Abstract
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.