Noncooperative dynamic games for Inventory applications: a consensus approach

We focus on a finite horizon noncooperative dynamic game where the stage cost of a single player associated to a decision is a monotonically nonincreasing function of the total number of players making the same decision. For the single-stage version of the game, we characterize Nash equilibria and derive a consensus protocol that makes the players converge to the unique Pareto optimal Nash equilibrium. Such an equilibrium guarantees the interests of the players and is also social optimal in the set of Nash equilibria. For the multi-stage version of the game, we present an algorithm that converges to Nash equilibria, unfortunately not necessarily Pareto optimal. The algorithm returns a sequence of joint decisions, each one obtained from the previous one by an unilateral improvement on the part of a single player. The sequence with which the players act is chosen a priori and may influence the Nash equilibrium to which the path converges. We also specialize the game to a multi-retailer inventory system, where competing retailers aim at coordinating their supply strategies in order to minimize their local costs.