# A Law of Large Numbers for Weighted Majority

*Olle Haggstrom* (),
*Gil Kalai* () and
*Elchanan Mossel* ()

Discussion Paper Series from The Federmann Center for the Study of Rationality, the Hebrew University, Jerusalem

**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.

**Keywords:** law of large numbers; voting power; influences; boolean functions; monotone simple games; aggregation of informations; the voting paradox (search for similar items in EconPapers)

**New Economics Papers:** this item is included in nep-cdm and nep-dcm

**Date:** 2004-06

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