In this study, we analyse the global stability of the equilibrium in a departure time choice problem using a game-theoretic approach that deals with atomic users. We first formulate the departure time choice problem as a strategic game in which atomic users select departure times to minimise their trip cost; we call this game the 'departure time choice game'. The concept of the epsilon-Nash equilibrium is introduced to ensure the existence of pure-strategy equilibrium corresponding to the departure time choice equilibrium in conventional fluid models. Then, we prove that the departure time choice game is a weakly acyclic game. By analysing the convergent better responses, we clarify the mechanisms of global convergence to equilibrium. This means that the epsilon-Nash equilibrium is achieved by sequential better responses of users, which are departure time changes to improve their own utility, in an appropriate order. Specifically, the following behavioural rules are important to ensure global convergence: (i) the adjustment of the departure time of the first user departing from the origin to the corresponding equilibrium departure time and (ii) the fixation of users to their equilibrium departure times in order (starting with the earliest). Using convergence mechanisms, we construct evolutionary dynamics under which global stability is guaranteed. We also investigate the stable and unstable dynamics studied in the literature based on convergence mechanisms, and gain insight into the factors influencing the different stability results. Finally, numerical experiments are conducted to demonstrate the theoretical results.

翻译：本研究采用博弈论方法处理原子用户，分析了出发时间选择问题中均衡的全局稳定性。我们首先将出发时间选择问题建模为一个策略博弈，其中原子用户选择出发时间以最小化其出行成本；我们将该博弈称为“出发时间选择博弈”。引入ε-纳什均衡概念，以确保存在与常规流体模型中出发时间选择均衡相对应的纯策略均衡。随后，我们证明了出发时间选择博弈是一个弱循环博弈。通过分析收敛的更好响应，我们阐明了向均衡全局收敛的机制。这意味着，在适当顺序下，用户依次通过改变出发时间以改善自身效用的更好响应，即可实现ε-纳什均衡。具体而言，以下行为规则对保证全局收敛至关重要：(i) 首个从起点出发的用户将其出发时间调整至对应的均衡出发时间，以及 (ii) 用户按照出发时间顺序（从最早者开始）固定至各自的均衡出发时间。基于收敛机制，我们构建了保证全局稳定性的进化动力学。我们还基于收敛机制研究了文献中讨论的稳定与不稳定动力学，并深入洞察了导致不同稳定性结果的影响因素。最后，通过数值实验验证了理论结果。