零决定策略的几何
The Geometry of Zero-Determinant Strategies
摘要：
零决定（ZD）策略的出现重塑了重复囚徒困境博弈中互惠与合作的研究。ZD策略的衍生物已经通过他们单方面强制执行他们自己的平均收益和他们的合作伙伴之间的线性关系的能力得到了证明。通常的做法是用成对收益的参数图中的直线来方便地表示这种关系。然而，很少有人注意到研究的实际几何的战略空间的所有容许ZD战略。在这里，我们的工作提供了直观的几何关系，不同类别的ZD策略，以及其特定的参数化的非平凡的几何解释。ZD策略的自适应动态进一步揭示了一般ZD策略和所谓的均衡器之间不可预见的联系，均衡器可以将任何合作者的收益设置为固定值。我们证明了形成超平面的均衡器类是临界平衡流形，其中只有一部分是稳定的。同一超平面也是合作增强区域的分界线，其中最佳响应是增加四个回报结果中的每一个的合作。我们的研究结果揭示了ZD策略的简单但优雅的几何形状，这是以前被忽视的。
英文摘要：
The advent of Zero-Determinant (ZD) strategies has reshaped the study of reciprocity and cooperation in the iterated Prisoner's Dilemma games. The ramification of ZD strategies has been demonstrated through their ability to unilaterally enforce a linear relationship between their own average payoff and that of their co-player. Common practice conveniently represents this relationship by a straight line in the parametric plot of pairwise payoffs. Yet little attention has been paid to studying the actual geometry of the strategy space of all admissible ZD strategies. Here, our work offers intuitive geometric relationships between different classes of ZD strategies as well as nontrivial geometric interpretations of their specific parameterizations. Adaptive dynamics of ZD strategies further reveals the unforeseen connection between general ZD strategies and the so-called equalizers that can set any co-player's payoff to a fixed value. We show that the class of equalizers forming a hyperplane is the critical equilibrium manifold, only part of which is stable. The same hyperplane is also a separatrix of the cooperation-enhancing region where the optimum response is to increase cooperation for each of the four payoff outcomes. Our results shed light on the simple but elegant geometry of ZD strategies that is previously overlooked.
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