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 (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 (2011) improved that upper bound; their proof employed a property of linear inequalities demonstrated by Muroga (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 (1894), 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 (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 (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. Lastly, we address roughly weighted simple games. Gvozdeva and Slinko (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.
翻译:本研究调查简单的游戏。 本领域的一个基本研究问题是确定一个简单的游戏是否必要和足够地具备必要和充分的条件,使一个简单的游戏成为加权多数游戏。 Taylor 和 Zwicker (1992年) 显示,一个简单的游戏如果存在有限规模的交易变换,而且只有在存在有限规模的交易变换的情况下,才没有加权,它们也为这种交易变换的规模提供了一个上限。 Gvozdeva 和 Slinko (2011年) 改进了这种交易变换的上限; 他们的证据采用了Muroga (1971年) 所展示的线性不平等的属性。 在本研究中,我们提供了一个新的证据,证明当某个特定简单游戏变换为非加权游戏时,交易变换成交易变换交易。 我们的变现使用法性变换交易变换(1894年) 显示一个非加权的变换交易变换的上限。 我们还讨论了一个加权游戏的整比重表示,我们最终展示了一个最起码的Slogoal 。