  In recent work by Bramoull\'{e} and Kranton, a model for the provision of
public goods on a network was presented and relations between equilibria of
such a game and properties of the network were established. This model was
further extended to include games with imperfect substitutability in
Bramoull\'{e} et al. The vast multiplicity of equilibria in such games along
with the drastic changes in equilibria with small changes in network structure,
makes it challenging for a system planner to estimate the maximum social
welfare of such a game or to devise interventions that enhance this welfare.
Our main results address this challenge by providing close approximations to
the maximum social welfare and the maximum aggregate play in terms of only
network characteristics such as the maximum degree and independence number. For
the special case when the underlying network is a tree, we derive formulae
which use only the number of nodes and their degrees. These results allow a
system planner to assess aggregate outcomes and design interventions for the
game, directly from the underlying graph structure, without enumerating all
equilibria of the game, thereby significantly simplifying the planner's
problem. A part of our results can be viewed as a logical extension of [7]
where the maximum weighted aggregate effort of the model in [2] was
characterized as the weighted independence number of the graph.
