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玩家采用（概率性的）混合策略。我们定义了有限概率分布的复杂度度量，并研究了在有理收益的有限双人$n\times n$博弈中实现纳什均衡所需的复杂度。我们的核心结果表明，存在一类博弈，其中两方玩家在其（唯一）纳什均衡中所采用的混合分布复杂度存在指数级与线性级的差距。这一差距导致玩家在使用已知的最优采样算法表示和采样相应分布时，所需的空间量存在非对称性。我们还建立了正规型博弈中纳什均衡复杂度的指数级上界和下界。这些结果凸显了（i）玩家能够采用任意混合策略这一假设的非平凡性，以及（ii）玩家在正规型博弈中实现纳什均衡可能需要的资源差异。