Properties of equilibria in transport problems with complex interactions between users

It is well known that uniqueness and stability are guaranteed properties of traffic equilibria in static user-equilibrium traffic assignment problems, if the link travel utilities are assumed to be strictly monotonically decreasing with respect to the link traffic volumes. However, these preferable properties may not necessarily hold in a wide range of transport problems with complex interactions, e.g. asymmetric interactions (including dynamic traffic assignment), social interactions, or with economies of scale. This study aims to investigate such solution properties of transport models with complex interactions between users. Generic formulations of models are considered in this study, both for utility functions and for the evolutionary dynamics relevant to the stability analysis. Such an analysis for a generic formulation is mathematically challenging due to the potential non-differentiability of the dynamical system, precluding the application of standard analyses for smooth systems. To address this issue, this study proposes a transport system with two alternatives and two user groups. While it is a simple model whose dynamics can be depicted on a plane, it also includes the core components of transport models, i.e. multiple choices and user-classes. This study classifies all possible formulations into nine cases with respect to the signs (i.e. positive or negative) of interactions between users. Then, the evolutionary dynamics of each case is mathematically analysed to examine stability of equilibria. Finally, the solution properties of each case is revealed. Multiple equilibria exist in many cases. In addition, cases with no stable equilibrium are also found, yet even in such cases we are able to characterise the circumstances in which the different kinds of unstable behaviour may arise.