Contextual dynamic pricing aims to set personalized prices based on sequential interactions with customers. At each time period, a customer who is interested in purchasing a product comes to the platform. The customer’s valuation for the product is a linear function of contexts, including product and customer features, plus some random market noise. The seller does not observe the customer’s true valuation, but instead needs to learn the valuation by leveraging contextual information and historic binary purchase feedback. Existing models typically assume full or partial knowledge of the random noise distribution. In this paper, we consider contextual dynamic pricing with unknown random noise in the linear valuation model. Our distribution-free pricing policy learns both the contextual function and the market noise simultaneously. A key ingredient of our method is a novel perturbed linear bandit framework, in which a modified linear upper confidence bound algorithm is proposed to balance the exploration of market noise and the exploitation of the current knowledge for better pricing. We establish the regret upper bound and a matching lower bound of our policy in the perturbed linear bandit framework and prove a sublinear regret bound in the considered pricing problem. Finally, we demonstrate the superior performance of our policy on simulations and a real-life auto loan data set. Funding: Y. Liu and W.W. Sun acknowledge support from the National Science Foundation Division of Social and Economic Sciences [Grant NSF-SES 2217440]. Supplemental Material: The supplementary material is available at https://doi.org/10.1287/moor.2023.1369 .
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Online Allocation and Pricing: Constant Regret via Bellman Inequalities
We develop a framework for designing simple and efficient policies for a family of online allocation and pricing problems that includes online packing, budget-constrained probing, dynamic pricing, and online contextual bandits with knapsacks. In each case, we evaluate the performance of our policies in terms of their regret (i.e., additive gap) relative to an offline controller that is endowed with more information than the online controller. Our framework is based on Bellman inequalities, which decompose the loss of an algorithm into two distinct sources of error: (1) arising from computational tractability issues, and (2) arising from estimation/prediction of random trajectories. Balancing these errors guides the choice of benchmarks, and leads to policies that are both tractable and have strong performance guarantees. In particular, in all our examples, we demonstrate constant-regret policies that only require resolving a linear program in each period, followed by a simple greedy action-selection rule; thus, our policies are practical as well as provably near optimal.
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- NSF-PAR ID:
- 10241023
- Date Published:
- Journal Name:
- Operations Research
- ISSN:
- 0030-364X
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1287/opre.2020.2061
