Minimizing Inequity in Facility Location Games

TL;DR

This work studies fairness in facility location games on the real line by introducing the maximum group effect (mge), a group-centric objective that bounds the worst-case weighted burden across predefined groups. It unifies several classical objectives by allowing two instantiations of the group cost, ${\tt wTGC}$ and ${\tt wMGC}$, and designs strategyproof mechanisms tailored to these metrics. For the single-facility case, it presents BALANCED (2-approx for wTGC) and MAJOR-PHANTOM (2-approx for wMGC), achieving tight bounds and unifying classical truthful mechanisms under a single fairness-aware framework. In the two-facility setting, ENDPOINT is shown to provide tight bounds for both wTGC and wMGC, while for $k\ge 3$ deterministic SP mechanisms cannot attain bounded approximations. The results bridge efficiency and fairness in FLGs and point to future work on randomized mechanisms and higher-dimensional extensions.

Abstract

This paper studies the problem of minimizing group-level inequity in facility location games on the real line, where agents belong to different groups and may act strategically. We explore a fairness-oriented objective that minimizes the maximum group effect introduced by Marsh and Schilling (1994). Each group's effect is defined as its total or maximum distance to the nearest facility, weighted by group-specific factors. We show that this formulation generalizes several prominent optimization objectives, including the classical utilitarian (social cost) and egalitarian (maximum cost) objectives, as well as two group-fair objectives, maximum total and average group cost. In order to minimize the maximum group effect, we first propose two novel mechanisms for the single-facility case, the BALANCED mechanism and the MAJOR-PHANTOM mechanism. Both are strategyproof and achieve tight approximation guarantees under distinct formulations of the maximum group effect objective. Our mechanisms not only close the existing gap in approximation bounds for group-fairness objectives identified by Zhou, Li, and Chan (2022), but also unify many classical truthful mechanisms within a broader fairness-aware framework. For the two-facility case, we revisit and extend the classical endpoint mechanism to our generalized setting and demonstrate that it provides tight bounds for two distinct maximum group effect objectives.

Figures (1)

• Figure 1: $\mathsf{mge}$: A Systematic Generalization of Objective Metrics.

Theorems & Definitions (30)

• Definition 2.1: Strategyproofness (SP)
• Definition 2.2: Approximation Ratio
• Proposition 3.1
• Theorem 3.2
• proof
• Corollary 3.3: ProcacciaT09
• Corollary 3.4
• Lemma 3.5
• Theorem 3.6
• proof