  The recent emergence of navigational tools has changed traffic patterns and
has now enabled new types of congestion-aware routing control like dynamic road
pricing. Using the fundamental diagram of traffic flows - applied in
macroscopic and mesoscopic traffic modeling - the article introduces a new
N-player dynamic routing game with explicit congestion dynamics. The model is
well-posed and can reproduce heterogeneous departure times and congestion spill
back phenomena. However, as Nash equilibrium computations are PPAD-complete,
solving the game becomes intractable for large but realistic numbers of
vehicles N. Therefore, the corresponding mean field game is also introduced.
Experiments were performed on several classical benchmark networks of the
traffic community: the Pigou, Braess, and Sioux Falls networks with
heterogeneous origin, destination and departure time tuples. The Pigou and the
Braess examples reveal that the mean field approximation is generally very
accurate and computationally efficient as soon as the number of vehicles
exceeds a few dozen. On the Sioux Falls network (76 links, 100 time steps),
this approach enables learning traffic dynamics with more than 14,000 vehicles.
