Optimally Selecting Representative Agents from a Metric Space

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.