Dynamic system optimal traffic assignment with atomic users: Convergence and stability

TL;DR

This paper analyzes the convergence and stability of dynamic system optimal traffic assignment with atomic users, demonstrating that certain response dynamics lead to stable and efficient traffic states, with implications for pricing schemes.

Contribution

It formulates DSO traffic assignment as a potential game and proves convergence and stability properties of response dynamics, also examining evolutionary pricing schemes.

Findings

Logit response dynamics lead to globally stable states.

Better/best response dynamics converge to local optima.

Evolutionary marginal cost pricing improves traffic efficiency and robustness.

Abstract

In this study, we analyse the convergence and stability of dynamic system optimal (DSO) traffic assignment with fixed departure times. We first formulate the DSO traffic assignment problem as a strategic game wherein atomic users select routes that minimise their marginal social costs, called a 'DSO game'. By utilising the fact that the DSO game is a potential game, we prove that a globally optimal state is a stochastically stable state under the logit response dynamics, and the better/best response dynamics converges to a locally optimal state. Furthermore, as an application of DSO assignment, we examine characteristics of the evolutionary implementation scheme of marginal cost pricing. Through theoretical comparison with a fixed pricing scheme, we found the following properties of the evolutionary implementation scheme: (i) the total travel time decreases smoother to an efficient traffic state as congestion externalities are perfectly internalised; (ii) a traffic state would reach a more efficient state as the globally optimal state is stabilised. Numerical experiments also suggest that these properties make the evolutionary scheme robust in the sense that they prevent a traffic state from going to worse traffic states with high total travel times.