Traffic user equilibrium and proportionality

We discuss the problem of proportionality and uniqueness for route flows in the classical traffic user equilibrium model. It is well-known that under appropriate assumptions the user equilibrium $(f,x)$ is unique in the link flow $x$ but typically not in the route flow $f$. We consider the concept of proportionality in detail and re-discuss the well-known relation between the so-called bypass proportionality and entropy maximization. We exhibit special proportionality conditions which uniquely determine the route flow $f$. The results are illustrated with some simple example networks