Price Competition in Networked Markets: How do monopolies impact social   welfare?
  We study the efficiency of allocations in large markets with a network structure where every seller owns an edge in a graph and every buyer desires a path connecting some nodes. While it is known that stable allocations in such settings can be very inefficient, the exact properties of equilibria in markets with multiple sellers are not fully understood even in single-source single-sink networks. In this work, we show that for a large class of natural buyer demand functions, we are guaranteed the existence of an equilibrium with several desirable properties. The crucial insight that we gain into the equilibrium structure allows us to obtain tight bounds on efficiency in terms of the various parameters governing the market, especially the number of monopolies M. All of our efficiency results extend to markets without the network structure.   While it is known that monopolies can cause large inefficiencies in general, our main results for single-source single-sink networks indicate that for several natural demand functions the efficiency only drops linearly with M. For example, for concave demand we prove that the efficiency loss is at most a factor 1+M/2 from the optimum, for demand with monotone hazard rate it is at most 1+M, and for polynomial demand the efficiency decreases logarithmically with M. In contrast to previous work that showed that monopolies may adversely affect welfare, our main contribution is showing that monopolies may not be as `evil' as they are made out to be; the loss in efficiency is bounded in many natural markets. Finally, we consider more general, multiple-source networks and show that in the absence of monopolies, mild assumptions on the network topology guarantee an equilibrium that maximizes social welfare. 
Elliot Anshelevich and Shreyas Sekar
cs.GT