Existence of Fair Resolute Voting Rules

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.