Computing Most Equitable Voting Rules

TL;DR

This paper develops quasipolynomial-time algorithms for computing the most equitable voting rules that satisfy fairness axioms, connecting the problem to graph isomorphism and automorphism complexities, and establishing new computational hardness results.

Contribution

It introduces efficient algorithms for fairness-preserving voting rules and links their complexity to graph isomorphism, providing the first GI/GA-complete results in computational social choice.

Findings

Quasipolynomial algorithms for equitable voting rules

Verification of fairness axioms is GI-complete

Tie-breaking method preserves fairness properties

Abstract

How to design fair and (computationally) efficient voting rules is a central challenge in Computational Social Choice. In this paper, we aim at designing efficient algorithms for computing most equitable rules for large classes of preferences and decisions, which optimally satisfy two fundamental fairness/equity axioms: anonymity (every voter being treated equally) and neutrality (every alternative being treated equally). By revealing a natural connection to the graph isomorphism problem and leveraging recent breakthroughs by Babai [2019], we design quasipolynomial-time algorithms that compute most equitable rules with verifications, which also compute verifications about whether anonymity and neutrality are satisfied at the input profile. Further extending this approach, we propose the canonical-labeling tie-breaking, which runs in quasipolynomial-time and optimally breaks ties to preserve anonymity and neutrality. As for the complexity lower bound, we prove that even computing verifications for most equitable rules is GI-complete (i.e., as hard as the graph isomorphism problem), and sometimes GA-complete (i.e., as hard as the graph automorphism problem), for many commonly studied combinations of preferences and decisions. To the best of our knowledge, these are the first problems in computational social choice that are known to be complete in the class GI or GA.