Take-it-or-leave-it contracts in many-to-many matching markets

We study a class of sequential non-revelation mechanisms in which hospitals make simultaneous take-it-or-leave-it offers to doctors. We prove that all pure strategy subgame perfect Nash equilibrium outcomes are stable, but the inclusion of contracts shrinks the set of equilibrium outcomes. Our analysis reveals the existence of an advantage in setting the terms of the relationship that is absent from the model without contracts. The mechanisms of this class are outcome equivalent and implement the set of stable allocations in subgame perfect Nash equilibrium when competitive rivalry is present. The equilibrium outcomes form a lattice when preferences are substitutable.