Arrow’s impossibility theorem

1. Arrow’s impossibility theorem EC-CS reading group Kenneth Arrow Journal of Political Economy, 1950

2. Social choice theory – a science of collective decision making • Aggregate individual preferences into a social preference • E.g, Voting • (individual preference  votes) • (social preference  president) • Aggregate in a “satisfactory” manner • Fair? • In a manner that fulfills pre-defined conditions

4. The easy case: 2-candidate • Fair properties • Unanimity • Everyone prefers a to b, then society must prefers a to b • E.g, dictatorship • Agent anonymous • Name of agent doesn’t matter • Permutation of agent same social order • Outcome anonymous • Reverse individual order  reverse of social order • Monotonicity • If W(>)= a>b, and >’ is a profile that prefers a more, then W(>’)= a>b

5. May’s theorem (1952) • A social welfare function satisfies all these properties iff it is a Majority rule • Majority rule prefers pair-wise comparison winner • Tie breaks alphabetically • Holds without unanimity • QED for 2-candidate case!

6. Failure of majority in 3-candidate: the Condorcet paradox • Consider the following situation • Individual 1’s vote: a>b>c • Individual 2’s vote: b>c>a • Individual 3’s vote: c>a>b • By majority rule, the society • prefers a over b • prefers b over c • prefers c over a • It is a cycle! • Majority is not well-defined • We must turn to other voting rules

9. Computer-aided proof of Arrow’s theorem[Tang and Lin, AAAI-08, AIJ-09] • Induction • Inductive case: If the negation (Unanimity, IIA, Nondictator) of the theorem holds in general (n agents, m candidates), then it holds in the base case (2 agents, 3 candidates) • Base case: Verify it doesn’t hold for 2 agents, 3 candidates by computer

11. Induction on # of agents Unanimous A function on N+1 agents IIA Non-dictatorial Unanimous A function on N agents IIA Nondictatorial

13. Construction CN(>1,>2,…,>n)=CN+1(>1,>2,…,>n,>1)

14. Induction on # of alternatives Unanimous A function on M+1 alter. IIA Non-dictatorial Unanimous A function on M alter. IIA Non-dictatorial

16. Construction • C{b,c}( )=C{a,b,c}( )

17. Base case

23. Discussion • Would the requirement of SWF be too restrictive? • SWF outputsa ranking of all candidates • We only care about the winner! • A voting rule: • a preference profile  a candidate • Would this relaxation yield some possibility?

24. Voting model • A set of agents • A set of alternatives • Vote: permutation of alternatives • Vote profiles: a vote from each agent • Social-choice function: • C: {profiles}  {candidates}

25. Muller-Satterthwaite theorem • Weak unanimity • An alternative that is dominated by another in every vote can’t be chosen • Monotonicity • C(>)=a • a weakly improves its relative ranking in >’ (wrt. >) • C(>’)=a • Dictatorship • C(>)=top(>i) for all >, for some i • Muller-Satterthwaite Theorem: for |O|≥3 • Weak unanimity+ Monotonicity  Dictatorship

26. Gibbard-Satterthwaite theorem

27. Proofs • Our induction proof for Arrow works just fine for both theorems! • Same induction • Same construction • Similar program for the base case • It works for two more important theorems • Maskin’s theorem for Nash implementation • Sen’s theorem for Paretianliberty

28. Follow-up research: circumvent Arrow • Weaken each conditions in Arrow • Weaken unanimity, IIA • Restrict domain • Arrow: set of all pref profiles • Black: Single-peaked pref • Majority is well defined on single-peaked pref.

29. Follow-up research: circumvent G-S • G-S says every onto and strategy-proof is dictatorial • However, it is sometimes hard to find a manipulation • There are quite a few voting rules where finding a manipulation is NP-hard • Borda, STV (AAAI-11, IJCAI-11 best paper )