We study the dynamic pricing problem where the demand function is nonparametric and H\"older smooth, and we focus on adaptivity to the unknown H\"older smoothness parameter $\beta$ of the demand function. Traditionally the optimal dynamic pricing algorithm heavily relies on the knowledge of $\beta$ to achieve a minimax optimal regret of $\widetilde{O}(T^{\frac{\beta+1}{2\beta+1}})$. However, we highlight the challenge of adaptivity in this dynamic pricing problem by proving that no pricing policy can adaptively achieve this minimax optimal regret without knowledge of $\beta$. Motivated by the impossibility result, we propose a self-similarity condition to enable adaptivity. Importantly, we show that the self-similarity condition does not compromise the problem's inherent complexity since it preserves the regret lower bound $\Omega(T^{\frac{\beta+1}{2\beta+1}})$. Furthermore, we develop a smoothness-adaptive dynamic pricing algorithm and theoretically prove that the algorithm achieves this minimax optimal regret bound without the prior knowledge $\beta$.

This article is part of a VB special issue. Read the full series here: How Data Privacy Is Transforming Marketing. Product pricing plays a critical role for every product, especially in ecommerce. According to Shopify, global ecommerce sales are expected to total $5.7 trillion worldwide in 2022. However, determining the right price for your goods and services can be tricky and requires large volumes of data to be effective: Should you use static prices, monitor those of your rivals or mix the two?

We propose a novel \textit{online regularization} scheme for revenue-maximization in high-dimensional dynamic pricing algorithms. The online regularization scheme equips the proposed optimistic online regularized maximum likelihood pricing (\texttt{OORMLP}) algorithm with three major advantages: encode market noise knowledge into pricing process optimism; empower online statistical learning with always-validity over all decision points; envelop prediction error process with time-uniform non-asymptotic oracle inequalities. This type of non-asymptotic inference results allows us to design safer and more robust dynamic pricing algorithms in practice. In theory, the proposed \texttt{OORMLP} algorithm exploits the sparsity structure of high-dimensional models and obtains a logarithmic regret in a decision horizon. These theoretical advances are made possible by proposing an optimistic online LASSO procedure that resolves dynamic pricing problems at the \textit{process} level, based on a novel use of non-asymptotic martingale concentration. In experiments, we evaluate \texttt{OORMLP} in different synthetic pricing problem settings and observe that \texttt{OORMLP} performs better than \texttt{RMLP} proposed in \cite{javanmard2019dynamic}.

The invention of price tag took place in the 1870's to maintain the fairness of everybody looking to buy the product they love. Dynamic pricing had always been the norm ever since human history. A century back even the ticket for a cinema was charged less for a matinee screening as compared to the usual popular evening shows. Born out of the '80's, dynamic pricing is now one of the most commonly used marketing techniques by several industries. Anyone old enough will remember the American Airlines' Super Saver fares online commercial where the airline played a major cutthroat with the fares.

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We study the problem of learning \emph{across} a sequence of price experiments for related products, focusing on implementing the Thompson sampling algorithm for dynamic pricing. We consider a practical formulation of this problem where the unknown parameters of the demand function for each product come from a prior that is shared across products, but is unknown a priori. Our main contribution is a meta dynamic pricing algorithm that learns this prior online while solving a sequence of non-overlapping pricing experiments (each with horizon $T$) for $N$ different products. Our algorithm addresses two challenges: (i) balancing the need to learn the prior (\emph{meta-exploration}) with the need to leverage the current estimate of the prior to achieve good performance (\emph{meta-exploitation}), and (ii) accounting for uncertainty in the estimated prior by appropriately "widening" the prior as a function of its estimation error, thereby ensuring convergence of each price experiment. We prove that the price of an unknown prior for Thompson sampling is negligible in experiment-rich environments (large $N$). In particular, our algorithm's meta regret can be upper bounded by $\widetilde{O}\left(\sqrt{NT}\right)$ when the covariance of the prior is known, and $\widetilde{O}\left(N^{\frac{3}{4}}\sqrt{T}\right)$ otherwise. Numerical experiments on synthetic and real auto loan data demonstrate that our algorithm significantly speeds up learning compared to prior-independent algorithms or a naive approach of greedily using the updated prior across products.

Machine learning is the technology behind any sophisticated dynamic pricing algorithm. These algorithms make optimal pricing decisions in real time, helping a business increase revenues or profits. In the case of a freemium mobile app, a dynamic pricing algorithm sets optimal prices for in-app purchases to increase revenues and engage price-sensitive customers. Dynamic pricing for mobile games and apps is a recent innovation. The majority of mobile apps are not using dynamic pricing algorithms today.