Punishment Effect of Prisoner Dilemma Game Based on a New Evolution Strategy Rule

We discuss the effect of the punishment in the prisoner's dilemma game. We propose a new evolution strategy rule which can reflect the external factor for both players in the evolution game. In general, if the punishment exists, the D(defection-defection) structure (i.e., both of the two players choose D-D strategy) which is the Nash equilibrium for the game can keep stable and never let the cooperation emerge. However, if a new evolution strategy rule is adopted, we can find that the D-D structure can not keep stable and it will decrease during the game from the simulations. In fact, the punishment mainly affects the C-D (cooperation-defection) structure in the network. After the fraction of the C-D structure achieved some levels, the punishment can keep the C-D structure stable and prevent it from transforming into C-C (cooperation-cooperation) structure. Moreover, in light of the stability of structure and the payoff of the individual gains, it can be found that the probability which is related to the payoff can affect the result of the evolution game.