Dynamic Pricing with External Information and Inventory Constraint

A merchant dynamically sets prices in each time period when selling a product over a finite time horizon with a given initial inventory. The merchant utilizes new external information that is observed at the beginning of each time period, whereas the demand function—how the external information and the price jointly impact that single-period demand distribution—is unknown. The merchant’s decision, setting price dynamically, serves dual roles to learn the unknown demand function and to balance inventory with an ultimate objective to maximize the expected cumulative revenue. The main objective of this work is to characterize and provide a full spectrum of relations between the order of optimal expected cumulative revenue achieved in three decision-making regimes: the merchant’s online decision-making regime, a clairvoyant regime with complete knowledge about the demand function, and a deterministic regime in which all the uncertainties are relaxed to the expectations. In the analyses, we derive an unconstrained representation of the optimality gap for generic constrained online learning problems, which renders tractable lower and upper bounds for the expected revenue achieved by dynamic pricing algorithms between different regimes. This analytical framework also inspires the design of two dual-based dynamic pricing algorithms for the clairvoyant and online regimes.