On the existence of a minimum integer representation for weighted voting systems

Josep Freixas () and Xavier Molinero ()

Annals of Operations Research, 2009, vol. 166, issue 1, 243-260

Abstract: A basic problem in the theory of simple games and other fields is to study whether a simple game (Boolean function) is weighted (linearly separable). A second related problem consists in studying whether a weighted game has a minimum integer realization. In this paper we simultaneously analyze both problems by using linear programming. For less than 9 voters, we find that there are 154 weighted games without minimum integer realization, but all of them have minimum normalized realization. Isbell in 1958 was the first to find a weighted game without a minimum normalized realization, he needed to consider 12 voters to construct a game with such a property. The main result of this work proves the existence of weighted games with this property with less than 12 voters. Copyright Springer Science+Business Media, LLC 2009

Keywords: Simple games; Weighted voting games; Minimal realizations; Minimum realization; Realizations with minimum sum (search for similar items in EconPapers)
Date: 2009
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Citations: View citations in EconPapers (28)

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DOI: 10.1007/s10479-008-0422-2

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