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

We study the contextual dynamic pricing problem where a firm sells products to $T$ sequentially arriving consumers that behave according to an unknown demand model. The firm aims to maximize its revenue, i.e. minimize its regret over a clairvoyant that knows the model in advance. The demand model is a generalized linear model (GLM), allowing for a stochastic feature vector in $\mathbb R^d$ that encodes product and consumer information. We first show that the optimal regret upper bound is of order $\sqrt{dT}$, up to a logarithmic factor, improving upon existing upper bounds in the literature by a $\sqrt{d}$ factor. This sharper rate is materialised by two algorithms: a confidence bound-type (supCB) algorithm and an explore-then-commit (ETC) algorithm. A key insight of our theoretical result is an intrinsic connection between dynamic pricing and the contextual multi-armed bandit problem with many arms based on a careful discretization. We further study contextual dynamic pricing under the local differential privacy (LDP) constraints. In particular, we propose a stochastic gradient descent based ETC algorithm that achieves an optimal regret upper bound of order $d\sqrt{T}/\epsilon$, up to a logarithmic factor, where $\epsilon>0$ is the privacy parameter. The regret upper bounds with and without LDP constraints are accompanied by newly constructed minimax lower bounds, which further characterize the cost of privacy. Extensive numerical experiments and a real data application on online lending are conducted to illustrate the efficiency and practical value of the proposed algorithms in dynamic pricing.