Online Price Competition under Generalized Linear Demands

We study 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 $λ_i(\mathbf{p}) = μ_i(\langle \boldsymbolθ_{i,0}, \mathbf{p} \rangle)$, with known increasing link $μ_i$ and unknown parameter $\boldsymbolθ_{i,0}$, 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. Our framework generalizes existing approaches that focus solely on linear demand models. We propose a novel decentralized policy, PML-GLUCB, that combines penalized MLE with an upper-confidence pricing rule, removing the need for coordinated exploration phases across sellers – which is integral to previous linear models – and accommodating both binary and real-valued demand observations. Relative to a dynamic benchmark policy, each seller achieves $O(N^{2}\sqrt{T}\log(T))$ regret, which essentially matches the optimal rate known in the linear setting. A significant technical contribution of our work is the development of a variant of the elliptical potential lemma – typically applied in single-agent systems – adapted to our competitive multi-agent environment.

💡 Research Summary

The paper studies a sequential pricing game with N sellers who simultaneously set prices for a homogeneous product over a finite horizon T. Unlike most prior work that assumes a monopolist or linear demand, each seller’s expected demand follows a generalized linear (single‑index) model
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