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Problem definition: We seek to provide an interpretable framework for segmenting users in a population for personalized decision making. Methodology/results: We propose a general methodology, market segmentation trees (MSTs), for learning market segmentations explicitly driven by identifying differences in user response patterns. To demonstrate the versatility of our methodology, we design two new specialized MST algorithms: (i) choice model trees (CMTs), which can be used to predict a user’s choice amongst multiple options, and (ii) isotonic regression trees (IRTs), which can be used to solve the bid landscape forecasting problem. We provide a theoretical analysis of the asymptotic running times of our algorithmic methods, which validates their computational tractability on large data sets. We also provide a customizable, open-source code base for training MSTs in Python that uses several strategies for scalability, including parallel processing and warm starts. Finally, we assess the practical performance of MSTs on several synthetic and real-world data sets, showing that our method reliably finds market segmentations that accurately model response behavior. Managerial implications: The standard approach to conduct market segmentation for personalized decision making is to first perform market segmentation by clustering users according to similarities in their contextual features and then fit a “response model” to each segment to model how users respond to decisions. However, this approach may not be ideal if the contextual features prominent in distinguishing clusters are not key drivers of response behavior. Our approach addresses this issue by integrating market segmentation and response modeling, which consistently leads to improvements in response prediction accuracy, thereby aiding personalization. We find that such an integrated approach can be computationally tractable and effective even on large-scale data sets. Moreover, MSTs are interpretable because the market segments can easily be described by a decision tree and often require only a fraction of the number of market segments generated by traditional approaches. Disclaimer: This work was done prior to Ryan McNellis joining Amazon. Funding: This work was supported by the National Science Foundation [Grants CMMI-1763000 and CMMI-1944428]. Supplemental Material: The online appendices are available at https://doi.org/10.1287/msom.2023.1195 . 

We consider a personalized pricing problem in which we have data consisting of feature information, historical pricing decisions, and binary realized demand. The goal is to perform off-policy evaluation for a new personalized pricing policy that maps features to prices. Methods based on inverse propensity weighting (including doubly robust methods) for off-policy evaluation may perform poorly when the logging policy has little exploration or is deterministic, which is common in pricing applications. Building on the balanced policy evaluation framework of Kallus (2018), we propose a new approach tailored to pricing applications. The key idea is to compute an estimate that minimizes the worst-case mean squared error or maximizes a worst-case lower bound on policy performance, where in both cases the worst-case is taken with respect to a set of possible revenue functions. We establish theoretical convergence guarantees and empirically demonstrate the advantage of our approach using a real-world pricing dataset. 

Price discrimination strategies, which offer different prices to customers based on differences in their valuations, have become common practice. Although it allows sellers to increase their profits, it also raises several concerns in terms of fairness (e.g., by charging higher prices (or denying access) to protected minorities in case they have higher (or lower) valuations than the general population). This topic has received extensive attention from media, industry, and regulatory agencies. In this paper, we consider the problem of setting prices for different groups under fairness constraints. We first propose four definitions: fairness in price, demand, consumer surplus, and no-purchase valuation. We prove that satisfying more than one of these fairness constraints is impossible even under simple settings. We then analyze the pricing strategy of a profit-maximizing seller and the impact of imposing fairness on the seller’s profit, consumer surplus, and social welfare. Under a linear demand model, we find that imposing a small amount of price fairness increases social welfare, whereas too much price fairness may result in a lower welfare relative to imposing no fairness. On the other hand, imposing fairness in demand or consumer surplus always decreases social welfare. Finally, no-purchase valuation fairness always increases social welfare. We observe similar patterns under several extensions and for other common demand models numerically. Our results and insights provide a first step in understanding the impact of imposing fairness in the context of discriminatory pricing. This paper was accepted by Jayashankar Swaminathan, operations management. Funding: A. N. Elmachtoub was supported by the Division of Civil, Mechanical and Manufacturing Innovation [Grants 1763000 and 1944428]. Supplemental Material: The data files and online appendix are available at https://doi.org/10.1287/mnsc.2022.4317 . 

We consider a fundamental pricing model in which a fixed number of units of a reusable resource are used to serve customers. Customers arrive to the system according to a stochastic process and, upon arrival, decide whether to purchase the service, depending on their willingness to pay and the current price. The service time during which the resource is used by the customer is stochastic, and the firm may incur a service cost. This model represents various markets for reusable resources, such as cloud computing, shared vehicles, rotable parts, and hotel rooms. In the present paper, we analyze this pricing problem when the firm attempts to maximize a weighted combination of three central metrics: profit, market share, and service level. Under Poisson arrivals, exponential service times, and standard assumptions on the willingness-to-pay distribution, we establish a series of results that characterize the performance of static pricing in such environments. In particular, although an optimal policy is fully dynamic in such a context, we prove that a static pricing policy simultaneously guarantees 78.9% of the profit, market share, and service level from the optimal policy. Notably, this result holds for any service rate and number of units the firm operates. Our proof technique relies on a judicious construction of a static price that is derived directly from the optimal dynamic pricing policy. In the special case in which there are two units and the induced demand is linear, we also prove that the static policy guarantees 95.5% of the profit from the optimal policy. Our numerical findings on a large test bed of instances suggest that the latter result is quite indicative of the profit obtained by the static pricing policy across all parameters. 

Many real-world analytics problems involve two significant challenges: prediction and optimization. Because of the typically complex nature of each challenge, the standard paradigm is predict-then-optimize. By and large, machine learning tools are intended to minimize prediction error and do not account for how the predictions will be used in the downstream optimization problem. In contrast, we propose a new and very general framework, called Smart “Predict, then Optimize” (SPO), which directly leverages the optimization problem structure—that is, its objective and constraints—for designing better prediction models. A key component of our framework is the SPO loss function, which measures the decision error induced by a prediction. Training a prediction model with respect to the SPO loss is computationally challenging, and, thus, we derive, using duality theory, a convex surrogate loss function, which we call the SPO+ loss. Most importantly, we prove that the SPO+ loss is statistically consistent with respect to the SPO loss under mild conditions. Our SPO+ loss function can tractably handle any polyhedral, convex, or even mixed-integer optimization problem with a linear objective. Numerical experiments on shortest-path and portfolio-optimization problems show that the SPO framework can lead to significant improvement under the predict-then-optimize paradigm, in particular, when the prediction model being trained is misspecified. We find that linear models trained using SPO+ loss tend to dominate random-forest algorithms, even when the ground truth is highly nonlinear. This paper was accepted by Yinyu Ye, optimization. Supplemental Material: Data and the online appendix are available at https://doi.org/10.1287/mnsc.2020.3922 

Increased availability of high-quality customer information has fueled interest in personalized pricing strategies, that is, strategies that predict an individual customer’s valuation for a product and then offer a price tailored to that customer. Although the appeal of personalized pricing is clear, it may also incur large costs in the forms of market research, investment in information technology and analytics expertise, and branding risks. In light of these trade-offs, our work studies the value of personalized pricing strategies over a simple single-price strategy. We first provide closed-form lower and upper bounds on the ratio between the profits of an idealized personalized pricing strategy (first-degree price discrimination) and a single-price strategy. Our bounds depend on simple statistics of the valuation distribution and shed light on the types of markets for which personalized pricing has little or significant potential value. Second, we consider a feature-based pricing model where customer valuations can be estimated from observed features. We show how to transform our aforementioned bounds into lower and upper bounds on the value of feature-based pricing over single pricing depending on the degree to which the features are informative for the valuation. Finally, we demonstrate how to obtain sharper bounds by incorporating additional information about the valuation distribution (moments or shape constraints) by solving tractable linear optimization problems. This paper was accepted by David Simchi-Levi, revenue management and market analytics.