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 $λ_i(\mathbf p) = μ_i(\langle \boldsymbol θ_{i,0}, \mathbf p \rangle)$, with known increasing link $μ_i$ and unknown parameter $\boldsymbol θ_{i,0}$, 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.

翻译：摘要：我们研究了市场中的序贯价格竞争问题，该市场包含$N$个卖家，每个卖家的定价决策均受竞争对手影响。具体而言，每个卖家$i$的需求函数遵循单指标模型$\lambda_i(\mathbf p) = \mu_i(\langle \boldsymbol \theta_{i,0}, \mathbf p \rangle)$，其中$\mu_i$为已知的递增链接函数，$\boldsymbol \theta_{i,0}$为未知参数，向量$\mathbf{p}$表示某一时刻所有卖家同时提供的价格向量。每个卖家仅能观测到自身已实现的需求（竞争对手不可观测）以及对手设定的价格。我们提出了一种新型去中心化策略PML-GLUCB，该策略将惩罚极大似然估计与上置信界定价规则相结合。我们的方法：（i）消除了对卖家间协调式前期探索阶段的需求——而这一机制是先前模型的核心要素——使得该方法更贴合现实市场条件；（ii）推广了仅适用于线性需求模型的现有方法；（iii）同时适用于二元和实值需求观测。相对于动态基准策略，每个卖家可实现$\widetilde{O}(\sqrt{T})$的遗憾值，这与线性设定下已知的最优速率相匹配。