## Optima and equilibria for traffic flow on a network of roads.

## Alberto Bressan

## Department of Mathematics, Penn State University

## Date and time:

Friday, March 15, 2013 - 1:00pm

## Abstract:

Daily traffic patterns are the result of a large number of individual decisions, where each driver chooses an optimal departure time and an optimal route to reach destination. From a mathematical perspective, traffic flow can be modeled by a family of conservation laws, describing the density of cars along each road. In addition, one can introduce a cost functional, accounting for the time that each driver spends on the road and a penalty for late arrival. In the case of a single road, under natural assumptions one can prove the existence of a unique globally optimal solution, minimizing the sum of the costs to all drivers. In a realistic situation, however, the actual traffic is better described as a Nash equilibrium, where no driver can lower his individual cost by changing his own departure time. For a single road, a characterization of the Nash solution can be provided, establishing its existence and uniqueness. It is interesting to compare the costs of the optimal and of the equilibrium solution. This yields indications on how to optimally design time-dependent fees to impose on toll roads.Using topological techniques, existence results have recently been extended to the general case of several groups of drivers, with different origins and destinations and different cost functions, traveling on a network of roads.An intriguing mathematical problem is to understand the dynamic stability of Nash equilibria. In this direction, some numerical experiments and conjectures will be discussed.