Minimizing Inequity in Facility Location Games

This paper studies the problem of minimizing group-level inequity in facility location games on the real line, where agents are partitioned into groups and may act strategically. While most classical objectives focus on minimizing total or maximum individual cost, we consider a fairness-driven objective that minimizes the maximum group effect (Marsh and Schilling 1994), where 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 Procaccia and Tennenholtz (2009); and two group-fair objectives, maximum total/average group cost, proposed by Zhou, Li, and Chan (2022). To minimize the maximum group effect objective, for single-facility case, we propose novel BALANCED mechanism and Major-Phantom mechanism, both of which are strategyproof and provide tight approximation guarantees when considering distinct maximum group effect objectives. Our mechanisms not only resolve the bound gap for group-fairness objectives studied by Zhou, Li, and Chan (2022) but also unify many classical truthful mechanisms with respect to classical optimization objectives. For the two-facility case, we revisit the classical endpoint mechanism in our more generalized scenario and show that the mechanism is powerful to provide tight bounds for two distinct maximum group effect objectives.