# Optimally Selecting Representative Agents from a Metric Space

arXiv: https://arxiv.org/abs/2608.29097 · HTML: https://mlsanity.com/paper/2608.29097/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 18 references that are in the map.
- First posted on arXiv 29 Aug 2026.

## Builds on (most cited first, 9 of 9)

- [Proportionally Fair Clustering](https://mlsanity.com/paper/1905.03674/) (2019)
- [Proportional Fairness in Clustering: A Social Choice Perspective](https://mlsanity.com/paper/2310.18162/) (2023)
- [Proportionally Representative Clustering](https://mlsanity.com/paper/2304.13917/) (2023)
- [Proportional Fairness in Non-Centroid Clustering](https://mlsanity.com/paper/2410.23273/) (2024)
- [Boosting Sortition via Proportional Representation](https://mlsanity.com/paper/2406.00913/) (2024)
- [Unifying Proportional Fairness in Centroid and Non-Centroid Clustering](https://mlsanity.com/paper/2601.00447/) (2026)
- [Plurality in Spatial Voting Games with constant β](https://mlsanity.com/paper/2005.04799/) (2020)
- [The Core in Max-Loss Non-Centroid Clustering Can Be Empty](https://mlsanity.com/paper/2511.19107/) (2025)
- [Fair Transit Stop Placement: A Clustering Perspective and Beyond](https://mlsanity.com/paper/2602.06776/) (2026)

## Abstract (arXiv)

This paper studies the problem of proportionally fair clustering, where the goal is to select k ``centers'' from a metric space that fairly represent a set of agents who also lie in the metric space. Specifically, we focus on finding a clustering satisfying a fairness property known as the Droop core. In the practical special case in which the set of feasible center locations contains every agent location, the previous best-known result guaranteed a (1 + √2)-approximation of the Droop core, while the best-known lower bound was 2. In this paper, we show that this lower bound is tight and that a clustering in the 2-Droop core always exists. Further, we show that such a clustering can be achieved by only selecting centers from locations in the metric space where an agent resides. We establish this using Scarf's theorem guaranteeing a nonempty core for balanced non-transferable utility games. This result has several interesting corollaries. Most notably, it resolves the β-plurality problem of Aronov et al. [2021] for general metric spaces. The main result of this paper was generated by mathttChatGPT-mathtt5.6-mathttSol through a series of interactions with the authors. The authors of this paper verified the generated proof and rewrote it for clarity.