解决计算近似两人纳什均衡的复杂性
Settling the complexity of computing approximate two-player Nash equilibria
摘要：
我们证明了存在一个常数$ε>0$，使得在PPAD的指数时间假设下，计算两个玩家（nXn）博弈中的$ε$ -近似纳什均衡需要准多项式时间，$n^{\log^{1-o(1)} n}$。这与Lipton、Markakis和Mehta [LMM 03]的算法相匹配（直到o（1）项）。我们的证明依赖于各种技术的概率可检查的证据（PCP）的研究，这是第一次，这样的想法被用于减少PPAD内的问题。在途中，我们还证明了新的硬度结果计算纳什均衡在游戏中有许多球员。特别是，我们表明，计算一个$ε$近似纳什均衡在游戏中的n个球员需要$2^{Ω(n)}$甲骨文查询的支付张量。这解决了Hart和Nisan [HN 13]、Babichenko [Bab 14]和Chen等人[CCT 15]提出的一个未决问题。事实上，我们对n人博弈的结果更强：它们与Babichenko等人最近引入的$(ε,δ)$ -WeakNash松弛相关。[BPR 16]。
英文摘要：
We prove that there exists a constant $ε>0$ such that, assuming the Exponential Time Hypothesis for PPAD, computing an $ε$-approximate Nash equilibrium in a two-player (nXn) game requires quasi-polynomial time, $n^{\log^{1-o(1)} n}$. This matches (up to the o(1) term) the algorithm of Lipton, Markakis, and Mehta [LMM03]. Our proof relies on a variety of techniques from the study of probabilistically checkable proofs (PCP); this is the first time that such ideas are used for a reduction between problems inside PPAD. En route, we also prove new hardness results for computing Nash equilibria in games with many players. In particular, we show that computing an $ε$-approximate Nash equilibrium in a game with n players requires $2^{Ω(n)}$ oracle queries to the payoff tensors. This resolves an open problem posed by Hart and Nisan [HN13], Babichenko [Bab14], and Chen et al. [CCT15]. In fact, our results for n-player games are stronger: they hold with respect to the $(ε,δ)$-WeakNash relaxation recently introduced by Babichenko et al. [BPR16].
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