The multi-class, multi-criterion traffic equilibrium and the efficiency of congestion pricing

We consider the effect of the so-called second-best tolls on the price of anarchy of the traffic equilibrium problem where there are multiple classes of users with a discrete set of values of time. We derive several bounds of the price of anarchy for this problem when the tolls are considered and not considered as part of the system cost, with the time-based criterion and the cost-based criterion, respectively. All the bounds give us useful information on the effect of the tolls, which can be used to design network toll schemes.