Traffic flow is modeled by a conservation law describing the density of cars. It is
assumed that each driver chooses his own departure time in order to minimize the sum of a
departure and an arrival cost. There are N groups of drivers, The
i-th group consists of κi
drivers, sharing the same departure and arrival costs
ϕi(t),ψi(t).
For any given population sizes
κ1,...,κn,
we prove the existence of a Nash equilibrium solution, where no driver can lower his own
total cost by choosing a different departure time. The possible non-uniqueness, and a
characterization of this Nash equilibrium solution, are also discussed.