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：我们考虑多个销售商在$T$个销售周期内的价格竞争问题。在每个周期中，销售商同时公布其定价（价格公开），随后观察到各自的需求量（不公开）。每个销售商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销售商的价格。我们提出一种采用半参数最小二乘估计的动态定价策略，并证明当销售商采用该策略时，其价格将以$O(T^{-1/7})$的速率收敛至完全信息条件下所能达到的纳什均衡价格。相较于动态基准策略，每个销售商将产生$O(T^{5/7})$的遗憾值。本工作的理论贡献在于：通过引入$s$-凹性概念证明了形状约束需求函数下均衡的存在性，并建立了所提出策略的遗憾界。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。我们的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更广泛研究做出了贡献。