Statistics Papers

Document Type

Journal Article

Date of this Version

7-2006

Publication Source

Advances in Applied Mathematics

Volume

37

Issue

1

Start Page

112

Last Page

123

DOI

10.1016/j.aam.2005.08.002

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.

Copyright/Permission Statement

© 2006. This manuscript version is made available under the CC-BY-NC-ND 4.0 license.

Comments

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.

Keywords

law of large numbers, voting power, influences, boolean functions, monotone simple games, aggregation of information, voting paradox

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Date Posted: 27 November 2017

This document has been peer reviewed.