Investigating the effectiveness of interaction-efficiency-driven strategy updating under progressive-interaction for the evolution of the prisoner's dilemma game

In reality, interactive friend is added may one after another, instead of interacting with all at first. This work has proposed a progressive-interaction behavior driven by consistent aspiration, in which individuals interact with replacement neighbor one after another. Furthermore, payoff and interaction-efficiency are two factors considered in a modified Fermi rule, which particularly emphasizes that individuals achieving identical aspirations with less interaction frequency per unit of time will have higher interaction-efficiency and are more likely to be imitated. Thus, this work introduces the interaction-efficiency-driven strategy updating rules under progressive-interaction behavior in a spatial prisoner's dilemma game. As shown by numerical simulations, the progressive-interaction significantly enhances cooperation more than traditional versions, which may be caused by a heterogeneous number of interacting neighbors. When the interaction-efficiency-driven strategy update is taken into account, the cooperation is elevated to a higher level. Additionally, a specific point is employed to simulate the relationship between the aspiration and the b. As indicated by results, different b values have appropriate aspiration levels to promote the system to achieve the optimal cooperative level. More specifically, an obvious phase transition phenomenon can be found for the promotion results of individual aspiration level on cooperation. Besides, the phase transition phenomenon is existent when we simulate the relationship between the aspiration and the parameter delta for a given b value. In short, this work puts forward a new perspective on understanding cooperative evolution.