AI 中文总结
本文在PPAD的PCP和ETH猜想下，证明近似纳什均衡与自由博弈价值估计的紧下界，确认经典算法在相关机制下的最优性，并回答开放问题。
AI 中文摘要
我们研究了在近似误差趋于零的机制下，计算近似纳什均衡和近似自由博弈价值的细粒度复杂性。在PPAD的PCP猜想和PPAD的ETH猜想下，我们证明计算2人N行动正规博弈中的ε-近似纳什均衡需要时间N^{(log(N)/ε^2)^{1-o(1)}}，从而表明经典的Lipton-Markakis-Mehta算法（2003年）在ε=ω(1/√N)的所有机制下都是最优的。虽然这种最优性在常数ε机制下已知（Rubinstein，2016年），但先前的工作只能在ε=o(1)机制下排除显著更小的运行时间N^{O(log(N)/ε)}。使用类似技术，我们随后在ETH下建立了自由博弈中ε-加性价值估计的类似紧下界N^{(log(N)/ε^2)^{1-o(1)}}，当ε≥2^{-o(√log N)}时，回答了Aaronson、Impagliazzo和Moshkovitz（2014年）的一个问题。
英文摘要
We study the fine-grained complexity of computing approximate Nash equilibria and approximating the value of free games in the regime where the approximation error vanishes. Under the PCP for PPAD and ETH for PPAD conjectures, we show that computing $\varepsilon$-approximate Nash equilibria in 2-player $N$-action normal-form games requires time $N^{(\log(N)/\varepsilon^2)^{1-o(1)}}$, thus showing that the classical Lipton-Markakis-Mehta algorithm (2003) is optimal through all regimes of $\varepsilon = ω(1/\sqrt{N})$. While such optimality was known in the constant-$\varepsilon$ regime (Rubinstein, 2016), previous work could only rule out significantly smaller running times of $N^{O(\log(N)/\varepsilon)}$ in the regime $\varepsilon = o(1)$. Using similar techniques, we then establish an analogous tight lower bound of $N^{(\log(N)/\varepsilon^2)^{1-o(1)}}$ under ETH for $\varepsilon$-additive value estimation in free games, when $\varepsilon \geq 2^{-o(\sqrt{\log N})}$, answering a question of Aaronson, Impagliazzo, and Moshkovitz (2014).
