Dynamic Pricing in High-dimensions

TL;DR

This paper introduces a dynamic pricing policy for high-dimensional product features that learns customer preferences over time, achieving near-optimal regret bounds in online marketplaces.

Contribution

It proposes the Regularized Maximum Likelihood Pricing (RMLP) algorithm, leveraging sparsity in high-dimensional choice models to minimize regret.

Findings

Achieves logarithmic regret in the number of time periods T.

Regret bound of O(s_0 log d log T), close to the theoretical lower bound.

Demonstrates effectiveness in high-dimensional online pricing scenarios.

Abstract

We study the pricing problem faced by a firm that sells a large number of products, described via a wide range of features, to customers that arrive over time. Customers independently make purchasing decisions according to a general choice model that includes products features and customers' characteristics, encoded as $d$-dimensional numerical vectors, as well as the price offered. The parameters of the choice model are a priori unknown to the firm, but can be learned as the (binary-valued) sales data accrues over time. The firm's objective is to minimize the regret, i.e., the expected revenue loss against a clairvoyant policy that knows the parameters of the choice model in advance, and always offers the revenue-maximizing price. This setting is motivated in part by the prevalence of online marketplaces that allow for real-time pricing. We assume a structured choice model, parameters of which depend on $s_0$ out of the $d$ product features. We propose a dynamic policy, called Regularized Maximum Likelihood Pricing (RMLP) that leverages the (sparsity) structure of the high-dimensional model and obtains a logarithmic regret in $T$. More specifically, the regret of our algorithm is of $O(s_0 \log d \cdot \log T)$. Furthermore, we show that no policy can obtain regret better than $O(s_0 (\log d + \log T))$.