Inventory Competition Under Dynamic Consumer Choice

Abstract

We analyze a model of inventory competition among n firms that provide competing, substitutable goods. Each firm chooses initial inventory levels for their good in a single period (newsboy-like) inventory model. Customers choose dynamically based on current availability, so the inventory levels at one firm affect the demand of all competing firms. This creates a strategic interaction among the firms' inventory decisions. Our work extends earlier work on variations of this problem by Karjalainen (1992), Lippman and McCardle (1997) and Parlar (1988). Specifically, we model demand in a more realistic way as a stochastic sequence of heterogeneous consumers who choose dynamically from among the available goods (or choose not to purchase) based on a utility maximization criterion. We also use a sample path analysis, so minimal assumptions are imposed on this demand process. We characterize the Nash equilibrium of the resulting stocking game and prove it is unique in the symmetric case. We show there is a bias toward overstocking caused by competition; specifically, reducing the quantity stocked at any equilibrium of the game increases total system profits, and at any joint-optimal set of stocking levels, each firm has an individual incentive to increase its own stock. For the symmetric case, we show that as the number of competing firms increases, the overstocking becomes so severe that total system (and individual firm) profits approach zero. Finally, we propose a stochastic gradient algorithm for computing equilibria and provide several numerical examples.