Distortion of Multi-Winner Elections on the Line Metric: The Polar Comparison Rule
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arXiv
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| Format: | Preprint |
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2024
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| author | Babashah, Negar Karimi, Hasti Seddighin, Masoud Shahkarami, Golnoosh |
| author_facet | Babashah, Negar Karimi, Hasti Seddighin, Masoud Shahkarami, Golnoosh |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_13720 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Distortion of Multi-Winner Elections on the Line Metric: The Polar Comparison Rule Babashah, Negar Karimi, Hasti Seddighin, Masoud Shahkarami, Golnoosh Computer Science and Game Theory 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. |
| title | Distortion of Multi-Winner Elections on the Line Metric: The Polar Comparison Rule |
| topic | Computer Science and Game Theory |
| url | https://arxiv.org/abs/2411.13720 |