纳什均衡的NP完全变式的不可逼近性
Inapproximability of NP-Complete Variants of Nash Equilibrium
摘要：
在Hazan和Krauthgamer最近的工作（SICOMP 2011）中，证明了在两人博弈中找到具有接近最优值的$\eps$近似纳什均衡与在随机图$G(n,1/2)$中找到大小为$O(\log n)$的隐藏集团一样困难。这就提出了一个问题，即在没有这些约束的情况下，近似纳什均衡是否也存在类似的难题。我们给出的证据表明，近最优值的约束，使问题明显困难：一个简单的算法找到一个最佳的1/2近似平衡，而严格优于1/2近似平衡是很难的隐藏集团问题。这是在对比无约束的问题，其中更复杂的算法，实现更好的近似，是已知的。与PPAD中的一般纳什均衡不同，最优（最大值）纳什均衡是NP难的。我们继续表明，最佳的纳什均衡只是几个已知的NP-困难的问题有关的纳什均衡，所有这些都有近似的变种，这是很难找到一个种植集团。特别是，我们证明了这对于以下问题的近似变体：找到一个纳什均衡值大于$η$（对于任何$η>0$，即使最好的纳什均衡值为$1-η$），找到第二个纳什均衡，并找到一个小支持的纳什均衡。最后，我们考虑了两人博弈中近似纯贝叶斯纳什均衡的复杂性。在这里，我们表明，一般的贝叶斯游戏的问题是NP难的。对于类型分布均匀的特殊情况，我们基于隐团问题给出了一个与硬度结果相匹配的拟多项式时间算法。
英文摘要：
In recent work of Hazan and Krauthgamer (SICOMP 2011), it was shown that finding an $\eps$-approximate Nash equilibrium with near-optimal value in a two-player game is as hard as finding a hidden clique of size $O(\log n)$ in the random graph $G(n,1/2)$. This raises the question of whether a similar intractability holds for approximate Nash equilibrium without such constraints. We give evidence that the constraint of near-optimal value makes the problem distinctly harder: a simple algorithm finds an optimal 1/2-approximate equilibrium, while finding strictly better than 1/2-approximate equilibria is as hard as the Hidden Clique problem. This is in contrast to the unconstrained problem where more sophisticated algorithms, achieving better approximations, are known. Unlike general Nash equilibrium, which is in PPAD, optimal (maximum value) Nash equilibrium is NP-hard. We proceed to show that optimal Nash equilibrium is just one of several known NP-hard problems related to Nash equilibrium, all of which have approximate variants which are as hard as finding a planted clique. In particular, we show this for approximate variants of the following problems: finding a Nash equilibrium with value greater than $η$ (for any $η>0$, even when the best Nash equilibrium has value $1-η$), finding a second Nash equilibrium, and finding a Nash equilibrium with small support. Finally, we consider the complexity of approximate pure Bayes Nash equilibria in two-player games. Here we show that for general Bayesian games the problem is NP-hard. For the special case where the distribution over types is uniform, we give a quasi-polynomial time algorithm matched by a hardness result based on the Hidden Clique problem.
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