Existence of Fair Resolute Voting Rules

2026-02-14

AI Summary

This study addresses the existence of voting rules that ensure equal influence for all voters in a two-candidate election that is symmetric—treating both candidates identically—and never results in a tie. Employing tools from cooperative game theory, combinatorics, and voting theory, the paper provides a complete characterization of the necessary and sufficient conditions for such fair rules under both the Shapley–Shubik and Banzhaf power indices. It establishes that, under the Shapley–Shubik index, a fair rule exists if and only if the number of voters \(n > 1\) and \(n\) is not a power of two. Under the Banzhaf index, fair rules exist for all \(n\) except \(n = 2, 4, 8\). The work further uncovers an intrinsic connection between these power indices and the structure of winning coalitions.

Technology Category

• Game Theory and Economic Paradigms: Social Choice / Voting
• Humans and AI: Voting
• Multiagent Systems: Mechanism Design

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Social Networks and Social Media: Fairness and bias in social network and social media analysis
• Security and Privacy: Large-scale security measurements

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.

Problem

• fair voting rules
• Shapley-Shubik index
• Banzhaf index
• voter influence
• electorate size

Innovation

• fair voting rules
• Shapley-Shubik index
• Banzhaf index
• voting power
• symmetric elections