Preference reasoning and aggregation: between AI and social choice

+

Preference reasoning and aggregation: between AI and social choice

Francesca Rossi University of Padova, Italy

+My wordle

PRICAI 2012 - Kuching, Malaysia

+Outline

n  Preferences

n  Collective decision making in multi-agent systems

n  Social choice

n  Computational social choice (CSS)

n  Some specific issues in CSS n  Computational concerns

n  Intractable manipulation n  Two preference formalisms

n  soft constraints, CP-nets n  Sequential voting n  Preference elicitation


+Why preferences?


¡  An intelligent system must be able to handle soft information l  different levels of preference or rejection l  several levels of tolerance l  vagueness l  imprecision

¡  Information may be non-crisp l  intrinsically: the world is not binary l  due to information which is only partially

available

+Preferences

n  Ubiquitous in real life n  I prefer Venice to Rome

n  A more tolerant way to set some constraints over the possible scenarios n  I prefer a blue car

n  Constraints can be used when we know what to accept or reject n  I don’t want to spend more than X

n  If all constraints, possibly n  no solution, or n  too many of them, all apparently equally good

n  Some problems are naturally modelled with preferences n  I don’t like meat, and I prefer fish to cheese

n  Constraints and preferences may be present in the same problem n  Configuration, timetabling, etc.



Example: University timetabling Professor Administra/on

I cannot teach on Wednesday afternoon. I prefer not to teach early in the morning, nor on Friday afternoon.

Lab C can fit only 120 students.

Better to not leave 1-hour holes in the day schedule.

Constraints

Constraints

Preferences

Preferences

+Several kinds of preferences


n Positive (degrees of acceptance) n  I like ice cream

n Negative (degrees of rejection) n  I don’t like strawberries

n Unconditional n  I prefer taking the bus

n Conditional n  I prefer taking the bus if it’s raining

n Multi-agent n  I like blue, my husband likes green, what color do we

buy the car?

+Two main ways to model preferences


n  Quantitative n  Numbers, or ordered set of objects

n  My preference for ice cream is 0.8, and for cake is 0.6 n  E.g., soft constraints

n  Qualitative n  Pairwise comparisons

n  Ice cream is better than cake n  E.g., CP-nets

n  Both very natural in some scenarios

n  Different expressive power

n  Different computational complexity for reasoning with them

+Desiderata for an AI preference framework


¡  Expressive power ¡  Compactness ¡  Efficiency ¡  Suitability for multi-agent settings

+Preferences for collective decision making in multi-agent systems

n Several agents

n Common set of possible decisions

n Each agent has its preferences over the possible decisions

n Goal: to choose one of the decisions, based on the preferences of the agents n Also a set of decisions, or a ranking over the decisions

n AI scenarios add: imprecision, uncertainty, complexity, etc.


+Example

n  Three friends need to decide what to cook for dinner

n 4 items (pasta, main, dessert, drink), 5 options for each

n Each friend has his/her own preferences over the meals

Agents = friends

Decisions = all possible dinners


+Another example:

n  Several time slots under consideration

n  Partecipants accepts or reject each time slot n  Very simple way to express preferences over time

slots n  Very little information communicated to the system

n  Collective choice: a single time slot n  The one with most acceptance votes from

participants


+Collective decision scenarios

n  IT enabled social environments n  People are connected all the time n  Social networks allow us to share a large amount of

information n  More and more, we want to exploit this information to

take collective decisions n  With our friends, colleagues, etc.

n  Also committees of agents n  Search engines n  Solvers n  Classifiers n  Product ranking agents, etc.


+How to compute a collective decision?

n  Let the agents vote by expressing their preferences over the possible decisions

n  Aggregate the votes to get a single decision

n  Let’s look at voting theory then!


+Voting theory (Social choice)

n  Voters

n  Candidates

n  Each voter expresses its preferences over the candidates

n  Goal: to choose one candidate (the winner), based on the voters’ preferences n  Also many candidates, or ranking

n  Rules (functions) to achieve the goal

n  Properties of the rules

n  Impossibility results


+ Some voting rules

n  Plurality n  Voting: one most preferred decision n  Selection: the decision preferred by the largest number of agents

n  Majority: like plurality, over 2 options

n  Borda (m options) n  Voting: rank over all options, m-i score of ith option n  Selection: option with greatest sum of scores

n  Approval (m options) n  Voting: approval of between 1 and m-1 options n  Selection: option with most votes n  This is what Doodle uses

n  Copeland n  Voting: ranking over all options n  Selection: option which wins most pairwise competitions

n  Cup n  Voting: ranking over all options n  Selection: winner in the agenda (tree of pairwise competitions)


+Plurality

n Ballot: 1 alternative

n Result: alternative(s) with the most vote(s)

n Example: n  6 voters

n  Candidates:

Profile Ballot Winner

+Borda

1 voter 1 voter 1 voter 1 voters 1 voter

3

2

1

0

rank

3

2

1

0

rank

3

2

1

0

rank

3

2

1

0

rank

3

2

1

0

rank

9

8

7

6

Borda Count

Winner

+Some desirable properties (1)

n  Condorcet-consistency n  If there is an option who beats every other option in pairwise

competitions, it is selected

n  Anonymity n  Result does not depend on who are the agents

n  Neutrality n  Result does not depend on which are the options

n  Monotonicity n  If an agent improves his preference for the winning option, this

option is still selected

n  Consistency n  If two sets of agents have the same result, this result is also

obtained by joining the two sets

n  Participation n  Given an agent, its addition to a profile leads to an equally or

more preferred result for this agent

n  Unanimity (efficiency) n  If all agents have the same top choice, it is selected


+ Some desirable properties(2)

n Non-dictatorship n There is no voter such that his top choice always

wins, regardless of the votes of other voters

n  Independence to Irrelevant Alternatives n  If choice X wins given a profile p, then Y cannot

win in any profile where the relation between X and Y is as in p

n Strategy-proofness n There is no profile and no agent such that the

agent would be better off lying


+Different properties for different voting rules

n  For the voting rules in our list: n  All are anonymous, neutral, non-dictatorial, and manipulable

n  All but Cup are efficient

n  Cup and Copeland are Condorcet-consistent

n  All but Cup and Copeland are consistent and participative

n  Only Approval is IIA


+Two classical impossibility results

n  Arrow’s theorem n  it is impossible to have unanimity + IIA + non-dictatoriality

n  Bad news, but IIA is a very strong property

n  Gibbard-Sattherwaite’s theorem n  it is impossible to have surjectivity + non-dictatoriality + strategy-

proofness

n  This are really bad news: we don’t want to give up surjectivity, nor non-dictatoriality!


+

Social choice scenarios vs. multi-agent systems

+Is social choice all we need for collective decision making in multi-agent systems?

After all …

n Voters = agents

n Candidates = decisions

n Preferences

n Winner = chosen decision

But …


+Main differences

n In multi-agent AI scenarios, we usually have n  Large sets of candidates (w.r.t. number of voters)

n  Combinatorial structure for candidate set

n  Knowledge representation formalisms to model preferences

n  Incomparability

n  Uncertainty, vagueness

n  Computational concerns


+Large set of candidates

n In AI scenarios, usually the set of decisions is much larger than the set of agents expressing preferences over the decisions n Many web pages, few search engines n Many solutions of a constraint problem, few

solvers


+Combinatorial structure for the set of decisions

n Combinatorial structure for the set of decisions

n Car (or PC, or camera) = several features, each with some instances

n Dinner example: n  Three friends need to decide what to cook for dinner n  4 items (pasta, main, dessert, drink) n  5 options for each è 54 = 625 possible dinners

n  In general: Cartesian product of several variable domains n  Variables = items of the menu, domain= 5 options


+Formalisms to model preferences compactly

n  Preference ordering over a large set of decisions è need to model them compactly n  Otherwise too much space and time to handle such preferences

n  Two examples: n  Soft constraints

n  CP-nets


+Incomparability

n Preferences do not always induce a total order

n Some items are naturally incomparable n Not because the information is missing

n  It depends also on the combinatorial structure (multi-criteria and Pareto dominance)

n To model uncertainty

n As a means to resolve conflicts

n Many AI formalisms to model preferences allow for partial orders


+Uncertainty, vagueness

n  Missing preferences n  Too costly to compute them

n  Privacy concerns

n  Ongoing elicitation process

n  Imprecise preferences n  Preferences coming from sensor data

n  Too costly to compute the exact preference

n  Estimates


+Computational concerns

n  We would like to avoid very costly ways to n  Model the preferences

n  Compute the winner

n  Reason with the agents’ preferences

n  On the other hand,we need a computational barrier against bad behaviours (such as manipulation)


+Computational social choice

n  Between multi-agent systems and social choice n  AI, economics, mathematics, political science, etc.

n  New concerns n  Preference modelling

n  Algorithms, complexity

n  Uncertainty, preference elicitation

n  Cross-fertilization in both directions


+

Computational concerns

+Computational concerns about voting rules

n  We want to avoid spending too much time to n  Elicit preferences

n  Compute the winner (winners, ranking)

n  On the other hand, we want a computational barrier against manipulation n  Given the impossibility result, we want to avoid rules which are

computationally easy to manipulate


+Manipulation

n A rule is manipulable if an agent can gain by lying about its preferences n Gain = obtain a result which is more preferred by

the agent n Strategy-proof rule: there is no incentive to

misrepresent the preferences

n Gibbard-Sattherwaite impossibility result n  With at least three agents and two candidates, it is

impossible for a voting rule to be at the same time surjective, strategy-proof, and non- dictatorial

n  We cannot give up non-dictatoriality and surjectivity!


+Intractable manipulation

n If manipulation is computationally intractable for F, then F might be considered resistant (albeit still not immune) to manipulation

n Most interesting for voting procedures for which winner determination is tractable n  complexity gap between manipulation (undesired

behaviour) and winner determination (desired functionality)


+Manipulability as a decision problem

Manipulability(F)

Instance: Set of ballots for all but one voter; alternative x.

Question: Is there a ballot for the final voter such that x wins?

How difficult it is to answer this question?

n  A manipulator would have to solve Manipulability(F) for all alternatives, in its preference ordering, to understand if there is a way he can get something better

n  If Manipulability(F) is computationally intractable, then manipulability may be considered less of a worry for procedure F.

n  Remark: We assume that the manipulator knows all the other ballots n  This unrealistic assumption is reasonable for intractability results: if manipulation is intractable

even under such favorable conditions, then all the better.


+Plurality is easy to manipulate

Manipulability(Plurality)ε P

n  Simply vote for x, the alternative to be made winner by means of manipulation. If manipulation is possible at all, this will work. Otherwise not.

n  In general: Manipulability(F) ε P for any rule F with polynomial winner determination and polynomial number of ballots.

PRICAI 2012 - Kuching, Malaysia [Bartholdi,Tovey,Trick,1989]

+ Manipulating Borda is easy

MANIPULABILITY(Borda) ε P

n  Place x (the alternative to be made winner through manipulation) at the top of your ballot.

n  Then inductively proceed as follows: Check if any of the remaining alternatives can be put next on the ballot without preventing x from winning. If yes, do so. If no, manipulation is impossible.


[Bartholdi,Tovey,Trick,1989]

+ Manipulating STV is difficult

n  MANIPULABILITY(STV) ε NP-complete

n  NP-membership is clear: checking whether a given ballot makes x win can be done in polynomial time (just try it, STV is polynomial to compute).

n  NP-hardness: by reduction from another NP-complete problem (3-Cover). The basic idea is to build a large election instance introducing all sorts of constraints on the ballot of the manipulator, such that finding a ballot meeting those constraints solves a given instance of 3-Cover.

PRICAI 2012 - Kuching, Malaysia [Bartholdi,Orlin, 1991]

+ Coalitional constructive manipulation

n  Manipulation by a coalition of agents

n  Constructive: to make some candidate win (not just to get a better result)


+ Weighted-coalitional constructive manipulation


+ Destructive manipulation

n  The goal is to make sure that a candidate does not win

(whoever else wins)‏

n  If constructive manipulation is easy then so is destructive manipulation

n Destructive manipulation can be easy even though constructive manipulation is hard

n Reverse does not hold n  E.g. Borda is polynomial to manipulate desctructively but NP-

hard constructively for 3 or more candidates for a weighted coalition


+Weighted-coalitional destructive manipulation


+

Formalisms to model preferences

Two examples: •  Soft constraints •  CP-nets

+ Modelling preferences compactly


n  Preference ordering: an ordering over the whole set of solutions (or candidates, or outcomes, …)

n  Solution space with a combinatorial structure è preferences over partial assignments of the decision variables, from which to generate the preference ordering over the solution space

+


n Variables {X1,…,Xn}=X

n Domains {D(X1),…,D(Xn)}=D

n Soft constraints n  each constraint involves some of the variables n  a preference is associated with each assignment of the

variables

n Set of preferences A n  Totally or partially ordered (induced by +)

n  a ≤ b iff a+b=b n  Combination operator (x) n  Top and bottom element (1, 0) n  Formally defined by a c-semiring <A,+,x,0,1>

Soft Constraints (the c-semiring framework)

+


Soft Constraints n  Soft constraint: a pair c=<f,con> where:

n  Scope: con={Xc1,…, Xc

k} subset of X n  Preference function : f: D(Xc

1)x…xD(Xck) → A

tuple (v1,…, vk) → p preference

n  Hard constraint: a soft constraint where for each tuple (v1,…, vk) f (v1,…, vk)=1 the tuple is allowed f (v1,…, vk)=0 the tuple is forbidden


Example: fuzzy constraints

Lunch time= 13 Meal = carne Wine = bianco Swimming time= 14

Decision A

pref(A)=min(0.3,0)=0

Lunch time = 12 Meal = pesce Wine = bianco Swimming time = 14

Decision B

pref(B)=min(1,1)=1

{12, 13} {14, 15}

Lunch /me

Swimming /me

(12, 15) à 1

(12, 14) à 1 (13, 14) à 0

(13, 15) à 1

{fish, meat} {white, red}

meal wine

(fish, red) à 0.8

(fish, white) à 1 (meat,white) à 0.3

(meat, red) à 0.7

Preference of a decision: minimal preference of its parts Aim: to find a decision with maximal preference Preference values: between 0 and 1

+


Soft Constraints n  A soft CSP induces an ordering over the solutions, from the ordering of the preference set

n  Totally ordered semiring è total order over solutions (possibly with ties)

n  Partially ordered semiring è total or partial order over solutions (possibly with ties)

n  Any ordering can be obtained!


Qualitative and conditional preferences

n Soft constraints model quantitatively unconditional preferences

n Many problems need statements like n “I like white wine if there is fish”

(conditional) n “I like white wine better than red wine”

(qualitative) n Quantitative è a level of preference for each

assignment of the variables in a soft constraint è possibly difficult to elicitate preferences from user

+Solution ordering

Main course Wine

fish white > red

meat red > white


fish>meat

peaches > strawberries

Main course

Fruit

Wine

Fish, white, peaches

Fish, red, peaches Fish, white, berries

Fish, red, berries meat, red, peaches

meat, red, berries meat, white, peaches

meat, white, berries

Op/mal solu/on

Soft constraints vs. CP-nets

all all some some difficult easy difficult easy

easy easy difficult difficult

difficult difficult for weighted, easy for fuzzy

difficult easy

difficult easy easy easy


Preference orderings

Find an optimal decision

Compare two decisions

Find the next best Decision

Check if a decision

is optimal

Soft CSPs Tree-like soft CSPS

CP-nets Acyclic CP-nets

+

Aggregating combinatorially-structured preferences

+Multiple issues n  Example:

n  3 referendum (yes/no)

n  Each voter has to give his preferences over triples of yes and no

n  Such as: YYY>NNN>YNY>YNN>etc.

n  With k issues, k-tuples (2k of such tuples if binary issues)

n  Not every voting rule will work well

n  Example: 13 voters, 3 binary issues: n  3 voters each vote for YNN, NYN, NNY

n  1 voter votes for YYY, YYN, YNY, NYY

n  No voter votes for NNN

n  If we use majority on each issue: the winner is NNN! n  Each issue has 7 out of 13 votes for N

+Maybe Plurality on tuples?

n  Ask each voter for her most preferred combination and apply the Plurality rule n  Avoids the paradox, computationally light

n  But … almost random decisions

n  Example: 10 binary issues, 20 voters è 210 = 1024 combinations to vote for but only 20 voters, so very high probability that no combination receives more than one vote è tie-breaking rule decides everything

n  Similar also for voting rules that use only a small part of the voters’ preferences (ex.: k-approval with small k)

+Other voting rules on tuples

n  Vote on combinations and use other voting rules that use the whole preference ordering on combinations

n  Avoids the arbitrariness problem of plurality

n  Not feasible when there are large domains

n  Example: n  Borda (needs the whole preference ordering)

n  6 binary issues è 26=64 possible combinations è each voter has to choose among 64! possible ballots

+Sequential voting

n  Main idea: Vote separately on each issue, but do so sequentially

n  This gives voters the opportunity to make their vote for one issue depend on the decisions on previous issues

+Sequential voting and Condorcet losers

n  Condorcet loser (CL): candidate that loses against any other candidate in a pairwise contest n  Plurality may choose a Condorcet loser

n  Thm.: Sequential plurality voting over binary issues never elects a Condorcet loser n  Proof: Consider the election for the final issue. The winning

combination cannot be a CL, since it wins at least against the other combination that was still possible after the penultimate election

n  [Lacy, Niou, J. of Theoretical Politics, 2000]

n  But no guarantee that sequential voting elects the Condorcet winner (Condorcet consistency).

+

A sequential approach to aggregate combinatorially-structured preferences

+

Agent 1 Agent 2 Agent 3

Dinner example, three agents, fuzzy constraints

Pasta

Drink

Pesto 1 Tom 0.7

(Pesto, Beer) 1 (Pesto,Wine) 0.5 (Tom ,Beer) 0.7 (Tom,Wine) 0.3

Beer 1 Wine 0.7

Pasta

Drink

Pesto 0.9 Tom 1


Beer 1 Wine 1

Pasta

Drink

Pesto 1 Tom 0.3

(Pesto, Beer) 1 (Pesto,Wine) 0.3 (Tom ,Beer) 0.3 (Tom,Wine) 1

Beer 1 Wine 1

+The sequential voting approach

n  For each variable n  compute an explicit profile over the variable domain n  apply a voting rule to this explicit profile n  add the information about the selected variable value

n  Similar approach used for CP-nets in [Lang, Xia, 2009]


+

Agent 1 Agent 2 Agent 3

Dinner example using plurality


Pasta

Drink

Pesto 1 Tom 0.7


Beer 1 Wine 0.7

Pasta

Drink

Pesto 0.9 Tom 1


Beer 1 Wine 1

Pasta

Drink

Pesto 1 Tom 0.3

(Pesto, Beer) 1 (Pesto,Wine) 0.3 (Tom ,Beer) 0.3 (Tom,Wine) 1

Beer 1 Wine 1

Plurality Pasta = Pesto

Pesto 1 Tom 0

Pesto 0.9 Tom 0

Pesto 1 Tom 0

(Pesto, Beer) 1 (Pesto,Wine) 0.5 (Tom ,Beer) 0 (Tom,Wine) 0



Beer 1 Wine 0.5

Beer 1 Wine 0.9

Beer 1 Wine 0.3

Plurality Drink = Beer

Winner

+Local vs. sequential properties

n  If each ri has the property, does the sequential rule have the property?

n  If some ri does not have the property, does the sequential rule not have it? n  If the sequential rule has a property, do all the ri have it?


Properties

Local to sequential Sequential to local

Condorcet consistency no yes

Anonymity yes yes

Neutrality no yes

Consistency yes yes

Participation no yes

Efficiency yes if single most preferred option for all agents

yes

Monotonicity yes yes

IIA no yes

Non-dictatorship yes yes

Strategy-proofness no yes


+The sequential approach behaves like the non-sequential one

n  Independently of the variable ordering

n  Independently of the amount of consensus among agents

n  Also on best and worst cases

n  … in our experimental setting


[Dalla Pozza, Pini, Rossi, Venable, IJCAI 2011] [Dalla Pozza, Rossi, Venable, ICAART 2011]

+

Sequential Voting with CP-nets

+Profiles via compatible CP-nets

n  n voters, voting by giving a CP-net each n  Same variables, different dependency graph and CP tables

n  Compatible CP-nets: there exists a linear order on the variables that is compatible with the dependency graph of all CP-nets (that is, it completes the DAG)

n  Then vote sequentially in this order

n  Thm.: Under these assumptions, sequential voting is Condorcet consistent if all local voting rules are n  (Lang and Xia, Math. Social Sciences, 2009)

ROVER 1 ROVER 2 ROVER 3

Example

WHERE

WHAT

Loc-A >Loc-B

WHERE

WHAT

Loc-B> Loc-A

St2>St1

WHERE

WHAT

Loc-A >Loc-B

Plurality WHERE =

Loc-A

Loc-A Loc-A Loc-A

Plurality

WHAT =

Image

Winner

3 Rovers must decide: •  Where to go: Location A or Location B •  What to do: Analyze a rock or Take a picture •  Which station to downlink the data to: Station 1 or Station 2

Image >Analyze

DLINK

St1 >St2

DLINK DLINK

St2>St1

Loc-A: Image > Analyze Loc-B: Analyze> Image

Plurality

DLINK = St2

Analyze >Image Image >Analyze

Loc-A: Analyze> Image Loc-B: Image> Analyze

+

Bribing CP-nets

+Bribery when voting with CP-nets

n Agents express their preferences via CP-nets

n External agent with a desired candidate p and a budget B

n Briber can ask voters to change their preferences

n Voters charge the briber

n Cost scheme describing the cost of bribing each voter n  Fixed cost (C-equal), number of flips (C-flip), number of flips with any cost per flip (C-any), flips of

more important vars count more (C-level), distance from current optimal and new optimal (C-dist)

Can the briber make p win by spending within budget B? How difficult it is for the briber to know this and understand what to do?


+Complexity results for bribery with CP-nets

Sequential Majority

Sequential Majority with weights

Plurality Veto K-Approval (IV)

Plurality Veto K-Approval* (DV, IV+DV)

C_EQUAL NP-complete NP-complete P P

C_FLIP P NP-complete P P

C_LEVEL P NP-complete P ?

C_ANY P NP-complete ? ?

C_DIST ? NP-complete P P

[Mattei, Rossi, Venable, Pini, AAMAS 2012]

+

Preference elicitation

Preference elicitation

n Some preferences may be missing

n Time consuming, costly, difficult, to elicit them all

n Want to terminate elicitation as soon as winner fixed


+Possible and necessary winners

n  Necessary winner n  However remaining votes are cast, he must win

n  Possible winner n  There is a way for remaining votes to be cast so that he wins

n  Closely connected to manipulation n  A is possible winner iff there is a constructive manipulation for A n  A is a necessary winner iff there is no destructive manipulation for A

n  Closely connected to preference elicitation n  Elicitation can only be terminated iff possible winners = necessary winner n  “Deciding elicitation is over” is in P => computing possible (and necessary)

winners is also in P


[Konczak and Lang, 2005] [Walsh, 2008]

+ Computing possible and necessary winners n  Consider specific voting rules n  Unweighted votes n  Arbitrary number of candidates

n For STV, computing possible winners is NP-hard, and necessary winners is coNP-hard

n Even NP-hard to approximate set of possible winners within constant factor in size

n Easy for many other rules


[Pini, Rossi, Venable, Walsh, IJCAI 2007] [Pini, Rossi, Venable, Walsh, AAMAS 2011] [Lang, Pini, Rossi, Salvagnin, Venable, Walsh, JAAMAS 2011] [Pini, Rossi, Venable, Walsh, AIJ 2011] [

+Manipulation can be allowed: iterative voting n  Once agents have voted, maybe some would want to change

their vote because the collective decision is not acceptable to them

n  Doodle allows for that

n  A form of legal manipulation

n  But we want to make sure this process converges n  At some point, all agent must be either satisfied with the result or

must be unable to change the result

n  Define manipulation moves (how to modify one agent’s vote) to assure convergence n  Plurality and veto converge, Borda does not

n  On going work for Copleand, Cup, Maximin PRICAI 2012 - Kuching, Malaysia

+Conclusions

n  Brief introduction to computational social choice: n  Between multi-agent systems and social choice

n  AI, economics, mathematics, etc.

n  New concerns

n  Preference modelling

n  Algorithms, complexity

n  Uncertainty, preference elicitation

n  Cross-fertilization in both directions


+If you want to know more

“A short introduction to preferences: between Artificial Intelligence and Social Choice”

F. Rossi, K. B. Venable, T. Walsh

Morgan & Claypool

2011

http://www.morganclaypool.com/


+


Waiting for all of you in Beijing next year for IJCAI 2013!

Technology

Preference reasoning and aggregation: between AI and social choice