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.