Multi-Class User Equilibria under Social Marginal Cost Pricing

In the congested cities of today, congestion pricing is a tempting alternative. With a single user class, already Beckmann et al. showed that “system optimal” traffic flows can be achieved by social marginal cost (SMC) pricing. However, different user classes can have wildly differing time values. Hence, when introducing tolls, one should consider multi-class user quilibria, where the classes have different time values. With SMC pricing, Netter claims that multi-class equilibrium problems cannot be stated as an optimization problems. We show that, depending on the formulation, the multi-class SMC-pricing equilibrium problem (with different time values) can be stated either as an asymmetric or as a symmetric equilibrium problem. In the latter case, the corresponding optimization problem is in general non-convex. For this non-convex problem, we devise descent methods of Frank-Wolfe type. We apply the methods and study a synthetic case based on Sioux Falls.