  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.
