Distortion of Multi-Winner Elections on the Line Metric: The Polar Comparison Rule

TL;DR

This paper introduces the Polar Comparison Rule for multi-winner elections on a line metric, achieving improved worst-case distortion bounds for selecting committees based on additive social costs.

Contribution

It proposes a new voting rule and provides tight bounds on its distortion, advancing understanding of metric distortion in multi-winner elections.

Findings

Achieves a maximum distortion of 2.33 for all committee sizes greater than 2.

Establishes tight bounds of 2.41 for k=2 and 2.33 for k=3.

Provides lower bounds on distortion depending on the parity of k.

Abstract

We study the problem of minimizing metric distortion in multi-winner elections, where a committee of size $k$ is selected from a set of candidates based on voters' ordinal preferences. We assume that voters and candidates are embedded on a line metric, and social cost is determined by the underlying metric distances. The distortion of a voting rule is the worst-case ratio between the social cost of the elected committee and an optimal committee. Previous work has focused on the $q$-cost model, in which a voter's cost is given by the distance to their $q$th closest committee member. Here, we study the additive cost, where a voter's cost is the sum of distances to all committee members. We introduce the Polar Comparison Rule and analyze its distortion under utilitarian additive cost. We show that it achieves a distortion of at most $2.33$ for all committee sizes $k>2$, improving upon the previously best-known upper bound of $3$. Moreover, for $k=2$ and $k=3$, we establish tight distortion bounds of $2.41$ and $2.33$, respectively. We also derive lower bounds that depend on the parity of $k$ and analyze the behavior of distortion for small and large committee sizes. Finally, we extend our results to the egalitarian additive cost.