Matching with regional constraints: An equivalence

In two-sided matching market, when the regional constraints are present, the deferred acceptance (DA) algorithm suffers from undesirable inefficiency due to the artificial allocation of the regional caps among hospitals. We show that, given preferences, there exist allocations that guarantee the efficiency of the DA algorithm. Furthermore, it is equivalent to the FDA algorithm developed by Kamada and Kojima (2015), which endows the latter with an interpretation as a tool for endogenous capacity design. Our proof applies the optimality within the matching with contracts (Hatfield and Milgrom 2005) framework, offering a broadly applicable method for establishing equivalence among DA-based mechanisms.