Matching algorithms have been classically studied with the goal of finding stable solutions. However, in many important societal problems, the degree of fairness in the matching assumes crucial importance, for instance when we have to match COVID-19 patients to care units. We study the problem of finding a stable many-to-one matching while satisfying fairness among all the agents with cardinal utilities. We consider various fairness definitions from fair allocation literature, such as envy-freeness (EF) and leximin optimal fairness. We find that EF and its weaker versions are incompatible with stability, even under a restricted setting with isometric utilities. We focus on leximin optimal fairness and show that finding such a matching is NP-Hard, even under isometric utilities. Next, we narrow our focus onto ranked isometric utilities and provide a characterisation for the space of stable matchings. We present a novel and efficient algorithm that finds the leximin optimal stable matching under ranked isometric utilities. To the best of our knowledge, we are the first to address the problem of finding a leximin optimally fair and stable matching.

中文翻译：

---

在多对一匹配中实现公平和稳定

匹配算法的经典研究旨在寻找稳定的解决方案。但是，在许多重要的社会问题中，匹配的公平程度至关重要，例如当我们必须将 COVID-19 患者与护理单位进行匹配时。我们研究了在满足所有具有基数效用的代理之间的公平性的同时找到稳定的多对一匹配的问题。我们考虑了公平分配文献中的各种公平定义，例如无嫉妒（EF）和 leximin 最优公平。我们发现 EF 及其较弱的版本与稳定性不兼容，即使在具有等距实用程序的受限设置下也是如此。我们专注于 leximin 最优公平性，并表明即使在等距效用下，找到这样的匹配也是 NP-Hard 的。下一个，我们将注意力集中在排序的等距效用上，并提供了稳定匹配空间的表征。我们提出了一种新颖有效的算法，该算法可在等距效用排序下找到 leximin 最优稳定匹配。据我们所知，我们是第一个解决寻找 leximin 最佳公平和稳定匹配问题的人。