Continuity, Uniqueness and Long-Term Behavior of Nash Flows Over Time

We consider a dynamic model of traffic that has received a lot of attention in the past few years. Users control infinitesimal flow particles aiming to travel from a source to destination as quickly as possible. Flow patterns vary over time, and congestion effects are modeled via queues, which form whenever the inflow into a link exceeds its capacity. Despite lots of interest, some very basic questions remain open in this model. We resolve a number of them: - We show uniqueness of journey times in equilibria. - We show continuity of equilibria: small perturbations to the instance or to the traffic situation at some moment cannot lead to wildly different equilibrium evolutions. - We demonstrate that, assuming constant inflow into the network at the source, equilibria always settle down into a "steady state" in which the behavior extends forever in a linear fashion. One of our main conceptual contributions is to show that the answer to the first two questions, on uniqueness and continuity, are intimately connected to the third. Our result also shows very clearly that resolving uniqueness and continuity, despite initial appearances, cannot be resolved by analytic techniques, but are related to very combinatorial aspects of the model. To resolve the third question, we substantially extend the approach of Cominetti et al., who show a steady-state result in the regime where the input flow rate is smaller than the network capacity.