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.

这项研究调查了简单的游戏。该领域的一个基本研究问题是确定一个简单博弈成为加权多数博弈的充分必要条件。Taylor 和 Zwicker (Proc Am Math Soc 115:1089–1094, 1992) 表明，当且仅当存在有限大小的交易变换时，简单游戏是无权重的。他们还提供了此类交易转换大小的上限（如果存在）。Gvozdeva 和 Slinko (Math Soc Sci 61:20–30, 2011) 改进了这个上限；他们的证明使用了 Muroga 证明的线性不等式的性质（阈值逻辑及其应用，1971）。在这项研究中，我们提供了一个新的证据，证明当给定的简单游戏是非加权的时，交易变换的存在。我们的证明使用 Farkas 引理 (1902)，并在交易转换的大小上产生改进的上限。我们还讨论了加权简单游戏的整数权重表示，改进了 Muroga 获得的边界（阈值逻辑及其应用，1971）。根据 Kurz 获得的计算结果（Ann Oper Res 196:361–369, 2012），我们表明当玩家数量小于或等于 5 时，我们对配额的限制是严格的。此外，我们讨论了在我们有最小获胜联盟和最大失败联盟的假设下找到整数权重表示的问题。特别是，我们展示了舍入方法的性能。最后，我们讨论粗略加权的简单游戏。Gvozdeva 和 Slinko（数学社会科学 61：20-30，2011）表明，当且仅当存在有效的非加权证明时，给定的简单游戏不会粗略加权。我们给出了一个有效的非加权证书长度的上限。我们还讨论了一个粗略加权的简单游戏的整数权重表示。