Personalized Pricing with Invalid Instrumental Variables: Identification, Estimation, and Policy Learning

Pricing based on individual customer characteristics is widely used to maximize sellers' revenues. We study how a firm can learn such a personalized pricing policy from historical transactions when unobserved factors affect both the prices it charged and the revenue it earned. Standard instrumental variable (IV) methods either focus on a discrete treatment space or require that the IV affect revenue only through price. This exclusion restriction is difficult to justify in pricing, where the most natural candidate IVs are price-setting levers that may also affect revenue directly. We propose PRINT (Personalized pRicing using Invalid iNsTrumental variables), a policy learning method for continuous prices that uses such candidate IVs. Relying on structural models of revenue and price, we show that the optimal pricing policy can be identified with invalid IV that shifts higher moments of historical prices, when the unobserved factors that drive customers' price sensitivity are separated from those used by the historical pricing rule. Based on this identification, which leads to conditional moment restrictions with generalized residual functions, we construct an adversarial min-max estimator that accommodates flexible machine learning models and establish regret bounds for the learned pricing policy. Simulation studies and an application to a U.S. online auto lender illustrate the method's performance. Our results give a principled way to learn personalized prices from observational data using imperfect price-setting levers, and clarify when doing so is justified.