Optimal Bayesian Price Fine-Tuning

Abstract

We consider the problem of optimally fine-tuning the price near a historically fixed price (incumbent price) to maximize the expected revenue over a finite horizon. The demand follows a linear Gaussian model in which the price sensitivity is unknown and can be learned in a Bayesian fashion as the sales data accumulate. We formulate the problem as a discrete-time Bayesian dynamic program and connect it to the Gauss-Weierstrass Transform through a reformulation. We prove that the optimal pricing policy has a simple and intuitively appealing structure: the optimal policy adjusts the myopic price away from the incumbent price, except when the latter two prices coincide and the precision of the posterior belief exceeds a threshold. Under some priors, a fixed price is optimal for the entire horizon. Further, we show that incomplete learning is absent from the optimal policy if the belief precision stays under a switching curve, but may emerge when the belief precision jumps above that curve, especially near the end of the horizon. We also develop an efficient algorithm using a technique called the fast Gauss transform, which enables us to obtain the optimal policy with only moderate computation.