Topological Conditions for Uniqueness of Equilibrium in Networks

Equilibrium flow in a physical network with a large number of users (e.g., transportation, communication, and computer networks) need not be unique if the costs of the network elements are not the same for all users. Such differences among users may arise if they are not equally affected by congestion or have different intrinsic preferences. Whether or not, for all assignments of strictly increasing cost functions, each user's equilibrium cost is the same in all Nash equilibria can be determined from the network topology. Specifically, this paper shows that in a two-terminal network, the equilibrium costs are always unique if and only if the network is one of several simple networks or consists of several such networks connected in series. The complementary class of all two-terminal networks with multiple equilibrium costs for some assignment of (user-specific) strictly increasing cost functions is similarly characterized by an embedded network of a particular simple type.