Analyzing Path Dependence in Final Modal User Equilibrium Using a Joint Trip-Length Distribution Framework

Abstract

Traditional transportation policy evaluations typically focus on the final state of User Equilibrium (UE), assuming it is unique and independent of the implementation process. However, in complex multimodal networks, this uniqueness is often undermined by cross-modal dynamics. This study provides a novel framework to model modal equilibrium by reformulating the problem through a joint probability distribution of trip-length tuples. By shifting the focus from individual origin-destination pairs to trip-length distributions, the framework significantly reduces computational complexity while maintaining spatial accuracy via a trip-based Macroscopic Fundamental Diagram (MFD). We demonstrate the utility of this approach on the standard problem of designing and implementing pricing zones in dense urban areas. Taking Lyon, France, as a case study, we simulate the phased rollout of congestion pricing to observe the system’s evolution. Our findings reveal that the final equilibrium state is inherently path-dependent: the specific sequence of implementation phases fundamentally alters the eventual modal split and network utilization levels. This research contributes both a computationally efficient methodology for multimodal modeling and empirical evidence that the sequence of policy interventions are as critical to urban planning as the final design itself.