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arXiv · 2411.13720

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

Negar Babashah·Hasti Karimi·Masoud Seddighin·Golnoosh Shahkarami
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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.

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Negar Babashah, Hasti Karimi, Masoud Seddighin, Golnoosh Shahkarami. 2026-02-17. Distortion of Multi-Winner Elections on the Line Metric: The Polar Comparison Rule. https://arxiv.org/abs/2411.13720

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