Congestion Pricing of Road Networks with Users Having Different Time Values

We study congestion pricing of road networks with users differing only in their time values. In particular, we analyze the marginal social cost (MSC) pricing, a tolling scheme that charges each user a penalty corresponding to the value of the delays inflicted on other users, as well as its implementation through fixed tolls. We show that the variational inequalities characterizing the corresponding equilibria can be stated in symmetric or nonsymmetric forms. The symmetric forms correspond to optimization problems, convex in the fixed-toll case and nonconvex in the MSC case, which hence may have multiple equilibria. The objective of the latter problem is the total value of travel time, which thus is minimized at the global optima of that problem. Implementing close-to-optimal MSC tolls as fixed tolls leads to equilibria with possibly non-unique class specific flows, but with identical close-to-optimal values of the total value of travel time. Finally we give an adaptation, to the MSC setting, of the Frank-Wolfe algorithm, which is further applied to some test cases, including Stockholm.