We consider price competition among multiple sellers over a selling horizon of $T$ periods. In each period, sellers simultaneously offer their prices (which are made public) and subsequently observe their respective demand (not made public). The demand function of each seller depends on all sellers' prices through a private, unknown, and nonlinear relationship. We propose a dynamic pricing policy that uses semi-parametric least-squares estimation and show that when the sellers employ our policy, their prices converge at a rate of $O(T^{-1/7})$ to the Nash equilibrium prices that sellers would reach if they were fully informed. Each seller incurs a regret of $O(T^{5/7})$ relative to a dynamic benchmark policy. A theoretical contribution of our work is proving the existence of equilibrium under shape-constrained demand functions via the concept of $s$-concavity and establishing regret bounds of our proposed policy. Technically, we also establish new concentration results for the least squares estimator under shape constraints. Our findings offer significant insights into dynamic competition-aware pricing and contribute to the broader study of non-parametric learning in strategic decision-making.

翻译：我们研究多个卖家在$T$个销售周期内的价格竞争问题。每个周期中，卖家同时公布其价格（公开可见），随后观察各自的需求量（不公开）。每个卖家的需求函数通过一种私有、未知且非线性的关系依赖于所有卖家的价格。我们提出一种采用半参数最小二乘估计的动态定价策略，并证明当卖家采用该策略时，其价格以$O(T^{-1/7})$的速率收敛至完全信息条件下可达到的纳什均衡价格。相较于动态基准策略，每个卖家产生的遗憾为$O(T^{5/7})$。本工作的理论贡献在于：通过$s$-凹性概念证明了形状约束需求函数下均衡的存在性，并建立了所提策略的遗憾界。在技术层面，我们还为形状约束下的最小二乘估计量建立了新的集中性结果。本研究为动态竞争感知定价提供了重要见解，并对战略决策中非参数学习的广泛研究作出了贡献。