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Social Indeterminacy


  • Gil Kalai



An extension of Condorcet's paradox by McGarvey (1953) asserts that for every asymmetric relation R on a finite set of candidates there is a strict-preferences voter profile that has the relation R as its strict simple majority relation. We prove that McGarvey's theorem can be extended to arbitrary neutral monotone social welfare functions which can be described by a strong simple game G if the voting power of each individual, measured by the it Shapley-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 is p > 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.

Suggested Citation

  • Gil Kalai, 2004. "Social Indeterminacy," Discussion Paper Series dp362, The Federmann Center for the Study of Rationality, the Hebrew University, Jerusalem.
  • Handle: RePEc:huj:dispap:dp362

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    Cited by:

    1. Joe Neeman, 2014. "A law of large numbers for weighted plurality," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 42(1), pages 99-109, January.
    2. Olle Haggstrom & Gil Kalai & Elchanan Mossel, 2004. "A Law of Large Numbers for Weighted Majority," Discussion Paper Series dp363, The Federmann Center for the Study of Rationality, the Hebrew University, Jerusalem.
    3. Beigman, Eyal, 2010. "Simple games with many effective voters," Games and Economic Behavior, Elsevier, vol. 68(1), pages 15-22, January.

    More about this item


    social choice; information aggregation; Arrow's theorem; simple games; the Shapley-Shubik power index; threshold phenomena;

    JEL classification:

    • D71 - Microeconomics - - Analysis of Collective Decision-Making - - - Social Choice; Clubs; Committees; Associations
    • D72 - Microeconomics - - Analysis of Collective Decision-Making - - - Political Processes: Rent-seeking, Lobbying, Elections, Legislatures, and Voting Behavior
    • C71 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Cooperative Games
    • D80 - Microeconomics - - Information, Knowledge, and Uncertainty - - - General

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