  Recently Bramoulle and Kranton presented a model for the provision of public
goods over a network and showed the existence of a class of Nash equilibria
called specialized equilibria wherein some agents exert maximum effort while
other agents free ride. We examine the efficiency, effort and cost of
specialized equilibria in comparison to other equilibria. Our main results show
that the welfare of a particular specialized equilibrium approaches the maximum
welfare amongst all equilibria as the concavity of the benefit function tends
to unity. For forest networks a similar result also holds as the concavity
approaches zero. Moreover, without any such concavity conditions, there exists
for any network a specialized equilibrium that requires the maximum weighted
effort amongst all equilibria. When the network is a forest, a specialized
equilibrium also incurs the minimum total cost amongst all equilibria. For
well-covered forest networks we show that all welfare maximizing equilibria are
specialized and all equilibria incur the same total cost. Thus we argue that
specialized equilibria may be considered as a refinement of the equilibrium of
the public goods game. We show several results on the structure and efficiency
of equilibria that highlight the role of dependants in the network.
