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We study price-of-anarchy type questions in two-sided markets with combinatorial consumers and limited supply sellers. Sellers own edges in a network and sell bandwidth at fixed prices subject to capacity constraints; consumers buy bandwidth between their sources and sinks so as to maximize their value from sending traffic minus the prices they pay to edges. We characterize the price of anarchy and price of stability in these "network pricing" games with respect to two objectives-the social value (social welfare) of the consumers, and the total profit obtained by all the sellers. In single-source single-sink networks we give tight bounds on these quantities based on the degree of competition, specifically the number of monopolistic edges, in the network. In multiple-source singlesink networks, we show that equilibria perform well only under additional assumptions on the network and demand structure. ⋆ Supported by NSF CAREER award CCF-0643763. ⋆⋆ Supported in part by NSF CAREER Award CCF-0448664, an ONR Young Investigator Award, and an Alfred P. Sloan Fellowship.

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The Internet is composed of multiple economically-independent service providers that sell bandwidth in their networks so as to maximize their own revenue. Users, on the other hand, route their traffic selfishly to maximize their own utility. How does this selfishness impact the efficiency of operation of the network? To answer this question we consider a two-stage network pricing game where service providers first select prices to charge on their links, and users then pick paths to route their traffic. We give tight bounds on the price of anarchy of the game with respect to social value-the total value obtained by all the traffic routed. Unlike recent work on network pricing, in our pricing game users do not face congestion costs; instead service providers must ensure that capacity constraints on their links are satisfied. Our model extends the classic Bertrand game in economics to network settings.