Dynamic Pricing and Demand Learning on a Large Network of Products: A PAC-Bayesian Approach

TL;DR

This paper develops a PAC-Bayesian approach for dynamic pricing in large product networks, enabling demand learning and regret minimization under various sparsity assumptions, with proven asymptotic optimality.

Contribution

It introduces a novel pricing-and-learning policy that leverages PAC-Bayesian methods and addresses different sparsity structures in product networks, achieving near-optimal regret bounds.

Findings

Policy achieves asymptotically optimal performance in N and T.

Pseudo-regret can be linear in N under certain sparsity conditions.

The approach handles large, complex product networks effectively.

Abstract

We consider a seller offering a large network of $N$ products over a time horizon of $T$ periods. The seller does not know the parameters of the products' linear demand model, and can dynamically adjust product prices to learn the demand model based on sales observations. The seller aims to minimize its pseudo-regret, i.e., the expected revenue loss relative to a clairvoyant who knows the underlying demand model. We consider a sparse set of demand relationships between products to characterize various connectivity properties of the product network. In particular, we study three different sparsity frameworks: (1) $L_0$ sparsity, which constrains the number of connections in the network, and (2) off-diagonal sparsity, which constrains the magnitude of cross-product price sensitivities, and (3) a new notion of spectral sparsity, which constrains the asymptotic decay of a similarity metric on network nodes. We propose a dynamic pricing-and-learning policy that combines the optimism-in-the-face-of-uncertainty and PAC-Bayesian approaches, and show that this policy achieves asymptotically optimal performance in terms of $N$ and $T$. We also show that in the case of spectral and off-diagonal sparsity, the seller can have a pseudo-regret linear in $N$, even when the network is dense.