Equilibrium of heterogeneous congestion control: Existence and uniqueness

When heterogeneous congestion control protocols that react to different pricing signals share the same network, the resulting equilibrium may no longer be interpreted as a solution to the standard utility maximization problem. We prove the existence of equilibrium in general multiprotocol networks under mild assumptions. For almost all networks, the equilibria are locally unique, finite, and odd in number. They cannot all be locally stable unless there is a globally unique equilibrium. Finally, we show that if the price mapping functions, which map link prices to effective prices observed by the sources, are sufficiently similar, then global uniqueness is guaranteed.