On the existence of a minimum integer representation for weighted voting systems
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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
CitacióFreixas, J.; Molinero, X. On the existence of a minimum integer representation for weighted voting systems. "Annals of operations research", Febrer 2009, vol. 166, núm. 1, p. 243-260.