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 Nash equilibria 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 in the (unique) Nash equilibrium of these games. This gap induces asymmetries in the amounts of space required by the players to represent and sample from the corresponding distributions using known state-of-the-art sampling algorithms. We also establish exponential upper and lower bounds on the complexity of Nash equilibria in normal-form games. These results highlight (i) the nontriviality of the assumption that players can play any mixed strategy and (ii) the disparity in resources that players may require to play Nash equilibria in normal-form games.

翻译：一般而言，正规型博弈中的纳什均衡可能要求参与者采用（概率性的）混合策略。我们定义了有限概率分布复杂度的度量，并研究了具有理性收益的有限两人$n\times n$博弈中达到纳什均衡所需的复杂度。核心结果表明，存在某些博弈，其中两个参与者在（唯一）纳什均衡中所使用的混合分布的复杂度之间存在指数级与线性级的差距。这种差距导致参与者使用已知最优采样算法表示及抽样相应分布所需的空间量出现非对称性。此外，我们建立了正规型博弈中纳什均衡复杂度的指数级上下界。这些结果突显了：（i）参与者可任意使用混合策略这一假设的非平凡性，以及（ii）参与者为达到正规型博弈纳什均衡所需资源的差异性。