Buyer–supplier games: Optimization over the core

In a buyer–supplier game, a special type of assignment game, a distinguished player, called the buyer, wishes to purchase some combinatorial structure. A set of players, called suppliers, offer various components of the structure for sale. Any combinatorial minimization problem can be transformed into a buyer–supplier game. While most previous work has been concerned with characterizing the core of buyer–supplier games, in this paper we study optimization over the set of core vectors. We give a polynomial time algorithm for optimizing over the core of any buyer–supplier game for which the underlying minimization problem is solvable in polynomial time. In addition, we show that it is hard to determine whether a given vector belongs to the core if the base minimization problem is not solvable in polynomial time. Finally, we introduce and study the concept of focus point price, which answers the question: If we are constrained to play in equilibrium, how much can we lose by playing the wrong equilibrium?