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

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

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

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!

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

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 9

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

Construction C N (> 1,> 2,…,> n )=C N+1 (> 1,> 2,…,> n, > 1 ) 13

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

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

Base case

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

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} 24

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 25

Gibbard-Satterthwaite theorem

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 Paretian liberty

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.

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 )