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.
翻译:我们为在双动作图形游戏中找到近似平衡的计算复杂性提供了完整的描述。 我们考虑了两个最受研究最周密的近似概念: $\varepsilon$-Nash equilibria (varepsilon$-NE) 和 $\varepsilon$-WSNE) 支持的 Nash equilibria ($\varepsilon$-WSNE) 。 我们证明计算一个$\varepsilon$-NE$( 0. 1美元) 对任何恒定的 $\ varepsilon < 1/2 美元来说都是PAD- 完成的, 而一个非常简单的算法( 即让所有玩家在两种动作之间统一组合) 产生1/2美元- NE。 另一方面,我们显示计算一个 $\ $\ varepsilon$- WSNE( $ 1美元), 而一个$ WSNENE($) 是微不足道的, 因为任何战略剖面图都是$ 1- WSNENE。 我们所有较低的游戏也立即适用于每个玩的图形游戏。