Many-to-one matching: Externalities and stability

In this paper, we establish sufficient conditions on the domain of preferences and agents' behavior in order to characterize the existence of stable assignments in many-to-one matching problems with externalities. The set of stable matchings depends on what agents believe other agents will do if they deviate. Such sets of reactions are called estimation functions or simply estimations. We show that, unless some restrictions would be imposed on agents' preferences, there is no constraint on agents' behavior that assures the existence of stable matchings. In addition, we introduce a condition on preferences called bottom q-substitutability that guarantees the existence of at least one stable matching when the set of estimations includes all possible matches. Finally, we analyze a notion of the core and its relation with the set of stable assignments.