Many-to-one matching markets exist in numerous different forms, such as college admissions, matching medical interns to hospitals for residencies, assigning housing to college students, and the classic firms and workers market. In the these markets, externalities such as complementarities and peer effects severely complicate the preference ordering of each agent. Further, research has shown that externalities lead to serious problems for market stability and for developing efficient algorithms to find stable matchings.In this paper we make the observation that peer effects are often the result of underlying social connections, and we explore a formulation of the many-to-one matching market where peer effects are derived from an underlying social network. Our model captures peer effects and complementarities using utility functions rather than traditional preference ordering. With this model and considering pairwise stability, we prove that stable matchings always exist and characterize the set of stable matchings in terms of social welfare. We also give distributed algorithms that are guaranteed to converge to a stable matching. To asses the competitive ratio of these algorithms and to more generally characterize the efficiency of matching markets with externalities, we prove general bounds on how far the welfare of the worst-case stable matching can be from the welfare of the optimal matching, and find that the structure of the social network (e.g. how well clustered the network is) plays a large role.

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