Computational Social Choice

1. Computational Social Choice Lirong Xia WINE-13 Tutorial Dec 11, 2013

2. 2011 UK Referendum • The second nationwide referendum in UK • 1stwas in 1975 • Member of Parliament election: Plurality rule  Alternative vote rule? • 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. Social Choice and Computer Science Computational thinking + optimization algorithms CS Social Choice 21th Century Strategic thinking + methods/principles of aggregation PLATO 4thC. B.C. LULL 13thC. BORDA 18thC. CONDORCET 18thC. ARROW 20thC. TURING et al. 20thC. PLATO et al. 4thC. B.C.---20thC.

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

8. 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.

9. A burgeoning area • Recently has been drawing a lot of attention • IJCAI-11: 15 papers, best paper • AAAI-11: 6 papers, best paper • AAMAS-11: 10 full papers, best paper runner-up • AAMAS-12 9 full papers, best student paper • EC-12: 3 papers • Workshop: COMSOC Workshop 06, 08, 10, 12, 14 • Courses: • Technical University Munich (Felix Brandt) • Harvard (Yiling Chen) • U. of Amsterdam (UlleEndriss) • RPI (Lirong Xia) • Book in progress: Handbook of Computational Social Choice

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

11. Two goals for social choice mechanisms GOAL1: democracy GOAL2: truth 1. Classical Social Choice 3. Statistical approaches 2. Computational aspects

12. Outline 1. Classical Social Choice 45 min 5 min 2.1 Computational aspects Part 1 55 min NP- Hard 30 min 2.2 Computational aspects Part 2 30 min NP- Hard 5 min 3. Statistical approaches 75 min NP- Hard

13. 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]

14. 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

15. Plurality with runoff • The election has two rounds • In the first round, all alternatives except the two with the highest plurality score drop out • In the second round, the alternative that is preferred by more voters wins • [used in Iran, France, North Carolina State] a > b > c > d a > d c > d > a >b d > a d >a • d > a > b > c • d > a • b > c > d >a d

16. Single transferable vote (STV) • Also called instant run-off voting or alternative vote • The election has m-1rounds, in each round, • The alternative with the lowest plurality score drops out, and is removed from all votes • The last-remaining alternative is the winner • [used in Australia and Ireland] a > b > c > d a > c > d a > c a > c c > d > a c > d > a >b c > a c > a • d > a > b > c • d > a > c • b > c > d >a • c > d >a a

17. 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

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

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

20. 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

21. Which axiom is more important? • Some of these axiomatic properties are not compatible with others • Food for thought: how to evaluate partial satisfaction of axioms?

22. 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

23. Another easy fact [Fishburn APSR-74] • Thm. No positional scoring rule is Condorcet consistent: • suppose s1 >s2 >s3 is the Condorcet winner > > 3 Voters CONTRADICTION > > 2 Voters : 3s1 + 2s2+ 2s3 < > > 1Voter : 3s1 + 3s2+ 1s3 > > 1Voter

24. Not-So-Easy facts • Arrow’s impossibility theorem • Google it! • Gibbard-Satterthwaite theorem • Next section • 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]

25. Food for thought • Can we quantify a voting rule’s satisfiability of these axiomatic properties? • Tradeoffs between satisfiability of axioms • Use computational techniques to design new voting rules • use AI techniques to automatically prove or discover new impossibility theorems [Tang&Lin AIJ-09]

26. Outline 1. Classical Social Choice 45 min 5 min 2.1 Computational aspects Part 1 55 min 15 min 2.2 Computational aspects Part 2 30 min 5 min 3. Statistical approaches 75 min

27. Computational axioms • Easy to compute: • the winner can be computed in polynomial time • Hard to manipulate: • computing a beneficial false vote is hard

28. Computational axioms • Easy to compute: • the winner can be computed in polynomial time • Hard to manipulate: • computing a beneficial false vote is hard

29. Which rule is easy to compute? • Almost all common voting rules, except • Kemeny: NP-hard [Bartholdi et al. 89], Θ2p-complete [Hemaspaandra et al. TCS-05] • Young: Θ2p-complete [Rothe et al. TCS-03] • Dodgson: Θ2p-complete [Hemaspaandra et al. JACM-97] • Slater: NP-complete [Hurdy EJOR-10] • Practical algorithms for Kemeny (also for others) • ILP [Conitzer, Davenport, & Kalagnanam AAAI-06] • Approximation [Ailon, Charikar, & Newman STOC-05] • PTAS [Kenyon-Mathieu and W. SchudySTOC-07] • Fixed-parameter analysis[Betzler et al. TCS-09]

30. Really easy to compute? • Easy to compute axiom: computing the winner takes polynomial time in the input size • input size: nmlogm • What if m is extremely large?

31. Combinatorial domains(Multi-issue domains) • The set of alternatives can be uniquely characterized by multiple issues • Let I={x1,...,xp} be the set of p issues • Let Di be the set of values that the i-th issue can take, then A=D1×... ×Dp • Example: • Issues={ Main course, Wine } • Alternatives={} ×{ }

32. Multiple referenda • In California, voters voted on 11 binary issues ( / ) • 211=2048 combinations in total • 5/11 are about budget and taxes • Prop.30 Increase sales and some income tax for education • Prop.38 Increase income tax on almost everyone for education

33. Other combinatorial domains • Belief merging [Gabbay et al. JLC-09] • Judgment aggregation [List and Pettit EP-02] merging operator K1 K2 … Kn

34. Computational axioms • Easy to compute: • the winner can be computed in polynomial time • Hard to manipulate: • computing a beneficial false vote is hard

35. Strategic behavior (of the agents) • Manipulation: an agent (manipulator) casts a vote that does not represent her true preferences, to make herself better off • A voting rule is strategy-proof if there is never a (beneficial) manipulation under this rule • How important strategy-proofness is as an desired axiomatic property? • compared to other axiomatic properties

36. Manipulation under plurality rule (ties are broken in favor of ) > > > > Alice Plurality rule > > Bob > > Carol

37. Any strategy-proof voting rule? • No reasonable voting rule is strategyproof • Gibbard-Satterthwaite Theorem [Gibbard Econometrica-73, Satterthwaite JET-75]:When there are at least three alternatives, no voting rules except dictatorships satisfy • non-imposition: every alternative wins for some profile • unrestricted domain: voters can use any linear order as their votes • strategy-proofness • Axiomatic characterization for dictatorships!

38. A few ways out • Relax non-dictatorship: use a dictatorship • Restrict the number of alternatives to 2 • Relax unrestricted domain: mainly pursued by economists • Single-peaked preferences: • Approval voting: A voter submit 0 or 1 for each alternative

39. Computational thinking • Use a voting rule that is too complicated so that nobody can easily predict the winner • Dodgson • Kemeny • The randomized voting rule used in Venice Republic for more than 500 years [Walsh&Xia AAMAS-12] • We want a voting rule where • Winner determination is easy • Manipulation is hard

40. Overview Manipulation is inevitable (Gibbard-Satterthwaite Theorem) Can we use computational complexity as a barrier? Why prevent manipulation? • Yes • May lead to very • undesirable outcomes Is it a strong barrier? • No How often? Other barriers? • Seems not very often • Limited information • Limited communication

41. Manipulation: A computational complexity perspective If it is computationallytoo hard for a manipulator to compute a manipulation, she is best off voting truthfully • Similar as in cryptography For which common voting rules manipulation is computationally hard? NP- Hard

42. Computing a manipulation • Initiated by [Bartholdi, Tovey, &Trick SCW-89b] • Votes are weighted or unweighted • Bounded number of alternatives [Conitzer, Sandholm, &Lang JACM-07] • Unweightedmanipulation: easy for most common rules • Weighted manipulation: depends on the number of manipulators • Unbounded number of alternatives (next few slides) • Assuming the manipulators have complete information!

43. Unweightedcoalitional manipulation (UCM) problem • Given • The voting rule r • The non-manipulators’ profile PNM • The number of manipulators n’ • The alternative c preferred by the manipulators • We are asked whether or not there exists a profile PM (of the manipulators) such that c is the winner of PNM∪PM under r

44. The stunningly big table for UCM

45. What can we conclude? • For some common voting rules, computational complexity provides some protection against manipulation • Is computational complexity a strong barrier? • NP-hardness is a worst-case concept

46. Probably NOT a strong barrier 1. Frequency of manipulability 2. Easiness of Approximation 3. Quantitative G-S AAMAS-14 workshop Computational Social Choice: Beyond the Worst Case

47. A first angle: frequency of manipulability • Non-manipulators’ votes are drawn i.i.d. • E.g. i.i.d. uniformly over all linear orders (the impartial culture assumption) • How often can the manipulators make c win? • Specific voting rules [Peleg T&D-79, Baharad&Neeman RED-02, Slinko T&D-02, Slinko MSS-04, Procaccia and Rosenschein AAMAS-07]

48. A general result[Xia&ConitzerEC-08a] • Theorem. For any generalized scoring rule • Including many common voting rules • Computational complexity is not a strong barrier against manipulation • UCM as a decision problem is easy to compute in most cases • The case of Θ(√n)has been studied experimentally in [Walsh IJCAI-09] All-powerful Θ(√n) # manipulators No power

49. A second angle: approximation • Unweighted coalitional optimization (UCO):compute the smallest number of manipulators that can make cwin • A greedy algorithm has additive error no more than 1 for Borda[Zuckerman, Procaccia, &Rosenschein AIJ-09]

50. An approximation algorithm for positional scoring rules[Xia,Conitzer,& Procaccia EC-10] • A polynomial-time approximation algorithm that works for all positional scoring rules • Additive error is no more than m-2 • Based on a new connection between UCO for positional scoring rules and a class of scheduling problems • Computational complexity is not a strong barrier against manipulation • The cost of successful manipulation can be easily approximated (for positional scoring rules)