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Social Indeterminacy

Author

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  • Gil Kalai

Abstract

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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    File URL: http://ratio.huji.ac.il/sites/default/files/publications/dp362.pdf
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    Citations

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    Cited by:

    1. Beigman, Eyal, 2010. "Simple games with many effective voters," Games and Economic Behavior, Elsevier, vol. 68(1), pages 15-22, January.
    2. Gil Kalai & Elchanan Mossel, 2015. "Sharp Thresholds for Monotone Non-Boolean Functions and Social Choice Theory," Mathematics of Operations Research, INFORMS, vol. 40(4), pages 915-925, October.
    3. 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.
    4. 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.
    5. Emilio De Santis & Fabio Spizzichino, 2023. "Construction of voting situations concordant with ranking patterns," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 46(1), pages 129-156, June.

    More about this item

    Keywords

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

    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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