The intricate geometry of zero-determinant strategies underlying evolutionary adaptation from extortion to generosity

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.