1 / 105

Computational Social Choice

Computational Social Choice. Lirong Xia. IJCAI-13 Tutorial Aug 4, 2013. A shameless advertisement…. About RPI The first technological institutes in English-speaking countries Graduate school rankings in US 47 th Computer Science 28 th Computer Engineering 17 th Applied Math

skylar
Download Presentation

Computational Social Choice

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Computational Social Choice Lirong Xia IJCAI-13 Tutorial Aug 4, 2013

  2. A shameless advertisement… • About RPI • The first technological institutes in English-speaking countries • Graduate school rankings in US • 47thComputer Science • 28th Computer Engineering • 17th Applied Math • I am generally interested in both theory and application of economics and computation. • Hiring highly motivated Ph.D. students • If interested, please send me an email (xial@cs.rpi.edu)

  3. 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

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

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

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

  7. 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.

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

  9. 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.

  10. 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 (2013 fall Lirong Xia) • Book in progress: Handbook of Computational Social Choice

  11. Flavor of this tutorial • High-level objectives for • design • evaluation • logic flow among research topics “Give a man a fish and you feed him for a day. Teach a man to fish and you feed him for a lifetime.” -----Chinese proverb • Plus some concrete results

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

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

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

  15. 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]

  16. 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

  17. 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, 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

  18. 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

  19. The Kemeny rule • Kendall tau distance • K(V,W)= # {different pairwise comparisons} • Kemeny(D)=argminW K(D,W)=argminWΣP∈DK(D,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

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

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

  22. 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

  23. 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?

  24. 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

  25. 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

  26. 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]

  27. 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]

  28. 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

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

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

  31. 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]

  32. 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?

  33. 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={} ×{ }

  34. 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

  35. Overview Combinatorial voting • Preference • representation • New voting rule • Evaluation

  36. Preference representation: CP-nets[Boutilier et al. JAIR-04] Variables:x,y,z. Graph CPTs This CP-net encodes the following partial order: x y z

  37. Sequential voting rules [Lang IJCAI-07] • Issues: main course, wine • Order: main course > wine • Local rules are majority rules • V1: > , : > , : > • V2: > , : > , : > • V3: > , : > , : > • Step 1: • Step 2: given , is the winner for wine • Winner: ( , )

  38. Research topics • How can we say that sequential voting is good? • computationally efficient • satisfies good axioms [Lang and Xia MSS-09] • need to worry about manipulation in the worst case [Xia, Conitzer &Lang EC-11] • Other compact languages • GAI network [Gonzales et al. AIJ-11] • TCP-net [Li et al. AAMAS-11] • Soft constraints [Pozza et al. IJCAI-11]

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

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

  41. 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

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

  43. 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!

  44. 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: • Range voting: A voter submit any natural number between 0 and 10 for each alternative • Approval voting: A voter submit 0 or 1 for each alternative

  45. 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

  46. 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

  47. 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

  48. 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!

  49. 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

  50. The stunningly big table for UCM

More Related