Optimal Bayesian Demand Learning over Short Horizons

Abstract

We investigate the optimal Bayesian dynamic pricing and demand learning policy over short selling horizons, where the pricing decisions are time‐sensitive. The seller fine‐tunes the price near an incumbent price in order to maximize the total revenue. The existing literature focuses on policies that are asymptotically optimal, that is, near optimal when the selling horizons are sufficiently long, but little is known about the optimal Bayesian policies, especially over short horizons. We formulate the problem as a finite‐horizon stochastic dynamic program and identify a connection between the optimality equations and the generalized Weierstrass transform (GWT). We fully characterize the structure of the Bayesian optimal policy for the linear Gaussian demand model and prove that the optimal policy adjusts the myopic price away from the incumbent price. A notable exception occurs when the two prices coincide and the precision of the posterior belief exceeds a threshold, in which case it is optimal to forgo learning and use a fixed‐price policy. Exploiting the structural results makes it possible to compute the optimal policy efficiently on an ordinary computer.