Optimism when winning and cautiousness when losing promote cooperation in the spatial prisoner’s dilemma game

In real societies, whether an individual participates in unpromising interactions sometimes is a probabilistic option instead of an all-or-nothing option, changing according to the risk of interactions. Inspired by this, we introduce an imitation-induced adjustment mechanism of individual willingness to interact into the spatial prisoner’s dilemma game, where an individual decreases its willingness to interact when it imitates others and increases when it is imitated by others. The adjustment value at each time interval is defined as the adjustment sensitivity of individual willingness to interact. Extensive simulation results show that the proposed mechanism can promote cooperation effectively, and a higher adjustment sensitivity has a more positive influence on the maintenance of cooperation. This promotion is mainly because the survived cooperators can resist the invasion of defectors, through interacting with defectors with lower willingness to interact around them. Finally, the generality of this promotion is testified in two additional scenarios.