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In a public goods game, every player chooses whether or not to produce a good that all neighboring players will have access to. We consider a setting in which the public good is indivisible, neighboring players are out-neighbors in a directed graph, and there is a capacity constraint on their number, $k$, that can benefit from the good. This means that each player makes a two-pronged decision: decide whether or not to produce and, conditional on producing, choose which $k$ out-neighbors to share access. We examine both pure and mixed Nash equilibria in the model from the perspective of existence, computation, and efficiency. We perform a comprehensive study for these three dimensions with respect to both sharing capacity ($k$) and the network structure (the underlying directed graph), and establish sharp complexity dichotomies for each.
