Weighted games without a unique minimal representation in integers

Abstract

Isbell in 1959 was the first to find a weighted game without a minimum integer realization in which the affected players do not play a symmetric role in the game. His example has 12 players in a weighted decisive game, i.e. a weighted game for which a coalition wins iff its complement loses. The goal of this article is to provide a procedure for weighted games that allows finding out what is the minimum number of players needed to get a weighted game without a minimum integer weighted representation in which the affected players do not play a symmetric role in the game. We prove, by means of an algorithm, that the minimum number of voters required is nine.