Optimal Online and Offline Algorithms for Contextual MNL with Applications to Assortment and Pricing

TL;DR

This paper develops new algorithms for joint assortment and pricing optimization under a contextual multinomial logit model, providing theoretical guarantees and bridging the gap between separate studies of assortment and pricing.

Contribution

It introduces a new demand estimation confidence region and proposes offline and online algorithms with improved regret bounds for joint assortment and pricing.

Findings

Offline algorithm has suboptimality governed by local information.

Online SupCB-type algorithm achieves near-minimax regret bounds.

Specialized bounds recover optimal rates for assortment-only or pricing-only cases.

Abstract

Selecting which products to display and at what prices is a central decision in retail and e-commerce operations. In many applications, these two choices must be made jointly under limited display capacity and uncertain customer demand. In this paper, we study the joint assortment and pricing problem under a price-based contextual multinomial logit model, where customer preferences depend on both product features and selling prices. Our analysis begins with the construction of a new confidence region for demand estimation under price-dependent features. Building on this result, we develop a pessimistic offline algorithm and SupCB-type online algorithms for joint assortment and pricing optimization. In the offline setting, we establish a suboptimality guarantee governed by local information around the optimal assortment-price pair, rather than by exact coverage of the optimal action. In the online setting, our SupCB-type algorithm improves the best previously known regret bound to $\widetilde{O}(W\sqrt{dT\log N}/L_0)$, and we also provide a computationally simpler Thompson-sampling alternative. When specialized to the assortment-only or pricing-only setting, our bounds recover the near-minimax-optimal rates established in those respective domains, thereby bridging the gap between the mature study on assortment optimization or dynamic pricing and the limited literature on their joint optimization.