Heterogeneous willingness induced by different states promotes the evolution of cooperation

In reality, the spread of strategies is often affected by individual willingness, which may be further influenced by the individual state. Herein, based on the spatial prisoner’s dilemma game, we propose a novel co-evolutionary model to explore the role of heterogeneous willingness induced by different states in the evolution of cooperation. In detail, we randomly set players in free or busy states, their states will remain constant throughout the evolution once being set. Within our model, the busy player has a quite small probability to teach one of his neighbors, and meanwhile, the busy neighbor will consider his learning willingness to imitate this strategy. However, the willingness of free player to teach or learn is not affected. Furthermore, we mainly discuss the impact of two different update schemes on the collective cooperation. For the Scheme I, the focal player [Formula: see text] will randomly select one of his neighbors [Formula: see text] as the teaching object, while for the Scheme II, the focal player [Formula: see text] tends to choose one of his neighbors with the lower payoff to teach his strategy. Through lots of numerical simulations, we find that there exists an optimal parameter [Formula: see text] and [Formula: see text] to promote the evolution of cooperation most effectively. In addition, the Scheme II performs better in alleviating social dilemmas.