Punishment Mechanism with Self-Adjusting Rules in Spatial Voluntary Public Goods Games

The phenomena of cooperation in animal and human society are ubiquitous, but the selfish outcome that no player contributes to the public good will lead to the "tragedy of the commons". The recent research shows that high punishment can improve the cooperation of the population. In this paper, we introduce a punishment mechanism into spatial voluntary public goods games with every individual only knowing his own payoff in each round. Using the self-adjusting rules, we find that the different cost for punishment can lead to different effects on the voluntary public goods games. Especially, when the cost for punishment is decreased, a higher contribution region will appear in the case of low r value. It means even for the low r value, individuals can form the contribution groups in large quantities to produce a more efficient outcome than that in moderate r value. In addition, we also find the players' memory can have effects on the average outcome of the population.