Stabilisation strategy for unstable transport systems under general evolutionary dynamics

Stability of equilibria in transport systems has been discussed for decades. Even in deterministic cases, where stochasticity is ignored, stability is not a general property; a counterexample has been found in (within-day) dynamic traffic assignment problems. Instability can be a source of uncertainty of travel time and although pricing may stabilise an unstable transport system, pricing is not always acceptable to the public. This study aims to develop a pricing strategy that stabilises a transport system with minimum tolls. We show that with our stabilising pricing system tolls are bounded above and converge to zero when the error in estimation of a no-toll equilibrium converges to zero. We then show that the proposed toll scheme stabilises a wide range of evolutionary dynamics. We also propose a heuristic procedure to minimise the toll level. The procedure can also be viewed as a method of finding a possibly unstable equilibrium solution of a transport system. This suggests that, while we have not provided a rigorous proof, we may be able to find an equilibrium solution of any transport problem including problems which arise in dynamic traffic assignment (DTA); in these DTA cases, how to construct a solution algorithm that always converges to an equilibrium solution is still an open question. The methods proposed here will be extended so that they apply in more realistic behavioural settings in future work

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

Estimating Markov Chain Mixing Times: Convergence Rate Towards Equilibrium of a Stochastic Process Traffic Assignment Model

Network equilibrium models have been extensively used for decades. The rationale for using equilibrium as a predictor is essentially that (i) a unique and globally stable equilibrium point is guaranteed to exist and (ii) the transient period over which a system adapts to a change is sufficiently short in time that it can be neglected. However, we find transport problems without a unique and stable equilibrium in the literature. Even if it exists, it is not certain how long it takes for the system to reach an equilibrium point after an external shock onto the transport system, such as infrastructure improvement and damage by a disaster. The day-to-day adjustment process must be analysed to answer these questions. Among several models, the Markov chain approach has been claimed to be the most general and flexible. It is also advantageous as a unique stationary distribution is guaranteed in mild conditions, even when a unique and stable equilibrium does not exist. In the present paper, we first aim to develop a methodology for estimating the Markov chain mixing time (MCMT), a worst-case assessment of the convergence time of a Markov chain to its stationary distribution. The main tools are coupling and aggregation, which enable us to analyse MCMTs in large-scale transport systems. Our second aim is to conduct a preliminary examination of the relationships between MCMTs and critical properties of the system, such as travellers’ sensitivity to differences in travel cost and the frequency of travellers’ revisions of their choices. Through analytical and numerical analyses, we found key relationships in a few transport problems, including those without a unique and stable equilibrium. We also showed that the proposed method, combined with coupling and aggregation, can be applied to larger transport models