Proportionality is an attractive fairness concept that has been applied to a range of problems including the facility location problem, a classic problem in social choice. In our work, we propose a concept called Strong Proportionality, which ensures that when there are two groups of agents at different locations, both groups incur the same total cost. We show that although Strong Proportionality is a well-motivated and basic axiom, there is no deterministic strategyproof mechanism satisfying the property. We then identify a randomized mechanism called Random Rank (which uniformly selects a number $k$ between $1$ to $n$ and locates the facility at the $k$'th highest agent location) which satisfies Strong Proportionality in expectation. Our main theorem characterizes Random Rank as the unique mechanism that achieves universal truthfulness, universal anonymity, and Strong Proportionality in expectation among all randomized mechanisms. Finally, we show via the AverageOrRandomRank mechanism that even stronger ex-post fairness guarantees can be achieved by weakening universal truthfulness to strategyproofness in expectation.

翻译：相称性是一个具有吸引力的公平性概念,适用于一系列问题,包括设施地点问题,这是社会选择中的一个典型问题。我们在工作中提出一个名为“强烈比例性”的概念,确保在不同地点有两组人员时,两组人员的费用总额相同。我们表明,虽然强烈比例性是一个动机良好和基本的轴心,但没有满足财产需要的确定性战略防范机制。然后我们确定一个随机机制,称为随机级(它统一选择了1美元至10美元之间的一个数字,并将设施设在美元至美元的最高代理地点),这符合预期的强烈比例性。我们的主要理论将随机级作为实现普遍真实性、普遍匿名和所有随机化机制之间期望的强烈比例性的独特机制。最后,我们通过OverandomRank机制表明,通过削弱普遍真实性以战略对预期的正确性,可以实现更强有力的事后公平性保障。