Contract Design and Stability in Matching Markets

We develop a model of many-to-many matching with contracts which subsumes as special cases many-to-many matching markets and buyer-seller markets with heterogeneous and indivisible goods. In our setting, in contrast to results for the setting of many-to-one matching with contracts, preference substitutability is sufficient and necessary to guarantee the existence of a nonempty lattice of stable allocations. Several fundamental structural results, such as the rural hospitals theorem, extend to our setting. We apply these results to derive a novel condition on preferences, weaker than substitutability, that is sufficient for the existence of a lattice of stable allocations in the context of many-to-one matching with contracts; extensions of this condition yield rural hospitals and strategy-proofness results. We also develop a theory of language for many-tomany matching markets with contracts, and show a natural tradeoff between expressiveness and stability: bundling makes a contractual language less expressive, but encourages substitutability of agents’ preferences over contracts. JEL Codes: C78, C62, D4, L14