFrom private attitude topublic opinion a dynamic theory of social impactrdquo Psychologi-cal Review vol 97 no 3 pp 362ndash376 1990

[5] G Deffuant S Huet and F Amblard ldquoAn individual-basedmodel of innovation diffusion mixing social value and individ-ual benefitrdquo American Journal of Sociology vol 110 no 4 pp1041ndash1069 2005

[6] G Deffuant D Neau F Amblard and G Weisbuch ldquoMixingbeliefs among interacting agentsrdquoAdvances in Complex Systems(ACS) vol 3 no 01n04 pp 87ndash98 2000

[7] J Rouchier P Tubaro and C Emery ldquoOpinion transmission inorganizations an agent-based modeling approachrdquo Computa-tional and Mathematical OrganizationTheory vol 20 no 3 pp252ndash277 2014

[8] M Meadows and D Cliff ldquoThe relative disagreement model ofopinion dynamics where do extremists come fromrdquo in Pro-ceedings of the International Workshop on Self-Organizing Sys-tems Lecture Notes in Computer Science pp 66ndash77 Springer2013

[9] W Jager and F Amblard ldquoUniformity bipolarization and plu-riformity captured as generic stylized behavior with an agent-based simulation model of attitude changerdquo Computational ampMathematical Organization Theory vol 10 no 4 pp 295ndash3032005

[10] Y-F Zhang H-Y Duan and Z-L Geng ldquoEvolutionary mech-anism of frangibility in social consensus system based onnegative emotions spreadrdquo Complexity vol 2017 Article ID4037049 8 pages 2017

[11] A Flache andMWMacy ldquoSmall worlds and cultural polariza-tionrdquoThe Journal of Mathematical Sociology vol 35 no 1-3 pp146ndash176 2011

[12] A Tsang and K Larson ldquoOpinion dynamics of skepticalagentsrdquo in Proceedings of the 13th International Conference onAutonomous Agents and Multiagent Systems AAMAS 2014 pp277ndash284 International Foundation forAutonomousAgents andMultiagent Systems France May 2014

[13] K M Hansen Deliberative democracy and opinion formationUniversity Press of Southern Denmark Odense 2004

[14] C R Sunstein ldquoThe law of group polarizationrdquo Journal ofPolitical Philosophy vol 10 no 2 pp 175ndash195 2002 WileyOnline Library

[15] S Moscovici and W Doise Dissensions et Consensus unetheorie generale des decisions collectives Presses Universitairesde France-PUF 1992

[16] J S Fishkin and R C Luskin ldquoExperimenting with a demo-cratic ideal deliberative polling and public opinionrdquo ActaPolitica vol 40 no 3 pp 284ndash298 2005

[17] R C Luskin J S Fishkin and S Iyengar Considered Opinionson US Foreign Policy Face-To-Face versus Online Delibera-tive Polling International Communication Association NewOrleans La USA 2004

[18] J Barabas ldquoHowdeliberation affects policy opinionsrdquoAmericanPolitical Science Review vol 98 no 4 pp 687ndash701 2004

[19] O Pryymak A Rogers and N R Jennings ldquoEfficient opinionsharing in large decentralised teamsrdquo in Proceedings of the 11thInternational Conference on Autonomous Agents andMultiagentSystems vol 1 pp 543ndash550 Spain June 2012

[20] J Rouchier and E Tanimura ldquoWhen overconfident agents slowdown collective learningrdquo Simulation vol 88 no 1 pp 33ndash492012

[21] D Miller ldquoDeliberative democracy and social choicerdquo PoliticalStudies vol 40 S1 pp 54ndash67 1992

[22] M Sherif and C I Hovland Social Judgment Assimilation andContrast Effects in Communication and Attitude Change YaleUniversity Press 1961

[23] P M Dung ldquoOn the acceptability of arguments and its funda-mental role in nonmonotonic reasoning logic programmingand n-person gamesrdquo Artificial Intelligence vol 77 no 2 pp321ndash357 1995

[24] M Caminada and G Pigozzi ldquoOn judgment aggregation inabstract argumentationrdquo Autonomous Agents and Multi-AgentSystems vol 22 no 1 pp 64ndash102 2011

[25] P Baroni M Caminada andM Giacomin ldquoAn introduction toargumentation semanticsrdquoThe Knowledge Engineering Reviewvol 26 no 4 pp 365ndash410 2011

[26] S Chambers ldquoDeliberative democratic theoryrdquo Annual Reviewof Political Science vol 6 no 1 pp 307ndash326 2003

[27] E Bonzon andNMaudet ldquoOn the outcomes of multiparty per-suasionrdquo in Proceedings of the 10th International Conference onAutonomous Agents and Multiagent Systems 2011 AAMAS 2011vol 1 pp 47ndash54 International Foundation for AutonomousAgents and Multiagent Systems Taiwan May 2011

[28] J Ranciere Disagreement Politics and Philosophy University ofMinnesota Press 1999

[29] T Bench-Capon and K Atkinson ldquoAbstract argumentation andvaluesrdquo in Argumentation in Artificial Intelligence pp 45ndash64Springer 2009

[30] M Caminada ldquoOn the issue of reinstatement in argumenta-tionrdquo in Proceedings of the European Workshop on Logics in

Complexity 31

Artificial Intelligence vol 4160 of Lecture Notes in ComputerScience pp 111ndash123 Springer Berlin Heidelberg

[31] GAVreeswijk ldquoAn algorithm to computeminimally groundedand admissible defence sets in argument systemsrdquo in Pro-ceedings of the 2006 conference on Computational Models ofArgument Proceedings of COMMA 2006 vol 144 pp 109ndash120IOS Press Amsterdam The Netherlands 2006

[32] C W Sherif M Kelly H L Rodgers et al ldquoPersonal involve-ment social judgment and actionrdquo Journal of Personality andSocial Psychology vol 27 no 3 pp 311ndash328 1973

[33] G Deffuant F Amblard G Weisbuch and T Faure ldquoHow canextremism prevail A study based on the relative agreementinteraction modelrdquo Journal of Artificial Societies and SocialSimulation vol 5 no 4 p 1 2002

[34] E Awad M W Caminada G Pigozzi M Podlaszewski andI Rahwan ldquoPareto optimality and strategy-proofness in groupargument evaluationrdquo Journal of Logic and Computation vol 27no 8 pp 2581ndash2609 2017

[35] D Grossi and G Pigozzi ldquoJudgment aggregation a primerrdquoSynthesis Lectures on Artificial Intelligence and Machine Learn-ing vol 8 no 2 pp 1ndash151 2014

[36] H Landemore and S E Page ldquoDeliberation and disagreementproblem solving prediction and positive dissensusrdquo PoliticsPhilosophy and Economics vol 14 no 3 pp 229ndash254 2015

[37] R Hegselmann and U Krause ldquoOpinion dynamics andbounded confidence models analysis and simulationrdquo Journalof Artificial Societies and Social Simulation vol 5 no 3 2002

[38] A Stefanelli and R Seidl ldquoModerate and polarized opinionsUsing empirical data for an agent-based simulationrdquo inProceed-ings of the Social Simulation Conference 2014

[39] S Gabbriellini and P Torroni ldquoA new framework for ABMsbased on argumentative reasoningrdquo in Advances in SocialSimulation vol 229 pp 25ndash36 Springer Berlin Germany 2014

[40] H Mercier and D Sperber ldquoWhy do humans reason Argu-ments for an argumentative theoryrdquo Behavioral and BrainSciences vol 34 no 2 pp 57ndash74 2011

[41] H V Parunak T C Belding R Hilscher and S BruecknerldquoModeling and managing collective cognitive convergencerdquoin In Proceedings of the 7th iNternational Joint Conference onAutonomous Agents and Multiagent Systems vol 3 pp 1505ndash1508 International Foundation for Autonomous Agents andMultiagent Systems 2008

[42] H Prakken ldquoFormal systems for persuasion dialoguerdquo TheKnowledge Engineering Review vol 21 no 2 pp 163ndash188 2006

[43] K Atkinson T Bench-Capon and P Mcburney ldquoA dialoguegame protocol for multi-agent argument over proposals foractionrdquoAutonomousAgents andMulti-Agent Systems vol 11 no2 pp 153ndash171 2005

[44] M Thimm and A J Garcıa ldquoClassification and strategicalissues of argumentation games on structured argumentationframeworksrdquo in Proceedings of the 9th International Conferenceon Autonomous Agents and Multiagent Systems vol 1 pp 1247ndash1254 International Foundation for Autonomous Agents andMultiagent Systems Canada 2010

[45] I Rahwan and K Larson ldquoMechanism design for abstractargumentationrdquo in Proceedings of the 7th International JointConference on Autonomous Agents and Multiagent SystemsAAMAS 2008 pp 1031ndash1038 International Foundation forAutonomous Agents and Multiagent Systems Portugal May2008

[46] H Prakken R Riveret A Rotolo and G Sartor ldquoHeuristics inArgumentation A Game-Theoretical Investigationrdquo EuropeanUniversity Institute 2008

[47] N Hadidi Y Dimopoulos and P Moraitis ldquoTactics and con-cessions for argumentation-based negotiationrdquo in Proceedingsof the COMMA vol 245 pp 285ndash296 2012

[48] N Oren and T J Norman ldquoArguing using opponent modelsrdquoin Proceedings of the International Workshop on Argumentationin Multi-Agent Systems pp 160ndash174 Springer 2009

[49] S Delre W Jager T Bijmolt and M Janssen ldquoTargeting andtiming promotional activities An agent-based model for thetakeoff of new productsrdquo Journal of Business Research vol 60no 8 pp 826ndash835 2007

[50] A Mirtabatabaei P Jia and F Bullo ldquoEulerian opinion dynam-ics with bounded confidence and exogenous inputsrdquo SIAMJournal on Applied Dynamical Systems vol 13 no 1 pp 425ndash446 2014

[51] V Barbet N Guiraud V Laperriere and J Rouchier Hagglingon Values Towards Consensus or Trouble 2019

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Submit your manuscripts atwwwhindawicom

26 Complexity

Other models which do not make this restrictive assumptioncan also be found in the literature [27 47] and derive fromthe class or family of opponent models [27 48] in which twoopposing sides attempt to win a dialogue Ourmodel is in theintersection of these two but the frame combining opiniondiffusion of the kind and argumentation remains new

The idea of mixing interpersonal influence and verticalcommunication however is not original It is described andimplemented in innovation diffusion models such as [5 49]In both vertical communication is modeled as exogenoustransparent informationmdashagents are aware of the existenceof an innovation thus triggering several processes of choiceand stabilization of opinions Also in [50] an Eulerian modelis implemented to show the effect of media and exogenousinformation on opinion distributions With respect to thispoint the originality of our work is in that the informationemitted as vertical communication is endogenous It is issuedfrom a deliberation model that agents shape on the basisof their opinions arguments and behavior In the spirit of[51] where the authors control for the design of verticalcommunication (to whom the vertical communication isaddressed its timing how it influences agents opinions) wecontrol for the process generating the information for ourown set of variables length of debates majority quotas andthe frequency at which discussions take place

6 Conclusion and Perspectives

The main objective of this article was to build a bridgewhere decision-making argumentation and opinion dif-fusion could come together We proposed a model thatcombined abstract argumentation theory and a boundedconfidence opinion diffusion model and showed to whatextent it could explain variance of opinions extremismcoherence in collective decisions and judgment accuracyon arguments given an ideal state of full information Thesecond objective was to show in what way governance pro-viding agents with arenas of discussion and deliberation wasimportant in addressing the success of collective decision-making processes and the quality of their outcomes

The model revealed that allowing for more deliberationtime though in lower frequency and allowing for widerparticipation in deliberation increased the variance of opin-ions and the proportion of extremists in a group Theseobservations are consistent with the combination of resultsfound in [9] and in [13 14] which stress that deliberationmay polarize groups and may have a meager effect on theshifts of opinions and inconsistent with [16] where it isargued that opinions tend to moderate after deliberationIt happens that deliberation alone did moderate opinionyet when integrated into a more complex system in whichindividuals were allowed to interact with one another andnot all that was deliberated was accepted its influence wasshadowed by other more individualistic dynamics

Undeniably the grounded semantics played an importantrole in the weak effect of deliberation since it models askeptical way of reasoning over arguments For agentsaccepting arguments thus updating their opinions accord-ingly happened not too often Nevertheless asking for more

deliberation did increase judgment accuracy as observed in[18] yet in a marginally decreasing fashion We showed thatvoting within the deliberation protocol not only increasedthe proportion of extremists and the variance of opinions ina group but also determined how coherent deliberation andvoting procedures were with each other Lastly we showedthat agents that are focused judged arguments better hadmore stable opinions and constituted a group with higherproportion of extremists than their naive counterparts

In terms of governance the model says that there is per-haps no trade-off between extremismand judgment accuracyInstead it asserts that the higher the variance of opinions thecloser the group gets to the correct decisions This situationmay be interpreted as Landermore and Pagersquos notion of ldquoposi-tive dissensusrdquo (in complex collective decision-making tasks)[36] as opposed to consensus as a normative requirementfor correct collective decision-making If a decider has toorganize deliberation to legitimize a public policy dependingon what kind of world he or she wants different set ofparameters may be chosen to account for different levels oflegitimacy and correctness For instance one may ask for adecision to be legitimate to respect a certain level of coherenceor of judgment accuracy and to be discussed by a sundryof different-opinionated peoples If so deliberation has tobe made more often more participants have to be includedin debates or in deliberation instances and not too manytime steps should be introduced in between two deliberationinstances So the model may capture normative ideas todefine correct and legitimate collective decisions and tomake recommendations on those grounds Requirements ofhigh consensual deliberative accuracy (judgment accuracy)legitimacy (coherence) and diversity (variance of opinions)are all taken into account and for such a reason the model israther successful in describing a sound process of collectivedecision-making with interpretable outcomes

61 Extensions of the Model and Other Ideas Our model canbe extended in many ways Value-based abstract argumen-tation frameworks (VAFs) for example as an extension ofDungrsquos argumentation framework provide a formal descrip-tion of the process of decision-making in which argumentsare given values and audiences (set of agents) ignore attacksbetween arguments on the basis of their preferences over val-ues Social abstract argumentation frameworks (SAFs) alsoprovide an interesting and seemingly convenient formalismfor our model In SAFs agents vote on the acceptability ofarguments and the resulting labeling is a combination of thearguments that are voted for and labeling-based semantics Inour model we apply the same rule yet only for one argumentand given precise agent-based votingmechanisms and votingrules which are eventually parametrized

To consider our model as a specification of a mixed VAFand SAF model may be of great interest for future workThis formalism can provide solid foundations to describeexplain interpret and eventually extend the results foundhere from an argument-based perspective On those groundsfuture work will focus on creating a hybrid argumentationframeworkmdashwith its semanticsmdashthat takes heed of bothvalues or principles and voting

Complexity 27

Concerning other possible extensions of the model anatural reflex would be to argue on the validity and therobustness of the mix between the bounded confidenceopinion diffusion models and deliberation Consequentiallyadding deliberation to different bounded confidence modelsof opinion diffusion such as the Hegelsmann-Krause model[37] or to models of social impact theory like in [4] canbe of great interest for future work We acknowledge that afiner understanding of argumentation frameworks randomlattices and their respective applications for this modelare necessary Testing for different labeling-based semanticsor even trading labeling-based semantics for scoring-basedsemantics (semantics that defines acceptability on scoresgiven to arguments) such as debate semantics and ranking-based semantics (semantics that yield a preorder) may beinteresting to consider in future work

More into modeling deliberation it is not inadequateto believe that agents may also advance proposals duringdeliberation and replace to some extent the proposal forwhich the deliberation is taking place Endogenizing theprocess that produces the type of argument to be deliberatedon may be interesting to explore either by having agentsstrategically replace proposals at certain moments duringthe debate or having the central authority attempt to chooseproposals such that the probability of having them acceptedis high Agents may also be thought to be autoorganizing(independent of a central authority) and concede on the sizeof the instances of deliberation To consider differences insemantics across agents or even endogenizing the semanticson how urgent or important the proposal that is beingdiscussed is can also be an interesting direction to take

Last but not least many nontrivial modifications of ourmodel are possible and most should include better-thoughtdeliberation protocols It may be interesting to design andobserve protocols in which deliberation only affects agentsthat are actually debating for example Trust network effectsmultidimensionality of opinions new processes of argumentexchange or learning among agents are notions to furtherdevelop in order to make the model more realistic and relaxsome unreliable assumptions In sum the model is to berefined and extended with the objective of either studyingconcrete cases of deliberative polling and opinion dynamicsor implementing more intuitive thought experiments

Appendix

We present below some of the pseudocode we used toimplement the model

Algorithms

(a) Algorithm 1 implements the dyadic influence in themodel It describes how two agents interact andupdate their opinions

(b) Algorithm 2 implements the deliberative influence inthe model which describes how agents update theiropinions in sight of the accepted deliberated proposalarguments

(c) Algorithm 3 is used to obtain the grounded labelingof an argumentation framework

(d) Algorithm 4 provides an overview of a run in thesimulations119872 is the number of proposal argumentsto discuss which we set to 100 in the model

Each object in the system is indexed by a119908ℎ119900 number Hencea set of agents is annotated by a set of natural numbers Weuse this fact in the implementation of the algorithm 119874119887119895119890119888119905119894denotes the object of 119908ℎ119900 number 119894 Agentsrsquo opinions (119900) arecalibrated so that 119900 isin [minus1 1]Data Availability

The program and data used to support the findings of thisstudy are available from the corresponding author uponrequest

Disclosure

A preliminary version of this work was presented anddiscussed at the conference Bridging the Gap between FormalArgumentation and Actual Human Reasoning on October 4-5 2018 at BochumGermanyThe authors thank the audiencefor their valuable comments and suggestions

Conflicts of Interest

The authors declare that they have no conflicts of interestregarding the publication of this article

Acknowledgments

The contribution of Gabriella Pigozzi was supported by theDeutsche Forschungsgemeinschaft (DFG) and the CzechScience Foundation (GACR) as part of the joint project FromShared Evidence to Group Attitudes (RO 45486-1)

Endnotes

1 In other words the superiority of the ldquoright answerrdquo orlabeling in our case will appear as such to all

2 All agents hold the same opinion3 By sensitivity we mean how much an agent is ready to

change her opinion on the sole fact that an informationalcue has been debated on

4 The assumption is strong but this trust among agentsmay be rooted from a genuine individual motivation toreach a democratic consensus

5 We adopted a rule of thumbs that says that if an agentis neutral with respect to the principle then she willvote for any proposal that is presented to her duringdeliberation

6 If the argument is not unique she chooses one atrandom

7 If there are more than one candidate the agent will favorthe argument at minimal distance from 119868

28 Complexity

Require Vector of values for parameters (119881 = (120583 119880 119879))Ensure Successful interactions among agents119896 larr997888 0119873119890 larr997888 0

while 119896 le |119873|2 do ⊳Fill119873119890 with |119873|2 agents119896 larr997888 119896 + 1119873119890 larr997888 119877119886119899119889119900119898(|119873| + 1) ⊳Pseudo-random generator Generates integer between 0 and |119873|119873119903 larr997888 0 120575119894119895 larr997888 0 ⊳Variable to store differences in opinionfor all 119894 isin 119873119890 doRandomly pick a number 119895 isin 119873(119873119890 cup 119873119903)119873119903 larr997888 119873119903 cup 119895 120575119894119895 larr997888 |119900119894 minus 119900119895|procedure dyadic-update(119894 119895) ⊳Two agents 119894 and 119895 discuss and update opinions1199001015840 larr997888 119900119894 (120583119895 119880119895 119879119895) larr997888 (120583119894 119880119894 119879119894) larr997888 (120583 119880 119879) ⊳Save agent irsquos opinion before updating and homogenize agents

if 120575119894119895 lt 119880119894 then119900119894 larr997888 120583119894119900119895 + (1 minus 120583119894)119900119894if 120575119894119895 gt 119879119894 then119900119894 larr997888 (1 + 120583119894)119900119894 minus 120583119894119900119895if 120575119894119895 lt 119880119895 then119900119895 larr997888 1205831198951199001015840 + (1 minus 120583119895)119900119895if 120575119894119895 gt 119879119895 then119900119895 larr997888 (1 + 120583119895)119900119895 minus 1205831198951199001015840

Algorithm 1 Dyadic social influence

Require Vector of values for parameters (119881 = (120574 119901119886 119901119903))Ensure Successful interaction between proposal arguments and agents

for all 119894 isin 119873 doprocedure Probability-Update(V119886 119901119886 119901119903) ⊳Update individual probabilities of change due to deliberated cues119909119894 larr997888 119901|V119886minus119900119894 |2119886 119910119894 larr997888 1199012|V119886minus119900119894 |119903119902 larr997888 0if L119886119887(119868) = 119894119899 then

procedure deliberative-update(119894 119886) ⊳Agent i argument 119886120574119894 larr997888 120574 119902 larr997888 119906 sim U(0 1) ⊳ 119906 is a realization of a uniform random variableif 119902 lt 119909119894 then119900119894 larr997888 120574119894119900119894 + (1 minus 120574119894)V119886break

if 119902 lt 119910119894(1 minus 119909119894) then119900119894 larr997888 (1 + 120574119894)119900119894 minus 120574119894V119886Algorithm 2 Deliberation influence

8 In terms of computational complexity naive agentsmaketheirmoves in119874(|A119894| log(|A119894|)) while focused agents doat least as bad

9 Tables have memory Therefore debates are not neces-sarily trees

10 If there are less than (119899119863 times |119873|)2 agents for any of bothgroups the CA summons all of them to the deliberationtable

11 Or careful consideration of the viewpoints of otherswhich implies that citizens keep an open mind and donot reject arguments outright

12 A ldquohealthyrdquo consensus refers to the case in which manyconsensual decisions are taken but opinion-wise agentsdo not agree with each other Quotes are used on theword healthy because some political science theorists

(eg [36]) believe that in certain situations agreement inopinions for deliberation is epistemologically damaging

13 Simulations on groups of |119873| = 800 and |119873| = 400agents shows that scaling the population has an effect onall themetrics but it is confounding depending onwhichcouples of attitudes structures are considered (119880 119879) Westick to |119873| = 400 as in [9] for comparability andsimulation running time

14 By balanced wemean with as many arguments with V119886 lt0 as with V119886 gt 0

15 Experiments on the system suggest that a bigger(smaller) argument pool 119872 = 1000 (119872 = 200) isassociated with a higher (lower) variance of opinions(and extremism) and higher (lower) judgment accuracyFor the rest of the observations 119872 seem to have noeffect We choose a middle value for 119872 Similarly theargument pool chosen to be balanced or uniformly

Complexity 29

Require Argumentation framework 119860119865 = (AR)Ensure Assignment of labels to arguments from the grounded labeling of 119860119865L(A)

L0(A) larr997888 (0 0 0)119896 larr997888 0do119894119899 L119896+1(A) larr997888 119894119899 L119896(A) cup 119886 | 119886 is not labeled 119894119899 L119896(A) 119886119899119889 forall119887 isin A 119904119905 119887 R 119886 then 119887 isin 119900119906119905 L119896(A)119900119906119905 L119896+1(A) larr997888 119900119906119905 L119896(A) cup 119886 | 119886 is not labeled 119894119899 L119896(A) 119886119899119889 exist119887 isin A 119904119905 119887 R 119886 and 119887 isin 119894119899 L119896+1(A)

while L119896+1(A) = L119896(A) = 0Algorithm 3 Solve argumentation frameworks (with grounded semantics)

Require Vector of values for parameters (119881)Ensure Vector of statistics (119880)

procedure Set-parameters(119881) ⊳Initialize parameters for scenarioprocedure Init-objs(119899119898 119896 ) ⊳Create119898 arguments 119899 agents and 119860119865120576procedure Solve-Argumentation-Framework(119860119865) ⊳Assigns the epistemic labelingL120576 toA119894 larr997888 1119905 larr997888 1while 119894 le 119872 do ⊳ 119872 the number of arguments to discuss119868119894 larr997888 119866119890119899 119886119903119892(1) ⊳Generates proposal argument 119868119860 119894119905 larr997888 (119868119894 0)L119886119887(119868119894) larr997888 119906119899119889 ⊳Initialize label for 119868119895 larr997888 1 ⊳Initialize debate counter 119895while ((L119886119887(119868) = 119906119899119889 or 119898 le 119895) and 119895 le 119898) doif 119905 equiv 0(119898119900119889 119905119863) then

procedure build-argumentation-framework(119899119863 119887119890ℎ119886V119894119900119903) ⊳Build the argument graph agents deliberate(119861119894119905 119878119894119905) larr997888 (119883 119884) sub (⋃119897isin119873119899119863 A119897119894119905 ⋃119897isin119873119899119863 R119897119894119905) ⊳ 119873119899119863 sub 119873 are agents sampled by the CA119860 119894119905 larr997888 119889(119860 119894119905 (119861119894119905 119878119894119905))procedure Solve-Argumentation-Framework(A119889) ⊳FindL(A119889)L119886119887(119868119894) larr997888 L119886119887119863(119868119894)119895 larr997888 119895 + 1

else119896 larr997888 1while 119896 le 119905119863 doprocedure dyadic-social-influence(120583 119879 119880) ⊳Agents discuss one-to-one119896 larr997888 119896 + 1119905 larr997888 119905 + 1

if 120572 gt 0 andL119886119887(119868119894) = 119900119906119905 then ⊳Agents vote for the proposalif (|119897 isin 119873|119900119897 times V119868119894 ge 0| ge 120572 times 119899) then

L119886119887(119868119894) larr997888 119894119899else

L119886119887(119868119894) larr997888 119900119906119905if L119886119887(119868119894) = 119894119899 thenprocedure Deliberation-influence(120574 119901119886 119901119903) ⊳Agents are influenced by the result of the decision process119894 larr997888 119894 + 1119905 larr997888 119905 + 1119880 larr997888 119862119900119898119901119906119905119890 119878119905119886119905119894119904119905119894119888119904(119881)

return 119880 ⊳Get the statisticsAlgorithm 4 Run for one scenario

generated does not seem to play a significant explanatoryrole on any of the metrics

16 Experiments show that themaximumnumber of debatesis not an important parameter even if it is an intuitiveone when considering decision-making processes Inpractice debates end after at most 119898 + 1 deliberation

steps and even then it happens rarely However to avoidunpleasant surprises like infinite debates on a completeargument graph containing the proposal argument weadded the constraint 119898 Its value is precisely 7 becausethe biggest minimum number of debates we studyis 6

30 Complexity

17 To ensure comparability with the deliberative andmixedopinion dynamics models on judgment accuracy andcoherence agents vote on proposal arguments every 119905119863times(119898+1) 119905119863 and119898does not explain variability in judgmentaccuracy nor in coherence

18 Student t-tests for independent samples are robust tothe violation of two hypothesis on the distribution ofthe samples normality and homogeneity of variancesas long as the sample size is big enough (central limittheorem) and if the difference in size of the comparedsamples is smallrespectively

19 Studentrsquos test version forwhen variances are not assumedequal and the difference in size of the compared samplesis big

20 We do not correct for homoskedasticity nor normality oferrors in the estimates

21 Balanced as the same number of runs per scenario orsuch that no significant correlations are found betweenparameters

References

[1] R Axelrod ldquoThe dissemination of culture a model withlocal convergence and global polarizationrdquo Journal of ConflictResolution vol 41 no 2 pp 203ndash226 1997

[2] D M Rousseau Reinforcing the MicroMacro Bridge Organiza-tionalThinking and Pluralistic Vehicles SAGEPublications SageLos Angeles Calif USA 2011

[3] S Galam and SMoscovici ldquoTowards a theory of collective phe-nomena consensus and attitude changes in groupsrdquo EuropeanJournal of Social Psychology vol 21 no 1 pp 49ndash74 1991

[4] A Nowak J Szamrej and B Latane ldquoFrom private attitude topublic opinion a dynamic theory of social impactrdquo Psychologi-cal Review vol 97 no 3 pp 362ndash376 1990

[5] G Deffuant S Huet and F Amblard ldquoAn individual-basedmodel of innovation diffusion mixing social value and individ-ual benefitrdquo American Journal of Sociology vol 110 no 4 pp1041ndash1069 2005

[6] G Deffuant D Neau F Amblard and G Weisbuch ldquoMixingbeliefs among interacting agentsrdquoAdvances in Complex Systems(ACS) vol 3 no 01n04 pp 87ndash98 2000

[7] J Rouchier P Tubaro and C Emery ldquoOpinion transmission inorganizations an agent-based modeling approachrdquo Computa-tional and Mathematical OrganizationTheory vol 20 no 3 pp252ndash277 2014

[8] M Meadows and D Cliff ldquoThe relative disagreement model ofopinion dynamics where do extremists come fromrdquo in Pro-ceedings of the International Workshop on Self-Organizing Sys-tems Lecture Notes in Computer Science pp 66ndash77 Springer2013

[9] W Jager and F Amblard ldquoUniformity bipolarization and plu-riformity captured as generic stylized behavior with an agent-based simulation model of attitude changerdquo Computational ampMathematical Organization Theory vol 10 no 4 pp 295ndash3032005

[10] Y-F Zhang H-Y Duan and Z-L Geng ldquoEvolutionary mech-anism of frangibility in social consensus system based onnegative emotions spreadrdquo Complexity vol 2017 Article ID4037049 8 pages 2017

[11] A Flache andMWMacy ldquoSmall worlds and cultural polariza-tionrdquoThe Journal of Mathematical Sociology vol 35 no 1-3 pp146ndash176 2011

[12] A Tsang and K Larson ldquoOpinion dynamics of skepticalagentsrdquo in Proceedings of the 13th International Conference onAutonomous Agents and Multiagent Systems AAMAS 2014 pp277ndash284 International Foundation forAutonomousAgents andMultiagent Systems France May 2014

[13] K M Hansen Deliberative democracy and opinion formationUniversity Press of Southern Denmark Odense 2004

[14] C R Sunstein ldquoThe law of group polarizationrdquo Journal ofPolitical Philosophy vol 10 no 2 pp 175ndash195 2002 WileyOnline Library

[15] S Moscovici and W Doise Dissensions et Consensus unetheorie generale des decisions collectives Presses Universitairesde France-PUF 1992

[16] J S Fishkin and R C Luskin ldquoExperimenting with a demo-cratic ideal deliberative polling and public opinionrdquo ActaPolitica vol 40 no 3 pp 284ndash298 2005

[17] R C Luskin J S Fishkin and S Iyengar Considered Opinionson US Foreign Policy Face-To-Face versus Online Delibera-tive Polling International Communication Association NewOrleans La USA 2004

[18] J Barabas ldquoHowdeliberation affects policy opinionsrdquoAmericanPolitical Science Review vol 98 no 4 pp 687ndash701 2004

[19] O Pryymak A Rogers and N R Jennings ldquoEfficient opinionsharing in large decentralised teamsrdquo in Proceedings of the 11thInternational Conference on Autonomous Agents andMultiagentSystems vol 1 pp 543ndash550 Spain June 2012

[20] J Rouchier and E Tanimura ldquoWhen overconfident agents slowdown collective learningrdquo Simulation vol 88 no 1 pp 33ndash492012

[21] D Miller ldquoDeliberative democracy and social choicerdquo PoliticalStudies vol 40 S1 pp 54ndash67 1992

[22] M Sherif and C I Hovland Social Judgment Assimilation andContrast Effects in Communication and Attitude Change YaleUniversity Press 1961

[23] P M Dung ldquoOn the acceptability of arguments and its funda-mental role in nonmonotonic reasoning logic programmingand n-person gamesrdquo Artificial Intelligence vol 77 no 2 pp321ndash357 1995

[24] M Caminada and G Pigozzi ldquoOn judgment aggregation inabstract argumentationrdquo Autonomous Agents and Multi-AgentSystems vol 22 no 1 pp 64ndash102 2011

[25] P Baroni M Caminada andM Giacomin ldquoAn introduction toargumentation semanticsrdquoThe Knowledge Engineering Reviewvol 26 no 4 pp 365ndash410 2011

[26] S Chambers ldquoDeliberative democratic theoryrdquo Annual Reviewof Political Science vol 6 no 1 pp 307ndash326 2003

[27] E Bonzon andNMaudet ldquoOn the outcomes of multiparty per-suasionrdquo in Proceedings of the 10th International Conference onAutonomous Agents and Multiagent Systems 2011 AAMAS 2011vol 1 pp 47ndash54 International Foundation for AutonomousAgents and Multiagent Systems Taiwan May 2011

[28] J Ranciere Disagreement Politics and Philosophy University ofMinnesota Press 1999

[29] T Bench-Capon and K Atkinson ldquoAbstract argumentation andvaluesrdquo in Argumentation in Artificial Intelligence pp 45ndash64Springer 2009

[30] M Caminada ldquoOn the issue of reinstatement in argumenta-tionrdquo in Proceedings of the European Workshop on Logics in

Complexity 31

Artificial Intelligence vol 4160 of Lecture Notes in ComputerScience pp 111ndash123 Springer Berlin Heidelberg

[31] GAVreeswijk ldquoAn algorithm to computeminimally groundedand admissible defence sets in argument systemsrdquo in Pro-ceedings of the 2006 conference on Computational Models ofArgument Proceedings of COMMA 2006 vol 144 pp 109ndash120IOS Press Amsterdam The Netherlands 2006

[32] C W Sherif M Kelly H L Rodgers et al ldquoPersonal involve-ment social judgment and actionrdquo Journal of Personality andSocial Psychology vol 27 no 3 pp 311ndash328 1973

[33] G Deffuant F Amblard G Weisbuch and T Faure ldquoHow canextremism prevail A study based on the relative agreementinteraction modelrdquo Journal of Artificial Societies and SocialSimulation vol 5 no 4 p 1 2002

[34] E Awad M W Caminada G Pigozzi M Podlaszewski andI Rahwan ldquoPareto optimality and strategy-proofness in groupargument evaluationrdquo Journal of Logic and Computation vol 27no 8 pp 2581ndash2609 2017

[35] D Grossi and G Pigozzi ldquoJudgment aggregation a primerrdquoSynthesis Lectures on Artificial Intelligence and Machine Learn-ing vol 8 no 2 pp 1ndash151 2014

[36] H Landemore and S E Page ldquoDeliberation and disagreementproblem solving prediction and positive dissensusrdquo PoliticsPhilosophy and Economics vol 14 no 3 pp 229ndash254 2015

[37] R Hegselmann and U Krause ldquoOpinion dynamics andbounded confidence models analysis and simulationrdquo Journalof Artificial Societies and Social Simulation vol 5 no 3 2002

[38] A Stefanelli and R Seidl ldquoModerate and polarized opinionsUsing empirical data for an agent-based simulationrdquo inProceed-ings of the Social Simulation Conference 2014

[39] S Gabbriellini and P Torroni ldquoA new framework for ABMsbased on argumentative reasoningrdquo in Advances in SocialSimulation vol 229 pp 25ndash36 Springer Berlin Germany 2014

[40] H Mercier and D Sperber ldquoWhy do humans reason Argu-ments for an argumentative theoryrdquo Behavioral and BrainSciences vol 34 no 2 pp 57ndash74 2011

[41] H V Parunak T C Belding R Hilscher and S BruecknerldquoModeling and managing collective cognitive convergencerdquoin In Proceedings of the 7th iNternational Joint Conference onAutonomous Agents and Multiagent Systems vol 3 pp 1505ndash1508 International Foundation for Autonomous Agents andMultiagent Systems 2008

[42] H Prakken ldquoFormal systems for persuasion dialoguerdquo TheKnowledge Engineering Review vol 21 no 2 pp 163ndash188 2006

[43] K Atkinson T Bench-Capon and P Mcburney ldquoA dialoguegame protocol for multi-agent argument over proposals foractionrdquoAutonomousAgents andMulti-Agent Systems vol 11 no2 pp 153ndash171 2005

[44] M Thimm and A J Garcıa ldquoClassification and strategicalissues of argumentation games on structured argumentationframeworksrdquo in Proceedings of the 9th International Conferenceon Autonomous Agents and Multiagent Systems vol 1 pp 1247ndash1254 International Foundation for Autonomous Agents andMultiagent Systems Canada 2010

[45] I Rahwan and K Larson ldquoMechanism design for abstractargumentationrdquo in Proceedings of the 7th International JointConference on Autonomous Agents and Multiagent SystemsAAMAS 2008 pp 1031ndash1038 International Foundation forAutonomous Agents and Multiagent Systems Portugal May2008

[46] H Prakken R Riveret A Rotolo and G Sartor ldquoHeuristics inArgumentation A Game-Theoretical Investigationrdquo EuropeanUniversity Institute 2008

[47] N Hadidi Y Dimopoulos and P Moraitis ldquoTactics and con-cessions for argumentation-based negotiationrdquo in Proceedingsof the COMMA vol 245 pp 285ndash296 2012

[48] N Oren and T J Norman ldquoArguing using opponent modelsrdquoin Proceedings of the International Workshop on Argumentationin Multi-Agent Systems pp 160ndash174 Springer 2009

[49] S Delre W Jager T Bijmolt and M Janssen ldquoTargeting andtiming promotional activities An agent-based model for thetakeoff of new productsrdquo Journal of Business Research vol 60no 8 pp 826ndash835 2007

[50] A Mirtabatabaei P Jia and F Bullo ldquoEulerian opinion dynam-ics with bounded confidence and exogenous inputsrdquo SIAMJournal on Applied Dynamical Systems vol 13 no 1 pp 425ndash446 2014

[51] V Barbet N Guiraud V Laperriere and J Rouchier Hagglingon Values Towards Consensus or Trouble 2019

Hindawiwwwhindawicom Volume 2018

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Submit your manuscripts atwwwhindawicom

Complexity 27

Concerning other possible extensions of the model anatural reflex would be to argue on the validity and therobustness of the mix between the bounded confidenceopinion diffusion models and deliberation Consequentiallyadding deliberation to different bounded confidence modelsof opinion diffusion such as the Hegelsmann-Krause model[37] or to models of social impact theory like in [4] canbe of great interest for future work We acknowledge that afiner understanding of argumentation frameworks randomlattices and their respective applications for this modelare necessary Testing for different labeling-based semanticsor even trading labeling-based semantics for scoring-basedsemantics (semantics that defines acceptability on scoresgiven to arguments) such as debate semantics and ranking-based semantics (semantics that yield a preorder) may beinteresting to consider in future work

More into modeling deliberation it is not inadequateto believe that agents may also advance proposals duringdeliberation and replace to some extent the proposal forwhich the deliberation is taking place Endogenizing theprocess that produces the type of argument to be deliberatedon may be interesting to explore either by having agentsstrategically replace proposals at certain moments duringthe debate or having the central authority attempt to chooseproposals such that the probability of having them acceptedis high Agents may also be thought to be autoorganizing(independent of a central authority) and concede on the sizeof the instances of deliberation To consider differences insemantics across agents or even endogenizing the semanticson how urgent or important the proposal that is beingdiscussed is can also be an interesting direction to take

Last but not least many nontrivial modifications of ourmodel are possible and most should include better-thoughtdeliberation protocols It may be interesting to design andobserve protocols in which deliberation only affects agentsthat are actually debating for example Trust network effectsmultidimensionality of opinions new processes of argumentexchange or learning among agents are notions to furtherdevelop in order to make the model more realistic and relaxsome unreliable assumptions In sum the model is to berefined and extended with the objective of either studyingconcrete cases of deliberative polling and opinion dynamicsor implementing more intuitive thought experiments

Appendix

We present below some of the pseudocode we used toimplement the model

Algorithms

(a) Algorithm 1 implements the dyadic influence in themodel It describes how two agents interact andupdate their opinions

(b) Algorithm 2 implements the deliberative influence inthe model which describes how agents update theiropinions in sight of the accepted deliberated proposalarguments

(c) Algorithm 3 is used to obtain the grounded labelingof an argumentation framework

(d) Algorithm 4 provides an overview of a run in thesimulations119872 is the number of proposal argumentsto discuss which we set to 100 in the model

Each object in the system is indexed by a119908ℎ119900 number Hencea set of agents is annotated by a set of natural numbers Weuse this fact in the implementation of the algorithm 119874119887119895119890119888119905119894denotes the object of 119908ℎ119900 number 119894 Agentsrsquo opinions (119900) arecalibrated so that 119900 isin [minus1 1]Data Availability

The program and data used to support the findings of thisstudy are available from the corresponding author uponrequest

Disclosure

A preliminary version of this work was presented anddiscussed at the conference Bridging the Gap between FormalArgumentation and Actual Human Reasoning on October 4-5 2018 at BochumGermanyThe authors thank the audiencefor their valuable comments and suggestions

Conflicts of Interest

The authors declare that they have no conflicts of interestregarding the publication of this article

Acknowledgments

The contribution of Gabriella Pigozzi was supported by theDeutsche Forschungsgemeinschaft (DFG) and the CzechScience Foundation (GACR) as part of the joint project FromShared Evidence to Group Attitudes (RO 45486-1)

Endnotes

1 In other words the superiority of the ldquoright answerrdquo orlabeling in our case will appear as such to all

2 All agents hold the same opinion3 By sensitivity we mean how much an agent is ready to

change her opinion on the sole fact that an informationalcue has been debated on

4 The assumption is strong but this trust among agentsmay be rooted from a genuine individual motivation toreach a democratic consensus

5 We adopted a rule of thumbs that says that if an agentis neutral with respect to the principle then she willvote for any proposal that is presented to her duringdeliberation

6 If the argument is not unique she chooses one atrandom

7 If there are more than one candidate the agent will favorthe argument at minimal distance from 119868

28 Complexity

Require Vector of values for parameters (119881 = (120583 119880 119879))Ensure Successful interactions among agents119896 larr997888 0119873119890 larr997888 0

while 119896 le |119873|2 do ⊳Fill119873119890 with |119873|2 agents119896 larr997888 119896 + 1119873119890 larr997888 119877119886119899119889119900119898(|119873| + 1) ⊳Pseudo-random generator Generates integer between 0 and |119873|119873119903 larr997888 0 120575119894119895 larr997888 0 ⊳Variable to store differences in opinionfor all 119894 isin 119873119890 doRandomly pick a number 119895 isin 119873(119873119890 cup 119873119903)119873119903 larr997888 119873119903 cup 119895 120575119894119895 larr997888 |119900119894 minus 119900119895|procedure dyadic-update(119894 119895) ⊳Two agents 119894 and 119895 discuss and update opinions1199001015840 larr997888 119900119894 (120583119895 119880119895 119879119895) larr997888 (120583119894 119880119894 119879119894) larr997888 (120583 119880 119879) ⊳Save agent irsquos opinion before updating and homogenize agents

if 120575119894119895 lt 119880119894 then119900119894 larr997888 120583119894119900119895 + (1 minus 120583119894)119900119894if 120575119894119895 gt 119879119894 then119900119894 larr997888 (1 + 120583119894)119900119894 minus 120583119894119900119895if 120575119894119895 lt 119880119895 then119900119895 larr997888 1205831198951199001015840 + (1 minus 120583119895)119900119895if 120575119894119895 gt 119879119895 then119900119895 larr997888 (1 + 120583119895)119900119895 minus 1205831198951199001015840

Algorithm 1 Dyadic social influence

Require Vector of values for parameters (119881 = (120574 119901119886 119901119903))Ensure Successful interaction between proposal arguments and agents

for all 119894 isin 119873 doprocedure Probability-Update(V119886 119901119886 119901119903) ⊳Update individual probabilities of change due to deliberated cues119909119894 larr997888 119901|V119886minus119900119894 |2119886 119910119894 larr997888 1199012|V119886minus119900119894 |119903119902 larr997888 0if L119886119887(119868) = 119894119899 then

procedure deliberative-update(119894 119886) ⊳Agent i argument 119886120574119894 larr997888 120574 119902 larr997888 119906 sim U(0 1) ⊳ 119906 is a realization of a uniform random variableif 119902 lt 119909119894 then119900119894 larr997888 120574119894119900119894 + (1 minus 120574119894)V119886break

if 119902 lt 119910119894(1 minus 119909119894) then119900119894 larr997888 (1 + 120574119894)119900119894 minus 120574119894V119886Algorithm 2 Deliberation influence

8 In terms of computational complexity naive agentsmaketheirmoves in119874(|A119894| log(|A119894|)) while focused agents doat least as bad

9 Tables have memory Therefore debates are not neces-sarily trees

10 If there are less than (119899119863 times |119873|)2 agents for any of bothgroups the CA summons all of them to the deliberationtable

11 Or careful consideration of the viewpoints of otherswhich implies that citizens keep an open mind and donot reject arguments outright

12 A ldquohealthyrdquo consensus refers to the case in which manyconsensual decisions are taken but opinion-wise agentsdo not agree with each other Quotes are used on theword healthy because some political science theorists

(eg [36]) believe that in certain situations agreement inopinions for deliberation is epistemologically damaging

13 Simulations on groups of |119873| = 800 and |119873| = 400agents shows that scaling the population has an effect onall themetrics but it is confounding depending onwhichcouples of attitudes structures are considered (119880 119879) Westick to |119873| = 400 as in [9] for comparability andsimulation running time

14 By balanced wemean with as many arguments with V119886 lt0 as with V119886 gt 0

15 Experiments on the system suggest that a bigger(smaller) argument pool 119872 = 1000 (119872 = 200) isassociated with a higher (lower) variance of opinions(and extremism) and higher (lower) judgment accuracyFor the rest of the observations 119872 seem to have noeffect We choose a middle value for 119872 Similarly theargument pool chosen to be balanced or uniformly

Complexity 29

Require Argumentation framework 119860119865 = (AR)Ensure Assignment of labels to arguments from the grounded labeling of 119860119865L(A)

L0(A) larr997888 (0 0 0)119896 larr997888 0do119894119899 L119896+1(A) larr997888 119894119899 L119896(A) cup 119886 | 119886 is not labeled 119894119899 L119896(A) 119886119899119889 forall119887 isin A 119904119905 119887 R 119886 then 119887 isin 119900119906119905 L119896(A)119900119906119905 L119896+1(A) larr997888 119900119906119905 L119896(A) cup 119886 | 119886 is not labeled 119894119899 L119896(A) 119886119899119889 exist119887 isin A 119904119905 119887 R 119886 and 119887 isin 119894119899 L119896+1(A)

while L119896+1(A) = L119896(A) = 0Algorithm 3 Solve argumentation frameworks (with grounded semantics)

Require Vector of values for parameters (119881)Ensure Vector of statistics (119880)

procedure Set-parameters(119881) ⊳Initialize parameters for scenarioprocedure Init-objs(119899119898 119896 ) ⊳Create119898 arguments 119899 agents and 119860119865120576procedure Solve-Argumentation-Framework(119860119865) ⊳Assigns the epistemic labelingL120576 toA119894 larr997888 1119905 larr997888 1while 119894 le 119872 do ⊳ 119872 the number of arguments to discuss119868119894 larr997888 119866119890119899 119886119903119892(1) ⊳Generates proposal argument 119868119860 119894119905 larr997888 (119868119894 0)L119886119887(119868119894) larr997888 119906119899119889 ⊳Initialize label for 119868119895 larr997888 1 ⊳Initialize debate counter 119895while ((L119886119887(119868) = 119906119899119889 or 119898 le 119895) and 119895 le 119898) doif 119905 equiv 0(119898119900119889 119905119863) then

procedure build-argumentation-framework(119899119863 119887119890ℎ119886V119894119900119903) ⊳Build the argument graph agents deliberate(119861119894119905 119878119894119905) larr997888 (119883 119884) sub (⋃119897isin119873119899119863 A119897119894119905 ⋃119897isin119873119899119863 R119897119894119905) ⊳ 119873119899119863 sub 119873 are agents sampled by the CA119860 119894119905 larr997888 119889(119860 119894119905 (119861119894119905 119878119894119905))procedure Solve-Argumentation-Framework(A119889) ⊳FindL(A119889)L119886119887(119868119894) larr997888 L119886119887119863(119868119894)119895 larr997888 119895 + 1

else119896 larr997888 1while 119896 le 119905119863 doprocedure dyadic-social-influence(120583 119879 119880) ⊳Agents discuss one-to-one119896 larr997888 119896 + 1119905 larr997888 119905 + 1

if 120572 gt 0 andL119886119887(119868119894) = 119900119906119905 then ⊳Agents vote for the proposalif (|119897 isin 119873|119900119897 times V119868119894 ge 0| ge 120572 times 119899) then

L119886119887(119868119894) larr997888 119894119899else

L119886119887(119868119894) larr997888 119900119906119905if L119886119887(119868119894) = 119894119899 thenprocedure Deliberation-influence(120574 119901119886 119901119903) ⊳Agents are influenced by the result of the decision process119894 larr997888 119894 + 1119905 larr997888 119905 + 1119880 larr997888 119862119900119898119901119906119905119890 119878119905119886119905119894119904119905119894119888119904(119881)

return 119880 ⊳Get the statisticsAlgorithm 4 Run for one scenario

generated does not seem to play a significant explanatoryrole on any of the metrics

16 Experiments show that themaximumnumber of debatesis not an important parameter even if it is an intuitiveone when considering decision-making processes Inpractice debates end after at most 119898 + 1 deliberation

steps and even then it happens rarely However to avoidunpleasant surprises like infinite debates on a completeargument graph containing the proposal argument weadded the constraint 119898 Its value is precisely 7 becausethe biggest minimum number of debates we studyis 6

30 Complexity

17 To ensure comparability with the deliberative andmixedopinion dynamics models on judgment accuracy andcoherence agents vote on proposal arguments every 119905119863times(119898+1) 119905119863 and119898does not explain variability in judgmentaccuracy nor in coherence

18 Student t-tests for independent samples are robust tothe violation of two hypothesis on the distribution ofthe samples normality and homogeneity of variancesas long as the sample size is big enough (central limittheorem) and if the difference in size of the comparedsamples is smallrespectively

19 Studentrsquos test version forwhen variances are not assumedequal and the difference in size of the compared samplesis big

20 We do not correct for homoskedasticity nor normality oferrors in the estimates

21 Balanced as the same number of runs per scenario orsuch that no significant correlations are found betweenparameters

References

[1] R Axelrod ldquoThe dissemination of culture a model withlocal convergence and global polarizationrdquo Journal of ConflictResolution vol 41 no 2 pp 203ndash226 1997

[2] D M Rousseau Reinforcing the MicroMacro Bridge Organiza-tionalThinking and Pluralistic Vehicles SAGEPublications SageLos Angeles Calif USA 2011

[3] S Galam and SMoscovici ldquoTowards a theory of collective phe-nomena consensus and attitude changes in groupsrdquo EuropeanJournal of Social Psychology vol 21 no 1 pp 49ndash74 1991

[4] A Nowak J Szamrej and B Latane ldquoFrom private attitude topublic opinion a dynamic theory of social impactrdquo Psychologi-cal Review vol 97 no 3 pp 362ndash376 1990

[5] G Deffuant S Huet and F Amblard ldquoAn individual-basedmodel of innovation diffusion mixing social value and individ-ual benefitrdquo American Journal of Sociology vol 110 no 4 pp1041ndash1069 2005

[6] G Deffuant D Neau F Amblard and G Weisbuch ldquoMixingbeliefs among interacting agentsrdquoAdvances in Complex Systems(ACS) vol 3 no 01n04 pp 87ndash98 2000

[7] J Rouchier P Tubaro and C Emery ldquoOpinion transmission inorganizations an agent-based modeling approachrdquo Computa-tional and Mathematical OrganizationTheory vol 20 no 3 pp252ndash277 2014

[8] M Meadows and D Cliff ldquoThe relative disagreement model ofopinion dynamics where do extremists come fromrdquo in Pro-ceedings of the International Workshop on Self-Organizing Sys-tems Lecture Notes in Computer Science pp 66ndash77 Springer2013

[9] W Jager and F Amblard ldquoUniformity bipolarization and plu-riformity captured as generic stylized behavior with an agent-based simulation model of attitude changerdquo Computational ampMathematical Organization Theory vol 10 no 4 pp 295ndash3032005

[10] Y-F Zhang H-Y Duan and Z-L Geng ldquoEvolutionary mech-anism of frangibility in social consensus system based onnegative emotions spreadrdquo Complexity vol 2017 Article ID4037049 8 pages 2017

[11] A Flache andMWMacy ldquoSmall worlds and cultural polariza-tionrdquoThe Journal of Mathematical Sociology vol 35 no 1-3 pp146ndash176 2011

[12] A Tsang and K Larson ldquoOpinion dynamics of skepticalagentsrdquo in Proceedings of the 13th International Conference onAutonomous Agents and Multiagent Systems AAMAS 2014 pp277ndash284 International Foundation forAutonomousAgents andMultiagent Systems France May 2014

[13] K M Hansen Deliberative democracy and opinion formationUniversity Press of Southern Denmark Odense 2004

[14] C R Sunstein ldquoThe law of group polarizationrdquo Journal ofPolitical Philosophy vol 10 no 2 pp 175ndash195 2002 WileyOnline Library

[15] S Moscovici and W Doise Dissensions et Consensus unetheorie generale des decisions collectives Presses Universitairesde France-PUF 1992

[16] J S Fishkin and R C Luskin ldquoExperimenting with a demo-cratic ideal deliberative polling and public opinionrdquo ActaPolitica vol 40 no 3 pp 284ndash298 2005

[17] R C Luskin J S Fishkin and S Iyengar Considered Opinionson US Foreign Policy Face-To-Face versus Online Delibera-tive Polling International Communication Association NewOrleans La USA 2004

[18] J Barabas ldquoHowdeliberation affects policy opinionsrdquoAmericanPolitical Science Review vol 98 no 4 pp 687ndash701 2004

[19] O Pryymak A Rogers and N R Jennings ldquoEfficient opinionsharing in large decentralised teamsrdquo in Proceedings of the 11thInternational Conference on Autonomous Agents andMultiagentSystems vol 1 pp 543ndash550 Spain June 2012

[20] J Rouchier and E Tanimura ldquoWhen overconfident agents slowdown collective learningrdquo Simulation vol 88 no 1 pp 33ndash492012

[21] D Miller ldquoDeliberative democracy and social choicerdquo PoliticalStudies vol 40 S1 pp 54ndash67 1992

[22] M Sherif and C I Hovland Social Judgment Assimilation andContrast Effects in Communication and Attitude Change YaleUniversity Press 1961

[23] P M Dung ldquoOn the acceptability of arguments and its funda-mental role in nonmonotonic reasoning logic programmingand n-person gamesrdquo Artificial Intelligence vol 77 no 2 pp321ndash357 1995

[24] M Caminada and G Pigozzi ldquoOn judgment aggregation inabstract argumentationrdquo Autonomous Agents and Multi-AgentSystems vol 22 no 1 pp 64ndash102 2011

[25] P Baroni M Caminada andM Giacomin ldquoAn introduction toargumentation semanticsrdquoThe Knowledge Engineering Reviewvol 26 no 4 pp 365ndash410 2011

[26] S Chambers ldquoDeliberative democratic theoryrdquo Annual Reviewof Political Science vol 6 no 1 pp 307ndash326 2003

[27] E Bonzon andNMaudet ldquoOn the outcomes of multiparty per-suasionrdquo in Proceedings of the 10th International Conference onAutonomous Agents and Multiagent Systems 2011 AAMAS 2011vol 1 pp 47ndash54 International Foundation for AutonomousAgents and Multiagent Systems Taiwan May 2011

[28] J Ranciere Disagreement Politics and Philosophy University ofMinnesota Press 1999

[29] T Bench-Capon and K Atkinson ldquoAbstract argumentation andvaluesrdquo in Argumentation in Artificial Intelligence pp 45ndash64Springer 2009

[30] M Caminada ldquoOn the issue of reinstatement in argumenta-tionrdquo in Proceedings of the European Workshop on Logics in

Complexity 31

Artificial Intelligence vol 4160 of Lecture Notes in ComputerScience pp 111ndash123 Springer Berlin Heidelberg

[31] GAVreeswijk ldquoAn algorithm to computeminimally groundedand admissible defence sets in argument systemsrdquo in Pro-ceedings of the 2006 conference on Computational Models ofArgument Proceedings of COMMA 2006 vol 144 pp 109ndash120IOS Press Amsterdam The Netherlands 2006

[32] C W Sherif M Kelly H L Rodgers et al ldquoPersonal involve-ment social judgment and actionrdquo Journal of Personality andSocial Psychology vol 27 no 3 pp 311ndash328 1973

[33] G Deffuant F Amblard G Weisbuch and T Faure ldquoHow canextremism prevail A study based on the relative agreementinteraction modelrdquo Journal of Artificial Societies and SocialSimulation vol 5 no 4 p 1 2002

[34] E Awad M W Caminada G Pigozzi M Podlaszewski andI Rahwan ldquoPareto optimality and strategy-proofness in groupargument evaluationrdquo Journal of Logic and Computation vol 27no 8 pp 2581ndash2609 2017

[35] D Grossi and G Pigozzi ldquoJudgment aggregation a primerrdquoSynthesis Lectures on Artificial Intelligence and Machine Learn-ing vol 8 no 2 pp 1ndash151 2014

[36] H Landemore and S E Page ldquoDeliberation and disagreementproblem solving prediction and positive dissensusrdquo PoliticsPhilosophy and Economics vol 14 no 3 pp 229ndash254 2015

[37] R Hegselmann and U Krause ldquoOpinion dynamics andbounded confidence models analysis and simulationrdquo Journalof Artificial Societies and Social Simulation vol 5 no 3 2002

[38] A Stefanelli and R Seidl ldquoModerate and polarized opinionsUsing empirical data for an agent-based simulationrdquo inProceed-ings of the Social Simulation Conference 2014

[39] S Gabbriellini and P Torroni ldquoA new framework for ABMsbased on argumentative reasoningrdquo in Advances in SocialSimulation vol 229 pp 25ndash36 Springer Berlin Germany 2014

[40] H Mercier and D Sperber ldquoWhy do humans reason Argu-ments for an argumentative theoryrdquo Behavioral and BrainSciences vol 34 no 2 pp 57ndash74 2011

[41] H V Parunak T C Belding R Hilscher and S BruecknerldquoModeling and managing collective cognitive convergencerdquoin In Proceedings of the 7th iNternational Joint Conference onAutonomous Agents and Multiagent Systems vol 3 pp 1505ndash1508 International Foundation for Autonomous Agents andMultiagent Systems 2008

[42] H Prakken ldquoFormal systems for persuasion dialoguerdquo TheKnowledge Engineering Review vol 21 no 2 pp 163ndash188 2006

[43] K Atkinson T Bench-Capon and P Mcburney ldquoA dialoguegame protocol for multi-agent argument over proposals foractionrdquoAutonomousAgents andMulti-Agent Systems vol 11 no2 pp 153ndash171 2005

[44] M Thimm and A J Garcıa ldquoClassification and strategicalissues of argumentation games on structured argumentationframeworksrdquo in Proceedings of the 9th International Conferenceon Autonomous Agents and Multiagent Systems vol 1 pp 1247ndash1254 International Foundation for Autonomous Agents andMultiagent Systems Canada 2010

[45] I Rahwan and K Larson ldquoMechanism design for abstractargumentationrdquo in Proceedings of the 7th International JointConference on Autonomous Agents and Multiagent SystemsAAMAS 2008 pp 1031ndash1038 International Foundation forAutonomous Agents and Multiagent Systems Portugal May2008

[46] H Prakken R Riveret A Rotolo and G Sartor ldquoHeuristics inArgumentation A Game-Theoretical Investigationrdquo EuropeanUniversity Institute 2008

[47] N Hadidi Y Dimopoulos and P Moraitis ldquoTactics and con-cessions for argumentation-based negotiationrdquo in Proceedingsof the COMMA vol 245 pp 285ndash296 2012

[48] N Oren and T J Norman ldquoArguing using opponent modelsrdquoin Proceedings of the International Workshop on Argumentationin Multi-Agent Systems pp 160ndash174 Springer 2009

[49] S Delre W Jager T Bijmolt and M Janssen ldquoTargeting andtiming promotional activities An agent-based model for thetakeoff of new productsrdquo Journal of Business Research vol 60no 8 pp 826ndash835 2007

[50] A Mirtabatabaei P Jia and F Bullo ldquoEulerian opinion dynam-ics with bounded confidence and exogenous inputsrdquo SIAMJournal on Applied Dynamical Systems vol 13 no 1 pp 425ndash446 2014

[51] V Barbet N Guiraud V Laperriere and J Rouchier Hagglingon Values Towards Consensus or Trouble 2019

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

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Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

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Dierential EquationsInternational Journal of

Volume 2018

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Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

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Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

28 Complexity

Require Vector of values for parameters (119881 = (120583 119880 119879))Ensure Successful interactions among agents119896 larr997888 0119873119890 larr997888 0

while 119896 le |119873|2 do ⊳Fill119873119890 with |119873|2 agents119896 larr997888 119896 + 1119873119890 larr997888 119877119886119899119889119900119898(|119873| + 1) ⊳Pseudo-random generator Generates integer between 0 and |119873|119873119903 larr997888 0 120575119894119895 larr997888 0 ⊳Variable to store differences in opinionfor all 119894 isin 119873119890 doRandomly pick a number 119895 isin 119873(119873119890 cup 119873119903)119873119903 larr997888 119873119903 cup 119895 120575119894119895 larr997888 |119900119894 minus 119900119895|procedure dyadic-update(119894 119895) ⊳Two agents 119894 and 119895 discuss and update opinions1199001015840 larr997888 119900119894 (120583119895 119880119895 119879119895) larr997888 (120583119894 119880119894 119879119894) larr997888 (120583 119880 119879) ⊳Save agent irsquos opinion before updating and homogenize agents

if 120575119894119895 lt 119880119894 then119900119894 larr997888 120583119894119900119895 + (1 minus 120583119894)119900119894if 120575119894119895 gt 119879119894 then119900119894 larr997888 (1 + 120583119894)119900119894 minus 120583119894119900119895if 120575119894119895 lt 119880119895 then119900119895 larr997888 1205831198951199001015840 + (1 minus 120583119895)119900119895if 120575119894119895 gt 119879119895 then119900119895 larr997888 (1 + 120583119895)119900119895 minus 1205831198951199001015840

Algorithm 1 Dyadic social influence

Require Vector of values for parameters (119881 = (120574 119901119886 119901119903))Ensure Successful interaction between proposal arguments and agents

for all 119894 isin 119873 doprocedure Probability-Update(V119886 119901119886 119901119903) ⊳Update individual probabilities of change due to deliberated cues119909119894 larr997888 119901|V119886minus119900119894 |2119886 119910119894 larr997888 1199012|V119886minus119900119894 |119903119902 larr997888 0if L119886119887(119868) = 119894119899 then

procedure deliberative-update(119894 119886) ⊳Agent i argument 119886120574119894 larr997888 120574 119902 larr997888 119906 sim U(0 1) ⊳ 119906 is a realization of a uniform random variableif 119902 lt 119909119894 then119900119894 larr997888 120574119894119900119894 + (1 minus 120574119894)V119886break

if 119902 lt 119910119894(1 minus 119909119894) then119900119894 larr997888 (1 + 120574119894)119900119894 minus 120574119894V119886Algorithm 2 Deliberation influence

8 In terms of computational complexity naive agentsmaketheirmoves in119874(|A119894| log(|A119894|)) while focused agents doat least as bad

9 Tables have memory Therefore debates are not neces-sarily trees

10 If there are less than (119899119863 times |119873|)2 agents for any of bothgroups the CA summons all of them to the deliberationtable

11 Or careful consideration of the viewpoints of otherswhich implies that citizens keep an open mind and donot reject arguments outright

12 A ldquohealthyrdquo consensus refers to the case in which manyconsensual decisions are taken but opinion-wise agentsdo not agree with each other Quotes are used on theword healthy because some political science theorists

(eg [36]) believe that in certain situations agreement inopinions for deliberation is epistemologically damaging

13 Simulations on groups of |119873| = 800 and |119873| = 400agents shows that scaling the population has an effect onall themetrics but it is confounding depending onwhichcouples of attitudes structures are considered (119880 119879) Westick to |119873| = 400 as in [9] for comparability andsimulation running time

14 By balanced wemean with as many arguments with V119886 lt0 as with V119886 gt 0

15 Experiments on the system suggest that a bigger(smaller) argument pool 119872 = 1000 (119872 = 200) isassociated with a higher (lower) variance of opinions(and extremism) and higher (lower) judgment accuracyFor the rest of the observations 119872 seem to have noeffect We choose a middle value for 119872 Similarly theargument pool chosen to be balanced or uniformly

Complexity 29

Require Argumentation framework 119860119865 = (AR)Ensure Assignment of labels to arguments from the grounded labeling of 119860119865L(A)

L0(A) larr997888 (0 0 0)119896 larr997888 0do119894119899 L119896+1(A) larr997888 119894119899 L119896(A) cup 119886 | 119886 is not labeled 119894119899 L119896(A) 119886119899119889 forall119887 isin A 119904119905 119887 R 119886 then 119887 isin 119900119906119905 L119896(A)119900119906119905 L119896+1(A) larr997888 119900119906119905 L119896(A) cup 119886 | 119886 is not labeled 119894119899 L119896(A) 119886119899119889 exist119887 isin A 119904119905 119887 R 119886 and 119887 isin 119894119899 L119896+1(A)

while L119896+1(A) = L119896(A) = 0Algorithm 3 Solve argumentation frameworks (with grounded semantics)

Require Vector of values for parameters (119881)Ensure Vector of statistics (119880)

procedure Set-parameters(119881) ⊳Initialize parameters for scenarioprocedure Init-objs(119899119898 119896 ) ⊳Create119898 arguments 119899 agents and 119860119865120576procedure Solve-Argumentation-Framework(119860119865) ⊳Assigns the epistemic labelingL120576 toA119894 larr997888 1119905 larr997888 1while 119894 le 119872 do ⊳ 119872 the number of arguments to discuss119868119894 larr997888 119866119890119899 119886119903119892(1) ⊳Generates proposal argument 119868119860 119894119905 larr997888 (119868119894 0)L119886119887(119868119894) larr997888 119906119899119889 ⊳Initialize label for 119868119895 larr997888 1 ⊳Initialize debate counter 119895while ((L119886119887(119868) = 119906119899119889 or 119898 le 119895) and 119895 le 119898) doif 119905 equiv 0(119898119900119889 119905119863) then

procedure build-argumentation-framework(119899119863 119887119890ℎ119886V119894119900119903) ⊳Build the argument graph agents deliberate(119861119894119905 119878119894119905) larr997888 (119883 119884) sub (⋃119897isin119873119899119863 A119897119894119905 ⋃119897isin119873119899119863 R119897119894119905) ⊳ 119873119899119863 sub 119873 are agents sampled by the CA119860 119894119905 larr997888 119889(119860 119894119905 (119861119894119905 119878119894119905))procedure Solve-Argumentation-Framework(A119889) ⊳FindL(A119889)L119886119887(119868119894) larr997888 L119886119887119863(119868119894)119895 larr997888 119895 + 1

else119896 larr997888 1while 119896 le 119905119863 doprocedure dyadic-social-influence(120583 119879 119880) ⊳Agents discuss one-to-one119896 larr997888 119896 + 1119905 larr997888 119905 + 1

if 120572 gt 0 andL119886119887(119868119894) = 119900119906119905 then ⊳Agents vote for the proposalif (|119897 isin 119873|119900119897 times V119868119894 ge 0| ge 120572 times 119899) then

L119886119887(119868119894) larr997888 119894119899else

L119886119887(119868119894) larr997888 119900119906119905if L119886119887(119868119894) = 119894119899 thenprocedure Deliberation-influence(120574 119901119886 119901119903) ⊳Agents are influenced by the result of the decision process119894 larr997888 119894 + 1119905 larr997888 119905 + 1119880 larr997888 119862119900119898119901119906119905119890 119878119905119886119905119894119904119905119894119888119904(119881)

return 119880 ⊳Get the statisticsAlgorithm 4 Run for one scenario

generated does not seem to play a significant explanatoryrole on any of the metrics

16 Experiments show that themaximumnumber of debatesis not an important parameter even if it is an intuitiveone when considering decision-making processes Inpractice debates end after at most 119898 + 1 deliberation

steps and even then it happens rarely However to avoidunpleasant surprises like infinite debates on a completeargument graph containing the proposal argument weadded the constraint 119898 Its value is precisely 7 becausethe biggest minimum number of debates we studyis 6

30 Complexity

17 To ensure comparability with the deliberative andmixedopinion dynamics models on judgment accuracy andcoherence agents vote on proposal arguments every 119905119863times(119898+1) 119905119863 and119898does not explain variability in judgmentaccuracy nor in coherence

18 Student t-tests for independent samples are robust tothe violation of two hypothesis on the distribution ofthe samples normality and homogeneity of variancesas long as the sample size is big enough (central limittheorem) and if the difference in size of the comparedsamples is smallrespectively

19 Studentrsquos test version forwhen variances are not assumedequal and the difference in size of the compared samplesis big

20 We do not correct for homoskedasticity nor normality oferrors in the estimates

21 Balanced as the same number of runs per scenario orsuch that no significant correlations are found betweenparameters

References

[1] R Axelrod ldquoThe dissemination of culture a model withlocal convergence and global polarizationrdquo Journal of ConflictResolution vol 41 no 2 pp 203ndash226 1997

[2] D M Rousseau Reinforcing the MicroMacro Bridge Organiza-tionalThinking and Pluralistic Vehicles SAGEPublications SageLos Angeles Calif USA 2011

[3] S Galam and SMoscovici ldquoTowards a theory of collective phe-nomena consensus and attitude changes in groupsrdquo EuropeanJournal of Social Psychology vol 21 no 1 pp 49ndash74 1991

[4] A Nowak J Szamrej and B Latane ldquoFrom private attitude topublic opinion a dynamic theory of social impactrdquo Psychologi-cal Review vol 97 no 3 pp 362ndash376 1990

[5] G Deffuant S Huet and F Amblard ldquoAn individual-basedmodel of innovation diffusion mixing social value and individ-ual benefitrdquo American Journal of Sociology vol 110 no 4 pp1041ndash1069 2005

[6] G Deffuant D Neau F Amblard and G Weisbuch ldquoMixingbeliefs among interacting agentsrdquoAdvances in Complex Systems(ACS) vol 3 no 01n04 pp 87ndash98 2000

[7] J Rouchier P Tubaro and C Emery ldquoOpinion transmission inorganizations an agent-based modeling approachrdquo Computa-tional and Mathematical OrganizationTheory vol 20 no 3 pp252ndash277 2014

[8] M Meadows and D Cliff ldquoThe relative disagreement model ofopinion dynamics where do extremists come fromrdquo in Pro-ceedings of the International Workshop on Self-Organizing Sys-tems Lecture Notes in Computer Science pp 66ndash77 Springer2013

[9] W Jager and F Amblard ldquoUniformity bipolarization and plu-riformity captured as generic stylized behavior with an agent-based simulation model of attitude changerdquo Computational ampMathematical Organization Theory vol 10 no 4 pp 295ndash3032005

[10] Y-F Zhang H-Y Duan and Z-L Geng ldquoEvolutionary mech-anism of frangibility in social consensus system based onnegative emotions spreadrdquo Complexity vol 2017 Article ID4037049 8 pages 2017

[11] A Flache andMWMacy ldquoSmall worlds and cultural polariza-tionrdquoThe Journal of Mathematical Sociology vol 35 no 1-3 pp146ndash176 2011

[12] A Tsang and K Larson ldquoOpinion dynamics of skepticalagentsrdquo in Proceedings of the 13th International Conference onAutonomous Agents and Multiagent Systems AAMAS 2014 pp277ndash284 International Foundation forAutonomousAgents andMultiagent Systems France May 2014

[13] K M Hansen Deliberative democracy and opinion formationUniversity Press of Southern Denmark Odense 2004

[14] C R Sunstein ldquoThe law of group polarizationrdquo Journal ofPolitical Philosophy vol 10 no 2 pp 175ndash195 2002 WileyOnline Library

[15] S Moscovici and W Doise Dissensions et Consensus unetheorie generale des decisions collectives Presses Universitairesde France-PUF 1992

[16] J S Fishkin and R C Luskin ldquoExperimenting with a demo-cratic ideal deliberative polling and public opinionrdquo ActaPolitica vol 40 no 3 pp 284ndash298 2005

[17] R C Luskin J S Fishkin and S Iyengar Considered Opinionson US Foreign Policy Face-To-Face versus Online Delibera-tive Polling International Communication Association NewOrleans La USA 2004

[18] J Barabas ldquoHowdeliberation affects policy opinionsrdquoAmericanPolitical Science Review vol 98 no 4 pp 687ndash701 2004

[19] O Pryymak A Rogers and N R Jennings ldquoEfficient opinionsharing in large decentralised teamsrdquo in Proceedings of the 11thInternational Conference on Autonomous Agents andMultiagentSystems vol 1 pp 543ndash550 Spain June 2012

[20] J Rouchier and E Tanimura ldquoWhen overconfident agents slowdown collective learningrdquo Simulation vol 88 no 1 pp 33ndash492012

[21] D Miller ldquoDeliberative democracy and social choicerdquo PoliticalStudies vol 40 S1 pp 54ndash67 1992

[22] M Sherif and C I Hovland Social Judgment Assimilation andContrast Effects in Communication and Attitude Change YaleUniversity Press 1961

[23] P M Dung ldquoOn the acceptability of arguments and its funda-mental role in nonmonotonic reasoning logic programmingand n-person gamesrdquo Artificial Intelligence vol 77 no 2 pp321ndash357 1995

[24] M Caminada and G Pigozzi ldquoOn judgment aggregation inabstract argumentationrdquo Autonomous Agents and Multi-AgentSystems vol 22 no 1 pp 64ndash102 2011

[25] P Baroni M Caminada andM Giacomin ldquoAn introduction toargumentation semanticsrdquoThe Knowledge Engineering Reviewvol 26 no 4 pp 365ndash410 2011

[26] S Chambers ldquoDeliberative democratic theoryrdquo Annual Reviewof Political Science vol 6 no 1 pp 307ndash326 2003

[27] E Bonzon andNMaudet ldquoOn the outcomes of multiparty per-suasionrdquo in Proceedings of the 10th International Conference onAutonomous Agents and Multiagent Systems 2011 AAMAS 2011vol 1 pp 47ndash54 International Foundation for AutonomousAgents and Multiagent Systems Taiwan May 2011

[28] J Ranciere Disagreement Politics and Philosophy University ofMinnesota Press 1999

[29] T Bench-Capon and K Atkinson ldquoAbstract argumentation andvaluesrdquo in Argumentation in Artificial Intelligence pp 45ndash64Springer 2009

[30] M Caminada ldquoOn the issue of reinstatement in argumenta-tionrdquo in Proceedings of the European Workshop on Logics in

Complexity 31

Artificial Intelligence vol 4160 of Lecture Notes in ComputerScience pp 111ndash123 Springer Berlin Heidelberg

[31] GAVreeswijk ldquoAn algorithm to computeminimally groundedand admissible defence sets in argument systemsrdquo in Pro-ceedings of the 2006 conference on Computational Models ofArgument Proceedings of COMMA 2006 vol 144 pp 109ndash120IOS Press Amsterdam The Netherlands 2006

[32] C W Sherif M Kelly H L Rodgers et al ldquoPersonal involve-ment social judgment and actionrdquo Journal of Personality andSocial Psychology vol 27 no 3 pp 311ndash328 1973

[33] G Deffuant F Amblard G Weisbuch and T Faure ldquoHow canextremism prevail A study based on the relative agreementinteraction modelrdquo Journal of Artificial Societies and SocialSimulation vol 5 no 4 p 1 2002

[34] E Awad M W Caminada G Pigozzi M Podlaszewski andI Rahwan ldquoPareto optimality and strategy-proofness in groupargument evaluationrdquo Journal of Logic and Computation vol 27no 8 pp 2581ndash2609 2017

[35] D Grossi and G Pigozzi ldquoJudgment aggregation a primerrdquoSynthesis Lectures on Artificial Intelligence and Machine Learn-ing vol 8 no 2 pp 1ndash151 2014

[36] H Landemore and S E Page ldquoDeliberation and disagreementproblem solving prediction and positive dissensusrdquo PoliticsPhilosophy and Economics vol 14 no 3 pp 229ndash254 2015

[37] R Hegselmann and U Krause ldquoOpinion dynamics andbounded confidence models analysis and simulationrdquo Journalof Artificial Societies and Social Simulation vol 5 no 3 2002

[38] A Stefanelli and R Seidl ldquoModerate and polarized opinionsUsing empirical data for an agent-based simulationrdquo inProceed-ings of the Social Simulation Conference 2014

[39] S Gabbriellini and P Torroni ldquoA new framework for ABMsbased on argumentative reasoningrdquo in Advances in SocialSimulation vol 229 pp 25ndash36 Springer Berlin Germany 2014

[40] H Mercier and D Sperber ldquoWhy do humans reason Argu-ments for an argumentative theoryrdquo Behavioral and BrainSciences vol 34 no 2 pp 57ndash74 2011

[41] H V Parunak T C Belding R Hilscher and S BruecknerldquoModeling and managing collective cognitive convergencerdquoin In Proceedings of the 7th iNternational Joint Conference onAutonomous Agents and Multiagent Systems vol 3 pp 1505ndash1508 International Foundation for Autonomous Agents andMultiagent Systems 2008

[42] H Prakken ldquoFormal systems for persuasion dialoguerdquo TheKnowledge Engineering Review vol 21 no 2 pp 163ndash188 2006

[43] K Atkinson T Bench-Capon and P Mcburney ldquoA dialoguegame protocol for multi-agent argument over proposals foractionrdquoAutonomousAgents andMulti-Agent Systems vol 11 no2 pp 153ndash171 2005

[44] M Thimm and A J Garcıa ldquoClassification and strategicalissues of argumentation games on structured argumentationframeworksrdquo in Proceedings of the 9th International Conferenceon Autonomous Agents and Multiagent Systems vol 1 pp 1247ndash1254 International Foundation for Autonomous Agents andMultiagent Systems Canada 2010

[45] I Rahwan and K Larson ldquoMechanism design for abstractargumentationrdquo in Proceedings of the 7th International JointConference on Autonomous Agents and Multiagent SystemsAAMAS 2008 pp 1031ndash1038 International Foundation forAutonomous Agents and Multiagent Systems Portugal May2008

[46] H Prakken R Riveret A Rotolo and G Sartor ldquoHeuristics inArgumentation A Game-Theoretical Investigationrdquo EuropeanUniversity Institute 2008

[47] N Hadidi Y Dimopoulos and P Moraitis ldquoTactics and con-cessions for argumentation-based negotiationrdquo in Proceedingsof the COMMA vol 245 pp 285ndash296 2012

[48] N Oren and T J Norman ldquoArguing using opponent modelsrdquoin Proceedings of the International Workshop on Argumentationin Multi-Agent Systems pp 160ndash174 Springer 2009

[49] S Delre W Jager T Bijmolt and M Janssen ldquoTargeting andtiming promotional activities An agent-based model for thetakeoff of new productsrdquo Journal of Business Research vol 60no 8 pp 826ndash835 2007

[50] A Mirtabatabaei P Jia and F Bullo ldquoEulerian opinion dynam-ics with bounded confidence and exogenous inputsrdquo SIAMJournal on Applied Dynamical Systems vol 13 no 1 pp 425ndash446 2014

[51] V Barbet N Guiraud V Laperriere and J Rouchier Hagglingon Values Towards Consensus or Trouble 2019

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Complexity 29

Require Argumentation framework 119860119865 = (AR)Ensure Assignment of labels to arguments from the grounded labeling of 119860119865L(A)

L0(A) larr997888 (0 0 0)119896 larr997888 0do119894119899 L119896+1(A) larr997888 119894119899 L119896(A) cup 119886 | 119886 is not labeled 119894119899 L119896(A) 119886119899119889 forall119887 isin A 119904119905 119887 R 119886 then 119887 isin 119900119906119905 L119896(A)119900119906119905 L119896+1(A) larr997888 119900119906119905 L119896(A) cup 119886 | 119886 is not labeled 119894119899 L119896(A) 119886119899119889 exist119887 isin A 119904119905 119887 R 119886 and 119887 isin 119894119899 L119896+1(A)

while L119896+1(A) = L119896(A) = 0Algorithm 3 Solve argumentation frameworks (with grounded semantics)

Require Vector of values for parameters (119881)Ensure Vector of statistics (119880)

procedure Set-parameters(119881) ⊳Initialize parameters for scenarioprocedure Init-objs(119899119898 119896 ) ⊳Create119898 arguments 119899 agents and 119860119865120576procedure Solve-Argumentation-Framework(119860119865) ⊳Assigns the epistemic labelingL120576 toA119894 larr997888 1119905 larr997888 1while 119894 le 119872 do ⊳ 119872 the number of arguments to discuss119868119894 larr997888 119866119890119899 119886119903119892(1) ⊳Generates proposal argument 119868119860 119894119905 larr997888 (119868119894 0)L119886119887(119868119894) larr997888 119906119899119889 ⊳Initialize label for 119868119895 larr997888 1 ⊳Initialize debate counter 119895while ((L119886119887(119868) = 119906119899119889 or 119898 le 119895) and 119895 le 119898) doif 119905 equiv 0(119898119900119889 119905119863) then

procedure build-argumentation-framework(119899119863 119887119890ℎ119886V119894119900119903) ⊳Build the argument graph agents deliberate(119861119894119905 119878119894119905) larr997888 (119883 119884) sub (⋃119897isin119873119899119863 A119897119894119905 ⋃119897isin119873119899119863 R119897119894119905) ⊳ 119873119899119863 sub 119873 are agents sampled by the CA119860 119894119905 larr997888 119889(119860 119894119905 (119861119894119905 119878119894119905))procedure Solve-Argumentation-Framework(A119889) ⊳FindL(A119889)L119886119887(119868119894) larr997888 L119886119887119863(119868119894)119895 larr997888 119895 + 1

else119896 larr997888 1while 119896 le 119905119863 doprocedure dyadic-social-influence(120583 119879 119880) ⊳Agents discuss one-to-one119896 larr997888 119896 + 1119905 larr997888 119905 + 1

if 120572 gt 0 andL119886119887(119868119894) = 119900119906119905 then ⊳Agents vote for the proposalif (|119897 isin 119873|119900119897 times V119868119894 ge 0| ge 120572 times 119899) then

L119886119887(119868119894) larr997888 119894119899else

L119886119887(119868119894) larr997888 119900119906119905if L119886119887(119868119894) = 119894119899 thenprocedure Deliberation-influence(120574 119901119886 119901119903) ⊳Agents are influenced by the result of the decision process119894 larr997888 119894 + 1119905 larr997888 119905 + 1119880 larr997888 119862119900119898119901119906119905119890 119878119905119886119905119894119904119905119894119888119904(119881)

return 119880 ⊳Get the statisticsAlgorithm 4 Run for one scenario

generated does not seem to play a significant explanatoryrole on any of the metrics

16 Experiments show that themaximumnumber of debatesis not an important parameter even if it is an intuitiveone when considering decision-making processes Inpractice debates end after at most 119898 + 1 deliberation

steps and even then it happens rarely However to avoidunpleasant surprises like infinite debates on a completeargument graph containing the proposal argument weadded the constraint 119898 Its value is precisely 7 becausethe biggest minimum number of debates we studyis 6

30 Complexity

17 To ensure comparability with the deliberative andmixedopinion dynamics models on judgment accuracy andcoherence agents vote on proposal arguments every 119905119863times(119898+1) 119905119863 and119898does not explain variability in judgmentaccuracy nor in coherence

18 Student t-tests for independent samples are robust tothe violation of two hypothesis on the distribution ofthe samples normality and homogeneity of variancesas long as the sample size is big enough (central limittheorem) and if the difference in size of the comparedsamples is smallrespectively

19 Studentrsquos test version forwhen variances are not assumedequal and the difference in size of the compared samplesis big

20 We do not correct for homoskedasticity nor normality oferrors in the estimates

21 Balanced as the same number of runs per scenario orsuch that no significant correlations are found betweenparameters

References

[1] R Axelrod ldquoThe dissemination of culture a model withlocal convergence and global polarizationrdquo Journal of ConflictResolution vol 41 no 2 pp 203ndash226 1997

[2] D M Rousseau Reinforcing the MicroMacro Bridge Organiza-tionalThinking and Pluralistic Vehicles SAGEPublications SageLos Angeles Calif USA 2011

[3] S Galam and SMoscovici ldquoTowards a theory of collective phe-nomena consensus and attitude changes in groupsrdquo EuropeanJournal of Social Psychology vol 21 no 1 pp 49ndash74 1991

[4] A Nowak J Szamrej and B Latane ldquoFrom private attitude topublic opinion a dynamic theory of social impactrdquo Psychologi-cal Review vol 97 no 3 pp 362ndash376 1990

[5] G Deffuant S Huet and F Amblard ldquoAn individual-basedmodel of innovation diffusion mixing social value and individ-ual benefitrdquo American Journal of Sociology vol 110 no 4 pp1041ndash1069 2005

[6] G Deffuant D Neau F Amblard and G Weisbuch ldquoMixingbeliefs among interacting agentsrdquoAdvances in Complex Systems(ACS) vol 3 no 01n04 pp 87ndash98 2000

[7] J Rouchier P Tubaro and C Emery ldquoOpinion transmission inorganizations an agent-based modeling approachrdquo Computa-tional and Mathematical OrganizationTheory vol 20 no 3 pp252ndash277 2014

[8] M Meadows and D Cliff ldquoThe relative disagreement model ofopinion dynamics where do extremists come fromrdquo in Pro-ceedings of the International Workshop on Self-Organizing Sys-tems Lecture Notes in Computer Science pp 66ndash77 Springer2013

[9] W Jager and F Amblard ldquoUniformity bipolarization and plu-riformity captured as generic stylized behavior with an agent-based simulation model of attitude changerdquo Computational ampMathematical Organization Theory vol 10 no 4 pp 295ndash3032005

[10] Y-F Zhang H-Y Duan and Z-L Geng ldquoEvolutionary mech-anism of frangibility in social consensus system based onnegative emotions spreadrdquo Complexity vol 2017 Article ID4037049 8 pages 2017

[11] A Flache andMWMacy ldquoSmall worlds and cultural polariza-tionrdquoThe Journal of Mathematical Sociology vol 35 no 1-3 pp146ndash176 2011

[12] A Tsang and K Larson ldquoOpinion dynamics of skepticalagentsrdquo in Proceedings of the 13th International Conference onAutonomous Agents and Multiagent Systems AAMAS 2014 pp277ndash284 International Foundation forAutonomousAgents andMultiagent Systems France May 2014

[13] K M Hansen Deliberative democracy and opinion formationUniversity Press of Southern Denmark Odense 2004

[14] C R Sunstein ldquoThe law of group polarizationrdquo Journal ofPolitical Philosophy vol 10 no 2 pp 175ndash195 2002 WileyOnline Library

[15] S Moscovici and W Doise Dissensions et Consensus unetheorie generale des decisions collectives Presses Universitairesde France-PUF 1992

[16] J S Fishkin and R C Luskin ldquoExperimenting with a demo-cratic ideal deliberative polling and public opinionrdquo ActaPolitica vol 40 no 3 pp 284ndash298 2005

[17] R C Luskin J S Fishkin and S Iyengar Considered Opinionson US Foreign Policy Face-To-Face versus Online Delibera-tive Polling International Communication Association NewOrleans La USA 2004

[18] J Barabas ldquoHowdeliberation affects policy opinionsrdquoAmericanPolitical Science Review vol 98 no 4 pp 687ndash701 2004

[19] O Pryymak A Rogers and N R Jennings ldquoEfficient opinionsharing in large decentralised teamsrdquo in Proceedings of the 11thInternational Conference on Autonomous Agents andMultiagentSystems vol 1 pp 543ndash550 Spain June 2012

[20] J Rouchier and E Tanimura ldquoWhen overconfident agents slowdown collective learningrdquo Simulation vol 88 no 1 pp 33ndash492012

[21] D Miller ldquoDeliberative democracy and social choicerdquo PoliticalStudies vol 40 S1 pp 54ndash67 1992

[22] M Sherif and C I Hovland Social Judgment Assimilation andContrast Effects in Communication and Attitude Change YaleUniversity Press 1961

[23] P M Dung ldquoOn the acceptability of arguments and its funda-mental role in nonmonotonic reasoning logic programmingand n-person gamesrdquo Artificial Intelligence vol 77 no 2 pp321ndash357 1995

[24] M Caminada and G Pigozzi ldquoOn judgment aggregation inabstract argumentationrdquo Autonomous Agents and Multi-AgentSystems vol 22 no 1 pp 64ndash102 2011

[25] P Baroni M Caminada andM Giacomin ldquoAn introduction toargumentation semanticsrdquoThe Knowledge Engineering Reviewvol 26 no 4 pp 365ndash410 2011

[26] S Chambers ldquoDeliberative democratic theoryrdquo Annual Reviewof Political Science vol 6 no 1 pp 307ndash326 2003

[27] E Bonzon andNMaudet ldquoOn the outcomes of multiparty per-suasionrdquo in Proceedings of the 10th International Conference onAutonomous Agents and Multiagent Systems 2011 AAMAS 2011vol 1 pp 47ndash54 International Foundation for AutonomousAgents and Multiagent Systems Taiwan May 2011

[28] J Ranciere Disagreement Politics and Philosophy University ofMinnesota Press 1999

[29] T Bench-Capon and K Atkinson ldquoAbstract argumentation andvaluesrdquo in Argumentation in Artificial Intelligence pp 45ndash64Springer 2009

[30] M Caminada ldquoOn the issue of reinstatement in argumenta-tionrdquo in Proceedings of the European Workshop on Logics in

Complexity 31

Artificial Intelligence vol 4160 of Lecture Notes in ComputerScience pp 111ndash123 Springer Berlin Heidelberg

[31] GAVreeswijk ldquoAn algorithm to computeminimally groundedand admissible defence sets in argument systemsrdquo in Pro-ceedings of the 2006 conference on Computational Models ofArgument Proceedings of COMMA 2006 vol 144 pp 109ndash120IOS Press Amsterdam The Netherlands 2006

[32] C W Sherif M Kelly H L Rodgers et al ldquoPersonal involve-ment social judgment and actionrdquo Journal of Personality andSocial Psychology vol 27 no 3 pp 311ndash328 1973

[33] G Deffuant F Amblard G Weisbuch and T Faure ldquoHow canextremism prevail A study based on the relative agreementinteraction modelrdquo Journal of Artificial Societies and SocialSimulation vol 5 no 4 p 1 2002

[34] E Awad M W Caminada G Pigozzi M Podlaszewski andI Rahwan ldquoPareto optimality and strategy-proofness in groupargument evaluationrdquo Journal of Logic and Computation vol 27no 8 pp 2581ndash2609 2017

[35] D Grossi and G Pigozzi ldquoJudgment aggregation a primerrdquoSynthesis Lectures on Artificial Intelligence and Machine Learn-ing vol 8 no 2 pp 1ndash151 2014

[36] H Landemore and S E Page ldquoDeliberation and disagreementproblem solving prediction and positive dissensusrdquo PoliticsPhilosophy and Economics vol 14 no 3 pp 229ndash254 2015

[37] R Hegselmann and U Krause ldquoOpinion dynamics andbounded confidence models analysis and simulationrdquo Journalof Artificial Societies and Social Simulation vol 5 no 3 2002

[38] A Stefanelli and R Seidl ldquoModerate and polarized opinionsUsing empirical data for an agent-based simulationrdquo inProceed-ings of the Social Simulation Conference 2014

[39] S Gabbriellini and P Torroni ldquoA new framework for ABMsbased on argumentative reasoningrdquo in Advances in SocialSimulation vol 229 pp 25ndash36 Springer Berlin Germany 2014

[40] H Mercier and D Sperber ldquoWhy do humans reason Argu-ments for an argumentative theoryrdquo Behavioral and BrainSciences vol 34 no 2 pp 57ndash74 2011

[41] H V Parunak T C Belding R Hilscher and S BruecknerldquoModeling and managing collective cognitive convergencerdquoin In Proceedings of the 7th iNternational Joint Conference onAutonomous Agents and Multiagent Systems vol 3 pp 1505ndash1508 International Foundation for Autonomous Agents andMultiagent Systems 2008

[42] H Prakken ldquoFormal systems for persuasion dialoguerdquo TheKnowledge Engineering Review vol 21 no 2 pp 163ndash188 2006

[43] K Atkinson T Bench-Capon and P Mcburney ldquoA dialoguegame protocol for multi-agent argument over proposals foractionrdquoAutonomousAgents andMulti-Agent Systems vol 11 no2 pp 153ndash171 2005

[44] M Thimm and A J Garcıa ldquoClassification and strategicalissues of argumentation games on structured argumentationframeworksrdquo in Proceedings of the 9th International Conferenceon Autonomous Agents and Multiagent Systems vol 1 pp 1247ndash1254 International Foundation for Autonomous Agents andMultiagent Systems Canada 2010

[45] I Rahwan and K Larson ldquoMechanism design for abstractargumentationrdquo in Proceedings of the 7th International JointConference on Autonomous Agents and Multiagent SystemsAAMAS 2008 pp 1031ndash1038 International Foundation forAutonomous Agents and Multiagent Systems Portugal May2008

[46] H Prakken R Riveret A Rotolo and G Sartor ldquoHeuristics inArgumentation A Game-Theoretical Investigationrdquo EuropeanUniversity Institute 2008

[47] N Hadidi Y Dimopoulos and P Moraitis ldquoTactics and con-cessions for argumentation-based negotiationrdquo in Proceedingsof the COMMA vol 245 pp 285ndash296 2012

[48] N Oren and T J Norman ldquoArguing using opponent modelsrdquoin Proceedings of the International Workshop on Argumentationin Multi-Agent Systems pp 160ndash174 Springer 2009

[49] S Delre W Jager T Bijmolt and M Janssen ldquoTargeting andtiming promotional activities An agent-based model for thetakeoff of new productsrdquo Journal of Business Research vol 60no 8 pp 826ndash835 2007

[50] A Mirtabatabaei P Jia and F Bullo ldquoEulerian opinion dynam-ics with bounded confidence and exogenous inputsrdquo SIAMJournal on Applied Dynamical Systems vol 13 no 1 pp 425ndash446 2014

[51] V Barbet N Guiraud V Laperriere and J Rouchier Hagglingon Values Towards Consensus or Trouble 2019

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

30 Complexity

17 To ensure comparability with the deliberative andmixedopinion dynamics models on judgment accuracy andcoherence agents vote on proposal arguments every 119905119863times(119898+1) 119905119863 and119898does not explain variability in judgmentaccuracy nor in coherence

18 Student t-tests for independent samples are robust tothe violation of two hypothesis on the distribution ofthe samples normality and homogeneity of variancesas long as the sample size is big enough (central limittheorem) and if the difference in size of the comparedsamples is smallrespectively

19 Studentrsquos test version forwhen variances are not assumedequal and the difference in size of the compared samplesis big

20 We do not correct for homoskedasticity nor normality oferrors in the estimates

21 Balanced as the same number of runs per scenario orsuch that no significant correlations are found betweenparameters

References

[1] R Axelrod ldquoThe dissemination of culture a model withlocal convergence and global polarizationrdquo Journal of ConflictResolution vol 41 no 2 pp 203ndash226 1997

[2] D M Rousseau Reinforcing the MicroMacro Bridge Organiza-tionalThinking and Pluralistic Vehicles SAGEPublications SageLos Angeles Calif USA 2011

[3] S Galam and SMoscovici ldquoTowards a theory of collective phe-nomena consensus and attitude changes in groupsrdquo EuropeanJournal of Social Psychology vol 21 no 1 pp 49ndash74 1991

[4] A Nowak J Szamrej and B Latane ldquoFrom private attitude topublic opinion a dynamic theory of social impactrdquo Psychologi-cal Review vol 97 no 3 pp 362ndash376 1990

[5] G Deffuant S Huet and F Amblard ldquoAn individual-basedmodel of innovation diffusion mixing social value and individ-ual benefitrdquo American Journal of Sociology vol 110 no 4 pp1041ndash1069 2005

[6] G Deffuant D Neau F Amblard and G Weisbuch ldquoMixingbeliefs among interacting agentsrdquoAdvances in Complex Systems(ACS) vol 3 no 01n04 pp 87ndash98 2000

[7] J Rouchier P Tubaro and C Emery ldquoOpinion transmission inorganizations an agent-based modeling approachrdquo Computa-tional and Mathematical OrganizationTheory vol 20 no 3 pp252ndash277 2014

[8] M Meadows and D Cliff ldquoThe relative disagreement model ofopinion dynamics where do extremists come fromrdquo in Pro-ceedings of the International Workshop on Self-Organizing Sys-tems Lecture Notes in Computer Science pp 66ndash77 Springer2013

[9] W Jager and F Amblard ldquoUniformity bipolarization and plu-riformity captured as generic stylized behavior with an agent-based simulation model of attitude changerdquo Computational ampMathematical Organization Theory vol 10 no 4 pp 295ndash3032005

[10] Y-F Zhang H-Y Duan and Z-L Geng ldquoEvolutionary mech-anism of frangibility in social consensus system based onnegative emotions spreadrdquo Complexity vol 2017 Article ID4037049 8 pages 2017

[11] A Flache andMWMacy ldquoSmall worlds and cultural polariza-tionrdquoThe Journal of Mathematical Sociology vol 35 no 1-3 pp146ndash176 2011

[12] A Tsang and K Larson ldquoOpinion dynamics of skepticalagentsrdquo in Proceedings of the 13th International Conference onAutonomous Agents and Multiagent Systems AAMAS 2014 pp277ndash284 International Foundation forAutonomousAgents andMultiagent Systems France May 2014

[13] K M Hansen Deliberative democracy and opinion formationUniversity Press of Southern Denmark Odense 2004

[14] C R Sunstein ldquoThe law of group polarizationrdquo Journal ofPolitical Philosophy vol 10 no 2 pp 175ndash195 2002 WileyOnline Library

[15] S Moscovici and W Doise Dissensions et Consensus unetheorie generale des decisions collectives Presses Universitairesde France-PUF 1992

[16] J S Fishkin and R C Luskin ldquoExperimenting with a demo-cratic ideal deliberative polling and public opinionrdquo ActaPolitica vol 40 no 3 pp 284ndash298 2005

[17] R C Luskin J S Fishkin and S Iyengar Considered Opinionson US Foreign Policy Face-To-Face versus Online Delibera-tive Polling International Communication Association NewOrleans La USA 2004

[18] J Barabas ldquoHowdeliberation affects policy opinionsrdquoAmericanPolitical Science Review vol 98 no 4 pp 687ndash701 2004

[19] O Pryymak A Rogers and N R Jennings ldquoEfficient opinionsharing in large decentralised teamsrdquo in Proceedings of the 11thInternational Conference on Autonomous Agents andMultiagentSystems vol 1 pp 543ndash550 Spain June 2012

[20] J Rouchier and E Tanimura ldquoWhen overconfident agents slowdown collective learningrdquo Simulation vol 88 no 1 pp 33ndash492012

[21] D Miller ldquoDeliberative democracy and social choicerdquo PoliticalStudies vol 40 S1 pp 54ndash67 1992

[22] M Sherif and C I Hovland Social Judgment Assimilation andContrast Effects in Communication and Attitude Change YaleUniversity Press 1961

[23] P M Dung ldquoOn the acceptability of arguments and its funda-mental role in nonmonotonic reasoning logic programmingand n-person gamesrdquo Artificial Intelligence vol 77 no 2 pp321ndash357 1995

[24] M Caminada and G Pigozzi ldquoOn judgment aggregation inabstract argumentationrdquo Autonomous Agents and Multi-AgentSystems vol 22 no 1 pp 64ndash102 2011

[25] P Baroni M Caminada andM Giacomin ldquoAn introduction toargumentation semanticsrdquoThe Knowledge Engineering Reviewvol 26 no 4 pp 365ndash410 2011

[26] S Chambers ldquoDeliberative democratic theoryrdquo Annual Reviewof Political Science vol 6 no 1 pp 307ndash326 2003

[27] E Bonzon andNMaudet ldquoOn the outcomes of multiparty per-suasionrdquo in Proceedings of the 10th International Conference onAutonomous Agents and Multiagent Systems 2011 AAMAS 2011vol 1 pp 47ndash54 International Foundation for AutonomousAgents and Multiagent Systems Taiwan May 2011

[28] J Ranciere Disagreement Politics and Philosophy University ofMinnesota Press 1999

[29] T Bench-Capon and K Atkinson ldquoAbstract argumentation andvaluesrdquo in Argumentation in Artificial Intelligence pp 45ndash64Springer 2009

[30] M Caminada ldquoOn the issue of reinstatement in argumenta-tionrdquo in Proceedings of the European Workshop on Logics in

Complexity 31

Artificial Intelligence vol 4160 of Lecture Notes in ComputerScience pp 111ndash123 Springer Berlin Heidelberg

[31] GAVreeswijk ldquoAn algorithm to computeminimally groundedand admissible defence sets in argument systemsrdquo in Pro-ceedings of the 2006 conference on Computational Models ofArgument Proceedings of COMMA 2006 vol 144 pp 109ndash120IOS Press Amsterdam The Netherlands 2006

[32] C W Sherif M Kelly H L Rodgers et al ldquoPersonal involve-ment social judgment and actionrdquo Journal of Personality andSocial Psychology vol 27 no 3 pp 311ndash328 1973

[33] G Deffuant F Amblard G Weisbuch and T Faure ldquoHow canextremism prevail A study based on the relative agreementinteraction modelrdquo Journal of Artificial Societies and SocialSimulation vol 5 no 4 p 1 2002

[34] E Awad M W Caminada G Pigozzi M Podlaszewski andI Rahwan ldquoPareto optimality and strategy-proofness in groupargument evaluationrdquo Journal of Logic and Computation vol 27no 8 pp 2581ndash2609 2017

[35] D Grossi and G Pigozzi ldquoJudgment aggregation a primerrdquoSynthesis Lectures on Artificial Intelligence and Machine Learn-ing vol 8 no 2 pp 1ndash151 2014

[36] H Landemore and S E Page ldquoDeliberation and disagreementproblem solving prediction and positive dissensusrdquo PoliticsPhilosophy and Economics vol 14 no 3 pp 229ndash254 2015

[37] R Hegselmann and U Krause ldquoOpinion dynamics andbounded confidence models analysis and simulationrdquo Journalof Artificial Societies and Social Simulation vol 5 no 3 2002

[38] A Stefanelli and R Seidl ldquoModerate and polarized opinionsUsing empirical data for an agent-based simulationrdquo inProceed-ings of the Social Simulation Conference 2014

[39] S Gabbriellini and P Torroni ldquoA new framework for ABMsbased on argumentative reasoningrdquo in Advances in SocialSimulation vol 229 pp 25ndash36 Springer Berlin Germany 2014

[40] H Mercier and D Sperber ldquoWhy do humans reason Argu-ments for an argumentative theoryrdquo Behavioral and BrainSciences vol 34 no 2 pp 57ndash74 2011

[41] H V Parunak T C Belding R Hilscher and S BruecknerldquoModeling and managing collective cognitive convergencerdquoin In Proceedings of the 7th iNternational Joint Conference onAutonomous Agents and Multiagent Systems vol 3 pp 1505ndash1508 International Foundation for Autonomous Agents andMultiagent Systems 2008

[42] H Prakken ldquoFormal systems for persuasion dialoguerdquo TheKnowledge Engineering Review vol 21 no 2 pp 163ndash188 2006

[43] K Atkinson T Bench-Capon and P Mcburney ldquoA dialoguegame protocol for multi-agent argument over proposals foractionrdquoAutonomousAgents andMulti-Agent Systems vol 11 no2 pp 153ndash171 2005

[44] M Thimm and A J Garcıa ldquoClassification and strategicalissues of argumentation games on structured argumentationframeworksrdquo in Proceedings of the 9th International Conferenceon Autonomous Agents and Multiagent Systems vol 1 pp 1247ndash1254 International Foundation for Autonomous Agents andMultiagent Systems Canada 2010

[45] I Rahwan and K Larson ldquoMechanism design for abstractargumentationrdquo in Proceedings of the 7th International JointConference on Autonomous Agents and Multiagent SystemsAAMAS 2008 pp 1031ndash1038 International Foundation forAutonomous Agents and Multiagent Systems Portugal May2008

[46] H Prakken R Riveret A Rotolo and G Sartor ldquoHeuristics inArgumentation A Game-Theoretical Investigationrdquo EuropeanUniversity Institute 2008

[47] N Hadidi Y Dimopoulos and P Moraitis ldquoTactics and con-cessions for argumentation-based negotiationrdquo in Proceedingsof the COMMA vol 245 pp 285ndash296 2012

[48] N Oren and T J Norman ldquoArguing using opponent modelsrdquoin Proceedings of the International Workshop on Argumentationin Multi-Agent Systems pp 160ndash174 Springer 2009

[49] S Delre W Jager T Bijmolt and M Janssen ldquoTargeting andtiming promotional activities An agent-based model for thetakeoff of new productsrdquo Journal of Business Research vol 60no 8 pp 826ndash835 2007

[50] A Mirtabatabaei P Jia and F Bullo ldquoEulerian opinion dynam-ics with bounded confidence and exogenous inputsrdquo SIAMJournal on Applied Dynamical Systems vol 13 no 1 pp 425ndash446 2014

[51] V Barbet N Guiraud V Laperriere and J Rouchier Hagglingon Values Towards Consensus or Trouble 2019

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Complexity 31

Artificial Intelligence vol 4160 of Lecture Notes in ComputerScience pp 111ndash123 Springer Berlin Heidelberg

[31] GAVreeswijk ldquoAn algorithm to computeminimally groundedand admissible defence sets in argument systemsrdquo in Pro-ceedings of the 2006 conference on Computational Models ofArgument Proceedings of COMMA 2006 vol 144 pp 109ndash120IOS Press Amsterdam The Netherlands 2006

[32] C W Sherif M Kelly H L Rodgers et al ldquoPersonal involve-ment social judgment and actionrdquo Journal of Personality andSocial Psychology vol 27 no 3 pp 311ndash328 1973

[33] G Deffuant F Amblard G Weisbuch and T Faure ldquoHow canextremism prevail A study based on the relative agreementinteraction modelrdquo Journal of Artificial Societies and SocialSimulation vol 5 no 4 p 1 2002

[34] E Awad M W Caminada G Pigozzi M Podlaszewski andI Rahwan ldquoPareto optimality and strategy-proofness in groupargument evaluationrdquo Journal of Logic and Computation vol 27no 8 pp 2581ndash2609 2017

[35] D Grossi and G Pigozzi ldquoJudgment aggregation a primerrdquoSynthesis Lectures on Artificial Intelligence and Machine Learn-ing vol 8 no 2 pp 1ndash151 2014

[36] H Landemore and S E Page ldquoDeliberation and disagreementproblem solving prediction and positive dissensusrdquo PoliticsPhilosophy and Economics vol 14 no 3 pp 229ndash254 2015

[37] R Hegselmann and U Krause ldquoOpinion dynamics andbounded confidence models analysis and simulationrdquo Journalof Artificial Societies and Social Simulation vol 5 no 3 2002

[38] A Stefanelli and R Seidl ldquoModerate and polarized opinionsUsing empirical data for an agent-based simulationrdquo inProceed-ings of the Social Simulation Conference 2014

[39] S Gabbriellini and P Torroni ldquoA new framework for ABMsbased on argumentative reasoningrdquo in Advances in SocialSimulation vol 229 pp 25ndash36 Springer Berlin Germany 2014

[40] H Mercier and D Sperber ldquoWhy do humans reason Argu-ments for an argumentative theoryrdquo Behavioral and BrainSciences vol 34 no 2 pp 57ndash74 2011

[41] H V Parunak T C Belding R Hilscher and S BruecknerldquoModeling and managing collective cognitive convergencerdquoin In Proceedings of the 7th iNternational Joint Conference onAutonomous Agents and Multiagent Systems vol 3 pp 1505ndash1508 International Foundation for Autonomous Agents andMultiagent Systems 2008

[42] H Prakken ldquoFormal systems for persuasion dialoguerdquo TheKnowledge Engineering Review vol 21 no 2 pp 163ndash188 2006

[43] K Atkinson T Bench-Capon and P Mcburney ldquoA dialoguegame protocol for multi-agent argument over proposals foractionrdquoAutonomousAgents andMulti-Agent Systems vol 11 no2 pp 153ndash171 2005

[44] M Thimm and A J Garcıa ldquoClassification and strategicalissues of argumentation games on structured argumentationframeworksrdquo in Proceedings of the 9th International Conferenceon Autonomous Agents and Multiagent Systems vol 1 pp 1247ndash1254 International Foundation for Autonomous Agents andMultiagent Systems Canada 2010

[45] I Rahwan and K Larson ldquoMechanism design for abstractargumentationrdquo in Proceedings of the 7th International JointConference on Autonomous Agents and Multiagent SystemsAAMAS 2008 pp 1031ndash1038 International Foundation forAutonomous Agents and Multiagent Systems Portugal May2008

[46] H Prakken R Riveret A Rotolo and G Sartor ldquoHeuristics inArgumentation A Game-Theoretical Investigationrdquo EuropeanUniversity Institute 2008

[47] N Hadidi Y Dimopoulos and P Moraitis ldquoTactics and con-cessions for argumentation-based negotiationrdquo in Proceedingsof the COMMA vol 245 pp 285ndash296 2012

[48] N Oren and T J Norman ldquoArguing using opponent modelsrdquoin Proceedings of the International Workshop on Argumentationin Multi-Agent Systems pp 160ndash174 Springer 2009

[49] S Delre W Jager T Bijmolt and M Janssen ldquoTargeting andtiming promotional activities An agent-based model for thetakeoff of new productsrdquo Journal of Business Research vol 60no 8 pp 826ndash835 2007

[50] A Mirtabatabaei P Jia and F Bullo ldquoEulerian opinion dynam-ics with bounded confidence and exogenous inputsrdquo SIAMJournal on Applied Dynamical Systems vol 13 no 1 pp 425ndash446 2014

[51] V Barbet N Guiraud V Laperriere and J Rouchier Hagglingon Values Towards Consensus or Trouble 2019

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MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom