We consider the obnoxious facility location problem (in which agents prefer the facility location to be far from them) and propose a hierarchy of distance-based proportional fairness concepts for the problem. These fairness axioms ensure that groups of agents at the same location are guaranteed to be a distance from the facility proportional to their group size. We consider deterministic and randomized mechanisms, and compute tight bounds on the price of proportional fairness. In the deterministic setting, we show that our proportional fairness axioms are incompatible with strategyproofness, and prove asymptotically tight $\epsilon$-price of anarchy and stability bounds for proportionally fair welfare-optimal mechanisms. In the randomized setting, we identify proportionally fair and strategyproof mechanisms that give an expected welfare within a constant factor of the optimal welfare. Finally, we prove existence results for two extensions to our model.

翻译：本文研究厌恶型设施选址问题（即智能体偏好设施位置远离自身），并提出基于距离的比例公平性概念层次结构。该公平性公理确保位于同一位置的智能体群体能够获得与其群体规模成比例的设施距离保障。我们分别研究确定性机制与随机化机制，并计算比例公平性的严格价格界。在确定性场景中，我们证明所提出的比例公平性公理与防策略性存在互斥性，并为比例公平的福利最优机制建立了渐近紧致的$\epsilon$-无政府价格与稳定价格边界。在随机化场景中，我们构建了兼具比例公平性与防策略性的机制，其期望福利与最优福利保持常数因子关系。最后，针对模型的两个扩展形式，我们证明了相应机制的存在性定理。