Arrow’s impossibility theorem

EC-CS reading groupKenneth 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

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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

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Induction on # of agents

A function on N+1 agents

Unanimous

IIA

Non-dictatorial

A function on N agents

Unanimous

IIA

Nondictatorial

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Construction

CN(>1,>2,…,>n)=CN+1(>1,>2,…,>n,>1)

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Induction on # of alternatives

A function on M+1 alter.

Unanimous

IIA

Non-dictatorial

A function on M alter.

Unanimous

IIA

Non-dictatorial

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Construction

• C{b,c}( )=C{a,b,c}( )

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?

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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}

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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

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 )