We focus on a simple, one-dimensional collective decision problem (often referred to as the facility location problem) and explore issues of strategyproofness and proportionality-based fairness. We introduce and analyze a hierarchy of proportionality-based fairness axioms of varying strength: Individual Fair Share (IFS), Unanimous Fair Share (UFS), Proportionality (as in Freeman et al, 2021), and Proportional Fairness (PF). For each axiom, we characterize the family of mechanisms that satisfy the axiom and strategyproofness. We show that imposing strategyproofness renders many of the axioms to be equivalent: the family of mechanisms that satisfy proportionality, unanimity, and strategyproofness is equivalent to the family of mechanisms that satisfy UFS and strategyproofness, which, in turn, is equivalent to the family of mechanisms that satisfy PF and strategyproofness. Furthermore, there is a unique such mechanism: the Uniform Phantom mechanism, which is studied in Freeman et al. (2021). We also characterize the outcomes of the Uniform Phantom mechanism as the unique (pure) equilibrium outcome for any mechanism that satisfies continuity, strict monotonicity, and UFS. Finally, we analyze the approximation guarantees, in terms of optimal social welfare and minimum total cost, obtained by mechanisms that are strategyproof and satisfy each proportionality-based fairness axiom. We show that the Uniform Phantom mechanism provides the best approximation of the optimal social welfare (and also minimum total cost) among all mechanisms that satisfy UFS.

翻译：我们聚焦于一个简单的一维集体决策问题（常称为设施选址问题），探讨防策略性与基于比例公平性的问题。我们引入并分析了一组强度递进的基于比例公平性的公理体系：个体公平份额（IFS）、一致公平份额（UFS）、比例性（Freeman等，2021年提出）和比例公平性（PF）。针对每条公理，我们刻画了满足该公理与防策略性的机制族。研究表明，施加防策略性约束使多数公理相互等价：满足比例性、一致性与防策略性的机制族等同于满足UFS与防策略性的机制族，进而也等同于满足PF与防策略性的机制族。此外，存在唯一的此类机制：即Freeman等（2021年）研究的统一幻影机制。我们进一步证明，对于任何满足连续性、严格单调性与UFS的机制，统一幻影机制产生的均衡结果是唯一的（纯策略）均衡结果。最后，我们分析了在满足防策略性及各比例公平性公理的机制下，关于最优社会福利与最小总成本的近似保证。研究表明，在所有满足UFS的机制中，统一幻影机制能提供最优社会福利（同时最小化总成本）的最佳近似比。