We study an off-policy contextual pricing problem in which the seller has access to samples of prices that customers were previously offered, whether they purchased at that price, and auxiliary features describing the customer and/or item being sold. This is in contrast to the well-studied setting in which samples of the customer’s valuation (willingness to pay) are observed. In our setting, the observed data are influenced by the previous pricing policy, and we do not know how customers would have responded to alternative prices. We introduce suitable loss functions for this setting that can be directly optimized to find an effective pricing policy with expected revenue guarantees without the need for estimation of an intermediate demand function. We focus on convex loss functions, which are especially important when linear pricing policies are preferred for interpretability. In such cases, the revenue optimization problem remains convex and tractable. Specifically, we propose generalized hinge and quantile pricing loss functions that price at a multiplicative factor of the conditional expected valuation or a particular quantile of the prices that sold despite the valuation data not being observed. We prove expected revenue bounds for these pricing policies when the valuation distribution is log-concave, and we provide generalization bounds for the finite sample case. Finally, we conduct simulations on both synthetic and real-world data to demonstrate that this approach is competitive with and, in some settings, outperforms state-of-the-art methods in contextual pricing. This paper was accepted by Vivek Farias, data science. Supplemental Material: The online appendix and data files are available at https://doi.org/10.1287/mnsc.2023.00122 .
Max W. Biggs (Tue,) studied this question.