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In transportation literature, User Equilibrium (UE) has been widely studied since early 1950’s, many studies of which define equilibrium flow of traffic for uncapacitated networks based on Wardrop’s first principle, implying also a Nash Equilibrium (NE). Although, in general, the two equilibria (UE and NE) are not explicitly the same, they are shown to be equivalent under special conditions, for uncapacitated UE, when volume-delay functions are separable, continuous, non-decreasing and non-negative. A good deal of research is devoted to explain UE in capacitated networks based on Wardrop’s first principle and the concept of generalized costs. However, UE for capacitated networks, even under the defined special conditions, is not equivalent to NE. This paper extends Wardrop’s first principle to explain UE in capacitated networks, which, under the same special conditions of uncapacitated networks, would represent an NE as well. Moreover, a complementarity equilibrium model is proposed for UE, based on an extension of Wardrop’s principle.

Wardrop’s First Principle: Extension for Capacitated Networks