Research Article Evolutionary Game Analysis of Competitive

Research ArticleEvolutionary Game Analysis of CompetitiveInformation Dissemination on Social NetworksAn Agent-Based Computational Approach

Qing Sun12 and Zhong Yao1

1School of Economics and Management Beihang University Beijing 100191 China2School of Computer Science and Engineering Beihang University Beijing 100191 China

Correspondence should be addressed to Zhong Yao iszhyaobuaaeducn

Received 22 August 2014 Accepted 5 December 2014

Academic Editor L W Zhang

Copyright copy 2015 Q Sun and Z Yao 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

Social networks are formed by individuals in which personalities utility functions and interaction rules are made as closeto reality as possible Taking the competitive product-related information as a case we proposed a game-theoretic modelfor competitive information dissemination in social networks The model is presented to explain how human factors impactcompetitive information dissemination which is described as the dynamic of a coordination game and playersrsquo payoff is defined by autility functionThenwedesign a computational system that integrates the agent the evolutionary game and the social networkTheapproach can help to visualize the evolution of of competitive information adoption and diffusion grasp the dynamic evolutionfeatures in information adoption game over time and explore microlevel interactions among users in different network structureunder various scenariosWe discuss several scenarios to analyze the influence of several factors on the dissemination of competitiveinformation ranging from personality of individuals to structure of networks

1 Introduction and Prior Work

The emergent and rapid development of online social net-working applications has changed the way in which bothconsumers and enterprises interact and collaborate with eachother Online social networking applications provide theengaged individuals with collaborative environment to shareinformation or ideas with their neighbors where the totaleffect is greater than the sum of individual effects In socialcommerce the dissemination of product-related informa-tion is affected by individualsrsquo actions which are stronglydetermined by their characteristics and often influenced bythe decisions of other individuals This kind of informationdiffusion on the social network including social advertise-ments word-of-mouth and comments finally influences thebuying behavior of the potential consumers The featuresand patterns of competitive information dissemination willaffect the product-related information spreading such ascompetitive advertisement and positive or negative word-of-mouth because of their commercial feature Therefore it is

important to study how tomodel and analyze the competitivediffusion through social network

Epidemic models have been widely adopted by res-earchers for information dissemination due to the analogybetween epidemics and the spread of information Theunderlying assumption of these models is that individualsadopt a new behavior with a probability when they interactwith others who have already adopted it [1] Gruhl et al [2]investigate the adoption of the classic Susceptible-Infected-Removed (SIR) model for information dissemination Yangand Leskovec [3] developed a linear influence model to focuson influence of individual node on the rate of disseminationthrough the implicit network Lu et al [4] propose amodifiedSIR model to describe the information diffusion in thesmall-world network proposing three different spreadingrules from the standard SIR model memory effects socialreinforcement and nonredundancy of contacts in whichthe influence of social network structure is considered andanalyzed These studies have macroscopically committed

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 679726 12 pageshttpdxdoiorg1011552015679726

2 Mathematical Problems in Engineering

to the description of information diffusion through socialnetworks

In recent years researchers gradually observe that gamebehaviors between individuals in social network being thefoundation of the social group behavior raise social dif-fusions which were always discussed in the microlevel inprior related research Micronodes in network constantlyadjust their strategy according to the behavior of other nodesthrough the dynamic game with other nodes to maximizetheir own interests and all the behaviours of individual nodefinally form the social network group behavior Thereforegame-theoretic models as a new perspective of interpretingsocial diffusion are increasingly adopted by computer sci-entists for analyzing network behaviors Game theory is aset of analytical tools and solution concepts which providesexplanatory and predicting power in interactive decision sit-uations when the aims goals and preferences of the partici-pating players are potentially in conflict [5] Unlike the classicgame theory which assumes that gaming exists betweentwo individuals and happens only one-off the evolutionarygame theory has opened up related hypothesis limitationsIt introduces the concept of time and space considering thegame as the summary of historical experience and individuallearning which makes it advantageous for social networkstudy Several papers have applied evolutionary game theorymethod to explain social diffusion Kostka et al [6] examinedthe dissemination of competing rumors in social networkusing concepts of game theory and location theorymodellingthe selection of starting nodes for the rumors as a strategygame Meier et al [7] presented a virus propagation gamemodel finding that the Windfall of Friendship does notincrease monotonically with stronger relationships Zinovievet al [8 9] adopted game theoretic models to understandhuman aspects of information dissemination in which per-sonalities of individuals are considered Jiang et al [10]examined the evolutionary process of knowledge sharingamong users in social network and designed a computationalexperimental system developing a mixed learning algorithmbased on individualrsquos historical game strategy neighborsrsquostrategy and information noise

Applying evolutionary game theory in social diffusionstudy presumes that the game process between individualsis not only determined by a single rival or individual butalso all individuals in the neighborhood Meanwhile it takesthe historical game experience for future game behavior intoconsideration The benefit of each individual is evaluated bythe accumulated result of gaming many times Individualsbenefit from the game with its neighbors at the same timethe benefit can be observed with comparisons to adjusttheir game strategies trying to achieve optimum benefitsand reach the overall game equilibrium Evolutionary gametheory also has advantages on competitive social behaviorsand social diffusions Alon et al [11] introduced a game-theoretic model of competitive dissemination of technolo-gies advertisements or influence through a social networkWang et al [12] proposed the stochastic game net model foranalyzing competitive network behaviors Takehara et al [13]introduced and studied a deterministicmodel for competitiveinformation diffusion on social networks In contrast tomany

other game theoretic models for the diffusion of informationand innovation [14 15] the model considered competitionbetween different innovations spreading instead of discussinga single one

This paper is an extension of recent works includingthe method suggested by Jiang et al [16] and the resultsdescribed by Yu et al [17] The dissemination of informationis modeled as the dynamic of coordination game in whichplayerrsquos payoff is defined by a utility function and several casesare analyzed to reach the conclusion that the spreading rate isinfluenced by characteristics of individuals and several otherfactors Different from themodels of competitive informationdiffusion introduced in prior work we adopt a frameworkfor describing competitive information dissemination basedon a game-theoretic model and multiagent-based dynamicsIn this framework we target several similar product-relatedpieces of information that competes with each other Con-sumers may participate in discussing adopting and spread-ing one of them Totally different from news or opinionsdissemination that is always described as the spreading of thevirus consumers always make product-related decision forthe utility motivation In general the utility is dynamicallydetermined by factors from the environment or from hispersonal reasons Meanwhile prior research has shown thatdifferent network structures topology has great impact onsocial commerce Therefore both sociological and psycho-logical characteristics are explicitly considered in our modelas the novelty In thismodel information passing intrinsicallyinvolves both sides considering their characteristics self-perceived knowledge brand loyalty and social conformitywhich further determine their decisions of whether or notto forward the information The decisions are also based onthe global properties of the network such as the knowledgedynamic all through the network These factors finally bringdifferent results of disseminations of different informationthat compete with each other Based on the dynamic gamemodel and strategy updating rules we analyze competitiveinformation propagation and the affecting factors

The remainder of this paper is organized as followsSection 2 presents the overview of competitive disseminationmodel based on evolutionary game theory social networkand multiagent theory and introduces the individualrsquos utilityfunction Section 3 discusses the dynamics and updatingrules for the model we build proposes the assumptionsand explains the model In Section 4 we use an agent-basedcomputational approach for simulation

2 Problem Statement

We propose a different global point of view regarding theincentives that govern the diffusion process Suppose we haveseveral firms that would like to advertise competitive prod-ucts via ldquoviral marketingrdquo Each firm initially targets a smallsubset of users in the hope that the social advertisementsabout their product would spread throughout the networkHowever a user that participates in discussing and spreadingproduct-related information is reluctant to participate inanother one

Mathematical Problems in Engineering 3

Table 1 Game payoff matrix

Player 119894119886 119887

Player 119895119886 119880

119886119886 119880119886119886

119880119886119887 119880119887119886

119887 119880119887119886 119880119886119887

119880119887119887 119880119887119887

For simplicity we consider two pieces of informationcompeting over the social network represented by infor-mation 119860 or 119861 They are about the same kind of prod-uct with different brands (eg Smartphones of iPhone orSamsung) Which information is adopted by the individualis determined by his self-perceived knowledge and herdmentality Information dissemination is caused by individ-ualsrsquo interaction with his neighbours for sharing knowledgeand opinion This interaction which arouses changes ofindividualsrsquo acceptance and preference could be consideredas a game based on information exchange Individuals updatetheir strategies according to their payoff and the influenceof their neighbours The change of individualrsquos informationlevel for the product affects the individualrsquos preference andaccordingly affects his choice of consumption

21 Evolutionary Game Model In the classic game-theoreticmodel the consumers are defined as players and each playerhas only one stateWe take each node in online social networkas players in game The game is played in period 119905 = 119899 119899 =

1 2 3 in online social networks that could be describedas an undirected graph

First of all we assume that all players engaged in theevolutionary game are pure strategists and each player couldchoose only one of the strategies strategy 119886 is participatingin discussion and spreading of information for information119860 (corporation) and strategy 119887 is participating in discussionand spreading of information 119861 (defection) Then the set ofavailable strategies for node 119894 could be described as two-dimensional vectors

119878119894= (

1

0) or (

0

1) (1)

where 119878119894is the strategy of node 119894 In each round the node

games with all of its neighbours (friends) and profit Thepayoff matrix 119872 is defined by the utility function 119880(119894)

illustrated in Table 1In Table 1 119880

119894119895 119894 119895 isin (119886 119887) represents the utility for the

player according to what strategy he chooses The evolution-ary game equilibrium can be detected by the dynamic repli-cation method [5 18] that can provide theoretical supportfor macroscopic decisions on information disseminationignoring network structure and environmental factorsThenthe expected cooperation (defection) benefit fromknowledgesharing can be expressed as

119880119894(119862) = 119901 lowast 119880

119886119886+ (1 minus 119901) lowast 119880

119886119887

119880119894(119863) = 119901 lowast 119880

119887119886+ (1 minus 119901) lowast 119880

119887119887

(2)

where 119901 is the percentage of nodes holding a cooperativeattitude (strategy 119886) 119880

119894(119862) and 119880

119894(119863) respectively stand for

the benefit that node 119894 get in current round of gameFrom the dynamic replication the average benefit of the

whole group is

119880 = 119901 lowast 119880119894(119862) + (1 minus 119901)119880

119894(119863) (3)

The state of the evolutionary system over time for differ-ent initial value of 119901 can be evaluated by the diffusion rate119865(119901)

119865 (119901)

=119889119901

119889119905= 119901 (119880

119894(119862) minus 119880)

= 119901 (1 minus 119901) [(119880119886119886minus 119880119886119887minus 119880119887119886+ 119880119887119887) lowast 119901 + (119880

119886119887minus 119880119887119887)]

(4)

Therefore the game has three possible solutions to reachequilibrium

119901 = 0 1119880119886119887minus 119880119887119887

119880119886119886minus 119880119886119887minus 119880119887119886+ 119880119887119887

119901lowast

=119880119886119887minus 119880119887119887

119880119886119886minus 119880119886119887minus 119880119887119886+ 119880119887119887

(5)

where 119901lowast is the value with which the group evolutionary willget the game equilibriumThis method can get the evolution-ary equilibrium independent of environmental factors andnetwork structure However the competitive disseminationwe discuss in this paper is more complexThe historical gamepath the impacts of environment and the interactive usersrsquodecision should be taken into consideration comprehensivelyIn this context the equilibrium cannot be simply and solelydetermined by the traditional dynamic replication methodIn this paper we choose themethod of computational simula-tion approach so as to get a more reasonable explanation anddescription for competitive information dissemination overthe social networks

22 Utility Definition The impact of other individualsrsquobehavior has been extensively studied in social psychologyand marketing In general many early past findings (Asch[19] and Schachter [20]) suggest that individuals have atendency to behave in accordance with group or social normsand behave negatively toward opinions that deviate fromthese norms Based upon this premise Deutsch and Gerard[21] further developed the distinction between normativesocial influence (pressure to conform to the expectationsof others) and informational influence (individualrsquos accep-tance of persuasive argument(s) of others) Meanwhile otherresearches [22 23] (McQuail [22] and Flanagin and Metzger[23]) proved that people expect two types of value to engage incommunity information value and social support We referto the conclusion and define the utility function when gamecontinues as follows

(1) Self-Perceived Knowledge Model People feel that themessage they spread can help others make informed decision


of purchase Product-related information is now being over-whelmed from a constantly increasing amount of advertisingMeanwhile it can also be obtained by interpersonal sources(Feick and Price [24]) Self-perceived knowledge level helpsto reduce the risk when making adopting and purchasingdecision In this paper we define 119896

119894isin [0 1] (119894 = 119860 or 119861)

as the quantitative measure of self-perceived knowledge forthe product of brand 119894 This kind of knowledge can either begained from mass media or word-of-mouth prevalent in thesocial networkThe quantity of 1 represents full knowledge ofthe product which means the player knows everything aboutthe background of the product The quantity of 0 representsno prior knowledge of the product which means the productis totally new to the player A value of 119896 isin (0 1) representspartial background knowledge of the product which meansthe player knows something about the product (such as ldquoIknow something about the Smartphone of this brandrdquo) Wedescribe the self-perceived knowledge level for both the twobrands of play 119894 as a vector 119870

119894= (119896119860

119896119861

) The consistentconclusion can easily be reached in the real social commercecircumstance because the more self-perceived knowledgea consumer has the lower risk and cost he will pay forconsumptions

Moreover we observe that knowledge will transfer withinplayers as game continues under some situations even newknowledgewill be created In order to describe the knowledgedynamics during the game process we assume that there aretwo independent kinds of knowledge each kind correspond-ing to one of the competitive pieces of information of productFor example 119896

119894and 119896

119895are referring to different products

They interact as the following rules

(1) Interacting with a player having different kinds ofknowledge does not increase knowledge 119896

119894and 119896119895are

different kinds of knowledge(2) Interacting with a player having no knowledge

increases knowledge 119896119894and 119896

119895are the same kind of

knowledge(3) Interacting with a player having full knowledge cre-

ates full knowledge 119896119894 119896119895are the same kind of

knowledge

The combined knowledge (marked as 119896(119894119895)

) is generatedby the following equation

119896(119894119895)

=

max (119896119894 119896119895) +min (1max (119896

119894+ 119896119895))

2 (6)

We define a transferring operator oplus to describe theknowledge transferring process among players when thegame is going The transferring equation is given below

119870119894119895119905+1

= 119870119894119905

oplus 119870119895119905 (7)

The above method was observed by Szabo and Fath [5]and Yu et al [17] with a slightly different definition

(2) Brand Commitment Model In marketing applicationsof social or group influence can be found across a wide

range of contexts [25] For example scholars (Arndt [26]and Gatignon and Robertson [27]) have often relied onsocial influence as a theoretical basis for studying WOMdynamics in the adoption of new products for understandingreference-grouprsquos influence on product and brand choice(Bearden and Etzel [28]) as well as polarization phenomenonin group decision-making (Ward and Reingen [29]) Socialidentity theory explained by Ellermers et al [30] and Tajfel[31] indicates that some people care about the success of thecompany they identify with Dutton et al [32] found thatpeople always voluntarily promote the company on whichthey have a positive view and become loyal to the brandonce they develop to self-identity Brand commitment is anenduring desire to maintain a relationship with a brand Itcan be perceived as a condition inwhich consumers are firmlyenchanted with a certain brand to the extent that there is nosecond choice In other words it implies brand loyalty

Therefore if an actor has a high level of commitment to abrand he or shewill tend to keep a stable relationshipwith thebrand The actions of person with high or low commitmentwill be different when they receive a negative message talkingabout a target brand For the sake of simplicity we define119887119894 (119894=119886 or 119887) as the quantitative measure of brand commitmentof the productThis kind of commitment can bring bias whenchoosing strategy A value of 119887 isin [0 1] represents playerrsquospreference or bias on some brand of product which meansthe player will definitely choose certain brand of the productand reject others no matter which one his neighbourschoose Under some situations we can even infer that theyare spokesman or discommender of particular brand Wedescribe the brand preference level of the two products forplayer 119894 as a vector 119861

119894= (119887119860

119887119861

)

(3) The Rational Choice Model of Conformity In socialcommerce applications of social or group influence can befound across a wide range of contexts prior results found byRyu and Han [25] and Arndt [26] suggest that individualsare susceptible to social influence and that they often behavein ways that conform to social norms or pressure The sameconclusion is also observed by Gatignon and Robertson [27]and Bearden and Etzel [28] Conformity is often meant torepresent a solution to the problem and attain or maintainsocial order that requires cooperationThese studies generallyfocus onmodelling the dynamics of norms in the perspectiveof cooperation With reference to the prior research weassume that the need for social conformity is associated withpopularity Popularitymeasures player 119894rsquos social influence anddominance It is one of the components in his utility functionFor the sake of simplicity we evaluate it with the number ofhis neighbours which ismodified from themethod suggestedby Yu et al [17]

119875119894= 120572

119899119894

119899max (8)

In this equation 119899max is the maximum number of neigh-bours of all players in the network and 119899

119894represents the

number of neighbours of player 119894 120572 = [0 1] is a controlling


parameter anddescribes the probability the playerwill chooseto assimilate For example if 120572 = 1 keeping conformity withhis neighbours brings benefit to the player 120572 = 0 indicatesthat the two playersrsquo choice conflicts with each other and thetotal benefit for both two players is 0 In that case he is morereasonable when making purchase decision

(4)The Utility Function Sociologists believe that how humanbeings interpret and accept things could be summarized tomore than one way [33] The first way is personal experienceand findings another one is the established agreements orbeliefs also some people rely on the behaviours of peoplein groups for they insist that believing what others sayis a very useful quality that can make people easily getalong and contributes a lot to stable social relations Inactual social activities these three ways are often combinedHowever giving different weights on them reflects differentpersonalities of people and different characteristics of socialbehaviour We believe that the purpose of a rational actor 119894 isto maximize her utility 119880

119894 Therefore 119880

119894can be defined as a

convex combination of information contribution and socialnorm contribution with coefficients 0 le 120588 120587 119896 le 1

119880119894= 120588119870119894+ 120587119861119894+ 119896119875119894 (9)

With reference to the method and the experienced datarecommended by Zinoviev and Duong [8] we use a set ofcoefficients 120588 120587 119896 to characterize a particular type of actorsFor example 120588 = 0 120587 = 1 and 119896 = 0 describe a type ofactors who have high loyalty to a special brand or those whorepresent the benefit of certain brand We call them ldquoswornfollowersrdquo 120587 = 0 120588 = 1 and 119896 = 0 probably correspond to acommunity of actors who believe that different self-perceivedknowledge level will uniquely determine the strategy-makingprocess we call them ldquoexpertsrdquo 120587 = 0 120588 = 0 and 119896 =

1 probably correspond to a community of actors who caremore about their reputation and conformity with the wholecommunity we call them ldquoconformistsrdquo Meanwhile 119895 isthe neighbour of 119894 in the network According to the utilitydefinition we can calculate the payoff matrix in detail

3 Analysis of Model

31 Assumption of the Model Product-related information isa kind of public information that can be got from variouschannels therefore we modify public goods game model andmake the following assumptions

(1) In social network the nodes are bounded rationalWe agreed that each node only interacts with itsneighbour nodes

(2) Benefit is defined by the payoff matrix Each nodeis trying to obtain maximum benefit by enhancingthe significance of its activities to improve the socialcapital in the community

(3) The evolution process of nodes in the network is influ-enced by various social factors Each node has to con-stantly adjust their strategy imitating the behavioursof its neighbours to improve the sociability withincommunity

Strategies in the evolutionary game tend to be influencedby other nodesrsquo opinion in the public information environ-mentTherefore the strategy for node 119894 could be described asa dimensional vector 119878

119894 This kind of game is carried out over

certain kinds of social network structures node 119894 interactswith all of his neighbors in each round and its payoff (119875

119894) can

be described as

119875119894= sum

119895isinΩ119894

119904119879

119894119872119904119895 (10)

whereΩ119894is the set of neighbors of node 119894 in the network and

119872 is the payoff matrix defined in Section 2

32 Learning and Updating Each node in the networkevolution needs to constantly adjust its strategy to imitate thebehaviors of its neighbors to improve sociability referring tovarious social factors Nodes 119894 and 119895 may also change theirstrategies because they contact public socialmedia or they areinfluenced by other environment factorsWhether to keep theold strategy or change to the new one is defined by updatingrules At the end of each stage of the evolutionary game allnodes can adjust their strategy according to the features andbenefits of its neighbours This kind of imitation processesformed a wide class of microscopic update rules The essenceof the imitation is that the node who has the opportunity torevise her strategy takes over one of her neighborrsquos strategywith some probability The imitation process covers tworespects whom to imitate and with what probability

There are several common strategy update rules describedin the dynamics of evolutionary games [5] (Majority rulesBest Response Dynamic etc) Among them this paperconsiders the typical rule of imitation Imitate If Better Thestandard procedure of the rule is to choose the node to imitateat random from the neighbourhood In the mean-field casethis can be interpreted as a random partner from the wholepopulation The imitation probability may depend on theinformation available for the node The rule can be differentif only the strategies used by the neighbours are knownor if both the strategies and their resulting last-round (oraccumulated) payoffs are available for inspection Node 119894withstrategy 119878

119894takes over the strategy 119878

119895of another node 119895 which

is chosen randomly from 119894rsquos neighbourhood if 119895rsquos strategy hasyielded higher payoff Otherwise the original strategy 119878

119894is

maintained If we denote the set of neighbours of node 119894 whohold the strategy 119878

119894byΩ119894(119904119894) sube Ω

119894(Ω119894(119886) + Ω

119894(119887) = Ω

119894) The

individualrsquos strategy transition rate from strategy 119904119894to strategy

1199041015840

119894could be written as

120596 (119904119894997888rarr 1199041015840

119894) =

120582

1003816100381610038161003816Ω1198941003816100381610038161003816

sum

119895isinΩ119894(1199041015840

119894)

120579 [119880119895minus 119880119894] (11)

where 120579 is the Heaviside function 120582 gt 0 is an arbitraryconstant and |Ω

119894| is the number of neighbors Imitation

rules are more realistic if they take into consideration theactual payoff difference between the original and the imitatedstrategies Proportional imitation does not allow for aninferior strategy to replace a more successful one Updaterules which forbid this are usually called payoff monotone


However payoff monotonicity is frequently broken in caseof bounded rationality Therefore a possible general form ofimitation more realistic could be described as follows

119875119903119894rarr 119895

=1

1 + exp [(119880119894minus 119880119895) 119870]

(12)

where 119875119903119894rarr 119895

represents the probability node 119894 and imitates thestrategy of its neighbour 119895 finally 119880

119895is the benefit generated

by node 119895 and it is the maximum among those accrued by itsneighbours while 119880

119894is the accumulated benefits of node 119894

and119870 is the information noise [34] representing the rationallevel of the nodes

The smaller119870 value is the more rational of the behaviouris If 119870 = 0 then the whole process can be describedas a completely rational game in which there is no errorwhen making decision 119870 = +infin indicates that gamestrategies choosing can be depicted as a pure randomprocesswhich is completely irrational 0 lt 119870 lt +infin indicates alimited rational game process namely the gamers adjust theirstrategies according to rules but there is a certain error

4 Computational Simulations

In this section we design the computational system simulatethe virtual community and evolutionary process collectnumerical results under different scenarios and make com-parisons through statistical analysis The simulation andexperiments are implemented by NetworksX 17 NetworkX isa graph theory and complex network modelling tool devel-oped with Python language containing built-in algorithmsfigure and complex network analysis modules which can beconveniently imported and executed for complex networkdata analysis simulation modelling and so forth

41 Formalize the Model Definition Let each node in thevirtual network be an agent and there are 119899 agents We definethe community network119873 as119873 = 119883NT 119880 119865 119905 in which

(1) 119883 is the set of agents and 119883 = agent1 agent2 agent119899 Each agent is a node in the network

(2) NT is the set of network types and NT =

smallword random scalefree(3) 119880 is the set of benefits derived by all agents in each

game round it is the accumulated value while gamingwith all its neighbours 119880 = 119880

1 1198802 119880

119899

(4) 119865 is the state transfer function The state of agent 119894 attime 119905 + 1 is a function of parameters as stated in (12)

42 Experimental System and the Default Parameters In ourstudy a node in the network is either in a cooperation(strategy 119886) or defection (strategy 119887) state We describe thediffusion of information for product 119860 (information 119860 forsimplicity in the later paper) in the network as

diffusion of information 119860 ()

=sum of the nodes holding strategy 119886

total number of nodeslowast 100

(13)

And we describe the diffusion of information for product119861 (information 119861 for simplicity) as




(14)

In the model we simulate individual strategy evolutionThe evolutions of the information spreading and acceptingconditions are studied trying to find out how to promoteor hinder certain preference of strategy from winning Weconduct several experiments on the competitive informationdissemination in a social network There are some famil-iar and natural models of social networks which couldbe applied in these experiments random graph scale-freenetwork and small-world network We applied the latter twodifferent networks for case study A scale-free network is arandom graphwhose degree distribution follows a power lawwhile a small-world network follows a random graph modelinwhichmost nodes can be reached fromevery other node bya small number of hops which is generally known as small-world phenomenon Many empirical networks are well-modelled by small-world networks such as social networkswikis and gene networks

We explore the interaction among all individuals overtime (100 simulation rounds in our case) The simulation isdivided into five procedures

(1) Set up the network structure define the game param-eters and the model parameters

(2) Graphically illustrate the network evolution and showhow the nodes change dynamically in the network

(3) In each step each participating individual updates thestrategy according to the rules of replication dynamicview (strategy)

(4) In each simulation time period take the evolutionarysteps in the periodic system that all the individuals ora fixed percentage of them reach or maintain a stablestate as the equilibrium of the evolution

(5) Display the simulation result and show the character-istics of the network

43 Experimental Results Thepopulation of our experimentsis set to 1000 The small-world network applied for ourexperiments is set as follows the number of connections pernode is set to 5 and the probability of connection to linkneighbours per node is set to 02 For the scale-free networkthe number of edges being added to the network each stepis set as 5 119896

119886 119896119887 119887119886 and 119887

119887are initialized using uniform

distribution on [0 1) throughout the experiment with slightdifference under different scenarios

(1) Results for Different Network Structures At the start ofthe experiments the two pieces of product information hasbeen equally adopted 119896

119886 119896119887 119887119886 and 119887

119887are initialized using

uniform distribution with the condition that 119860 dominate indissemination during gamingWe assume that the maximum


Diffusion of ADiffusion of B

Diff

usio

n of

info

rmat

ion

Gaming steps

11

10

09

08

07

06

05

04

03

02

01

00

minus01

minus020 10 20 30 40

(a) Information diffusions in small-world networks


Diff

usio

n of

info

rmat

ion

Gaming steps

11

10

09

08

07

06

05

04

03

02

01

00

minus01

minus020 10 20 30 40

(b) Information diffusions in scale-free networks

Figure 1 of adoption for competitive product-related information under different network structure

value of 119896119886is larger than the one of 119896

119887 for simplicity to

guarantee the condition 119896119886and 119896119887are initialized on [02 07]

and [0 02) and vary following the knowledge dynamic rulesdefined in Section 2 119887

119886and 119887119887follow uniform distribution on

the interval [0 1) 120587 = 03 120588 = 03 119896 = 04 and 119870 = 08 Theevolutionary steps in each round are set to 50

Following the conditions of this scenario we find thedominant strategy reaches the whole population or reaches astable dominant winning status (the equilibrium) in differentgaming steps over different network structure (see Figure 1)In the scale-free network the dynamic can reach a stablestatus more easily

Compared to the small-world networks in scale-freenetwork the majority of individuals have only one high-degree neighbour There will be more references for him toevaluate his neighboursrsquo payoff providing him with a higherprobability to successfully imitate a better strategy Then theneighbours of him are apt to imitate his strategy because ofhis high-degree status in the social networks After severalgame rounds the individual game is changed to group gamewith several high-degree players as the core nodes It isobvious that the number of groups is much less than thenumber of nodes who participate in the game thus the Scale-free networks can reach equilibrium faster than small-worldmodel and meanwhile shows higher sensitivity assumed by alarger amplitude around the equilibrium

(2) Results for Different Personalities Our analysis continueswith how the spreading of competitive information differsaccording to the weight of acceptance change that affectsagentsrsquo utility function In the prior model we classify thecases into different personalities

(a) The Influence of Individual Personality Differences onGame Equilibrium We use a particular set of coefficients

120588 120587 119896 to characterize different types of agents At the start ofexperiments 119896

119860 119896119861 119887119860 and 119887


distribution on the internal [0 1) and they evolve accordingto the knowledge dynamic rules described in Section 2Referring to the empirical data [17] (Yu et al 2012) 119870 =

001 We ran the experiment in three different cases whichcover population with different personalities In the first case(Figure 2) the agents have high desire for knowledge and lowdesire for brand loyalty and reputation we set the parameters(120587 120588 119896) distinguishing personalities as (08 01 01) whichwe called ldquoexpertsrdquo In the second case the agents have lowexpectation for reputation and knowledge and strong desirefor loyalty to a certain brand we set the parameter (120587 120588 119896)as (01 08 01) and we called them ldquosworn followersrdquo In thelast case the agents have great desire for social conformityand influence we set the parameter (120587 120588 119896) as (01 01 08)and call them ldquoconformistrdquo

In this experiment the gaming steps are set to 50 ineach simulation round and the initial of information 119860

is set to 80 to get a more definite comparison We carryout experiments in 100 simulating rounds totally for the twodifferent kinds of network structure

Comparing experimental results under three differentscenarios a phenomenon could be observed in commonthat the average steps before reaching the equilibrium inscale-free network is generally less than that in the small-world network which get the consistent conclusion withexperiment (a) The convergence of dynamic in scale-freenetwork is more obvious with slight fluctuation while thedissemination in the small-world network distribute morescattered

As one can see from the figures in all scenarios thewhole network rapidly converges to a stable distribution Inthe network of ldquoexpertsrdquo the average full convergence takesmore time than in the network of ldquosworn followersrdquo and


Equi

libriu

m ti

me

The small-world networksThe scale-free networks

49

44

39

34

29

24

19

14

9

4

minus1

Stimulation roundminus5 5 15 25 35 45 55 65 75 85 95

(a) Information diffusions among ldquoExpertsrdquo

Equi

libriu

m ti

me


49

44

39

34

29

24

19

14

9

4

minus1


(b) Information diffusions among ldquoSworn followersrdquo

Equi

libriu

m ti

me


49

44

39

34

29

24

19

14

9

4

minus1


(c) Information diffusions among ldquoConformistsrdquo

Figure 2 Different convergence times due to different settings of (120587 120588 119896) 119896119860 119896119861 119887119860 and 119887

119861are initialized on [0 1) The values of (120587 120588 119896)

are from (a) to (c) (08 01 01) (01 08 01) and (01 01 01) representing experts sworn followers and conformists

ldquoconformistrdquo Also the ldquoexpertrdquo network always has someunsubstantial numbers of people doubting the diffused factsperformance in the sharp fluctuations Generally speakinginformation dissemination will proceed to the stable statemost quickly in the scale-free network with ldquoconformistrdquo inthe majority

Local heterogeneity behaviour preference can explainthe phenomenon which could be sharply reduced over thenetwork of ldquosworn followersrdquo and ldquoconformistrdquo A compatibleopinion or preference could be more easily reached beingexplained by fewer steps before the equilibriumduring evolu-tion Each agent considers all the strategies their neighbourstake and keep consistent with the overall opinion easilyHowever when ldquoexpertsrdquo take a strategy accidental shadows

have more chance to happen which increase the uncertaintyof spreading result

(b)The Influence of Preference Benefit on InformationDissemi-nation Large proportion of a particular personality dominantin the population indicates a certain psychological tendencyin social networks For example the psychology of ldquoFirstImpressionrdquo leads to a majority of ldquosworn followersrdquo to aspecial brand or product The psychology of ldquoHerd effectrdquoleads to plenty of ldquoconformistsrdquo Meanwhile the psychologyof ldquoExperientialismrdquo makes most agents in the network morerational and specialistic

In addition a piece of information has its own life cycleexperiencing the process of production diffusion decay and


Game steps = 20

Game steps = 50

Game steps = 100

Preference of Aminus01 00 01 02 03 04 05 06 07 08 09 10 11

minus01

00

01

02

03

04

05

06

07

08

09

10

11

The p

roba

bilit

y of

Aw

inni

ng

(a) Probability for winning of information 119860 with yield change of 119860rsquospreference benefit

Game steps = 20

Game steps = 50

Game steps = 100

The a

vera

ge d

iffus

ion

rate

ofA

Preference of A

49

44

39

34

29

24

19

14

9

4

minus1

minus01 00 01 02 03 04 05 06 07 08 09 10 11

(b) Average diffusion rate of information 119860 with yield change of 119860rsquospreference benefit

Figure 3 Under different evolutionary time the diffusion states of information 119860 with yield change of 119860rsquos preference benefit (relative to 119861)The values of 119887

119860and 119887119861are well-distributed on [0 1)with step size 005 119896

119860 119896119861are initialized using uniform distribution on the internal [0 1)

death which cannot be absolutely fixed Therefore eachround of evolutionary game cannot continue without timelimitingThus we simulate the winning probability of certaininformation for evolutionary process in limited time periodstrying to reveal the dissemination of competitive informationfrom a more practical perspective

As is shown in Figure 3 the gaming steps are the gamingsteps are respectively set to 20 50 and 100 the initial ofinformation 119860 is set to relatively low rate as 1 Taking thesmall-world network under the psychological state of ldquoFirstImpressionrdquo for instance we carry out experiments in 100rounds for different evolution time trying to observe thechange on winning probability and diffusion rate of informa-tion 119860 due to different information preference benefits

The winning probability of 119860 generally increases with itspreference benefit and the evolution time When the benefitis lower than 01 comparing to 119861 there is no advantage for 119860to win There is a sharp rising between the preference values01 and 02 After that the probability fluctuates at a relativelyhigh level Following the rule of ldquoImitate If Betterrdquo a verysmall group of initial adoption can get a high win probabilityThe diffusion rate of 119860 also shows an ascending trend alongwith the preference benefit increasing when the benefit islower than 02 Then the average diffusion rate keeps stableMeanwhile it can be concluded that information diffusebecomes lower after 50 game steps which means in mostrounds of simulation the minority of population for infor-mation 119860 will quickly occupy the whole population becauseof the enough preference benefit and knowledge dynamic

(3) Results with Different Initial Settings We now discuss theimpact of the initial of information settings The initial

percentage of people adopting information119860 is set at 01 05and 08 respectively 119896

119886and 119896

119887are initialized on [02 07]

and [0 02) varied by the knowledge dynamic rules definedin Section 2 119887

119886and 119887

119887follow uniform distribution on the

interval [0 1) 120587 = 03 120588 = 03 119896 = 04 and 119870 = 08 Theevolution steps in each round are set to 50 The simulationround is set to 100 and the value gained at each step iscalculated to the average

The results indicate that the initial support of informationwill also have a major impact on the diffusion process Ifinformation 119860 in the initial state of the system has a higheradoptive rating (119901 gt 05) the chance of information 119860 winswill be further enhanced As to the lower initial adoptiverating of information119860 (119901 lt 05) the networkwill takemuchmore time to reach an equilibrium as shown in Figure 4 Inthe network structure of small-world information 119860 will failto win when its initial adoptive rating is set to 01 Wheninformation 119860rsquos initial adoptive rating is increasing thediffusion rate of 119860 raises sharply indicating that information119860 is more likely to reach the whole population Under thestructure of scale-free network the phase change process ismuch smoother It can be inferred that scale-free networksprovide more heterogeneous topology structure in whichinformation 119860 has a higher probability of winning Thusscale-free networks could be seen as a factor which decreaseswith the winning odds of preferred information for a certainproduct

(4) Results with Incomplete Information We finally examinethe impact of information noise on the probability of 119860rsquosadoption and disseminationThe information noise levels areset at 119896 = 001 10 and 10000 respectively


p = 10p = 50p = 80

Diff

usio

n of

A

Time

1100

1000

900

800

700

600

500

400

300

200

100

0

minus100minus5 0 5 10 15 20 25 30 35 40

(a) Diffusion of information 119860 under different initial setting in small-world networks

p = 10p = 50p = 80

Diff

usio

n of

A

Time

1100

1000

900

800

700

600

500

400

300

200

100

0

minus100minus5 0 5 10 15 20 25 30 35 40

(b) Diffusion of information 119860 under different initial setting in scale-free networks

Figure 4 Diffusion of information under different initial setting through different network structure The value of 119901 is 01 05 and 08

The initial percentage of people adopting information 119860is set at 01 to make a more significant comparison 119896

119886and

119896119887are initialized on [02 07] and [0 02) and vary following

the knowledge dynamic rules defined in Section 2 119887119886and 119887119887

follow uniform distribution on the interval [0 1) 120587 = 03120588 = 03 and 119896 = 04 The evolution steps in each simulationtime period are set to 50 The simulation round is set to 100and in each simulation period the mean value is referred

The results in Figure 5 reveal that individualrsquos irrationalselection of strategies will also affect the results of dissemi-nation The smaller 119870rsquos value is the more rational behaviourcan be observed The preferred information will take theadvantage more easily over the network It indicates that fora limited rational game process namely the game playersadjust their strategies according to certain rules the evolutionwill reach an equilibrium under the circumstance of lowknowledge noiseTherefore in the process of irrational gameit is hard for a certain piece of information to win moreaudience than the other even though the former one hasan obvious advantage of initial share The reason is that theincreasing of noise brings uncertainty of strategy evolutionwhen ldquoImitate If Betterrdquo updating rule plays a role It canalso easily reach the conclusion again that scale-free networksstructure could weaken the winning probability of preferredinformation

5 Conclusion and Future Work

An evolutionary game model based on stochastic strategyupdating dynamic for competitive product-related informa-tion dissemination in social network is presented in thispaper Several important implications can be drawn from thework

Firstly compared with the Small-world networks Scale-free networks are a kind of better structure for informationdissemination In the case that the overall average knowledgelevel is relatively high the advantageous information in scale-free networksmore easily defeat the others because of the lessdifference between individuals in this network structure

Secondly different virtual communities (with differ-ent network structures) should adopt different macrolevelmanagerial strategies for instance customized rewardingand penalty mechanisms Different incentives and penaltiescan influence userrsquos choice of product-related informationspreading decision and affect the performance of informationsharing

Furthermore results of the competitive information dis-semination are affected by the initial self-perceived knowl-edge distribution of products and brand preference amongindividuals which means that in order to make moreindividuals prefer a brand than the other the company shouldprovide more information to indirectly promote individualsrsquoacceptance of the product for example through the massmedia

Finally different personalities and mentalities in thevirtual community lead to different dissemination processof competitive product-related information Individualsrsquo irra-tional selection of strategies also affects the results of dissem-ination

Other models as we know including SIR Model MarkovModel and Random Petri-Net Model only explain theinformation spreading in individual behaviour level Theycould not preferably reflect the individual spreading behaviorevolution in the context of time change network structureand interaction with others Game theory is powerful tool ininformation dissemination behavior description Our work


K = 001

K = 10

K = 10000

Prop

ortio

n of

AAd

optio

n11

09

07

05

03

01

minus01

Time1 6 11 16 21 26 31 36 41 46

(a) The trend of change for information 119860 adoption in evolution oversmall-world networks

K = 001

K = 10

K = 10000

Prop

ortio

n of

AAd

optio

n

11

09

07

05

03

01

minus01

Time1 6 11 16 21 26 31 36 41 46

(b) The trend of change for information 119860 adoption in evolution overscale-free networks

Figure 5 The impact of information noise on the of information adoption in evolutionary process the value of 119870 is 001 10 and 1000

reveals several interesting conclusions as to the nature ofcompetitive information dissemination employing an agent-based simulationmethod to explore the evolution of informa-tion adoption and dissemination behavior in social networksWe hope that this paper would provide some new insightsinto the research of competitive information

This paper has some limitations and we will improve ourmodel from the aspects as follows

(1) We assume that the network structure is fixed anddid not consider the evolution of the network struc-tures themselves including not only homogeneousnetwork but also heterogeneous network containingseveral different types of players

(2) Theutility parameters are endogenous to some extentTo evaluate how the endogeneity impacts the person-ality and to what extent it will influence the choice ofinformation adoption is a problem we will discuss inthe future work

(3) Large number of real data should be used to supportthe model as a supplement of the simulation

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported by the National Natural ScienceFoundation of China (NSFC 71271012 and NSFC 71332003)

References

[1] A Montanaria and A Saberi ldquoThe spread of innovationsin social networksrdquo Proceedings of the National Academy ofSciences of the United States of America vol 107 no 47 pp20196ndash20201 2010

[2] D Gruhl R Guha D L Nowell and A Tomkins ldquoInformationdiffusion through blogspacerdquo in Proceedings of the 13th Interna-tional Conference onWorldWideWeb (WWW rsquo04) pp 491ndash5012004

[3] J Yang and J Leskovec ldquoModeling information diffusion inimplicit networksrdquo in Proceedings of the 10th IEEE InternationalConference on Data Mining (ICDM rsquo10) pp 599ndash608 IEEESydney Australia December 2010

[4] L Lu D-B Chen andT Zhou ldquoThe small world yields themosteffective information spreadingrdquoNew Journal of Physics vol 13Article ID 123005 10 pages 2011

[5] G Szabo and G Fath ldquoEvolutionary games on graphsrdquo PhysicsReports A vol 446 no 4ndash6 pp 97ndash216 2007

[6] J Kostka Y A Oswald and R Wattenhofer ldquoWord of mouthrumor dissemination in social networksrdquo in Structural Informa-tion and Communication Complexity vol 5058 of Lecture Notesin Computer Science pp 185ndash196 Springer Berlin Germany2008

[7] D Meier Y A Oswald S Schmid and R Wattenhofer ldquoOnthe windfall of friendship inoculation strategies on social net-worksrdquo in Proceedings of the 9th ACM Conference on ElectronicCommerce (EC rsquo08) pp 294ndash301 July 2008

[8] D Zinoviev and V Duong ldquoA game theoretical approachto broadcast information diffusion in social networksrdquo inProceedings of the 44th Annual Simulation Symposium pp 47ndash52 2011

[9] D Zinoviev V Duong and H G Zhang ldquoA game theoreticalapproach to modeling information dissemination in socialnetworksrdquo in Proceedings of the SUMMER 4th International


Conference on Knowledge Generation Communication andManagement vol 1 pp 407ndash412 2010

[10] G Jiang F Ma J Shang and P Y Chau ldquoEvolution ofknowledge sharing behavior in social commerce an agent-based computational approachrdquo Information Sciences vol 278pp 250ndash266 2014

[11] N Alon M Feldman A D Procaccia and M TennenholtzldquoA note on competitive diffusion through social networksrdquoInformation Processing Letters vol 110 no 6 pp 221ndash225 2010

[12] Y Wang J Li K Meng C Lin and X Cheng ldquoModelingand security analysis of enterprise network using attack-defensestochastic game Petri netsrdquo Security and Communication Net-works vol 6 no 1 pp 89ndash99 2013

[13] R Takehara M Hachimori and M Shigeno ldquoA comment onpure-strategy Nash equilibria in competitive diffusion gamesrdquoInformation Processing Letters vol 112 no 3 pp 59ndash60 2012

[14] J R Davis Z Goldman E N Koch et al ldquoEquilibria andefficiency loss in games on networksrdquo InternetMathematics vol7 no 3 pp 178ndash205 2011

[15] A Galeotti S Goyal M O Jackson F Vega-Redondo and LYariv ldquoNetwork gamesrdquoTheReview of Economic Studies vol 77no 1 pp 218ndash244 2010

[16] G Jiang B Hu and Y Wang ldquoAgent-based simulation ofcompetitive and collaborative mechanisms for mobile servicechainsrdquo Information Sciences vol 180 no 2 pp 225ndash240 2010

[17] J Yu Y Wang J Li H Shen and X Cheng ldquoAnalysisof competitive information dissemination in social networkbased on evolutionary game modelrdquo in Proceedings of the 2ndInternational Conference on Cloud and Green Computing (CGCrsquo12) pp 748ndash753 Xiangtan China November 2012

[18] J Hofbauer and K Sigmund ldquoEvolutionary game dynamicsrdquoBulletin of the AmericanMathematical Society vol 40 no 4 pp479ndash519 2003

[19] S E Asch ldquoOpinions and social pressurerdquo Scientific Americanvol 193 no 5 pp 31ndash35 1955

[20] S Schachter ldquoDeviation rejection and communicationrdquo Jour-nal of Abnormal and Social Psychology vol 46 no 2 pp 190ndash207 1951

[21] M Deutsch and H B Gerard ldquoA study of normative andinformational social influences upon individual judgmentrdquoJournal of Abnormal and Social Psychology vol 51 no 3 pp629ndash636 1955

[22] D McQuail Mass Communication Theory An IntroductionSage Publications Thousand Oaks Calif USA 2nd edition1987

[23] A J Flanagin and M J Metzger ldquoInternet use in the contem-porary media environmentrdquo Human Communication Researchvol 27 no 1 pp 153ndash181 2001

[24] L F Feick and L L Price ldquoThe market maven a diffuser ofmarketplace informationrdquo Journal of Marketing vol 51 pp 83ndash97 1987

[25] G Ryu and J K Han ldquoWord-of-mouth transmission in settingswith multiple opinions the impact of other opinions on WOMlikelihood and valencerdquo Journal of Consumer Psychology vol 19no 3 pp 403ndash415 2009

[26] J Arndt ldquoRole of product-related conversations in the diffusionof a new productrdquo Journal of Marketing Research vol 4 no 3pp 291ndash295 1967

[27] H Gatignon and T S Robertson ldquoA propositional inventory fordiffusion researchrdquo Journal of Consumer Research vol 11 pp849ndash867 1985

[28] W O Bearden and M J Etzel ldquoReference group influences onproduct and brand purchase decisionsrdquo Journal of ConsumerResearch vol 9 no 2 pp 183ndash194 1982

[29] J C Ward and P H Reingen ldquoSociocognitive analysis of groupdecision making among consumersrdquo Journal of ConsumerResearch vol 17 pp 245ndash265 1990

[30] N Ellermers R Spears and B Doosje Social Identity Black-well Oxford UK 1999

[31] H E Tajfel Differentiation between Social Groups Studies inthe Social Psychology of Intergroup Relations Academic PressOxford UK 1978

[32] J E Dutton JMDukerich andCVHarquail ldquoOrganizationalimages and member identificationrdquo Administrative ScienceQuarterly vol 39 no 2 pp 239ndash263 1994

[33] E R Babbie The Practice of Social Research WadsworthPublishing Co Inc 11th edition 2007

[34] W-B Du X-B Cao M-B Hu H-X Yang and H ZhouldquoEffects of expectation and noise on evolutionary gamesrdquoPhysica A Statistical Mechanics and its Applications vol 388no 11 pp 2215ndash2220 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


to the description of information diffusion through socialnetworks

In recent years researchers gradually observe that gamebehaviors between individuals in social network being thefoundation of the social group behavior raise social dif-fusions which were always discussed in the microlevel inprior related research Micronodes in network constantlyadjust their strategy according to the behavior of other nodesthrough the dynamic game with other nodes to maximizetheir own interests and all the behaviours of individual nodefinally form the social network group behavior Thereforegame-theoretic models as a new perspective of interpretingsocial diffusion are increasingly adopted by computer sci-entists for analyzing network behaviors Game theory is aset of analytical tools and solution concepts which providesexplanatory and predicting power in interactive decision sit-uations when the aims goals and preferences of the partici-pating players are potentially in conflict [5] Unlike the classicgame theory which assumes that gaming exists betweentwo individuals and happens only one-off the evolutionarygame theory has opened up related hypothesis limitationsIt introduces the concept of time and space considering thegame as the summary of historical experience and individuallearning which makes it advantageous for social networkstudy Several papers have applied evolutionary game theorymethod to explain social diffusion Kostka et al [6] examinedthe dissemination of competing rumors in social networkusing concepts of game theory and location theorymodellingthe selection of starting nodes for the rumors as a strategygame Meier et al [7] presented a virus propagation gamemodel finding that the Windfall of Friendship does notincrease monotonically with stronger relationships Zinovievet al [8 9] adopted game theoretic models to understandhuman aspects of information dissemination in which per-sonalities of individuals are considered Jiang et al [10]examined the evolutionary process of knowledge sharingamong users in social network and designed a computationalexperimental system developing a mixed learning algorithmbased on individualrsquos historical game strategy neighborsrsquostrategy and information noise

Applying evolutionary game theory in social diffusionstudy presumes that the game process between individualsis not only determined by a single rival or individual butalso all individuals in the neighborhood Meanwhile it takesthe historical game experience for future game behavior intoconsideration The benefit of each individual is evaluated bythe accumulated result of gaming many times Individualsbenefit from the game with its neighbors at the same timethe benefit can be observed with comparisons to adjusttheir game strategies trying to achieve optimum benefitsand reach the overall game equilibrium Evolutionary gametheory also has advantages on competitive social behaviorsand social diffusions Alon et al [11] introduced a game-theoretic model of competitive dissemination of technolo-gies advertisements or influence through a social networkWang et al [12] proposed the stochastic game net model foranalyzing competitive network behaviors Takehara et al [13]introduced and studied a deterministicmodel for competitiveinformation diffusion on social networks In contrast tomany

other game theoretic models for the diffusion of informationand innovation [14 15] the model considered competitionbetween different innovations spreading instead of discussinga single one

This paper is an extension of recent works includingthe method suggested by Jiang et al [16] and the resultsdescribed by Yu et al [17] The dissemination of informationis modeled as the dynamic of coordination game in whichplayerrsquos payoff is defined by a utility function and several casesare analyzed to reach the conclusion that the spreading rate isinfluenced by characteristics of individuals and several otherfactors Different from themodels of competitive informationdiffusion introduced in prior work we adopt a frameworkfor describing competitive information dissemination basedon a game-theoretic model and multiagent-based dynamicsIn this framework we target several similar product-relatedpieces of information that competes with each other Con-sumers may participate in discussing adopting and spread-ing one of them Totally different from news or opinionsdissemination that is always described as the spreading of thevirus consumers always make product-related decision forthe utility motivation In general the utility is dynamicallydetermined by factors from the environment or from hispersonal reasons Meanwhile prior research has shown thatdifferent network structures topology has great impact onsocial commerce Therefore both sociological and psycho-logical characteristics are explicitly considered in our modelas the novelty In thismodel information passing intrinsicallyinvolves both sides considering their characteristics self-perceived knowledge brand loyalty and social conformitywhich further determine their decisions of whether or notto forward the information The decisions are also based onthe global properties of the network such as the knowledgedynamic all through the network These factors finally bringdifferent results of disseminations of different informationthat compete with each other Based on the dynamic gamemodel and strategy updating rules we analyze competitiveinformation propagation and the affecting factors

The remainder of this paper is organized as followsSection 2 presents the overview of competitive disseminationmodel based on evolutionary game theory social networkand multiagent theory and introduces the individualrsquos utilityfunction Section 3 discusses the dynamics and updatingrules for the model we build proposes the assumptionsand explains the model In Section 4 we use an agent-basedcomputational approach for simulation

2 Problem Statement

We propose a different global point of view regarding theincentives that govern the diffusion process Suppose we haveseveral firms that would like to advertise competitive prod-ucts via ldquoviral marketingrdquo Each firm initially targets a smallsubset of users in the hope that the social advertisementsabout their product would spread throughout the networkHowever a user that participates in discussing and spreadingproduct-related information is reluctant to participate inanother one



Player 119894119886 119887

Player 119895119886 119880

119886119886 119880119886119886

119880119886119887 119880119887119886

119887 119880119887119886 119880119886119887

119880119887119887 119880119887119887





119878119894= (

1

0) or (

0

1) (1)






119880119894(119862) = 119901 lowast 119880

119886119886+ (1 minus 119901) lowast 119880

119886119887

119880119894(119863) = 119901 lowast 119880

119887119886+ (1 minus 119901) lowast 119880

119887119887

(2)


119894(119862) and 119880



whole group is

119880 = 119901 lowast 119880119894(119862) + (1 minus 119901)119880

119894(119863) (3)


119865 (119901)

=119889119901

119889119905= 119901 (119880

119894(119862) minus 119880)


119886119887minus 119880119887119887)]

(4)


119901 = 0 1119880119886119887minus 119880119887119887

119880119886119886minus 119880119886119887minus 119880119887119886+ 119880119887119887

119901lowast

=119880119886119887minus 119880119887119887

119880119886119886minus 119880119886119887minus 119880119887119886+ 119880119887119887

(5)






119894isin [0 1] (119894 = 119860 or 119861)


119894= (119896119860

119896119861



119894and 119896




119894and 119896119895are






knowledge



119896(119894119895)

=

max (119896119894 119896119895) +min (1max (119896

119894+ 119896119895))

2 (6)


119870119894119895119905+1

= 119870119894119905

oplus 119870119895119905 (7)





119894= (119887119860

119887119861

)


119875119894= 120572

119899119894

119899max (8)







119894 Therefore 119880



119880119894= 120588119870119894+ 120587119861119894+ 119896119875119894 (9)



3 Analysis of Model








119894) can

be described as

119875119894= sum

119895isinΩ119894

119904119879

119894119872119904119895 (10)








119894is


119894byΩ119894(119904119894) sube Ω

119894(Ω119894(119886) + Ω

119894(119887) = Ω

119894) The


1199041015840


120596 (119904119894997888rarr 1199041015840

119894) =

120582

1003816100381610038161003816Ω1198941003816100381610038161003816

sum

119895isinΩ119894(1199041015840

119894)

120579 [119880119895minus 119880119894] (11)






119875119903119894rarr 119895

=1

1 + exp [(119880119894minus 119880119895) 119870]

(12)

where 119875119903119894rarr 119895














1 1198802 119880

119899






(13)





(14)









119886 119896119887 119887119886 and 119887




119886 119896119887 119887119886 and 119887





Diff

usio

n of

info

rmat

ion

Gaming steps

11

10

09

08

07

06

05

04

03

02

01

00

minus01

minus020 10 20 30 40



Diff

usio

n of

info

rmat

ion

Gaming steps

11

10

09

08

07

06

05

04

03

02

01

00

minus01

minus020 10 20 30 40














119860 119896119861 119887119860 and 119887









Equi

libriu

m ti

me


49

44

39

34

29

24

19

14

9

4

minus1



Equi

libriu

m ti

me


49

44

39

34

29

24

19

14

9

4

minus1



Equi

libriu

m ti

me


49

44

39

34

29

24

19

14

9

4

minus1












Game steps = 20

Game steps = 50

Game steps = 100


minus01

00

01

02

03

04

05

06

07

08

09

10

11

The p

roba

bilit

y of

Aw

inni

ng


Game steps = 20

Game steps = 50

Game steps = 100

The a

vera

ge d

iffus

ion

rate

ofA

Preference of A

49

44

39

34

29

24

19

14

9

4

minus1

minus01 00 01 02 03 04 05 06 07 08 09 10 11










119886and 119896



119886and 119887






p = 10p = 50p = 80

Diff

usio

n of

A

Time

1100

1000

900

800

700

600

500

400

300

200

100

0

minus100minus5 0 5 10 15 20 25 30 35 40


p = 10p = 50p = 80

Diff

usio

n of

A

Time

1100

1000

900

800

700

600

500

400

300

200

100

0

minus100minus5 0 5 10 15 20 25 30 35 40




119886and













K = 001

K = 10

K = 10000

Prop

ortio

n of

AAd

optio

n11

09

07

05

03

01

minus01

Time1 6 11 16 21 26 31 36 41 46


K = 001

K = 10

K = 10000

Prop

ortio

n of

AAd

optio

n

11

09

07

05

03

01

minus01

Time1 6 11 16 21 26 31 36 41 46










Acknowledgment


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Player 119894119886 119887

Player 119895119886 119880

119886119886 119880119886119886

119880119886119887 119880119887119886

119887 119880119887119886 119880119886119887

119880119887119887 119880119887119887





119878119894= (

1

0) or (

0

1) (1)






119880119894(119862) = 119901 lowast 119880

119886119886+ (1 minus 119901) lowast 119880

119886119887

119880119894(119863) = 119901 lowast 119880

119887119886+ (1 minus 119901) lowast 119880

119887119887

(2)


119894(119862) and 119880



whole group is

119880 = 119901 lowast 119880119894(119862) + (1 minus 119901)119880

119894(119863) (3)


119865 (119901)

=119889119901

119889119905= 119901 (119880

119894(119862) minus 119880)


119886119887minus 119880119887119887)]

(4)


119901 = 0 1119880119886119887minus 119880119887119887

119880119886119886minus 119880119886119887minus 119880119887119886+ 119880119887119887

119901lowast

=119880119886119887minus 119880119887119887

119880119886119886minus 119880119886119887minus 119880119887119886+ 119880119887119887

(5)






119894isin [0 1] (119894 = 119860 or 119861)


119894= (119896119860

119896119861



119894and 119896




119894and 119896119895are






knowledge



119896(119894119895)

=

max (119896119894 119896119895) +min (1max (119896

119894+ 119896119895))

2 (6)


119870119894119895119905+1

= 119870119894119905

oplus 119870119895119905 (7)





119894= (119887119860

119887119861

)


119875119894= 120572

119899119894

119899max (8)







119894 Therefore 119880



119880119894= 120588119870119894+ 120587119861119894+ 119896119875119894 (9)



3 Analysis of Model








119894) can

be described as

119875119894= sum

119895isinΩ119894

119904119879

119894119872119904119895 (10)








119894is


119894byΩ119894(119904119894) sube Ω

119894(Ω119894(119886) + Ω

119894(119887) = Ω

119894) The


1199041015840


120596 (119904119894997888rarr 1199041015840

119894) =

120582

1003816100381610038161003816Ω1198941003816100381610038161003816

sum

119895isinΩ119894(1199041015840

119894)

120579 [119880119895minus 119880119894] (11)






119875119903119894rarr 119895

=1

1 + exp [(119880119894minus 119880119895) 119870]

(12)

where 119875119903119894rarr 119895














1 1198802 119880

119899






(13)





(14)









119886 119896119887 119887119886 and 119887




119886 119896119887 119887119886 and 119887





Diff

usio

n of

info

rmat

ion

Gaming steps

11

10

09

08

07

06

05

04

03

02

01

00

minus01

minus020 10 20 30 40



Diff

usio

n of

info

rmat

ion

Gaming steps

11

10

09

08

07

06

05

04

03

02

01

00

minus01

minus020 10 20 30 40














119860 119896119861 119887119860 and 119887









Equi

libriu

m ti

me


49

44

39

34

29

24

19

14

9

4

minus1



Equi

libriu

m ti

me


49

44

39

34

29

24

19

14

9

4

minus1



Equi

libriu

m ti

me


49

44

39

34

29

24

19

14

9

4

minus1












Game steps = 20

Game steps = 50

Game steps = 100


minus01

00

01

02

03

04

05

06

07

08

09

10

11

The p

roba

bilit

y of

Aw

inni

ng


Game steps = 20

Game steps = 50

Game steps = 100

The a

vera

ge d

iffus

ion

rate

ofA

Preference of A

49

44

39

34

29

24

19

14

9

4

minus1

minus01 00 01 02 03 04 05 06 07 08 09 10 11










119886and 119896



119886and 119887






p = 10p = 50p = 80

Diff

usio

n of

A

Time

1100

1000

900

800

700

600

500

400

300

200

100

0

minus100minus5 0 5 10 15 20 25 30 35 40


p = 10p = 50p = 80

Diff

usio

n of

A

Time

1100

1000

900

800

700

600

500

400

300

200

100

0

minus100minus5 0 5 10 15 20 25 30 35 40




119886and













K = 001

K = 10

K = 10000

Prop

ortio

n of

AAd

optio

n11

09

07

05

03

01

minus01

Time1 6 11 16 21 26 31 36 41 46


K = 001

K = 10

K = 10000

Prop

ortio

n of

AAd

optio

n

11

09

07

05

03

01

minus01

Time1 6 11 16 21 26 31 36 41 46










Acknowledgment


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119894isin [0 1] (119894 = 119860 or 119861)


119894= (119896119860

119896119861



119894and 119896




119894and 119896119895are






knowledge



119896(119894119895)

=

max (119896119894 119896119895) +min (1max (119896

119894+ 119896119895))

2 (6)


119870119894119895119905+1

= 119870119894119905

oplus 119870119895119905 (7)





119894= (119887119860

119887119861

)


119875119894= 120572

119899119894

119899max (8)







119894 Therefore 119880



119880119894= 120588119870119894+ 120587119861119894+ 119896119875119894 (9)



3 Analysis of Model








119894) can

be described as

119875119894= sum

119895isinΩ119894

119904119879

119894119872119904119895 (10)








119894is


119894byΩ119894(119904119894) sube Ω

119894(Ω119894(119886) + Ω

119894(119887) = Ω

119894) The


1199041015840


120596 (119904119894997888rarr 1199041015840

119894) =

120582

1003816100381610038161003816Ω1198941003816100381610038161003816

sum

119895isinΩ119894(1199041015840

119894)

120579 [119880119895minus 119880119894] (11)






119875119903119894rarr 119895

=1

1 + exp [(119880119894minus 119880119895) 119870]

(12)

where 119875119903119894rarr 119895














1 1198802 119880

119899






(13)





(14)









119886 119896119887 119887119886 and 119887




119886 119896119887 119887119886 and 119887





Diff

usio

n of

info

rmat

ion

Gaming steps

11

10

09

08

07

06

05

04

03

02

01

00

minus01

minus020 10 20 30 40



Diff

usio

n of

info

rmat

ion

Gaming steps

11

10

09

08

07

06

05

04

03

02

01

00

minus01

minus020 10 20 30 40














119860 119896119861 119887119860 and 119887









Equi

libriu

m ti

me


49

44

39

34

29

24

19

14

9

4

minus1



Equi

libriu

m ti

me


49

44

39

34

29

24

19

14

9

4

minus1



Equi

libriu

m ti

me


49

44

39

34

29

24

19

14

9

4

minus1












Game steps = 20

Game steps = 50

Game steps = 100


minus01

00

01

02

03

04

05

06

07

08

09

10

11

The p

roba

bilit

y of

Aw

inni

ng


Game steps = 20

Game steps = 50

Game steps = 100

The a

vera

ge d

iffus

ion

rate

ofA

Preference of A

49

44

39

34

29

24

19

14

9

4

minus1

minus01 00 01 02 03 04 05 06 07 08 09 10 11










119886and 119896



119886and 119887






p = 10p = 50p = 80

Diff

usio

n of

A

Time

1100

1000

900

800

700

600

500

400

300

200

100

0

minus100minus5 0 5 10 15 20 25 30 35 40


p = 10p = 50p = 80

Diff

usio

n of

A

Time

1100

1000

900

800

700

600

500

400

300

200

100

0

minus100minus5 0 5 10 15 20 25 30 35 40




119886and













K = 001

K = 10

K = 10000

Prop

ortio

n of

AAd

optio

n11

09

07

05

03

01

minus01

Time1 6 11 16 21 26 31 36 41 46


K = 001

K = 10

K = 10000

Prop

ortio

n of

AAd

optio

n

11

09

07

05

03

01

minus01

Time1 6 11 16 21 26 31 36 41 46










Acknowledgment


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119894 Therefore 119880



119880119894= 120588119870119894+ 120587119861119894+ 119896119875119894 (9)



3 Analysis of Model








119894) can

be described as

119875119894= sum

119895isinΩ119894

119904119879

119894119872119904119895 (10)








119894is


119894byΩ119894(119904119894) sube Ω

119894(Ω119894(119886) + Ω

119894(119887) = Ω

119894) The


1199041015840


120596 (119904119894997888rarr 1199041015840

119894) =

120582

1003816100381610038161003816Ω1198941003816100381610038161003816

sum

119895isinΩ119894(1199041015840

119894)

120579 [119880119895minus 119880119894] (11)






119875119903119894rarr 119895

=1

1 + exp [(119880119894minus 119880119895) 119870]

(12)

where 119875119903119894rarr 119895














1 1198802 119880

119899






(13)





(14)









119886 119896119887 119887119886 and 119887




119886 119896119887 119887119886 and 119887





Diff

usio

n of

info

rmat

ion

Gaming steps

11

10

09

08

07

06

05

04

03

02

01

00

minus01

minus020 10 20 30 40



Diff

usio

n of

info

rmat

ion

Gaming steps

11

10

09

08

07

06

05

04

03

02

01

00

minus01

minus020 10 20 30 40














119860 119896119861 119887119860 and 119887









Equi

libriu

m ti

me


49

44

39

34

29

24

19

14

9

4

minus1



Equi

libriu

m ti

me


49

44

39

34

29

24

19

14

9

4

minus1



Equi

libriu

m ti

me


49

44

39

34

29

24

19

14

9

4

minus1












Game steps = 20

Game steps = 50

Game steps = 100


minus01

00

01

02

03

04

05

06

07

08

09

10

11

The p

roba

bilit

y of

Aw

inni

ng


Game steps = 20

Game steps = 50

Game steps = 100

The a

vera

ge d

iffus

ion

rate

ofA

Preference of A

49

44

39

34

29

24

19

14

9

4

minus1

minus01 00 01 02 03 04 05 06 07 08 09 10 11










119886and 119896



119886and 119887






p = 10p = 50p = 80

Diff

usio

n of

A

Time

1100

1000

900

800

700

600

500

400

300

200

100

0

minus100minus5 0 5 10 15 20 25 30 35 40


p = 10p = 50p = 80

Diff

usio

n of

A

Time

1100

1000

900

800

700

600

500

400

300

200

100

0

minus100minus5 0 5 10 15 20 25 30 35 40




119886and













K = 001

K = 10

K = 10000

Prop

ortio

n of

AAd

optio

n11

09

07

05

03

01

minus01

Time1 6 11 16 21 26 31 36 41 46


K = 001

K = 10

K = 10000

Prop

ortio

n of

AAd

optio

n

11

09

07

05

03

01

minus01

Time1 6 11 16 21 26 31 36 41 46










Acknowledgment


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119875119903119894rarr 119895

=1

1 + exp [(119880119894minus 119880119895) 119870]

(12)

where 119875119903119894rarr 119895














1 1198802 119880

119899






(13)





(14)









119886 119896119887 119887119886 and 119887




119886 119896119887 119887119886 and 119887





Diff

usio

n of

info

rmat

ion

Gaming steps

11

10

09

08

07

06

05

04

03

02

01

00

minus01

minus020 10 20 30 40



Diff

usio

n of

info

rmat

ion

Gaming steps

11

10

09

08

07

06

05

04

03

02

01

00

minus01

minus020 10 20 30 40














119860 119896119861 119887119860 and 119887









Equi

libriu

m ti

me


49

44

39

34

29

24

19

14

9

4

minus1



Equi

libriu

m ti

me


49

44

39

34

29

24

19

14

9

4

minus1



Equi

libriu

m ti

me


49

44

39

34

29

24

19

14

9

4

minus1












Game steps = 20

Game steps = 50

Game steps = 100


minus01

00

01

02

03

04

05

06

07

08

09

10

11

The p

roba

bilit

y of

Aw

inni

ng


Game steps = 20

Game steps = 50

Game steps = 100

The a

vera

ge d

iffus

ion

rate

ofA

Preference of A

49

44

39

34

29

24

19

14

9

4

minus1

minus01 00 01 02 03 04 05 06 07 08 09 10 11










119886and 119896



119886and 119887






p = 10p = 50p = 80

Diff

usio

n of

A

Time

1100

1000

900

800

700

600

500

400

300

200

100

0

minus100minus5 0 5 10 15 20 25 30 35 40


p = 10p = 50p = 80

Diff

usio

n of

A

Time

1100

1000

900

800

700

600

500

400

300

200

100

0

minus100minus5 0 5 10 15 20 25 30 35 40




119886and













K = 001

K = 10

K = 10000

Prop

ortio

n of

AAd

optio

n11

09

07

05

03

01

minus01

Time1 6 11 16 21 26 31 36 41 46


K = 001

K = 10

K = 10000

Prop

ortio

n of

AAd

optio

n

11

09

07

05

03

01

minus01

Time1 6 11 16 21 26 31 36 41 46










Acknowledgment


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Diff

usio

n of

info

rmat

ion

Gaming steps

11

10

09

08

07

06

05

04

03

02

01

00

minus01

minus020 10 20 30 40



Diff

usio

n of

info

rmat

ion

Gaming steps

11

10

09

08

07

06

05

04

03

02

01

00

minus01

minus020 10 20 30 40














119860 119896119861 119887119860 and 119887









Equi

libriu

m ti

me


49

44

39

34

29

24

19

14

9

4

minus1



Equi

libriu

m ti

me


49

44

39

34

29

24

19

14

9

4

minus1



Equi

libriu

m ti

me


49

44

39

34

29

24

19

14

9

4

minus1












Game steps = 20

Game steps = 50

Game steps = 100


minus01

00

01

02

03

04

05

06

07

08

09

10

11

The p

roba

bilit

y of

Aw

inni

ng


Game steps = 20

Game steps = 50

Game steps = 100

The a

vera

ge d

iffus

ion

rate

ofA

Preference of A

49

44

39

34

29

24

19

14

9

4

minus1

minus01 00 01 02 03 04 05 06 07 08 09 10 11










119886and 119896



119886and 119887






p = 10p = 50p = 80

Diff

usio

n of

A

Time

1100

1000

900

800

700

600

500

400

300

200

100

0

minus100minus5 0 5 10 15 20 25 30 35 40


p = 10p = 50p = 80

Diff

usio

n of

A

Time

1100

1000

900

800

700

600

500

400

300

200

100

0

minus100minus5 0 5 10 15 20 25 30 35 40




119886and













K = 001

K = 10

K = 10000

Prop

ortio

n of

AAd

optio

n11

09

07

05

03

01

minus01

Time1 6 11 16 21 26 31 36 41 46


K = 001

K = 10

K = 10000

Prop

ortio

n of

AAd

optio

n

11

09

07

05

03

01

minus01

Time1 6 11 16 21 26 31 36 41 46










Acknowledgment


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Equi

libriu

m ti

me


49

44

39

34

29

24

19

14

9

4

minus1



Equi

libriu

m ti

me


49

44

39

34

29

24

19

14

9

4

minus1



Equi

libriu

m ti

me


49

44

39

34

29

24

19

14

9

4

minus1












Game steps = 20

Game steps = 50

Game steps = 100


minus01

00

01

02

03

04

05

06

07

08

09

10

11

The p

roba

bilit

y of

Aw

inni

ng


Game steps = 20

Game steps = 50

Game steps = 100

The a

vera

ge d

iffus

ion

rate

ofA

Preference of A

49

44

39

34

29

24

19

14

9

4

minus1

minus01 00 01 02 03 04 05 06 07 08 09 10 11










119886and 119896



119886and 119887






p = 10p = 50p = 80

Diff

usio

n of

A

Time

1100

1000

900

800

700

600

500

400

300

200

100

0

minus100minus5 0 5 10 15 20 25 30 35 40


p = 10p = 50p = 80

Diff

usio

n of

A

Time

1100

1000

900

800

700

600

500

400

300

200

100

0

minus100minus5 0 5 10 15 20 25 30 35 40




119886and













K = 001

K = 10

K = 10000

Prop

ortio

n of

AAd

optio

n11

09

07

05

03

01

minus01

Time1 6 11 16 21 26 31 36 41 46


K = 001

K = 10

K = 10000

Prop

ortio

n of

AAd

optio

n

11

09

07

05

03

01

minus01

Time1 6 11 16 21 26 31 36 41 46










Acknowledgment


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Game steps = 20

Game steps = 50

Game steps = 100


minus01

00

01

02

03

04

05

06

07

08

09

10

11

The p

roba

bilit

y of

Aw

inni

ng


Game steps = 20

Game steps = 50

Game steps = 100

The a

vera

ge d

iffus

ion

rate

ofA

Preference of A

49

44

39

34

29

24

19

14

9

4

minus1

minus01 00 01 02 03 04 05 06 07 08 09 10 11










119886and 119896



119886and 119887






p = 10p = 50p = 80

Diff

usio

n of

A

Time

1100

1000

900

800

700

600

500

400

300

200

100

0

minus100minus5 0 5 10 15 20 25 30 35 40


p = 10p = 50p = 80

Diff

usio

n of

A

Time

1100

1000

900

800

700

600

500

400

300

200

100

0

minus100minus5 0 5 10 15 20 25 30 35 40




119886and













K = 001

K = 10

K = 10000

Prop

ortio

n of

AAd

optio

n11

09

07

05

03

01

minus01

Time1 6 11 16 21 26 31 36 41 46


K = 001

K = 10

K = 10000

Prop

ortio

n of

AAd

optio

n

11

09

07

05

03

01

minus01

Time1 6 11 16 21 26 31 36 41 46










Acknowledgment


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










p = 10p = 50p = 80

Diff

usio

n of

A

Time

1100

1000

900

800

700

600

500

400

300

200

100

0

minus100minus5 0 5 10 15 20 25 30 35 40


p = 10p = 50p = 80

Diff

usio

n of

A

Time

1100

1000

900

800

700

600

500

400

300

200

100

0

minus100minus5 0 5 10 15 20 25 30 35 40




119886and













K = 001

K = 10

K = 10000

Prop

ortio

n of

AAd

optio

n11

09

07

05

03

01

minus01

Time1 6 11 16 21 26 31 36 41 46


K = 001

K = 10

K = 10000

Prop

ortio

n of

AAd

optio

n

11

09

07

05

03

01

minus01

Time1 6 11 16 21 26 31 36 41 46










Acknowledgment


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










K = 001

K = 10

K = 10000

Prop

ortio

n of

AAd

optio

n11

09

07

05

03

01

minus01

Time1 6 11 16 21 26 31 36 41 46


K = 001

K = 10

K = 10000

Prop

ortio

n of

AAd

optio

n

11

09

07

05

03

01

minus01

Time1 6 11 16 21 26 31 36 41 46










Acknowledgment


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Evolutionary Game Analysis of Competitive