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.

中文翻译：

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原子飞行器模型的出发时间选择问题的稳定性分析


在这项研究中，我们使用处理原子用户的博弈论方法分析了出发时间选择问题中平衡的全局稳定性。我们首先将出发时间选择问题表述为一个战略博弈，其中原子用户选择出发时间以最小化他们的行程成本;我们称这个游戏为 '出发时间选择游戏'。引入 epsilon-Nash 均衡的概念是为了确保存在对应于传统流体模型中出发时间选择均衡的纯策略均衡。然后，我们证明出发时间选择博弈是一个弱无环博弈。通过分析收敛的更好响应，我们阐明了全局收敛到平衡的机制。这意味着 epsilon-Nash 均衡是通过用户以适当的顺序进行更好的响应来实现的，这些响应是离开时间的变化以提高他们自己的效用。具体来说，以下行为规则对于确保全局收敛非常重要：（i） 将第一个离开起点的用户的离开时间调整为相应的均衡离开时间，以及 （ii） 按顺序将用户固定到他们的均衡离开时间（从最早开始）。使用收敛机制，我们构建了保证全球稳定性的进化动力学。我们还研究了文献中基于收敛机制研究的稳定和不稳定动力学，并深入了解影响不同稳定性结果的因素。最后，通过数值实验对理论结果进行了验证。