Matching with Contracts: Comment

Hatfield and Milgrom (2005) present a unified model of matching with contracts phrased in terms of hospitals and doctors, which subsumes the standard two-sided matching and some package auction models. They show that a stable allocation exists if contracts are substitutes for each hospital. They further claim that if a hospital's preferences violate the substitutes condition, there exist singleton preferences for the other hospitals and doctors such that no stable allocation exists. We show this last claim does not hold in general. We further present a weaker condition that is necessary to guarantee the existence of stable allocations.