Technical Note—A Monge Sequence-Based Approach to Characterize the Competitive Newsvendor Problem

We revisit the stochastic inventory game in which n players compete by setting their individual inventory levels in a market with stockout-based demand substitution. Because of specific tractability issues, the prior literature has largely focused on versions of this competitive newsvendor problem with assumptions on the number of players and their substitution behavior. In this note, we develop an approach to solve instances of this problem with any number of players and multistage spillovers of unsatisfied demand. We (i) establish that for multistage stockout-based substitution models explored in the literature, the search (substitution) behavior of customers can be replicated using a Monge sequence; (ii) obtain the first-order conditions that can be then used to determine equilibrium inventory levels; and (iii) discuss other structural properties of the solution based on Bottleneck Monge matrices. Special cases of our approach provide the well-known equilibrium results for two newsvendors.