Tight Inapproximability for Graphical Games

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 uniformly between their actions) yields 1/2-NE. On other hand, we show ε-WSNE 1, 1-WSNE trivial to achieve, because strategy profile 1-WSNE. All our lower bounds immediately also apply games with more actions per player.