Stable matching is a fundamental problem studied both in economics and computer science. The task is to find a matching between two sides of agents that have preferences over who they want to be matched with. A matching is stable if no pair of agents prefer each other over their current matches. The deferred acceptance algorithm of Gale and Shapley solves this problem in polynomial time. Further, it is a mechanism: the proposing side in the algorithm is always incentivised to report their preferences truthfully. The deferred acceptance algorithm has a natural interpretation as a distributed algorithm (and thus a distributed mechanism). However, the algorithm is slow in the worst case and it is known that the stable matching problem cannot be solved efficiently in the distributed setting. In this work we study a natural special case of the stable matching problem where all agents on one side share common preferences. We show that in this case the deferred acceptance algorithm does yield a fast and truthful distributed mechanism for finding a stable matching. We show how algorithms for sampling random colorings can be used to break ties fairly and extend the results to fractional stable matching.

翻译：稳定匹配是经济学和计算机科学中共同研究的一个基本问题。其任务是寻找代理双方之间的一种匹配，双方对希望与谁匹配具有偏好。当不存在任何一对代理更倾向于彼此而非当前匹配对象时，该匹配是稳定的。Gale-Shapley 的延迟接受算法能够在多项式时间内解决这一问题。此外，该算法是一种机制：算法中的提议方始终有激励真实报告其偏好。延迟接受算法具有一种自然的分布式算法（进而也是分布式机制）解释。然而，该算法在最坏情况下运行缓慢，且已知稳定匹配问题无法在分布式环境中高效求解。在本工作中，我们研究稳定匹配问题的一个自然特例：其中一方的所有代理共享共同偏好。我们证明，在此情况下，延迟接受算法确实能够为寻找稳定匹配提供一种快速且真实的分布式机制。我们还展示了如何利用随机着色采样算法来公平地打破平局，并将结果扩展到分数稳定匹配。