Trading transforms of non-weighted simple games and integer weights of weighted simple games

Abstract

This paper is concerned with simple games. One of the fundamental questions regarding simple games is that of what makes a simple game a weighted majority game. Taylor and Zwicker (1992) showed that a simple game is non-weighted if and only if there exists a trading transform of finite size. They also provided an upper bound on the size of such a trading transform, if it exists. Gvozdeva and Slinko ( 2009) improved on that upper bound. Their proof employs a property of linear inequalities demonstrated by Muroga (1971). We provide a new proof of the existence of a trading transform when a given simple game is non-weighted. Our proof employs Farkas' lemma (1894), and yields an improved upper bound on the length of a trading transform. We also discuss an integer weights representation of a weighted simple game, and improve on the bounds obtained by Muroga (1971). We show that our bounds are tight when the number of players is less than or equal to five, based on the computational results obtained by Kurz (2012). We deal with the problem of finding a minimum integer weights representation under the assumption that we have minimal winning coalitions and maximal losing coalitions. We propose a polynomial time probabilistic algorithm for finding an approximate solution.