Truth-Revealing Social Choice

1. Truth-Revealing Social Choice Lirong Xia ADT-15 Tutorial

2. 2011 UK Referendum • Member of Parliament election: Plurality rule  Alternative vote? • 68% No vs. 32% Yes

3. Ordinal Preference Aggregation: Social Choice A profile > > A B C Alice social choice mechanism > > A C A B Bob > > C B A Carol

4. Ranking pictures [PGM+ AAAI-12] . . . . . . . . . . . . . . . . . > > . . . . . . . . . . . A C B C C B A B > > A B > > … Turker1 Turker2 Turkern

5. Social choice Profile R1 R1* social choice mechanism R2 R2* Outcome … … Rn Rn* Ri, Ri*: full rankings over a set A of alternatives

6. Applications: real world • People/agents often have conflicting preferences, yet they have to make a joint decision

7. Applications: academic world • Multi-agent systems [Ephrati and Rosenschein91] • Recommendation systems [Ghoshet al. 99] • Meta-search engines [Dwork et al. 01] • Belief merging [Everaereet al. 07] • Human computation (crowdsourcing) [Mao et al. AAAI-13] • etc.

8. How to design a good social choice mechanism? What is being “good”?

9. Two goals for social choice mechanisms GOAL1: democracy GOAL2: truth Axiomatic social choice THIS TUTORIAL

10. Outline • Axiomatic social choice • The Condorcet Jury Theorem (CJT) • Break • Four directions of extending CJT • Beyond CJT: the objective decision-making perspective

11. Flavor of this tutorial • Research questions + Basic models • tip of the iceberg • More references • Survey by Nitzan and Paroush (online): Collective Decision Making and Jury Theorem • Survey by Gerlinga et al. [2005]: Information acquisition and decision making in committees: A survey • My personal summary, send me an email

12. Computational social choice • Joerg’s text book • Handbook of Computational Social Choice

13. Outline • Axiomatic social choice • The Condorcet Jury Theorem (CJT) • Break • Four directions of extending CJT • Beyond CJT: the objective decision-making perspective

14. Common voting rules(what has been done in the past two centuries) • Mathematically, a social choice mechanism (voting rule) is a mapping from {All profiles} to {outcomes} • an outcome is usually a winner, a set of winners, or a ranking • m : number of alternatives (candidates) • n : number of agents (voters) • D=(P1,…,Pn) a profile • Positional scoring rules • A score vectors1,...,sm • For each vote V, the alternative ranked in the i-th position gets si points • The alternative with the most total points is the winner • Special cases • Borda, with score vector (m-1, m-2, …,0) • Plurality, with score vector (1,0,…,0) [Used in the US]

15. An example • Three alternatives {c1, c2, c3} • Score vector(2,1,0) (=Borda) • 3 votes, • c1 gets 2+1+1=4, c2 gets 1+2+0=3, c3 gets 0+0+2=2 • The winner is c1 2 1 0 2 1 0 2 1 0

16. The Kemeny rule • Kendall tau distance • K(V,W)= # {different pairwise comparisons} • Kemeny(D)=argminW K(D,W)=argminWΣP∈DK(P,W) • For single winner, choose the top-ranked alternative in Kemeny(D) • [Has a statistical interpretation] K( b ≻c≻a,a≻b≻c ) = 2 1 1

17. …and many others • Approval, Baldwin, Black, Bucklin, Coombs, Copeland, Dodgson, maximin, Nanson, Range voting, Schulze, Slater, ranked pairs, etc…

18. Q: How to evaluate rules in terms of achieving democracy? • A: Axiomatic approach

19. Axiomatic approach(what has been done in the past 50 years) • Anonymity: names of the voters do not matter • Fairness for the voters • Non-dictatorship: there is no dictator, whose top-ranked alternative is always the winner • Fairness for the voters • Neutrality: names of the alternatives do not matter • Fairness for the alternatives • Consistency: if r(D1)∩r(D2)≠ϕ, then r(D1∪D2)=r(D1)∩r(D2) • Condorcet consistency: if there exists a Condorcet winner, then it must win • A Condorcet winner beats all other alternatives in pairwise elections • Easy to compute: winner determination is in P • Computational efficiency of preference aggregation • Hard to manipulate: computing a beneficial false vote is hard

20. Which axiom is more important? • Some of these axiomatic properties are not compatible with others

21. An easy fact • Theorem. For voting rules that selects a single winner, anonymity is not compatible with neutrality • proof: > > Alice > > Bob ≠ W.O.L.G. Anonymity Neutrality

22. Not-So-Easy facts • Arrow’s impossibility theorem • Google it! • Gibbard-Satterthwaite theorem • Google it! • Axiomatic characterization • Template: A voting rule satisfies axioms A1, A2, A2  if it is rule X • If you believe in A1 A2 A3 are the most desirable properties then X is optimal • (anonymity+neutrality+consistency+continuity) positional scoring rules[Young SIAMAM-75] • (neutrality+consistency+Condorcet consistency) Kemeny[Young&LevenglickSIAMAM-78]

23. Outline • Axiomatic social choice • The Condorcet Jury Theorem (CJT) • Break • Four directions of extending CJT • Beyond CJT: the objective decision-making perspective

24. The Condorcet Jury theorem (CJT) [Condorcet 1785, Laplace 1812] • Given • two alternatives {a,b}. • competence 0.5<p<1, • Suppose • agents’ signals are i.i.d. conditioned on the ground truth • w/p p, the same as the ground truth • w/p 1-p, different from the ground truth • agents truthfully report their signals • The majority rule reveals ground truth as n→∞

25. Why CJT is important? • It Justifies the democracy and wisdom of the crowd • It “lays, among other things, the foundations of the ideology of the democratic regime” [Paroush SCW-98]

26. Proof • Group competence • Pr(maj(Pn)=a|a) • Pn: ni.i.d. votes given ground truth a • Random variableXj: takes 1 w/p p, 0 otherwise • encoding whether signal=ground truth • Σj=1nXj /nconverges to p in probability (Law of Large Numbers)

27. Three parts of CJT The group competence • is higher than that of any single agent • increases in the group size n • goes to 1 as n→∞

28. Proof of competence monotonicity • From 2k to 2k+1 • The extra vote breaks ties with higher probability in favor of the ground truth • k@a+k@b • From 2k+1to 2k+2 • (k+1)@a+k@b(k+1)@a+(k+1)@b • k@a+(k+1)@b(k+1)@a+(k+1)@b p (k+1)@a+k@b k@a+(k+1)@b 1-p

29. Limitations of CJT more than two? • Given • two alternatives {a,b}. • competence 0.5<p<1, • Suppose • agents’ signals are i.i.d. conditioned on the ground truth • w/p p, the same as the ground truth • w/p 1-p, different from the ground truth • agents truthfully report their signals • The majority rule reveals ground truth as n→∞ heterogeneous agents? dependent agents? strategic agents? other rules?

30. Outline • Axiomatic social choice • The Condorcet Jury Theorem (CJT) • Break • Four directions of extending CJT • Beyond CJT: the objective decision-making perspective

31. Extensions • Dependent agents • Heterogeneous agents • Strategic agents • More than two alternatives

32. An active area Myerson Shapley&Grofman Social Choice and Welfare American Political Science Review Games and Economic Behavior Mathematical Social Sciences Theory and Decision Public Choice Econometrica+ JET MSS special issue on ADT-15

33. Extensions • Dependent agents • Heterogeneous agents • Strategic agents • More than two alternatives

34. Does CJT hold for dependent agents? The group competence • is higher than that of any single agent • Not always (mimicking one leader) • increases in the group size n • Not always (mimicking one leader) • goes to 1 as n→∞ • Yes for some dependency models [Berg 92; Ladha 92, 93; Peleg&Zamir 12]

35. Dependent agents • Positive correlations • agents are likely to receive similar signals even conditioned on the ground truth • Negative correlations • agents are likely to receive different signals • Conjecture: Positive correlations reduces group competence • positively correlated agents effectively reduces the number of agents

36. Opinion leader model[Boland et al. 89] • One leader (Y), 2k followers (X1,…, X2k), same competence p • Pr(Y=1) = Pr(Xj=1)=p • Xj’sare independent conditioned on Y • Correlation r2 • Pr(Xj=1|Y=1) = p+r(1-p) • Pr(Xj=0|Y=0) =(1-p) +rp • Theorem. In the opinion leader model • when p>0.5 the group competence decreases in r • when p<0.5 the group competence increases in r • when p=0.5 the group competence does not change in r

37. Common evidence model[Boland et al. 89] • One common evidence (E), 2k+1 agents (X1,…, X2k+1), same competence p • Pr(E=1) = Pr(Xj=1)=p • Xj’sare independent conditioned on E • Correlation r2 • Pr(Xj=1|E=1) = p+r(1-p) • Pr(Xj=0|E=0) =(1-p) +rp • Theorem. In the common evidence model • when p>0.5 the group competence decreases in r • when p<0.5 the group competence increases in r • when p=0.5 the group competence does not change in r

38. Common evidence model[Dietrich and List 2004] G E • Ground truth G • Common evidence E • Given any ideal vote function f: EG • Competence pe=Pr(Xj=f(e)|e) • Theorem. The majority rule converges to f(e) as n→∞ … X1 Xn

39. Extensions • Dependent agents • Heterogeneous agents • Strategic agents • More than two alternatives

40. Does CJT hold for heterogeneous agents? The group competence • is higher than that of any single agent • Not always (1, 0.9.0.8,…) • increases in the group size n • Not always (1, 0.9.0.8,…) • goes to 1 as n→∞ • not always: pj=0.5+1/n • Yes under some condition [Berend&Paroush, 1998]

41. Group competence for heterogeneous agents • Independent signals • Agent j’s competence is pj • Theorem[Berend&Paroush, 1998]. CJT holds if and only if • , or • for every sufficiently large n,

42. Competence monotonicity[Berend&Sapir 05] • Given the competence {p1,…,pn} of nagents where pj≥0.5 • Ml: average competence of m randomly chosen agents • Theorem [Berend&Sapir 05]. For two alternatives and all l≤n-1 • Ml ≤ Ml+1if m is even • Ml = Ml+1if m is odd

43. Optimal voting rule for two alternatives • Theorem [Shapley and Grofman 1984]. Given the competence {p1,…,pn} of n agents, the maximum likelihood estimator is the weighted majority voting with • Proof. Suppose the ground truth is a, the log likelihood of the profile is

44. Extensions • Dependent agents • Heterogeneous agents • Strategic agents • More than two alternatives

45. Does CJT hold for strategic agents? The group competence • is higher than that of any single agent • Not always (same-vote equilibrium) • increases in the group size n • Not always (same-vote equilibrium) • goes to 1 as n→∞ • Yes for some models and informative equilibrium

46. Strategic voting • Common interest Bayesian voting game [Austen-Smith&Banks APSR-96] • two alternatives {a, b}, two signals {A,B}, a prior, Pr(signal|truth), • pa=Pr(signal=A|truth=a) • pb=Pr(signal=B|truth=b) • agents have the same utility function U(outcome, ground truth) =1 iff outcome = ground truth • sincere voting: vote for the alternative with the highest posterior probability • informative voting: vote for the signal • strategic voting: vote for the alternative with the highest expected utility

47. Timeline of the game • Nature chooses a ground truth g • Every agent j receives a signal sj~Pr(sj|g) • Every agent computes the posterior distribution (belief) over the ground truth using Bayesian’s rule • Every agent chooses a vote to maximizes her expected utility according to her belief • The outcome is computed by the voting rule

48. High level example • Two signals, two voters • Model: Pr( | ) = Pr( | ) = p>0.5 p 1-p Posterior: p p 1-p 1-p The other signal: Truthful agent: + my vote , winner: half/half half/half utility for voting : 1 0 0.5 0.5

49. Sincere voting = informative voting? • Setting • Two alternatives {a, b}, two signals {A,B} • Three agents • pa=0.8, pb=0.6 • Uniform prior: Pr(a)=0.1, Pr(b)=0.9 • An agent receives A • Informative voting: a • posterior probability: • 0.1*0.8@a vs. 0.9*0.4@b • sincere voting: b

50. Sincere voting = strategic voting? • Setting • Two alternatives {a, b}, two signals {A,B} • Three agents • pa=0.8, pb=0.6 • Uniform prior: Pr(a)=Pr(b)=0.5 • An agent receives A, other two agents are sincere/informative • Informative voting: a • posterior probability: 2/3@a+1/3@b • sincere voting: a • probability of a tie (other two agents’ votes are {a, b}) • 0.32|a, 0.48|b • Expected utility for voting a: 0.32*2/3 • Expected utility for voting b: 0.48*1/3 • Strategic voting: a