Bayesian Dynamic Pricing and Subscription Period Selection with Unknown Customer Utility

Abstract

We consider a provider offering a subscription service to customers over a planning horizon. The customers decide whether to subscribe according to a utility model. The provider has a prior belief about the customer utility model and updates its belief based on the transaction data of new customers and the usage data of existing subscribers. The provider aims to minimize its regret—the expected profit loss relative to a clairvoyant who knows the customer utility model. To analyze regret, we first study the clairvoyant’s full-information problem, noting that the resulting dynamic program suffers from the curse of dimensionality. We characterize the optimal policy for the full-information problem via a customer-centric approach that balances the provider’s immediate and future profits from a customer. When the provider does not have full information, we find that the simple and commonly used certainty-equivalence policy exhibits poor performance. We illustrate that this can be due to incomplete or slow learning but can also occur because of offering a suboptimal contract with a long subscription period in the beginning. We develop an adaptive learning policy, namely the information-threshold policy, that focuses on learning until the provider’s accumulated information exceeds a chosen threshold. We show that this policy achieves asymptotically optimal performance with its regret growing logarithmically in the planning horizon. Our results indicate that offering a long subscription period could be costly when the provider knows little about the customers’ usage and the service cost is uncertain.

Keywords: dynamic pricing, subscription period, subscription dynamics, demand learning, cost learning