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 .

In this paper, we examine a data‐driven optimization approach to making optimal decisions as evaluated by a trained random forest, where these decisions can be constrained by an arbitrary polyhedral set. We model this optimization problem as a mixed‐integer linear program. We show this model can be solved to optimality efficiently using pareto‐optimal Benders cuts for ensembles containing a modest number of trees. We consider a random forest approximation that consists of sampling a subset of trees and establish that this gives rise to near‐optimal solutions by proving analytical guarantees. In particular, for axis‐aligned trees, we show that the number of trees we need to sample is sublinear in the size of the forest being approximated. Motivated by this result, we propose heuristics inspired by cross‐validation that optimize over smaller forests rather than one large forest and assess their performance on synthetic datasets. We present two case studies on a property investment problem and a jury selection problem. We show this approach performs well against other benchmarks while providing insights into the sensitivity of the algorithm's performance for different parameters of the random forest.

Problem definition: We present a data-driven study of the secondary ticket market. In particular, we are primarily concerned with accurately estimating price sensitivity for listed tickets. In this setting, there are many issues including endogeneity, heterogeneity in price sensitivity for different tickets, binary outcomes, and nonlinear interactions between ticket features. Our secondary goal is to highlight how this estimation can be integrated into a prescriptive trading strategy for buying and selling tickets in an active marketplace. Academic/practical relevance: We present a novel method for demand estimation with heterogeneous treatment effect in the presence of confounding. In practice, we embed this method within an optimization framework for ticket reselling, providing the ticket reselling platform with a new framework for pricing tickets on its platform. Methodology: We introduce a general double/orthogonalized machine learning method for classification problems. This method allows us to isolate the causal effects of price on the outcome by removing the conditional effects of the ticket and market features. Furthermore, we introduce a novel loss function that can be easily incorporated into powerful, off-the-shelf machine learning algorithms, including gradient boosted trees. We show how, in the presence of hidden confounding variables, instrumental variables can be incorporated. Results: Using a wide range of synthetic data sets, we show this approach beats state-of-the-art machine learning and causal inference approaches for estimating treatment effects in the classification setting. Furthermore, using National Basketball Association ticket listings from the 2014–2015 season, we show that probit models with instrumental variables, previously used for price estimation of tickets in the resale market, are significantly less accurate and potentially misspecified relative to our proposed approach. Through pricing simulations, we show our proposed method can achieve an 11% return on investment by buying and selling tickets, whereas existing techniques are not profitable. Managerial implications: The knowledge of how to price tickets on its platform offers a range of potential opportunities for our collaborator, both in terms of understanding sellers on their platform and in developing new products to offer them. History: This paper has been accepted as part of the 2019 Manufacturing & Service Operations Management Practice-Based Research Competition. Funding: This work was supported by the National Science Foundation [Grant CMMI-1563343]. Supplemental Material: The online appendices are available at https://doi.org/10.1287/msom.2021.1065 .