Departure Time Choice with Parametric Heterogeneity: Equilibrium and Instability

2026-04-15

AI Summary

This study investigates the stability of equilibria in day-to-day learning dynamics for commuters’ departure time choice under parameter heterogeneity. Building on the bottleneck model, it incorporates a continuous distribution of scheduling preference parameters and assumes a monotonic relationship between penalties for early and late arrival. The analysis focuses on daily adjustment processes satisfying local pressure and order-preservation conditions. By combining equilibrium analysis, fixed-point theory, and dynamical systems stability criteria, the paper rigorously demonstrates—despite the intuitive expectation that heterogeneity enhances stability—that all such day-to-day dynamics are inherently unstable once parameter heterogeneity is introduced. The work not only establishes the existence and uniqueness of the equilibrium solution but also reveals the counterintuitive yet pervasive instability of these dynamics, offering significant theoretical insights for modeling commuter behavior in transportation systems.

Technology Category

• Planning, Routing, and Scheduling: Scheduling under Uncertainty
• Game Theory and Economic Paradigms: Equilibrium
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements

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.

Problem

• departure time choice
• equilibrium instability
• schedule delay
• day-to-day dynamics
• bottleneck model

Innovation

• parametric heterogeneity
• departure time choice
• equilibrium instability
• day-to-day dynamics
• bottleneck model