图对策的紧不可逼近性
Tight Inapproximability for Graphical Games
摘要：
我们提供了一个完整的表征的计算复杂性找到近似平衡两个动作的图形游戏。我们考虑两个最好的近似概念：$\varepsilon$ -纳什均衡（$\varepsilon$ -NE）和$\varepsilon$ -良好支持的纳什均衡（$\varepsilon$ -WSNE），其中$\varepsilon \in [0,1]$。我们证明了计算$\varepsilon$ -NE是PPAD-完全的任何常数$\varepsilon < 1/2$，而一个非常简单的算法（即，让所有的球员之间的两个行动均匀混合）产生一个$1/2$ -NE。另一方面，我们表明，计算$\varepsilon$是PPAD-完全的任何常数$\varepsilon < 1$，而一个$1$ -WSNE是平凡的实现，因为任何战略配置文件是一个$1$ -WSNE。我们所有的下限也立即适用于每个玩家有两个以上动作的图形游戏。
英文摘要：
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.
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