Substitutes and Stability for Many-to-Many Matching With Contracts

Several weakened substitutability conditions have been proved to be sufficient for stability in many-to-one matching with contracts and to be vital for understanding real-life applications. We examine the roles of these weakened conditions in many-to-many matching with contracts, in particular, weak observable substitutability and weak observable substitutability across doctors. We prove new existence theorems by a modified cumulative offer algorithm. It is remarkable that size monotonicity at the offer-proposing side is essential for the existence result under weak observable substitutability across doctors