Dynamic pricing with finite price sets: a non-parametric approach

We study price optimization of perishable inventory over multiple, consecutive selling seasons in the presence of demand uncertainty. Each selling season consists of a finite number of discrete time periods, and demand per time period is Bernoulli distributed with price-dependent parameter. The set of feasible prices is finite, and the expected demand corresponding to each price is unknown to the seller, whose objective is to maximize cumulative expected revenue. We propose an algorithm that estimates the unknown parameters in a learning phase, and in each subsequent season applies a policy determined as the solution to a sample dynamic program, which modifies the underlying dynamic program by replacing the unknown parameters by the estimate. Revenue performance is measured by the regret: the expected revenue loss relative to the optimal attainable revenue under full information. For a given number of seasons n, we show that if the number of seasons allocated to learning is asymptotic to (n2logn)1/3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(n^2\log n)^{1/3}$$\end{document}, then the regret is of the same order, uniformly over all unknown demand parameters. An extensive numerical study that compares our algorithm to six benchmarks adapted from the literature demonstrates the effectiveness of our approach.