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.

翻译：本文研究在实数线上最小化群体层面不公平性的设施选址博弈问题，其中代理属于不同群体且可能采取策略行为。我们探讨了以公平为导向的目标，即最小化Marsh与Schilling（1994）提出的最大群体效应。每个群体的效应定义为该群体到最近设施的总距离或最大距离，并通过群体特定因子加权。我们证明该公式推广了多个重要的优化目标，包括经典的功利主义（社会成本）和平等主义（最大成本）目标，以及两种群体公平目标——最大群体总成本与平均群体成本。为最小化最大群体效应，我们首先针对单设施情形提出两种新颖机制：BALANCED机制与MAJOR-PHANTOM机制。两者均满足策略防护性，并在不同最大群体效应目标公式下达到紧致的近似保证。我们的机制不仅弥补了Zhou、Li与Chan（2022）指出的群体公平目标近似界现有空白，还将许多经典真实机制统一于更广泛的公平感知框架中。针对双设施情形，我们重新审视并扩展经典端点机制至广义设定，证明其能为两种不同的最大群体效应目标提供紧致界。