Consensus in Noncooperative Dynamic Games: a Multi-Retailer Inventory Application

We focus on Nash equilibria and Pareto optimal Nash equilibria for a finite horizon noncooperative dynamic game with a special structure of the stage cost. We study the existence of the above solutions by proving that the game is a potential game. For the single-stage version of the game, we characterize the above solutions 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. We also specialize the game to a multi-retailer inventory system.