Randomized Robust Price Optimization

The robust multi-product pricing problem is to determine the prices of a collection of products so as to maximize the worst-case revenue, where the worst case is taken over an uncertainty set of demand models that the firm expects could be realized in practice. A tacit assumption in this approach is that the pricing decision is a deterministic decision: the prices of the products are fixed and do not vary. In this paper, we consider a randomized approach to robust pricing, where a decision maker specifies a distribution over potential price vectors so as to maximize its worst-case revenue over an uncertainty set of demand models. We formally define this problem - the randomized robust price optimization problem - and analyze when a randomized price scheme performs as well as a deterministic scheme versus when it yields a benefit. We also propose solution methods for obtaining an optimal randomization scheme over a discrete set of candidate price vectors and show how these methods are applicable for common demand models, such as the linear, semi-log and log-log demand models. We numerically compare the randomized and deterministic approaches on a variety of synthetic and real problem instances; on instances derived from a real grocery retail scanner dataset, we show that the improvement in worst-case revenue can be as high as 92%. Using the same grocery retail scanner dataset, we also show that the randomized approach can produce price prescriptions that achieve higher out-of-sample revenue than the nominal and deterministic robust approaches.