Uniqueness of the Nash Equilibrium in Convex Routing Games: Topological Conditions

We consider the problem of non-cooperative routing in a congested network with atomic users, where each user wishes to minimize the cost of its own flow. Cost functions may differ among users, and are required to satisfy standard convexity and monotonicity conditions. A characterization is given of the class of two-terminal network topologies for which the Nash equilibrium is always unique. We further address the uniqueness issue for the mixed Nash-Wardrop equilibrium problem.