# Preprint 2011-005

# Optima and equilibria for a model of traffic flow

## Alberto Bressan and Ke Han

**Abstract:**
The paper is concerned with the
Lighthillâ€“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 time and on the arrival time
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 solution.
Numerical simulations show that,
by adding a time-dependent toll at the entrance of a highway,
in some cases one can actually reduce the cost to each driver
in the corresponding Nash equilibrium solution.