Substitutes for nonunitary many-to-many matching with contracts

We consider many-to-many matching with contracts where each agent can sign multiple contracts with the same agent. In other words, the unitarity condition is not assumed in this model. We offer a weaker substitutability condition called substitutability across doctors, which is a generalization of unilateral substitutability. We show that if every hospital’s choice function satisfies substitutability across doctors and every doctor’s choice function is responsive, then a stable outcome exists. We establish this result by generalizing the cumulative offer process. Moreover, we propose an even weaker substitutability condition that incorporates observability of doctors’ contract offers during the algorithm. The same algorithm guarantees the existence of a stable outcome even when substitutability across doctors is replaced with the weaker condition. We additionally present a specific class of choice functions satisfying the weaker condition that can be applied to a temporary staffing market, particularly in which temp agencies may assign multiple temporary workers to each client.