Minimax Optimality in Contextual Dynamic Pricing with General Valuation Models

We study contextual dynamic pricing, where a decision maker posts personalized prices based on observable contexts and receives binary purchase feedback indicating whether the customer’s valuation exceeds the price. Each valuation is modeled as an unknown latent function of the context, corrupted by independent and identically distributed market noise from an unknown distribution. Relying only on Lipschitz continuity of the noise distribution and bounded valuations, we propose a minimax-optimal algorithm. To accommodate the unknown distribution, our method discretizes the relevant noise range to form a finite set of candidate prices, then applies layered data partitioning to obtain confidence bounds substantially tighter than those derived via the elliptical potential lemma. A key advantage is that estimation bias in the valuation function cancels when comparing upper confidence bounds, eliminating the need to know the Lipschitz constant. The framework extends beyond linear models to general function classes through offline regression oracles. Our regret analysis depends solely on the oracle’s estimation error, typically governed by the statistical complexity of the class. These techniques yield a regret upper bound matching the minimax lower bound up to logarithmic factors. Furthermore, we refine these guarantees under additional structures—for example, linear valuation models, second-order smoothness, sparsity, and known noise distribution or observable valuations—and compare our bounds and assumptions with prior dynamic-pricing methods. Finally, numerical experiments corroborate the theory and show clear improvements over benchmark methods.

Funding: X. Gong’s research is generously supported by the National Natural Science Foundation of China [Grant 72501238]. W. You’s research is generously supported by the Hong Kong Research Grants Council [Grant GRF 16212823]. J. Zhang’s research is generously supported by the Hong Kong Research Grants Council [Theme-based Research Project T32-615/24-R].

Supplemental Material: All supplemental materials, including the code, data, and files required to reproduce the results are available at https://doi.org/10.1287/opre.2025.1779.