Substitution, Complementarity, and Stability

We provide necessary and sufficient conditions for a non-empty core in many-to-one assignment games. When players on the "many" side (buyers) are substitutes with respect to any given player on the other side (firms), we show that non-emptiness requires an additional condition that limits the competition among the buyers. When buyers are complements with respect to any given firm, a sufficient condition for non-emptiness is that buyers also be complements with respect to all of the firms, collectively. A necessary condition is that no firm can he guaranteed a profit when the core is non-empty.