Certainty Equivalent Pricing Under Sales-Dependent and Inventory-Dependent Demand

Abstract

When underlying demand is uncertain and follows a complex stochastic process, pricing problems are difficult to solve. In such cases, certainty equivalent (CE) policies, based on solving the deterministic relaxation of a stochastic pricing problem, can be used as practical alternatives. CE policies have lighter computational and informational requirements compared to solving the optimal problem. This is particularly true when the firm does not have complete information about the underlying demand distribution. While the effectiveness of CE pricing policies has been theoretically studied in some settings (e.g, independent demand, continuous price changes), the performance of CE policies are not known in general. This paper analyzes the performance of CE policies in a pricing problem (for a given inventory level) where future demand depends on sales and inventory and the firm has limited opportunities to change price. We show that CE policies are asymptotically optimal: as the problem scale (denoted by m) becomes large, the percentage regret decreases at the rate of O (m^(-1/2)). We also extend the result to the joint pricing and (initial) inventory problem. Our numerical results are even more promising. Even in non-asymptotic settings (small scaling factor and a few price changes), CE policies perform well and often result in revenues that are only a few percentage points lower than optimal. In addition, we compare the benefit of a closed-loop CE policy over an open-loop CE policy and provide a theoretical comparison between the two policies.