Context-Based Dynamic Pricing with Separable Demand Models

Motivated by the empirical evidence observed from the real-world dataset, this paper studies context-based dynamic pricing with separable demand models. Consider a seller selling products over a finite horizon of $T$ periods and facing an unknown expected demand function that admits a separable structure of the form $f(p)+g(x)$, where $p\in\mathbb{R}$ and $x\in\mathbb{R}^d$ denote the product's price and features respectively. The seller does not know the exact form of $f(p)$ or $g(x)$, but can dynamically adjust prices in each period based on the observed features and random demands. The seller's objective is to maximize the $T$-period expected revenue. We systematically characterize the statistical complexity of the online learning problem under different structures of $f(p)$ and $g(x)$. Specifically, we study three different models: (i) $f(p)$ is linear and $g(x)$ is generally β-Hölder continuous; (ii) $f(p)$ is generally kth-order smooth and $g(x)$ is linear; and (iii) $f(p)$ is generally kth-order smooth and $g(x)$ is generally β-Hölder continuous. For each demand model, we design an online learning algorithm with a provable regret upper bound, and establish a regret lower bound that must be incurred by any algorithm. Except for a corner case in the third model, the regret upper bound of our algorithm matches the corresponding lower bound, which enables us to characterize the optimal regret rate for the learning problem under each model. Our results reveal the fundamental differences in the optimal regret rates when $f(p)$ or $g(x)$ is associated with different structural properties. The numerical results demonstrate that our learning algorithms are more effective than benchmark algorithms for all the three models, and also show the effects of the parameters associated with $f(p)$ and $g(x)$ on the algorithm's empirical regret. Finally, we consider the extension to the setting where the effects of contexts on demands are also separable, and show that the regret can be further reduced.