The ϵ-equilibrium in transportation networks

Abstract

Abstract The principle of traffic equilibrium (Wardrop, 1952) demands that the trip makers have enough knowledge of transportation networks, which needs to be attained by accumulating experiences for a long time. Because of the complexity of traffic situation in the networks, the route choice decisions for trip makers are, however, not always optimal objectively. As a consequence, it is impossible to strictly attain Wardrop equilibrium. In this paper, a new concept of equilibrium, ϵ-equilibrium is proposed under the realistic assumption, i.e., the fuzzy road information and random route choices. ϵ-equilibrium in the network is discussed by adopting fuzzy set theory, mixed stochastic Games and Brouwer fixed point theorem in topological space. The conclusion is that there is ϵ-equilibrium in the real transportation network and this ϵ-equilibrium has some deviations from the strict Wardrop equilibrium. The notion of ϵ-equilibrium extends the traditional notion of network equilibrium. It shows that traffic flow in the network converges to a region rather than to the points.