Dynamic Pricing with a Poisson Bandit Model

Abstract

Suppose that one of two prices for the same product must be posted every day. Under each price, the demand function is described by a compound Poisson process with possibly unknown parameters. The objective is to sequentially post daily prices so as to maximize the total expected, possibly discounted gross revenue over a finite pricing horizon. To effectively balance between understanding the demand function and achieving economic revenues, we formulate the optimal pricing problem with a bandit model and characterize the solution by means of stochastic dynamic programming. When there is only one unknown demand function in the model, the optimal pricing decision is determined by a pricing index, whose limit is the Gittins index. These index values also demonstrate that it may be worth sacrificing some immediate payoff for the benefit of information gathering and better-informed decisions in the future. Moreover, the optimal stopping solution is derived and the myopic strategy is shown not to be optimal in general. When both demand functions are unknown, a version of the play-the-winner pricing rule is derived.