The warning effect of persistent defection strategy promotes cooperation in spatial prisoner’s dilemma game

In reality, leaders like to punish the employees who repeat mistakes in order to better manage their teams, and establish the prestige of the leaders. As the proverb goes, “kill chickens and warn monkeys”. This is also easy to promote cooperation among team members. Similarly, individuals who continuously make mistakes are less likely to be imitated by individuals around them. Motivated by those realities, based on PDG model, we propose the warning effect of persistent defection strategy evolutionary mechanism, in which discount punishment of the imitation probability is given to the player who imitates the neighbor player who continuously adopts the defection strategy. Here we set the discount punishment threshold $H$ for the number of continuous defection strategy. If the number of continuous defection times of the imitated player reaches the threshold $H$, the imitating player will be given a discount punishment of imitation probability. The proposed evolutionary mechanism is more consistent with real-world situations. For instance, in the real world, people like to imitate the words and actions of their neighbors, however, when the imitated person has persistent bad behavior in the recent past, and especially when he/she is warned by the relevant department or agency, the probability of his/her words and actions being imitated decreases. Simulation and analysis show that the proposed evolutionary mechanism can better promote cooperation than the traditional PDG model. We also find that increasing the discount punishment factor $\alpha$ makes it easier to promote cooperation of evolutionary systems. Besides, the smaller the threshold $H$, the easier it is to promote cooperation of evolutionary systems. This also shows that the earlier the discount punishment factor acts on the evolutionary systems, the easier it is to promote the evolution of cooperation.