Capacity games with supply function competition

We introduce a general model for suppliers competing for a buyer’s procurement business. The buyer faces uncertain demand, and there is a requirement to reserve capacity in advance of knowing the demand. Each supplier has costs that are two-dimensional, with some capacity costs incurred prior to production and some production costs incurred at the time of delivery. These costs are general functions of quantity, and this naturally leads us to a supply function competition framework in which each supplier offers a schedule of prices and quantities. We show that there is an equilibrium of a particular form: the buyer makes a reservation choice that maximizes the overall supply chain profit, each supplier makes a profit equal to their marginal contribution to the supply chain, and the buyer takes the remaining profit. This is a natural equilibrium for the suppliers to coordinate on, since no supplier can do better in any other equilibrium. These results make use of a submodularity property for the supply chain optimal profits as a function of the suppliers available and build on the assumption that the buyer breaks a tie in favor of the solutions that give the largest supply chain profit. We demonstrate the applications of our model in three operations management problems: a newsvendor problem with unreliable suppliers, a portfolio procurement problem with supply options and a spot market, and a bundling problem with nonsubstitutable products