Introduction
This chapter is devoted to the analysis of a two-sided matching market that consists of two sets of non-overlapping agents. The major objective here is to discuss the possibility of matching a set of agents with another set of agents. For instance, in a marriage problem, a set of men and a set of women need to be matched in pairs.
Such a market differs from a standard commodity market in which market price determines whether a person is a buyer or a seller. For example, a person may be a buyer of a good at some price and a seller of another good at some other price—the market is not two-sided. Additional examples of matching problems include: firms have to be matched with workers, hospitals have to be matched with interns, colleges have to admit students and football players require matching with clubs. ‘The term matching refers to the bilateral nature of exchange in these markets—for example, if I work for some firm, then that firm employs me’ (Roth and Sotomayor 1990, p.1). These markets are definitely different from markets for goods in which a person may be buyer of one good (say, potato) and a seller of another good (say, rice).
The matching theory is a leading area in economic theory because of its importance and also because of the difficulties involved in the allocation of indivisible resources. The appropriate tools for analysis are linear programming and combinatorics. In recent years, it has become quite popular because of applications game theory to study matching problems.
One very important problem in the analysis of matching problems is stability. The problem is to find a stable matching between two sets of agents given a set of preferences for each agent. An allocation where no person will make any gain from a further exchange is called stable. In their pioneering contribution, Gale and Shapley (1962) defined a matching problem and the concept of stable matching. They also showed that stable matchings always exist and suggested an algorithm for computing stable matchings.
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