Vincent Crawford (1991), "Comparative Statics in Matching Markets" Journal of Economic Theory, 54: 389-400.
Perhaps the most important advantage of the matching approach is its robustness to heterogeneity. A traditional competitive equilibrium cannot exist in general unless the goods traded in each market are homogeneous, because all goods in the same market must sell at the same price. A traditional model of a labor market with the degree of heterogeneity normally encountered therefore has the structure of a multi-market general equilibrium model. But because the markets in such a model are very thin, the usual arguments in support of price-taking are strained. The theory of matching markets replaces this collection of thin markets with a single market game, in which the terms of partnerships are determined endogenously, along with the matching, via negotiations between prospective partners. Gale and Shapley's notion of stability(*), suitable generalized, formalizes the idea of competition, and thereby makes it possible to evaluate the robustness of traditional competitive analysis to heterogeneity. (Stable outcomes in matching markets can in fact be viewed as traditional competitive equilibria when prices are allowed to reflect the differences between matches; see, for example, Shapley and Shubik, 1972(**))
The author, Vince Crawford, who is known as a leading researcher in game theory has written a few influential papers on matching theory. Especially, the following two are of great importance since they initiated the area of (many-to-one) matching with monetary transfers.
"Job Matching with Heterogeneous Firms and Workers"
with Elsie Marie Knoer, Econometrica, Vol. 49(2): 437-450, 1981.
"Job Matching, Coalition Formation, and Gross Substitutes"
with Alexander S. Kelso, Jr., Econometrica, Vol. 50(6): 1483-1504, 1982.
* Gale and Shapley (1962) "College Admissions and the Stability of Marriage" American Mathematics Monthly, 69: 9-15.
** Shapley and Shubik (1972) "The Assignment Game. 1. The Core" International Journal of Game Theory, 1: 111-130.