Price Competition in Networked Markets: How Do Monopolies Impact Social Welfare?

Abstract

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 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 buyer demand functions, equilibrium always exists and allocations can often be close to optimal. In the process, we characterize the structure and properties of equilibria using techniques from min-cost flows, and obtain tight bounds on efficiency in terms of the various parameters governing the market, especially the number of monopolies M.

Although 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 $1+\frac{M}{2}$ 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. 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.

This work was partially supported by NSF awards CCF-1101495 and CNS-1218374.