Online Price Competition under Generalized Linear Demands

TL;DR

This paper introduces a decentralized pricing policy for sellers in competitive markets with generalized demand models, achieving near-optimal regret without coordinated exploration.

Contribution

It proposes PML-GLUCB, a novel decentralized algorithm that handles generalized linear demand models and removes the need for front-loaded exploration phases.

Findings

Achieves  regret, matching optimal linear demand results.

Removes the need for coordinated exploration, aligning with realistic market conditions.

Generalizes existing linear demand models to broader generalized linear models.

Abstract

We study a sequential price competition among $N$ sellers, each influenced by the pricing decisions of their rivals. Specifically, the demand function for each seller $i$ follows the single index model $\lambda_i(\mathbf p) = \mu_i(\langle \boldsymbol \theta_{i,0}, \mathbf p \rangle)$, with known increasing link $\mu_i$ and unknown parameter $\boldsymbol \theta_{i,0}$, where the vector $\mathbf{p}$ denotes the vector of prices offered by all the sellers simultaneously at a given instant. Each seller observes only their own realized demand - unobservable to competitors - and the prices set by rivals. We propose a novel decentralized policy, PML-GLUCB, that combines penalized MLE with an upper-confidence pricing rule. Our approach (i) \emph{removes the need for coordinated front-loaded exploration phases across sellers} - which is integral to previous models - making our method aligned with realistic market conditions; (ii) generalizes existing approaches that focus solely on linear demand models; (iii) accommodates both binary and real-valued demand observations. Relative to a dynamic benchmark policy, each seller achieves $\widetilde{O}(\sqrt{T})$ regret, which matches the optimal rate known in the linear setting.