  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.
