  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.
