Nash equilibria for a model of traffic flow with several groups of drivers

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 κ_i drivers, sharing the same departure and arrival costs φ_i(t),ψ_i(t). For any given population sizes κ₁,...,κ_n, 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.