On Markov Equilibria in Dynamic Inventory Competition

We provide a review of the types of equilibria typically found in operations management inventory papers and a discussion on when the commonly used stationary infinite-horizon (open-loop) equilibrium may be sufficient for study. We focus particularly on order-up-to and basestock equilibria in the context of inventory duopolies. We give conditions under which the stationary infinite-horizon equilibrium is also a Markov perfect (closed-loop) equilibrium. These conditions are then applied to three specific duopolies. The first application is one with stockout-based substitution, where the firms face independent direct demand but some fraction of a firm’s lost sales will switch to the other firm. The second application is one where shelf-space display stimulates primary demand and reduces demand for the other firm’s product. The final application is one where the state variables represent goodwill rather than inventory. These specific problems have been previously studied in both the single period and/or stationary infinite-horizon (open-loop) settings but not in Markov perfect (closed-loop) settings. Under the Markov perfect setting, a variety of interesting dynamics may occur, including that there may be a so-called commitment value to inventory.