纳什、康利与计算：博弈动力学中的不可能性与不完整性
Nash, Conley, and Computation: Impossibility and Incompleteness in Game Dynamics
摘要：
在什么条件下，重复博弈的参与人的行为会收敛到纳什均衡？如果假设参与者的行为是离散时间或连续时间规则，从而将当前的混合策略配置文件映射到下一个，这将成为动力系统理论中的一个问题。我们应用这一理论，特别是链递归，吸引子，和康利指数的概念，证明了一个一般的不可能的结果：存在游戏，任何动态是必然的起点，不结束在一个纳什均衡。我们还证明了一个更强的结果$ε$ -近似纳什均衡：有游戏，使没有游戏动态可以收敛（在适当的意义上）到$ε$ -纳什均衡，事实上，这样的游戏集有积极的措施。进一步的数值结果表明，这适用于零和$0.09$之间的任何$ε$。我们的研究结果表明，虽然纳什均衡的概念（及其计算启发的近似）是普遍适用于所有的游戏，他们也从根本上不完整的长期行为的预测，无论选择的动态。
英文摘要：
Under what conditions do the behaviors of players, who play a game repeatedly, converge to a Nash equilibrium? If one assumes that the players' behavior is a discrete-time or continuous-time rule whereby the current mixed strategy profile is mapped to the next, this becomes a problem in the theory of dynamical systems. We apply this theory, and in particular the concepts of chain recurrence, attractors, and Conley index, to prove a general impossibility result: there exist games for which any dynamics is bound to have starting points that do not end up at a Nash equilibrium. We also prove a stronger result for $ε$-approximate Nash equilibria: there are games such that no game dynamics can converge (in an appropriate sense) to $ε$-Nash equilibria, and in fact the set of such games has positive measure. Further numerical results demonstrate that this holds for any $ε$ between zero and $0.09$. Our results establish that, although the notions of Nash equilibria (and its computation-inspired approximations) are universally applicable in all games, they are also fundamentally incomplete as predictors of long term behavior, regardless of the choice of dynamics.
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