In an earlier work (Kamada and Kojima 2015), we introduced “matching with constraints,” a two-sided matching problem in which the market is subject to feasibility constraints. That paper proposed and analyzed a possible solution to this problem and discussed various real-market applications of our results. The present paper reorganizes some of the findings from our more recent effort, based on Kamada and Kojima (2016a, b).

Many real matching markets are subject to distributional constraints. These constraints often take the form of restrictions on the numbers of agents on one side of the market matched to certain subsets on the other side. Real-life examples include restrictions on regions in medical matching, academic master's programs in graduate admission, and state-financed seats for college admission. Motivated by these markets, we study design of matching mechanisms under distributional constraints. We show that existing matching mechanisms suffer from inefficiency and instability, and propose a mechanism that is better in terms of efficiency, stability, and incentives while respecting the distributional constraints.

Real matching markets are subject to constraints. For example, the Japanese government introduced a new medical matching system in 2009 that imposes a “regional cap” in each of its 47 prefectures, which regulates the total number of medical residents who can be employed in each region. Based on Kamada and Kojima (2011), this paper studies matching markets with such constraints by examining in great detail the Japanese medical matching market. Specifically, we show that the new system introduced in 2009 has problems in terms of stability and strategy-proofness, and provide an alternative mechanism that does better.