The theory of two-sided matching has been extensively developed and applied to many real-life application domains. As the theory has been applied to increasingly diverse types of environments, researchers and practitioners have encountered various forms of distributional constraints. Arguably, the most general class of distributional constraints would be hereditary constraints; if a matching is feasible, then any matching that assigns weakly fewer students at each college is also feasible. However, under general hereditary constraints, it is shown that no strategyproof mechanism exists that simultaneously satisfies fairness and weak nonwastefulness, which is an efficiency (students' welfare) requirement weaker than nonwastefulness. We propose a new strategyproof mechanism that works for hereditary constraints called the Multi-Stage Generalized Deferred Acceptance mechanism (MS-GDA). It uses the Generalized Deferred Acceptance mechanism (GDA) as a subroutine, which works when distributional constraints belong to a well-behaved class called hereditary M$^\natural$-convex set. We show that GDA satisfies several desirable properties, most of which are also preserved in MS-GDA. We experimentally show that MS-GDA strikes a good balance between fairness and efficiency (students' welfare) compared to existing strategyproof mechanisms when distributional constraints are close to an M$^\natural$-convex set.

翻译：双边匹配理论已得到广泛发展，并应用于许多现实场景。随着该理论被应用于日益多样化的环境类型，研究人员和实践者遇到了各种形式的分配约束。可以说，最一般的分配约束类别是遗传约束：如果一个匹配是可行的，那么任何在各院校分配的学生数量不高于该匹配的匹配也是可行的。然而，在一般遗传约束下，已证明不存在同时满足公平性和弱非浪费性的策略证明机制，其中弱非浪费性是一种比非浪费性更弱的效率（学生福利）要求。我们提出了一种新的适用于遗传约束的策略证明机制，称为多阶段广义延迟接受机制（MS-GDA）。它使用广义延迟接受机制（GDA）作为子程序，后者适用于分配约束属于一类称为遗传M$^\natural$-凸集的性质良好的约束的情况。我们证明GDA满足若干理想性质，其中大部分性质在MS-GDA中也得以保持。实验表明，当分配约束接近M$^\natural$-凸集时，与现有策略证明机制相比，MS-GDA在公平性和效率（学生福利）之间取得了良好的平衡。