Dynamic Pricing with External Information and Inventory Constraint

A merchant sells a product over a selling season of T time periods in presence of a limited inventory. The merchant observes new external information at the beginning of each time period and then sets a price for that time period. Initially, the merchant does not know the distribution of the external information and the demand function, i.e., how the external information and price jointly impact the demand distribution in a single time period. The seller's decision, setting price dynamically, serves dual roles to learn the unknown demand function and to balance inventory, with an ultimate goal to maximize the expected revenue over the selling season. We characterize and prove a full spectrum of relations between the optimal 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 all the external information in advance, and a deterministic regime in which the demand function and all the uncertainties are revealed at the beginning. 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 history-dependent dynamic pricing algorithms for the clairvoyant regime and the online regime. Numerical experiments are conducted to demonstrate the performances of our algorithms