Departure Time Choice with Parametric Heterogeneity: Equilibrium and Instability

TL;DR

This paper investigates the stability of a modified departure time choice model with heterogeneity, proving that certain day-to-day dynamics are inherently unstable despite more realistic assumptions.

Contribution

It introduces a heterogeneity-inclusive model and provides a formal proof that common dynamic learning processes are unstable within this framework.

Findings

Existence and uniqueness of equilibrium are verified.

All studied day-to-day dynamics satisfying certain conditions are proven unstable.

The model's heterogeneity does not stabilize the equilibrium under typical dynamics.

Abstract

Vickrey's classic single-bottleneck departure time choice equilibrium model exhibits instability under many plausible day-to-day learning dynamics. Such instability is not observed in reality -- does this difference stem from the day-to-day dynamics or from one of the simplifying assumptions of the basic model? This paper explores a variant of the basic model with a continuous distribution of schedule delay parameters which we intuitively expect to have more favorable stability properties. To attain tractability we assume a monotonic relationship between earliness and lateness parameters. We first verify the existence and uniqueness of the equilibrium solution for this model. We then study a broad class of day-to-day dynamics satisfying local pressure and order preservation conditions. Our main contribution is a formal proof that, surprisingly, all such day-to-day dynamics in this context are unstable.