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.

翻译：我们研究了多赢家选举中最小化度量扭曲的问题，即根据投票者的序数偏好从候选集合中选出规模为$k$的委员会。我们假设投票者和候选人嵌入在线度量中，社会成本由底层度量距离决定。投票规则的扭曲是指当选委员会的社会成本与最优委员会社会成本之间的最坏情况比率。先前的研究主要关注$q$成本模型，其中投票者的成本由其到委员会中第$q$近成员的距离给出。本文研究加法成本模型，其中投票者的成本为其到所有委员会成员距离的总和。我们提出了极性比较规则，并分析了其在功利主义加法成本下的扭曲。我们证明对于所有委员会规模$k>2$，该规则的扭曲至多为$2.33$，改进了先前已知的最佳上界$3$。此外，对于$k=2$和$k=3$，我们分别建立了紧致的扭曲下界$2.41$和$2.33$。我们还推导了依赖于$k$奇偶性的下界，并分析了小规模和大规模委员会下扭曲的行为特征。最后，我们将结果推广到平均主义加法成本模型。