Distortion of multi-winner elections on the line metric: The polar comparison rule

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 qth 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 upper bound of almost 2.33 for all committee sizes k > 4, 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.