Resumen
We extend the proportional swap system by Smith (1984a) and the network tatonnement process by Friesz et al. (1994) to formulate the day-to-day evolution of departure time choice under bounded rationality in the bottleneck model. It is shown that the stationary points of the doubly dynamical systems exist and are equivalent to the boundedly rational user equilibrium (BRUE) points. We propose a dynamic pricing policy implemented in the day-to-day evolution process. Despite that the stationary points correspond to a set of BRUE points and the day-to-day departure rate without pricing may not converge to the set of BRUE points, the pricing policy can drive the day-to-day departure rate to reach a new equilibrium state, which is system-optimal (SO). The convergence to the SO state is theoretically proved. Finally, by numerical examples, we demonstrate the properties of the doubly dynamical systems and the effectiveness of the pricing policy.