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Globally Optimal and Nash Equilibrium Solutions for Traffic Flow on Networks

We consider a conservation law model of traffic flow on a network of roads, where drivers choose their departure times in order to minimize the sum of a departure cost and an arrival cost. Drivers can have different origins and destinations, and different cost functions. Under natural assumptions, two main results have been established: (i) the existence of a globally optimal solution, minimizing the sum of the costs to all drivers, and (ii) the existence of a Nash equilibrium solution, where no driver can lower his own cost by changing his departure time or the route taken to reach destination. In the special case of one single road, the global optimum and the Nash equilibrium are uniquely determined.