Equilibria and price of anarchy in parallel relay networks with node pricing

We study pricing games in single-layer relay networks where the source routes traffic selfishly according to the strategic bids made by relays. Each relay´s bid includes a charging function and a proposed traffic share. Relays aim to maximize their individual profit from forwarding traffic. We show that the socially optimal traffic allocation can always be induced by an equilibrium where no relay can increase its profit by unilaterally changing its bids. Inefficient equilibria arise due to the monopolistic pricing power of a superior relay. This lead to a finite price of anarchy if marginal cost functions are concave, and an unbounded price of anarchy when the marginal cost functions are convex.