Last-Iterate Convergence in No-Regret Learning: Games with Reference Effects Under Logit Demand

作者：Mengzi Amy Guo, Donghao Ying, Javad Lavaei, Zuo‐Jun Max Shen · 发表于：Management Science · 年份：2025 · DOI：10.1287/mnsc.2023.03464 · 被引用次数：3 · 研究领域：Auction Theory and Applications、Economic Policies and Impacts、Economic theories and models

This work examines the behaviors of the online projected gradient ascent (OPGA) algorithm and its variant in a repeated oligopoly price competition under reference effects. In particular, we consider that multiple firms engage in a multiperiod price competition, where consecutive periods are linked by the reference price update and each firm has access only to its own first-order feedback. Consumers assess their willingness to pay by comparing the current price against the memory-based reference price, and their choices follow the multinomial logit (MNL) model. We use the notion of stationary Nash equilibrium (SNE), defined as the fixed point of the equilibrium pricing policy, to simultaneously capture the long-run equilibrium and stability. We first study the loss-neutral reference effects and show that if the firms employ the OPGA algorithm—adjusting the price using the first-order derivatives of their log-revenues—the price and reference price paths attain last-iterate convergence to the unique SNE, thereby guaranteeing the no-regret learning and market stability. Moreover, with appropriate step-sizes, we prove that this algorithm exhibits a convergence rate of [Formula: see text] in terms of the squared distance and achieves a constant dynamic regret. Despite the simplicity of the algorithm, its convergence analysis is challenging due to the model lacking typical properties such as strong monotonicity and variational stability that are ordinarily used for the convergence ...