Evolution dynamics with the switching strategy of punishment and expulsion in the spatial public goods game

Punishment and expulsion are two different strategies to promote cooperation with altruistic sanctions. Of them, expulsion is usually considered more severe and effective than punishment as a strategy to expel a traitor from the field directly. However, frequent expulsions will reduce the number of neighbours around the expellers, with no one left to play subsequent games. Inspired by the fact that people usually accept more tolerant punishment measures for most violators, we introduced a switching strategy of punishing most betrayers and expelling a few betrayers in the spatial public goods game. Unlike the traditional expulsion mechanism, the switching strategy allows switching strategists to balance the relationship between quickly promoting cooperation and maintaining group stability. Our simulation proves that the switching strategy based on the number of defectors can significantly promote cooperation; when the difficulty intensity is minimal, in particular, it can stimulate the emergence of cooperation. In addition, we also determine the optimal switching threshold and the optimal population density under this mechanism. Our work provides a more realistic perspective on the choice of punishment and expulsion, which may help to deepen the understanding of the choice of punishment measures when the number of defectors is high or low.