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Co-evolutionary dynamics with alterable updating rules in the prisoner’s dilemma game

The emergence of cooperation has attracted lots of attention in various areas, ranging from sociology to robotics. Evolutionary game theory provides a suitable way to study the evolution of cooperative agents in selfish populations. In this work, we focus on the co-evolution of strategies and updating mechanisms in the prisoner’s dilemma game. Three mechanisms are available in these heterogeneous systems, where the agent updates its strategy with the classic Moran updating, or the imitation updating, or the aspiration-driven updating. Notably, the innovative feature of the aspiration-driven rule allows the agents to choose other rules even the whole population adopts the aspiration rule at a certain time. Results indicate that the unify of imitation rules can be always achieved in the final populations. Moreover, the speed of rule’s unification is always greater than the speed of strategy’s fixation. Remarkably, the decrease of the initial number of Moran agents makes the fixation of defection slow down, which is conducive to cooperation.