Among two-candidate elections that treat the candidates symmetrically and never result in a tie, which voting rules are fair? A natural requirement is that each voter exerts an equal influence over the outcome, i.e., is equally likely to swing the election one way or the other. A voter's influence has been formalized in two canonical ways: the Shapley-Shubik (1954) index and the Banzhaf (1964) index. We consider both indices, and ask: Which electorate sizes admit a fair voting rule (under the respective index)? For an odd number $n$ of voters, simple majority rule is an example of a fair voting rule. However, when $n$ is even, fair voting rules can be challenging to identify, and a diverse literature has studied this problem under different notions of fairness. Our main results completely characterize which values of $n$ admit fair voting rules under the two canonical indices we consider. For the Shapley-Shubik index, a fair voting rule exists for $n>1$ if and only if $n$ is not a power of $2$. For the Banzhaf index, a fair voting rule exists for all $n$ except $2$, $4$, and $8$. Along the way, we show how the Shapley-Shubik and Banzhaf indices relate to the winning coalitions of the voting rule, and compare these indices to previously considered notions of fairness.

翻译：在对称对待候选人且从不产生平局的双候选人选举中，哪些投票规则是公平的？一个自然的要求是每位选民对结果施加同等影响力，即其改变选举结果的可能性相同。选民影响力的形式化定义存在两种经典方式：Shapley-Shubik（1954）指数与Banzhaf（1964）指数。我们同时考察这两种指数，并探究：在相应指数下，哪些选民规模允许存在公平投票规则？当选民数$n$为奇数时，简单多数规则即是公平投票规则的实例。然而当$n$为偶数时，公平投票规则往往难以构建，现有文献已从不同公平性概念出发对该问题进行了广泛研究。我们的核心成果完整刻画了在两种经典指数下，哪些$n$值允许存在公平投票规则。对于Shapley-Shubik指数，当且仅当$n$不是2的幂次时，存在公平投票规则（$n>1$）。对于Banzhaf指数，除$n=2,4,8$外，所有$n$值均存在公平投票规则。研究过程中，我们揭示了Shapley-Shubik指数和Banzhaf指数与投票规则获胜联盟的关联，并将这些指数与先前研究的公平性概念进行了比较。