Dynamic Probabilistic Selling When Customers Have Boundedly Rational Expectations

Problem definition: The existing literature on probabilistic or opaque selling has largely focused on understanding why it is attractive to firms. In this paper, we intend to answer a follow-up question: How should opaque selling be managed in a firm’s operations over time? Academic/practical relevance: Answering this question is relevant yet complex, because in practice (i) the profitability of opaque selling depends on how customers respond to the firm’s product-offering strategies and (ii) the firm’s strategies have to be responsive to customers’ purchasing decisions to maximize its total profit. Methodology: We develop a simple game-theoretic framework to capture the dynamic nature of the problem in multiple periods when customers boundedly rationally expect the firm’s strategies through anecdotal reasoning. We characterize the firm’s optimal pricing and product-offering policy. Results: We find that offering the high-value product with a high probability followed by a lower probability is typically optimal over time. We finally analyze several model extensions, such as different numbers of customers, multiple anecdotes, infinitely many periods, and limited inventory, and show the robustness of our results. Managerial implications: We demonstrate the value of using a dynamic probabilistic selling policy and prove that our dynamic policy can double the firm’s profit compared with using the static policy proposed in the existing literature. In a dynamic programming model, we prove that a cycle policy oscillating between two product-offering probabilities is typically optimal in the steady state over infinitely many periods.