We provide a complete characterization for the computational complexity of finding approximate equilibria in two-action graphical games. consider two most well-studied approximation notions: ε-Nash (ε-NE) and ε-well-supported Nash (ε-WSNE), where ε is [0,1]. prove that computing an ε-NE PPAD-complete any constant smaller than 1/2, while very simple algorithm (namely, letting all players mix uni...