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.

中文翻译：

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加权投票游戏中权力与比例的最坏情况界限与应用假名操纵

加权投票游戏适用于多种多代理设置。它们使权力指数的形式化成为可能，这些指数量化了参与者的联盟权力。我们采用一种新颖的方法来研究这些游戏中大小玩家的力量。我们将小（大）玩家建模为拥有单（多）票。大玩家的总相对权力是通过他们的投票比例来衡量的。对于这个比率，我们显示了 Shapley-Shubik 和 Deegan-Packel 指数的小常数最坏情况界限。与此形成鲜明对比的是，Banzhaf 指数的这个比率是无限的。作为一个应用，我们定义了一个假名策略范式游戏，其中每个大玩家都可以在假身份之间分配投票，并研究其各种属性。总之，我们的结果为球员大小的影响提供了基础，