Priority service and max-min fairness

We study a pricing scheme for networks which use priorities to provide differentiated quality of service. We consider the situation where users are free to choose the priority of their traffic, but are charged accordingly. We model this situation as a non-cooperative game, where users behave in a selfish manner and choose an allocation of priorities to packets to optimize their own net benefit. We show that there exists an unique equilibrium for this game and the bandwidth allocation in equilibrium is weighted max-min fair.