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