Annals of Operations Research

, 166:243 | Cite as

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

  • Josep Freixas
  • Xavier Molinero


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.


Simple games Weighted voting games Minimal realizations Minimum realization Realizations with minimum sum 


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Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Dept. de Matemàtica Aplicada 3, Escola Politècnica Superior d’Enginyeria de ManresaUniversitat Politècnica de CatalunyaManresaSpain
  2. 2.Dept. de Llenguatges i Sistemes Informàtics, Escola Politècnica Superior d’Enginyeria de ManresaUniversitat Politècnica de CatalunyaManresaSpain

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