Social Indeterminacy
Econometrica · 2004 · Volume 72 · Issue 5
DOI: 10.1111/j.1468-0262.2004.00543.x
Abstract
An extension of Condorcet's paradox by McGarvey (1953) asserts that for every asymmetric relationRon a finite set of candidates there is a strict-preferences voter profile that has the relationRas its strict simple majority relation. We prove that McGarvey's theorem can be extended to arbitrary neutral monotone social welfare functions that can be described by a strong simple gameGif the voting power of each individual, measured by theShapley–Shubik power index, is sufficiently small. Our proof is based on an extension to another classic result concerning the majority rule. Condorcet studied an election between two candidates in which the voters' choices are random and independent and the probability of a voter choosing the first candidate isp>1/2. Condorcet's jury theorem asserts that if the number of voters tends to infinity then the probability that the first candidate will be elected tends to one. We prove that this assertion extends to a sequence of arbitrary monotone strong simple games if and only if the maximum voting power for all individuals tends to zero.
Keywords
Social choice, information aggregation, Arrow's theorem, simple games, the Shapley–Shubik power index, threshold phenomena
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