Multimodal Dynamic Pricing

Abstract

We consider a single product dynamic pricing with demand learning. The candidate prices belong to a wide range of a price interval; the modeling of the demand functions is nonparametric in nature, imposing only smoothness regularity conditions. One important aspect of our model is the possibility of the expected reward function to be nonconcave and indeed multimodal, which leads to many conceptual and technical challenges. Our proposed algorithm is inspired by both the Upper-Confidence-Bound algorithm for multiarmed bandit and the Optimism-in-the-Face-of-Uncertainty principle arising from linear contextual bandits. The multiarmed bandit formulation arises from local-bin approximation of an unknown continuous demand function, and the linear contextual bandit formulation is then applied to obtain more accurate local polynomial approximators within each bin. 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 proposed algorithm is [Formula: see text], which is shown to be optimal via the development of information theoretical lower bounds. We also show that in special cases, such as strongly concave or infinitely smooth reward functions, our algorithm achieves an [Formula: see text] regret, matching optimal regret established in previous works. Finally, we present computational results that verify the effectiveness of our method in numerical simulations. This paper was accepted by J. George Shanthikumar, big data analytics.

Towards Agnostic Feature-based Dynamic Pricing: Linear Policies vs Linear Valuation with Unknown Noise

“…, φ k (u)] as input, and outputs v = η x. A similar linearization idea in non-feature pricing has been adopted in Wang et al (2021) and achieves optimal regrets. However, it is still unknown whether their methods can be applied to this feature-based LV problem.…”

“…The crux of pricing is to learn the demand curve (i.e., the noise distribution in our LP problem) from Boolean-censored feedback. Wang et al (2021) concludes existing results and characterizes the impact of different assumptions on the demand curve on the minimax regret. The problem of contextual dynamic pricing is more challenging mainly because we need to learn the valuation parameter θ * and the noise distribution jointly.…”

“…Non-Contextual Dynamic Pricing. Dynamic pricing was extensively studied under the singleproduct (non-contextual) setting (Kleinberg and Leighton, 2003;Zeevi, 2009, 2012;Wang et al, 2014;Besbes and Zeevi, 2015;Chen et al, 2019;Wang et al, 2021). The crux of pricing is to learn the demand curve (i.e., the noise distribution in our LP problem) from Boolean-censored feedback.…”

“…In a dynamic pricing process, a seller presents prices for the products and adjusts these prices according to customers' feedback (i.e., whether they decide to buy or not) to maximize the revenue. Existing works on the single-product pricing problem (Kleinberg and Leighton, 2003;Wang et al, 2021) assume that customers make decisions only according to the comparisons between prices and their own (random) valuations, and the goal is to find out a best fixed price that maximizes the (expected) revenue. In general, the single-product pricing problem has been well studied under a variety of assumptions.…”

On Dynamic Pricing with Covariates

“…den Boer and Keskin (2020) consider a piece-wise linear (discontinuous) demand function and the model therein might point to a direction for non-Lipschitzness. Besides, the consideration of a multimodal revenue function (Wang et al 2021) in dynamic pricing under the arbitrary covariates is not addressed in this work.…”