Continuous spatial public goods game with self and peer punishment based on particle swarm optimization

How cooperative behavior emerges and evolves in human society remains a puzzle. It has been observed that the sense of guilt rooted from free-riding and the sense of justice for punishing the free-riders are prevalent in the real world. Inspired by this observation, two punishment mechanisms have been introduced in the spatial public goods game which are called self-punishment and peer punishment respectively in this paper. In each situation, we have introduced a corresponding parameter to describe the level of individual tolerance or social tolerance. For each individual, whether to punish others or whether it will be punished by others depends on the corresponding tolerance parameter. We focus on the effects of the two kinds of tolerance parameters on the cooperation of the population. The particle swarm optimization (PSO)-based learning rule is used to describe the strategy updating process of individuals. We consider both of the memory and the imitation in our model. Via simulation experiments, we find that both of the two punishment mechanisms could facilitate the promotion of cooperation to a large extent. For the self-punishment and for most parameters in the peer punishment, the smaller the tolerance parameter, the more conducive it is to promote cooperation. These results can help us to better understand the prevailing phenomenon of cooperation in the real world.