Co-operation and Phase Behavior under the Mixed Updating Rules

We present a model by considering two updating rules when the agents play prisoner's dilemma on a square lattice. Agents can update their strategies by referencing one of his neighbors of higher payoffs under the imitation updating rule or directly replaced by one of his neighbors according to the death-birth updating rule. The frequency of co-operation is related to the probability q of occurrence of the imitation updating or the death-birth updating and the game parameter b. The death-birth updating rule favors the co-operation while the imitation updating rule favors the defection on the lattice, although both rules suppress the co-operation in the well-mixed population. Therefore a totally co-operative state may emerge when the death-birth updating is involved in the evolution when b is relatively small. We also obtain a phase diagram on the q–b plane. There are three phases on the plane with two pure phases of a totally co-operative state and a totally defective state and a mixing phase of mixed strategies. Based on the pair approximation, we theoretically analyze the phase behavior and obtain a quantitative agreement with the simulation results.