A Law of Large Numbers for Weighted Majority

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Statistics Papers
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law of large numbers
voting power
influences
boolean functions
monotone simple games
aggregation of information
voting paradox
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Statistics and Probability
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Häggström, Olle
Kalai, Gil
Mossel, Elchanan
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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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2006-07-01
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Advances in Applied Mathematics
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At the time of publication, author Elchanan Mossel was affiliated with the University of California, Berkeley. Currently, he is a faculty member at the Statistics Department at the University of Pennsylvania.
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