Public goods games in directed networks

Public goods games in undirected networks are generally known to have pure Nash equilibria, which easy find. In contrast, we prove that, directed networks, a broad range of public intractable equilibrium problems: The existence equilibria is NP-hard decide, and mixed PPAD-hard We define general utility games, complexity dichotomy result for finding PPAD-completeness proof equilibria. Even the divisible variant problem, where prove, PPAD-complete. Finally, when treewidth network appropriately bounded, that polynomial-time algorithms possible.