Abstract
We complete the characterization of the computational complexity of equilibrium in public goods games on graphs. In this model, each vertex represents an agent deciding whether to produce a public good, with utility defined by a "best-response pattern" determining the best response to any number of productive neighbors. We prove that the equilibrium problem is NP-complete for every finite non-monotone best-response pattern. This answers the open problem of [Gilboa and Nisan, 2022], and completes the answer to a question raised by [Papadimitriou and Peng, 2021], for all finite best-response patterns.