No-Regret Learning for Stackelberg Equilibrium Computation in Newsvendor Pricing Games

TL;DR

The paper tackles dynamic pricing in a two‑agent Stackelberg setting with unknown demand parameters. It combines Newsvendor pricing theory with online learning, specifically contextual linear bandits via OFUL, to learn demand parameters and compute best responses under optimism, yielding sublinear Stackelberg regret and convergence to an approximate Stackelberg equilibrium. Key contributions include proving unique SE under perfect information, deriving finite‑time regret bounds for both leader and follower, and introducing LNPG with theoretical guarantees and empirical validation against a UCB baseline. The work enables sample‑efficient learning in supply chains with demand uncertainty and inventory risk, and offers a framework for extending to richer demand models and feature sets in future work.

Abstract

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.

Figures (8)

• Figure 1: The Newsvendor Pricing Game. In this Stackelberg game, there a logistics network between a supplier (leader) and retailer (follower), where utility functions are not necessarily supermodular, the supplier issues a wholesale price $a$, and the retailer issues a purchase quantity $b$, and a retail price $p$ in response.
• Figure 2: Linear demand uncertainty.
• Figure 3: Confidence ball.
• Figure 4: Denotes the change in the optimistic order amount $b_a$ as improved estimates of the demand function are obtained.
• Figure 5: Episodic reward and Stackelberg (leader) regret performance across an Newsvendor pricing game, comparing Algorithm \ref{['alg:se-newsv']} (LNPG) against UCB. All shaded areas, denoting confidence intervals, are within a quarter quantile. (Parameters used $\kappa=3, \theta_0 = 18, \theta_1 = 7$.)

Theorems & Definitions (26)

• Lemma 2.1
• Theorem 1
• Theorem 2
• Lemma 3.1
• Theorem 3
• Lemma 3.2
• Lemma 3.3
• Theorem 4
• Theorem 5
• Corollary 3.1