Dynamic inventory competition with stockout-based substitution

This paper continues the stream of literature observed in Olsen and Parker (2008) where retailers compete under a Markov equilibrium solution concept. In this presentation, we consider a duopoly where retailers compete by providing inventory under the circumstances where unsatisfied customers may seek satisfaction elsewhere or leave. A very general framework is formulated to address a variety of customer avenues when stock is unavailable. We find a base-stock inventory policy is the equilibrium policy in the infinite horizon (open loop) under several mild conditions; this model's solution is known as an equilibrium in stationary strategies (ESS). We consequently determine conditions under which the parsimonious base-stock policy is the Markov equilibrium (closed loop) in a discrete-time dynamic game for a general time horizon, coinciding with the ESS base-stock levels. Importantly, when these conditions do not apply, we have counterexamples where a firm has a unilateral incentive to deviate from the ESS, stocking at a higher level. These examples demonstrate a value of inventory commitment, where the retailer may extract a benefit over multiple periods through committing to a higher stocking level and forcing her rival to understock. Our conclusion is that when the Markov solution is base-stock, it coincides with the ESS, but other Markov solutions also exist.