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

Abstract

This study investigates simple games. A fundamental research question in this field is to determine necessary and sufficient conditions for a simple game to be a weighted majority game. Taylor and Zwicker (Proc Am Math Soc 115:1089–1094, 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 (Math Soc Sci 61:20–30, 2011) improved that upper bound; their proof employed a property of linear inequalities demonstrated by Muroga (Threshold logic and its applications, 1971). In this study, 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 (1902), and yields an improved upper bound on the size of a trading transform. We also discuss an integer-weight representation of a weighted simple game, improving the bounds obtained by Muroga (Threshold logic and its applications, 1971). We show that our bound on the quota is tight when the number of players is less than or equal to five, based on the computational results obtained by Kurz (Ann Oper Res 196:361–369, 2012). Furthermore, we discuss the problem of finding an integer-weight representation under the assumption that we have minimal winning coalitions and maximal losing coalitions. In particular, we show a performance of a rounding method. Finally, we address roughly weighted simple games. Gvozdeva and Slinko (Math Soc Sci 61:20–30, 2011) showed that a given simple game is not roughly weighted if and only if there exists a potent certificate of non-weightedness. We give an upper bound on the length of a potent certificate of non-weightedness. We also discuss an integer-weight representation of a roughly weighted simple game.