Contextual Dynamic Pricing: Algorithms, Optimality, and Local Differential Privacy Constraints

We study contextual dynamic pricing problems where a firm sells products to T sequentially-arriving consumers, behaving according to an unknown demand model. The firm aims to minimize its regret over a clairvoyant that knows the model in advance. The demand follows a generalized linear model (GLM), allowing for stochastic feature vectors in Rd encoding product and consumer information. We first show the optimal regret is of order dT, up to logarithmic factors, improving existing upper bounds by a d factor, achieved by an explore-then-commit (ETC) algorithm. We further study contextual dynamic pricing under local differential privacy (LDP) constraints. We propose a stochastic gradient descent-based ETC algorithm achieving regret upper bounds of order dT/ϵ, up to logarithmic factors, where ϵ>0 is the privacy parameter. The upper bounds with and without LDP constraints are matched by newly constructed minimax lower bounds, characterizing costs of privacy. Moreover, we extend our study to dynamic pricing under mixed privacy constraints, which naturally bridges private and non-private dynamic pricing. We propose a two-stage ETC algorithm and show that it improves the privacy-utility tradeoff by efficiently leveraging public data. Its optimality is further established via a newly-derived minimax lower bound. To our knowledge, this is the first time such setting is studied in the dynamic pricing literature. Extensive numerical experiments and real data applications are conducted to illustrate the efficiency and practical value of our algorithms. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.