Achieving Coordination in Non-Cooperative Joint Replenishment Games
Junjie Luo, Changjun Wang
TL;DR
The paper addresses decentralized coordination in the Joint Replenishment Problem by introducing Weighted Proportional Sharing (WPS) rules that allocate the major setup cost $K_0$ among participating retailers via fixed weights. It proves that every WPS rule yields a payoff-dominant Nash equilibrium that can be computed efficiently, and then develops practical rules, notably WPS-h with weights $H_i=(1/2)h_id_i$, achieving a PoS of at most 1.25, and WPS-d that relies only on publicly observable demand rates with PoS bounded by a subpolynomial function of information-variance. The analysis also establishes fundamental efficiency limits under private information, showing PoS lower bounds of at least 1.05 (when $K_i$ are private) and $\Omega(\sqrt{\log \gamma_d})$ in the private holding-cost setting, while WPS-$\hat{h}$ provides robustness to estimation errors. Overall, the work advances decentralized inventory coordination by offering theoretically sound and practically implementable cost-sharing rules that approach centralized efficiency under realistic information constraints.
Abstract
We analyze an infinite-horizon deterministic joint replenishment model from a non-cooperative game-theoretical approach. In this model, a group of retailers can choose to jointly place an order, which incurs a major setup cost independent of the group, and a minor setup cost for each retailer. Additionally, each retailer is associated with a holding cost. Our objective is to design cost allocation rules that minimize the long-run average system cost while accounting for the fact that each retailer independently selects its replenishment interval to minimize its own cost. We introduce a class of cost allocation rules that distribute the major setup cost among the associated retailers in proportion to their predefined weights. For these rules, we establish a monotonicity property of agent better responses, which enables us to prove the existence of a payoff dominant pure Nash equilibrium that can also be computed efficiently. We then analyze the efficiency of these equilibria by examining the price of stability (PoS), the ratio of the best Nash equilibrium's system cost to the social optimum, across different information settings. In particular, our analysis reveals that one rule, which leverages retailers' own holding cost rates, achieves a near-optimal PoS of 1.25, while another rule that does not require access to retailers' private information also yields a favorable PoS.