  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 stochastically stable under the logit response
dynamics, and the better/best response dynamics converges to locally optimal
states. 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.
