The Metric Distortion of Multiwinner Voting

TL;DR

This paper extends the metric distortion framework to multiwinner voting, revealing a trichotomy in the distortion behavior based on parameters, with implications for selecting committees in metric spaces.

Contribution

It introduces a novel analysis of metric distortion in multiwinner voting, uncovering a surprising trichotomy based on the parameters k and q.

Findings

Distortion is unbounded when q ≤ k/3.

Distortion is asymptotically linear in the number of agents when k/3 < q ≤ k/2.

Distortion is constant when q > k/2.

Abstract

We extend the recently introduced framework of metric distortion to multiwinner voting. In this framework, $n$ agents and $m$ alternatives are located in an underlying metric space. The exact distances between agents and alternatives are unknown. Instead, each agent provides a ranking of the alternatives, ordered from the closest to the farthest. Typically, the goal is to select a single alternative that approximately minimizes the total distance from the agents, and the worst-case approximation ratio is termed distortion. In the case of multiwinner voting, the goal is to select a committee of $k$ alternatives that (approximately) minimizes the total cost to all agents. We consider the scenario where the cost of an agent for a committee is her distance from the $q$-th closest alternative in the committee. We reveal a surprising trichotomy on the distortion of multiwinner voting rules in terms of $k$ and $q$: The distortion is unbounded when $q \leq k/3$, asymptotically linear in the number of agents when $k/3 < q \leq k/2$, and constant when $q > k/2$.