In general, Nash equilibria in normal-form games may require players to play (probabilistically) mixed strategies. We define a measure of the complexity of finite probability distributions and study the complexity required to play NEs in finite two player $n\times n$ games with rational payoffs. Our central results show that there exist games in which there is an exponential vs. linear gap in the complexity of the mixed distributions that the two players play at (the unique) NE. This gap induces gaps in the amount of space required to represent and sample from the corresponding distributions using known state-of-the-art sampling algorithms. We also establish upper and lower bounds on the complexity of any NE in the games that we study. These results highlight (i) the nontriviality of the assumption that players can any mixed strategy and (ii) the disparities in resources that players may require to play NEs in the games that we study.

翻译：一般而言，标准形式博弈中的纳什均衡可能要求参与者采取（概率性）混合策略。我们定义了有限概率分布复杂度的度量，并研究了在有理收益的有限两人n×n博弈中达到纳什均衡所需的复杂度。核心结果表明，存在这样的博弈：两个参与者在（唯一）纳什均衡中采取的混合分布的复杂度之间存在指数级与线性级的差距。这一差距导致在使用当前最先进的采样算法表示和采样相应分布时所需空间量的差异。同时，我们亦为所研究博弈中任何纳什均衡的复杂度建立了上界和下界。这些结果凸显了（i）参与者能够采取任意混合策略这一假设的非平凡性，以及（ii）我们在所研究博弈中参与者为达到纳什均衡可能需要的资源差异。