Consensus in inventory games

This paper studies design, convergence, stability and optimality of a distributed consensus protocol for n-player repeated non cooperative games under incomplete information. Information available to each player concerning the other players' strategies evolves in time. At each stage (time period), the players select myopically their best binary strategy on the basis of a payoff, defined on a single stage, monotonically decreasing with the number of active players. The game is specialized to an inventory application, where fixed costs are shared among all retailers, interested in reordering or not from a common warehouse. As information evolves in time, the number of active players changes too, and then each player adjusts its strategy at each stage based on the updated information. In particular, the authors focus on Pareto optimality as a measure of coordination of reordering strategies, proving that there exists a unique Pareto optimal Nash equilibrium which verifies certain stability conditions. The main contribution of the paper is the design of a consensus protocol allowing the distributed convergence of the strategies to the unique Pareto optimal Nash equilibrium. Results may also be extended to externality games, pollution/congestion games, and cost-sharing games, with the only constraint of being the strategies binary and with threshold structure.