Optimizing exclusivity agreements in a three-stage procurement game

We consider a scenario in which two companies compete for limited capacity among a set of suppliers, where one of the companies may purchase exclusivity rights from any of the suppliers. This results in a three-stage game in which the first company chooses which exclusivity rights to purchase, the second company attempts to satisfy its demand using the capacity of remaining suppliers, and the first company then satisfies its demand from the remaining capacity. The goal of the first company is to minimize its total procurement costs while the goal of the second company is to maximize the first company's minimum procurement costs. The problem is complicated by the presence of concave quantity discounts offered by suppliers. We formulate a three-stage mixed-integer program that is well suited to decomposition techniques and develop a cutting-plane algorithm for its solution.