  In the Binary Networked Public Goods game, every player needs to decide if
she participates in a public project whose utility is shared equally by the
community. We study the problem of deciding if there exists a pure strategy
Nash equilibrium (PSNE) in such games. The problem is already known to be
NP-complete. We provide fine-grained analysis of this problem under the lens of
parameterized complexity theory. We consider various natural graph parameters
and show either W[1]-hardness or exhibit an FPT algorithm. We finally exhibit
some special graph classes, for example path, cycle, bi-clique, complete graph,
etc., which always have a PSNE if the utility function of the players are fully
homogeneous.
