  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.
