Sufficient conditions for the uniqueness of the solution to the Stochastic User Equilibrium problem

Referring to additive demand models, the Stochastic User Equilibrium can be formulated in the reduced space of travel alternative generalized costs as a fixed point problem defined on R , where n is the number of travel alternatives minus the number of user classes. Through the analysis of an equivalent non-linear program, a new sufficient condition for the uniqueness of the solution to a generic unconstrained fixed point problem is introduced. The application of this result to the Stochastic User Equilibrium problem permits to derive a weaker sufficient condition for the uniqueness of the equilibrium than that requiring the Jacobian of the arc cost function to be positive semi definite. Specifically, it is shown that the problem admits only one solution also if this Jacobian becomes positive definite when summed up to the identity matrix multiplied by a positive scalar, whose value is strictly related to the Jacobian of the demand function. Some elementary cases are presented where the above condition is employed in order to characterize the equilibrium. This way it is shown that the lack of definiteness of the arc cost function Jacobian is not a taboo for establishing the uniqueness of the solution, and the problem of how much this condition may be violated can and should be addressed with reference to each specific instance of traffic assignment.