Non-revelation mechanisms for many-to-many matching: Equilibria versus stability

We study the subgame perfect Nash equilibria of the induced game. We prove that stable matchings are equilibrium outcomes if all A-agents' preferences are substitutable. We also show that the implementation of the set of stable matchings is closely related to the quotas of the A-agents. In particular, implementation holds when A-agents' preferences are substitutable and their quotas are non-binding.