Optima and equilibria for a model of traffic flow

The paper is concerned with the Lighthill-Whitham model of traffic flow, where the density of cars is described by a scalar conservation law. A cost functional is introduced, depending on the departure and arrival times of each driver. Under natural assumptions, we prove the existence of a unique globally optimal solution, minimizing the total cost to all drivers. This solution contains no shocks and can be explicitly described. We also prove the existence of a Nash equilibrium solution, where no driver can lower his individual cost by changing his own departure time. A characterization of the Nash solution is provided, establishing its uniqueness. Some explicit examples are worked out, comparing the costs of the optimal and the equilibrium solutions. The analysis also yields a strategy for optimal toll pricing.