Existence of Fair Resolute Voting Rules

TL;DR

This work characterizes when resolute two-candidate voting rules can be fair under the Shapley-Shubik and Banzhaf indices. It connects fairness to the structure of winning coalitions, introducing unbiasedness as a strong condition that implies both indices are balanced across voters. The authors show unbiased voting rules exist exactly when $n$ is not an even power of two, by constructing complementary balanced families via combinatorial designs; these exist whenever $\binom{n}{n/2}$ is divisible by 4. They further show Shapley-Shubik-fair rules share the same necessary condition, while Banzhaf-fair rules exist for all $n$ except $2$, $4$, and $8$, with detailed case analyses for the small excluded cases. Overall, the results reveal that fair, resolute voting rules are far more prevalent than naive expectations, especially under Banzhaf fairness, and provide concrete design-based methods to construct them.

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.

Figures (5)

• Figure 1: A fair, non-anonymous voting rule
• Figure 2: A fair rule with an even number of voters
• Figure 3: A (non-fair) voting rule
• Figure 4: Relation between fairness notions
• Figure 5: A triangular prism

Theorems & Definitions (55)

• theorem 1.1
• theorem 1.2
• lemma 2.1: Winning Coalitions are Intersecting
• proof
• proposition 2.2
• proposition 3.1
• proof
• corollary 3.2
• proof
• proposition 3.3