Existence of Fair Resolute Voting Rules

TL;DR

This paper characterizes for which electorate sizes fair voting rules exist under the Shapley-Shubik and Banzhaf indices, revealing that fairness depends on specific numerical properties of the number of voters.

Contribution

It provides a complete characterization of electorate sizes admitting fair voting rules under two canonical influence indices, extending understanding of fairness in voting systems.

Findings

Fair voting rules exist for all odd n.

For the Shapley-Shubik index, fair rules exist iff n is not a power of 2.

For the Banzhaf index, fair rules exist iff n is not 2, 4, or 8.

Abstract

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.