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 $q$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.

💡 Research Summary

The paper studies metric distortion in multi‑winner elections when both voters and candidates lie on a line (the 1‑Euclidean setting). The social cost is the additive (utilitarian) sum of distances from each voter to every member of the selected committee. The authors introduce a new deterministic voting rule, the Polar Comparison Rule (PCR), and analyze its worst‑case distortion.

The rule works as follows. For a committee of size two, it first picks the top‑ranked candidate of the median voter. Then it identifies the closest candidate on each side of the median that has not yet been chosen. The two “polar” candidates are compared by counting how many voters prefer one over the other; the rule selects the more popular one while giving a bias toward the side opposite the already chosen candidate. For size three, the same idea is applied a second time to add a third member. For general k, the authors iteratively apply the k=3 and k=2 procedures, which yields a committee that always contains candidates from both sides of the median.

Using a combination of structural observations (removing Pareto‑dominated candidates, determining the exact order of remaining candidates) and a novel “voter‑movement” technique (moving all voters toward a pivot point while stopping at rule‑specific obstacles), the authors bound the distortion of PCR. They prove:

• For any k > 2, distortion ≤ 7/3 ≈ 2.33, improving the previous best known bound of 3.
• For k = 2, distortion = 1 + √2 ≈ 2.41, and for k = 3, distortion = 7/3 ≈ 2.33; both bounds are tight.
• When k ≡ 1 (mod 3) the bound becomes 7/3 + 4(√2 − 4/3)/k, and when k ≡ 2 (mod 3) it becomes 7/3 + 2(√2 − 4/3)/k.

The paper also establishes lower bounds that depend on the parity of k. For odd k < m/2, distortion ≥ 2 + 1/k; for even k < m/2, distortion ≥ 1 + √(1 + 2/k); and for large committees (k ≥ m/2) distortion ≥ 1 + (m − k)/(3k − m). These lower bounds are shown by constructing two instances with identical preference profiles that any rule must treat differently.

Finally, the authors extend their analysis to the egalitarian additive cost (maximal voter cost). They prove that any committee consisting of candidates located between the two extreme voters achieves distortion exactly 2, and they give a simple rule that always selects such a committee.

Overall, the Polar Comparison Rule provides a simple, polynomial‑time method that dramatically reduces metric distortion for multi‑winner elections on the line, achieving near‑optimal performance for a wide range of committee sizes. The techniques introduced (Pareto‑domination pruning, voter‑movement with obstacles) may be useful for other metric‑based voting problems and suggest several directions for future work, including randomization, higher‑dimensional metrics, and alternative social cost functions.