A Stackelberg Regret Minimizing Framework for Online Learning in Newsvendor Pricing Games

We introduce the application of online learning in a Stackelberg game pertaining to a system with two learning agents in a dyadic exchange network, consisting of a supplier and retailer, specifically where the parameters of the demand function are unknown. In this game, the supplier is the first-moving leader, and must determine the optimal wholesale price of the product. Subsequently, the retailer who is the follower, must determine both the optimal procurement amount and selling price of the product. In the perfect information setting, this is known as the classical price-setting Newsvendor problem, and we prove the existence of a unique Stackelberg equilibrium when extending this to a two-player pricing game. In the framework of online learning, the parameters of the reward function for both the follower and leader must be learned, under the assumption that the follower will best respond with optimism under uncertainty. A novel algorithm based on contextual linear bandits with a measurable uncertainty set is used to provide a confidence bound on the parameters of the stochastic demand. Consequently, optimal finite time regret bounds on the Stackelberg regret, along with convergence guarantees to an approximate Stackelberg equilibrium, are provided.