Equilibria for Networks with Lower Semicontinuous Costs: With an Application to Congestion Pricing

Network equilibrium models traditionally rely on the assumption of continuous travel cost functions. However, there are times when this assumption is not appropriate. For example, some proposed congestion pricing schemes make use of discontinuous step-function tolls. Hence, the purpose of this paper is to explore the existence of network equilibria in the presence of discontinuities. In particular it is shown that when such costs are at least lower semicontinuous, a behaviorally meaningful notion of user equilibrium can still be defined which reduces to Wardrop equilibrium in the continuous case. In addition, it is shown that such equilibria are guaranteed to exist under fairly general conditions. Finally, these results are applied to construct a class of lower semicontinuous congestion pricing schemes which ensure the existence of such equilibria.