具有有限随机性的重复匹配便士
Repeated Matching Pennies with Limited Randomness
摘要：
我们考虑一个重复的匹配便士游戏中，玩家有有限的随机访问。在这个n阶段博弈中，达到（唯一的）纳什均衡需要n个随机比特。是否存在使用少于n个随机硬币的纳什均衡？我们的主要结果如下：我们给出了近似均衡的充分刻画，证明了对于[0，1]中的任意e，博弈有e-纳什均衡当且仅当双方都有（1 - e）n个随机硬币。当参与人在多项式时间内运行时，纳什均衡存在当且仅当单向函数存在。可以在随机性和运行时间之间进行权衡。特别地，在合理的假设下，如果我们只给一个参与者O（log n）个随机硬币，但允许他在任意多项式时间内运行，并且我们限制他的对手在时间n^k内运行，对于某个固定的k，那么我们可以维持一个纳什均衡。当博弈进行无限轮且效用时间折现时，在合理的假设下，我们可以将达到e-纳什均衡所需的随机性减少到n，其中n是在一般情况下达到近似纳什均衡所需的随机硬币数量。
英文摘要：
We consider a repeated Matching Pennies game in which players have limited access to randomness. Playing the (unique) Nash equilibrium in this n-stage game requires n random bits. Can there be Nash equilibria that use less than n random coins? Our main results are as follows: We give a full characterization of approximate equilibria, showing that, for any e in [0, 1], the game has a e-Nash equilibrium if and only if both players have (1 - e)n random coins. When players are bound to run in polynomial time, Nash equilibria can exist if and only if one-way functions exist. It is possible to trade-off randomness for running time. In particular, under reasonable assumptions, if we give one player only O(log n) random coins but allow him to run in arbitrary polynomial time and we restrict his opponent to run in time n^k, for some fixed k, then we can sustain an Nash equilibrium. When the game is played for an infinite amount of rounds with time discounted utilities, under reasonable assumptions, we can reduce the amount of randomness required to achieve a e-Nash equilibrium to n, where n is the number of random coins necessary to achieve an approximate Nash equilibrium in the general case.
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