Weighted voting games apply to a wide variety of multi-agent settings. They enable the formalization of power indices which quantify the coalitional power of players. We take a novel approach to the study of the power of big vs.~small players in these games. We model small (big) players as having single (multiple) votes. The aggregate relative power of big players is measured w.r.t.~their votes proportion. For this ratio, we show small constant worst-case bounds for the Shapley-Shubik and the Deegan-Packel indices. In sharp contrast, this ratio is unbounded for the Banzhaf index. As an application, we define a false-name strategic normal form game where each big player may split its votes between false identities, and study its various properties. Together, our results provide foundations for the implications of players' size, modeled as their ability to split, on their relative power.
翻译:加权投票游戏适用于各种各样的多试剂设置。 它们可以使能量化玩家联盟力量的力量指数正规化。 我们对这些游戏中大对小玩家的实力的研究采取了新颖的方法。 我们把小(大)玩家的模型做成是单张( 多张)的票。 大玩家的总相对实力被测量到 w.r.t.~他们的投票比例。 对于这个比例, 我们为Shapley- Shubik 和 Deegan- Packel 指数展示了小的、 不变的最坏的界限。 相反, 这个比率没有被定在Banzhaf 指数上。 作为应用程序, 我们定义了一个假名的普通战略游戏, 每个大玩家可以将选票分成错误身份, 并研究其各种属性。 我们的结果共同为玩家大小的影响提供了基础, 以他们分裂能力为模型, 其相对实力的大小影响为模型。