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A Law of Large Numbers for Weighted Majority

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Author Info

  • Olle Haggstrom

    ()

  • Gil Kalai

    ()

  • Elchanan Mossel

    ()

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    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.

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    File URL: http://ratio.huji.ac.il/sites/default/files/publications/dp363.pdf
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    Bibliographic Info

    Paper provided by The Center for the Study of Rationality, Hebrew University, Jerusalem in its series Discussion Paper Series with number dp363.

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    Length: 16 pages
    Date of creation: Jun 2004
    Date of revision:
    Handle: RePEc:huj:dispap:dp363

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    Related research

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

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    References

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    1. Paul Milgrom & Robert J. Weber, 1981. "A Theory of Auctions and Competitive Bidding," Discussion Papers 447R, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    2. Gil Kalai, 2004. "Social Indeterminacy," Econometrica, Econometric Society, vol. 72(5), pages 1565-1581, 09.
    3. Gil Kalai, 2004. "Social Indeterminacy," Discussion Paper Series dp362, The Center for the Study of Rationality, Hebrew University, Jerusalem.
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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 Center for the Study of Rationality, Hebrew University, Jerusalem.

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