Distributionally Robust Monopoly Pricing: Switching from Low to High Prices in Volatile Markets

Abstract

Problem definition: Traditional monopoly pricing assumes sellers have full information about consumer valuations. We consider monopoly pricing under limited information when a seller only knows the mean, variance, and support of the valuation distribution. The objective is to maximize expected revenue by selecting the optimal fixed price. Methodology/results: We adopt a distributionally robust framework, in which the seller considers all valuation distributions that comply with the limited information. We formulate a maximin problem that seeks to maximize expected revenue for the worst case valuation distribution. The minimization problem that identifies the worst case valuation distribution is solved using primal-dual methods and, in turn, leads to an explicitly solvable maximization problem. This yields a closed-form optimal pricing policy and a new fundamental principle prescribing when to use low and high robust prices. Managerial implications: We show that the optimal policy switches from low to high prices when variance becomes sufficiently large, yielding significant performance gains compared with existing robust prices that generally decay with market uncertainty. This presents guidelines for when the seller should switch from targeting mass markets to niche markets. Similar guidelines are obtained for delay-prone services with rational utility-maximizing customers, underlining the universality and wide applicability of the novel pricing policy. Funding: This research was supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO), Vici [Grant 202.068]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2024.0952 .