We study long-run market stability for repeated price competitions between two firms, where consumer demand depends on firms' posted prices and consumers' price expectations called reference prices. Consumers' reference prices vary over time according to a memory-based dynamic, which is a weighted average of all historical prices. We focus on the setting where firms are not aware of demand functions and how reference prices are formed but have access to an oracle that provides a measure of consumers' responsiveness to the current posted prices. We show that if the firms run no-regret algorithms, in particular, online mirror descent(OMD), with decreasing step sizes, the market stabilizes in the sense that firms' prices and reference prices converge to a stable Nash Equilibrium (SNE). Interestingly, we also show that there exist constant step sizesunder which the market stabilizes. We further characterize the rate of convergence to the SNE for both decreasing and constant OMD step sizes.

中文翻译：

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消费者参考效应下价格竞争中的无悔学习

我们研究了两家公司之间重复价格竞争的长期市场稳定性，其中消费者需求取决于公司公布的价格和消费者的价格预期，称为参考价格。消费者的参考价格根据基于记忆的动态随时间变化，这是所有历史价格的加权平均值。我们专注于企业不了解需求函数以及参考价格是如何形成的，但可以访问提供消费者对当前发布价格响应的度量的预言机。我们表明，如果公司运行无后悔算法，特别是在线镜像下降 (OMD)，随着步长的减小，市场会稳定，因为公司的价格和参考价格会收敛到稳定的纳什均衡 (SNE)。有趣的是，我们还表明存在稳定的市场稳定步长。我们进一步表征了 OMD 步长减小和不变时 SNE 的收敛速度。