Date of this Version
Advances in Applied Mathematics
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.
© 2006. This manuscript version is made available under the CC-BY-NC-ND 4.0 license.
law of large numbers, voting power, influences, boolean functions, monotone simple games, aggregation of information, voting paradox
Häggström, O., Kalai, G., & Mossel, E. (2006). A Law of Large Numbers for Weighted Majority. Advances in Applied Mathematics, 37 (1), 112-123. http://dx.doi.org/10.1016/j.aam.2005.08.002
Date Posted: 27 November 2017
This document has been peer reviewed.