Multi-Modal Dynamic Pricing

37 Pages Posted: 26 Nov 2019 Last revised: 20 Jul 2020

See all articles by Yining Wang

Yining Wang

University of Florida - Warrington College of Business Administration

Boxiao Chen

University of Illinois at Chicago - College of Business Administration

David Simchi-Levi

Massachusetts Institute of Technology (MIT) - School of Engineering

Date Written: November 18, 2019

Abstract

We consider a stylistic question of dynamic pricing of a single product with demand learning. The candidate prices belong to a wide range of price interval, and the modeling of the demand functions is nonparametric in nature, imposing only smoothness regularity conditions. One important aspect of our modeling is the possibility of the expected reward function to be non-convex and indeed multi-modal, which leads to many conceptual and technical challenges. Our proposed algorithm is inspired by both the Upper-Confidence-Bound (UCB) algorithm for multi-armed bandit and the Optimism-in-Face-of-Uncertainty (OFU) principle arising from linear contextual bandits. Through rigorous regret analysis, we demonstrate that our proposed algorithm achieves optimal worst-case regret over a wide range of smooth function classes. More specifically, for k-times smooth functions and T selling periods, the regret of our propose algorithm is O(T^{(k+1)/(2k+1)}), which is shown to be optimal via information theoretical lower bounds. We also show that in special cases such as strongly concave or infinitely smooth reward functions, our algorithm achieves an O(sqrt{T}) regret matching optimal regret established in previous works. Finally, we present numerical results which verify the effectiveness of our method in numerical simulations.

Keywords: multi-modal reward function, dynamic pricing, nonparametric learning, asymptotic analyses

Suggested Citation

Wang, Yining and Chen, Boxiao and Simchi-Levi, David, Multi-Modal Dynamic Pricing (November 18, 2019). Available at SSRN: https://ssrn.com/abstract=3489355 or http://dx.doi.org/10.2139/ssrn.3489355

Yining Wang

University of Florida - Warrington College of Business Administration ( email )

Gainesville, FL 32611
United States

Boxiao Chen (Contact Author)

University of Illinois at Chicago - College of Business Administration ( email )

601 S Morgan St
Chicago, IL 60607
United States

David Simchi-Levi

Massachusetts Institute of Technology (MIT) - School of Engineering ( email )

MA
United States

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