# The Metric Distortion of Multiwinner Voting

arXiv: https://arxiv.org/abs/2201.13332 · HTML: https://mlsanity.com/paper/2201.13332/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 31 Jan 2022.

## Built on by (most cited first, 7 of 7)

- [Proportional Fairness in Clustering: A Social Choice Perspective](https://mlsanity.com/paper/2310.18162/) (2023)
- [Plurality Veto: A Simple Voting Rule Achieving Optimal Metric Distortion](https://mlsanity.com/paper/2206.07098/) (2022)
- [Best of Both Distortion Worlds](https://mlsanity.com/paper/2305.19453/) (2023)
- [Breaking the Metric Voting Distortion Barrier](https://mlsanity.com/paper/2306.17838/) (2023)
- [Proportional Representation in Metric Spaces and Low-Distortion Committee Selection](https://mlsanity.com/paper/2312.10369/) (2023)
- [Full Proportional Justified Representation](https://mlsanity.com/paper/2501.12015/) (2025)
- [Compatibility of Max and Sum Objectives for Committee Selection and k-Facility Location](https://mlsanity.com/paper/2507.17063/) (2025)

## Abstract (arXiv)

We extend the recently introduced framework of metric distortion to multiwinner voting. In this framework, n agents and m alternatives are located in an underlying metric space. The exact distances between agents and alternatives are unknown. Instead, each agent provides a ranking of the alternatives, ordered from the closest to the farthest. Typically, the goal is to select a single alternative that approximately minimizes the total distance from the agents, and the worst-case approximation ratio is termed distortion. In the case of multiwinner voting, the goal is to select a committee of k alternatives that (approximately) minimizes the total cost to all agents. We consider the scenario where the cost of an agent for a committee is her distance from the q-th closest alternative in the committee. We reveal a surprising trichotomy on the distortion of multiwinner voting rules in terms of k and q: The distortion is unbounded when q ≤ k/3, asymptotically linear in the number of agents when k/3 < q ≤ k/2, and constant when q > k/2.