Abstract: 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 $R^d$ encoding product and consumer information. We first show the optimal regret is of order $\sqrt{dT}$, up to logarithmic factors, improving existing upper bounds by a $\sqrt{d}$ factor. This optimal rate is materialized by two algorithms: a confidence bound-type algorithm and an explore-then-commit (ETC) algorithm. A key insight is an intrinsic connection between dynamic pricing and contextual multi-armed bandit problems with many arms with a careful discretization. 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 $d\sqrt{T}/\epsilon$, up to logarithmic factors, where $\epsilon>0$ 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.

Bio：蒋斐宇，复旦大学管理学院统计与数据科学系副教授，博士毕业于清华大学，主要研究领域为时间序列分析、变点分析、金融计量经济学、在线学习等，在Biometrika、JASA、JRSSB、JMLR、JOE和MS等期刊发表多篇论文，其研究得到国家自然科学基金青年科学基金(B类、C类)和上海市扬帆计划的支持。