Optimal traffic conditions in two-link parallel networks with polynomial costs

In the classical nonatomic routing game model on a simple network, we seek to determine for how many different levels of demand the equilibrium distribution of selfish users on the network can coincide with the optimal one. To this end, we study the Price of Anarchy as a function of the traffic inflow, in the special case of two-link parallel networks and polynomial costs. We obtain, in the most simple cases, some sharp bounds on the number of solutions.