Relating Metric Distortion and Fairness of Social Choice Rules

TL;DR

This paper explores the relationship between metric distortion and fairness in social choice rules, showing that certain rules like STV and Recursive Copeland have bounded distortion and fairness ratios, linking efficiency and fairness.

Contribution

It establishes a close connection between distortion and fairness ratios, extending these notions to set-based rules and providing bounds for various voting rules.

Findings

STV has an $O( ext{log } m)$ fairness ratio.

Recursive Copeland achieves distortion 5 and fairness ratio 7.

Distortion and fairness are within an additive factor of 2 for any rule.

Abstract

One way of evaluating social choice (voting) rules is through a utilitarian distortion framework. In this model, we assume that agents submit full rankings over the alternatives, and these rankings are generated from underlying, but unknown, quantitative costs. The \emph{distortion} of a social choice rule is then the ratio of the total social cost of the chosen alternative to the optimal social cost of any alternative; since the true costs are unknown, we consider the worst-case distortion over all possible underlying costs. Analogously, we can consider the worst-case \emph{fairness ratio} of a social choice rule by comparing a useful notion of fairness (based on approximate majorization) for the chosen alternative to that of the optimal alternative. With an additional metric assumption -- that the costs equal the agent-alternative distances in some metric space -- it is known that the Copeland rule achieves both a distortion and fairness ratio of at most 5. For other rules, only bounds on the distortion are known, e.g., the popular Single Transferable Vote (STV) rule has distortion $O(\log m)$, where $m$ is the number of alternatives. We prove that the distinct notions of distortion and fairness ratio are in fact closely linked -- within an additive factor of 2 for any voting rule -- and thus STV also achieves an $O(\log m)$ fairness ratio. We further extend the notions of distortion and fairness ratio to social choice rules choosing a \emph{set} of alternatives. By relating the distortion of single-winner rules to multiple-winner rules, we establish that Recursive Copeland achieves a distortion of 5 and a fairness ratio of at most 7 for choosing a set of alternatives.