Most Equitable Voting Rules

TL;DR

This paper explores the design of voting rules that maximize fairness by optimally balancing anonymity and neutrality, introducing a novel refinement concept and an efficient algorithm for complex preference settings.

Contribution

It introduces a new notion of most equitable refinements, characterizes conditions for the ANR impossibility, and develops a polynomial-time algorithm for computing the MFP tie-breaking.

Findings

Characterized conditions for the ANR impossibility in large-agent settings.

Proposed the most-favorable-permutation (MFP) tie-breaking method.

Developed a polynomial-time algorithm for MFP with full rankings.

Abstract

In social choice theory, anonymity (all agents being treated equally) and neutrality (all alternatives being treated equally) are widely regarded as ``minimal demands'' and ``uncontroversial'' axioms of equity and fairness. However, the ANR impossibility -- there is no voting rule that satisfies anonymity, neutrality, and resolvability (always choosing one winner) -- holds even in the simple setting of two alternatives and two agents. How to design voting rules that optimally satisfy anonymity, neutrality, and resolvability remains an open question. We address the optimal design question for a wide range of preferences and decisions that include ranked lists and committees. Our conceptual contribution is a novel and strong notion of most equitable refinements that optimally preserves anonymity and neutrality for any irresolute rule that satisfies the two axioms. Our technical contributions are twofold. First, we characterize the conditions for the ANR impossibility to hold under general settings, especially when the number of agents is large. Second, we propose the most-favorable-permutation (MFP) tie-breaking to compute a most equitable refinement and design a polynomial-time algorithm to compute MFP when agents' preferences are full rankings.