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

Replicating cash flows of multiple agents in game-theoretic settings tends to be a challenging task. In this paper, we consider the competitive newsvendor game where multiple newsvendors choose inventory levels before demand arrival and the unmet demand of each newsvendor spills over to multiple other newsvendors. We show that this spillover behavior and the resulting cash flows of each newsvendor can be replicated within a transportation problem after assigning artificial costs on spillover behavior. This replication provides an opportunity to study structural properties of the problem, as well as determine the equilibrium of the game. This paradigm of using artificial costs within an optimization framework to replicate agents’ cash flows can be used in many other games as well. 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.