Problem
Research questions and friction points this paper is trying to address.
Nash Equilibrium
Free Games
Fine-Grained Complexity
The Fine-Grained Complexity of Approximate Nash Equilibrium and Free Games
🤖 AI Summary
研究通过PCP和ETH猜想证明了在两人N行动标准游戏中计算ε-近似纳什均衡的复杂性下界，并对自由游戏的价值估计建立了类似下界。
📝 Abstract
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 $\varepsilon = \omega(1/\sqrt{N})$. 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).
Innovation
Methods, ideas, or system contributions that make the work stand out.
Fine-Grained Complexity
Approximate Nash Equilibrium
Free Games
PCP for PPAD
ETH for PPAD
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