Designing Matching Mechanisms under Constraints: An Approach from Discrete Convex Analysis

We consider two-sided matching problems where agents on one side of the market (hospitals) are required to satisfy certain distributional constraints. We show that when the preferences and constraints of the hospitals can be represented by an M-natural-concave function, (i) the generalized Deferred Acceptance mechanism is strategyproof for doctors, (ii) it produces the doctor-optimal stable matching, and (iii) its time complexity is proportional to the square of the number of possible contracts. Furthermore, we provide sufficient conditions for representation by an M-natural-concave function. These conditions are applicable to various existing works and enable new applications as well, thereby providing a recipe for developing desirable mechanisms in practice.