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This paper studies a dynamic pricing problem under model misspecification. To characterize model misspecification, we adopt the ε-contamination model—the most fundamental model in robust statistics and machine learning. In particular, for a selling horizon of length T, the online ε-contamination model assumes that demands are realized according to a typical unknown demand function only for [Formula: see text] periods. For the rest of [Formula: see text] periods, an outlier purchase can happen with arbitrary demand functions. The challenges brought by the presence of outlier customers are mainly due to the fact that arrivals of outliers and their exhibited demand behaviors are completely arbitrary, therefore calling for robust estimation and exploration strategies that can handle any outlier arrival and demand patterns. We first consider unconstrained dynamic pricing without any inventory constraint. In this case, we adopt the Follow-the-Regularized-Leader algorithm to hedge against outlier purchase behavior. Then, we introduce inventory constraints. When the inventory is insufficient, we study a robust bisection-search algorithm to identify the clearance price—that is, the price at which the initial inventory is expected to clear at the end of T periods. Finally, we study the general dynamic pricing case, where a retailer has no clue whether the inventory is sufficient or not. In this case, we design a meta-algorithm that combines the previous two policies. All algorithms are fully adaptive, without requiring prior knowledge of the outlier proportion parameter ε. Simulation study shows that our policy outperforms existing policies in the literature.
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This content will become publicly available on March 1, 2026
Offline Feature-Based Pricing Under Censored Demand: A Causal Inference Approach
Problem definition: We study a feature-based pricing problem with demand censoring in an offline, data-driven setting. In this problem, a firm is endowed with a finite amount of inventory and faces a random demand that is dependent on the offered price and the features (from products, customers, or both). Any unsatisfied demand that exceeds the inventory level is lost and unobservable. The firm does not know the demand function but has access to an offline data set consisting of quadruplets of historical features, inventory, price, and potentially censored sales quantity. Our objective is to use the offline data set to find the optimal feature-based pricing rule so as to maximize the expected profit. Methodology/results: Through the lens of causal inference, we propose a novel data-driven algorithm that is motivated by survival analysis and doubly robust estimation. We derive a finite sample regret bound to justify the proposed offline learning algorithm and prove its robustness. Numerical experiments demonstrate the robust performance of our proposed algorithm in accurately estimating optimal prices on both training and testing data. Managerial implications: The work provides practitioners with an innovative modeling and algorithmic framework for the feature-based pricing problem with demand censoring through the lens of causal inference. Our numerical experiments underscore the value of considering demand censoring in the context of feature-based pricing. Funding: The research of E. Fang is partially supported by the National Science Foundation [Grants NSF DMS-2346292, NSF DMS-2434666] and the Whitehead Scholarship. The research of C. Shi is partially supported by the Amazon Research Award. Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2024.1061 .
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- PAR ID:
- 10581612
- Publisher / Repository:
- INFORMS
- Date Published:
- Journal Name:
- Manufacturing & Service Operations Management
- Volume:
- 27
- Issue:
- 2
- ISSN:
- 1523-4614
- Page Range / eLocation ID:
- 535 to 553
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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