Matching with Contracts: Comment

摘要

The theory of two-sided matching markets has interested researchers for its theoretical appeal and its relevance to the design of real-world institutions. The medical residency matching mechanism in the United States, the National Resident Matching Program (NRMP), and student assignment systems in New York City and Boston are important mechanisms designed by economists using the theory. Hatfield and Paul R. Milgrom (2005) present a unified framework of matching with contracts, which includes the two-sided matching and package auction models as well as a simplified version of a labor market model of Alexander Kelso and Vincent P. Crawford (1982) as special cases. They introduce the substitutes condition of contracts, which is an adaptation of the substitutability condition in the matching literature (Roth and Marilda A. Oliveira Sotomayor 1990) to matching with contracts. They show that there exists a stable allocation if contracts are substitutes for hospitals, and that a number of other results in matching generalize to problems with contracts. They further claim that the substitutes condition on individual hospitals' preferences is necessary to guarantee the existence of a stable allocation for all possible singleton preferences of others: that is, if contracts are not substitutes for a hospital, then there exists a preference profile of other hospitals with single job openings and doctors such that there exists no stable allocation.