Efficiency and Braess´ Paradox under pricing in general networks

We study the flow control and routing decisions of self-interested users in a general congested network where a single profit-maximizing service provider sets prices for different paths in the network. We define an equilibrium of the user choices. We then define the monopoly equilibrium (ME) as the equilibrium prices set by the service provider and the corresponding user equilibrium. We analyze the networks containing different types of user utilities: elastic or inelastic. For a network containing inelastic user utilities, we show the flow allocations at the ME and the social optimum are the same. For a network containing elastic user utilities, we explicitly characterize the ME and study its performance relative to the user equilibrium at 0 prices and the social optimum that would result from centrally maximizing the aggregate system utility. We also define Braess´ Paradox for a network involving pricing and show that Braess´ Paradox does not occur under monopoly prices.