Inventory Control in a Duopoly: A Dynamic Non-Cooperative Game-Theoretic Approach

We consider a discrete-time infinite horizon game in which two suppliers of the same item compete for a single customer. In each time period, the customer chooses randomly one of the two suppliers and demands from him a random number of items. The probability of choosing one or the other supplier reflects the supplier’s goodwill level and depends on the service that the customer received in his previous order. The supplier chosen by the customer must deliver all the items of the demand immediately. If he does not have all the items in stock, then he must order the missing items and deliver them to the customer in the next period. After the items of the demand that are in stock are delivered to the customer, each supplier must decide how much to order for the next period, given his as well as his competitor’s inventory and goodwill level at the end of the current period. The goal of each supplier is to maximize his long-term expected average profit, which equals his sales revenue minus his ordering and inventory holding costs. We formulate this problem as an infinite dynamic non-cooperative two-player game with imperfect information, since in each period each player knows the competitors ’ state (inventory and goodwill level) but not the competitor’s control (order quantity). We use value iteration to solve the resulting intertwined dynamic programming equations characterizing the optimal ordering policy for each supplier, where in each iteration, we solve a Nash equilibrium problem. We demonstrate this methodology with a numerical example and we comment on the results