Bayesian Dynamic Pricing with Unknown Customer Willingness-to-Pay and Limited Inventory

We consider a dynamic pricing problem in which the seller sells a limited amount of inventory over a short time horizon. The distribution of customer willingness-to-pay is unknown, and the seller learns about the distribution from observing customer purchase decisions. Such a problem arises in practice when certain unique assets are put for sale (e.g., selling a house), but the problem is known to be both analytically and computationally challenging. In this paper, we seek to derive new insights, solution bounds, and heuristics for the problem. We formulate the problem as a Bayesian dynamic program and transform it with an unnormalized prior. Based on the unnormalized prior, we first show that the optimal price for this problem is always finite, which motivates us to search for the optimal solution via the first-order condition. To this end, we prove a generalized envelope theorem for the Bayesian dynamic program, and derive an explicit expression of the first-order derivative of the dynamic program objective function. This derivative expression reveals new insights about the intertwined effects of left censoring, right censoring, and limited inventory in the problem. It also enables us to derive new, easy-to-compute solution bounds and two derivative-based heuristics for the problem. The first heuristic approximates the derivative by a weighted average of its upper and lower bounds, and the second heuristic uses an easy-to-compute open-loop policy as a surrogate for the approximate evaluation of the derivative function. Numerical studies show that both heuristics perform well, with the second heuristic consistently outperforming existing heuristics in the literature. Overall, our paper prescribes a new, derivative-based approach to tackle the dynamic pricing problem with unknown customer willingness-to-pay, limited inventory, and short horizon. Our proposed solutions can help managers to achieve near optimal revenue performance for this challenging problem