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

• 成果类型: Article; Early Access
• 署名作者: Guo, Mengzi Amy; Ying, Donghao; Lavaei, Tavad; Shen, Zuo-. Tun Max
• 署名单位: University of California System; University of California Berkeley; University of Hong Kong; University of Hong Kong
• 刊物名称: MANAGEMENT SCIENCE
• ISSN/ISSBN: 0025-1909
• DOI: 10.1287/mnsc.2023.03464
• 发表日期: 2025
• 关键词: last-iterate convergence; Price competition; Reference effect; Multinomial logit model

摘要

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 Oe(1/t2) 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 analysis of online games. The inherent asymmetry nature of reference effects motivates the exploration beyond lossneutrality. When loss-averse reference effects are introduced, we propose a variant of the original algorithm named the conservative-OPGA (C-OPGA) to handle the nonsmooth revenue functions and show that the price and reference price achieve last-iterate convergence root fiffi to the set of SNEs with the rate of O(1/ t ). Finally, we demonstrate the practicality and robustness of OPGA and C-OPGA by theoretically showing that these algorithms can also adapt to firm-differentiated step-sizes and inexact gradients.