Co-evolving prisoner's dilemma: Performance indicators and analytic approaches

Understanding the intrinsic relation between the dynamical processes in a co-evolving network and the necessary ingredients in formulating a reliable theory is an important question and a challenging task. Using two slightly different definitions of performance indicator in the context of a co-evolving prisoner's dilemma game, it is shown that very different cooperative levels result and theories of different complexity are required to understand the key features. When the payoff per opponent is used as the indicator (Case A), non-cooperative strategy has an edge and dominates in a large part of the parameter space formed by the cutting-and-rewiring probability and the strategy imitation probability. When the payoff from all opponents is used (Case B), cooperative strategy has an edge and dominates the parameter space. Two distinct phases, one homogeneous and dynamical and another inhomogeneous and static, emerge and the phase boundary in the parameter space is studied in detail. A simple theory assuming an average competing environment for cooperative agents and another for non-cooperative agents is shown to perform well in Case A. The same theory, however, fails badly for Case B. It is necessary to include more spatial correlation into a theory for Case B. We show that the local configuration approximation, which takes into account of the different competing environments for agents with different strategies and degrees, is needed to give reliable results for Case B. The results illustrate that formulating a proper theory requires both a conceptual understanding of the effects of the adaptive processes in the problem and a delicate balance between simplicity and accuracy.