Tight Inapproximability for Graphical Games

TL;DR

This paper characterizes the computational difficulty of finding approximate Nash equilibria in two-action graphical games, establishing tight bounds on the complexity for different approximation levels and showing PPAD-completeness results.

Contribution

It provides a complete complexity classification for approximate equilibria in two-action graphical games, including tight bounds and simple algorithms for specific approximation thresholds.

Findings

Computing an ε-Nash equilibrium is PPAD-complete for ε<1/2.

A uniform random strategy yields a 1/2-NE.

Computing an ε-well-supported Nash equilibrium is PPAD-complete for ε<1.

Abstract

We provide a complete characterization for the computational complexity of finding approximate equilibria in two-action graphical games. We consider the two most well-studied approximation notions: $\varepsilon$-Nash equilibria ($\varepsilon$-NE) and $\varepsilon$-well-supported Nash equilibria ($\varepsilon$-WSNE), where $\varepsilon \in [0,1]$. We prove that computing an $\varepsilon$-NE is PPAD-complete for any constant $\varepsilon < 1/2$, while a very simple algorithm (namely, letting all players mix uniformly between their two actions) yields a $1/2$-NE. On the other hand, we show that computing an $\varepsilon$-WSNE is PPAD-complete for any constant $\varepsilon < 1$, while a $1$-WSNE is trivial to achieve, because any strategy profile is a $1$-WSNE. All of our lower bounds immediately also apply to graphical games with more than two actions per player.