Matching Markets: Theory and Practice

It has been almost half a century since David Gale and Lloyd Shapley published their pathbreaking paper “College admissions and the stability of marriage” in American Mathematical Monthly . It is hard to know whether Gale and Shapley expected the literature they initiated to be used to improve lives of masses of people all around the world. We are very fortunate to see that this is happening today. The model Gale and Shapley presented is very simple. A number of boys and girls have preferences for each other and would like to be matched. The question Gale and Shapley were especially interested in was whether there is a “stable” way to match each boy with a girl so that no unmatched pair can later find out that they can both do better by matching each other. They found that there indeed is such a stable matching, and they presented a deferred acceptance algorithm that achieves this objective. Versions of this algorithm are used today to match hospitals with residents and students with public schools in New York City and Boston. In 1974, Lloyd Shapley and Herbert Scarf published a somewhat related paper, “On Cores and Indivisibility,” in the first issue of the Journal of Mathematical Economics . Their model was arguably the simplest exchange economy one could think of. Each agent comes to the market with one indivisible good and seeks to trade it for possibly more preferred ones that might be brought by other agents. In their simple model agents are restricted to consume only one good. They were interested in whether a core allocation exists in their model. They showed that there indeed