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A Law of Large Numbers for Weighted Majority

Author

Listed:
  • Olle Haggstrom

    ()

  • Gil Kalai

    ()

  • Elchanan Mossel

    ()

Abstract

Consider 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 which he derived from the weak law of large numbers asserts that if the number of voters tends to infinity then the probability that the first candidate will be elected tends to one. The notion of influence of a voter or its voting power is relevant for extensions of the weak law of large numbers for voting rules which are more general than simple majority. In this paper we point out two different ways to extend the classical notions of voting power and influences to arbitrary probability distributions. The extension relevant to us is the “effect” of a voter, which is a weighted version of the correlation between the voter’s vote and the election’s outcomes. We prove an extension of the weak law of large numbers to weighted majority games when all individual effects are small and show that this result does not apply to any voting rule which is not based on weighted majority.

Suggested Citation

  • 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.
  • Handle: RePEc:huj:dispap:dp363
    as

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    File URL: http://ratio.huji.ac.il/sites/default/files/publications/dp363.pdf
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    References listed on IDEAS

    as
    1. Milgrom, Paul R & Weber, Robert J, 1982. "A Theory of Auctions and Competitive Bidding," Econometrica, Econometric Society, vol. 50(5), pages 1089-1122, September.
    2. repec:cup:apsrev:v:82:y:1988:i:04:p:1231-1244_19 is not listed on IDEAS
    3. Gil Kalai, 2004. "Social Indeterminacy," Discussion Paper Series dp362, The Federmann Center for the Study of Rationality, the Hebrew University, Jerusalem.
    4. Gil Kalai, 2004. "Social Indeterminacy," Econometrica, Econometric Society, vol. 72(5), pages 1565-1581, September.
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    Citations

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    Cited by:

    1. Gil Kalai & Muli Safra, 2005. "Threshold Phenomena and Influence," Levine's Bibliography 666156000000000683, UCLA Department of Economics.
    2. Gil Kalai & Shmuel Safra, 2005. "Threshold Phenomena and Influence, with Some Perspectives from Mathematics, Computer Science, and Economics," Discussion Paper Series dp398, The Federmann Center for the Study of Rationality, the Hebrew University, Jerusalem.

    More about this item

    Keywords

    law of large numbers; voting power; influences; boolean functions; monotone simple games; aggregation of informations; the voting paradox;

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