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，销售商同时公布其价格（价格信息为公开），随后观察各自的需求（需求信息不公开）。每个销售商的需求函数通过私有的、未知的非线性关系依赖于所有销售商的价格。我们提出一种动态定价策略，该策略采用半参数最小二乘估计方法，并证明当销售商采用该策略时，其价格以$O(T^{-1/7})$的速率收敛于完全信息条件下可达到的纳什均衡价格。相对于动态基准策略，每个销售商产生$O(T^{5/7})$的遗憾值。本研究的理论贡献在于：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，并确定了所提策略的遗憾界。在技术层面，我们还建立了形状约束下最小二乘估计量的新集中性结果。研究结果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作出了贡献。