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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. [ABSTRACT FROM AUTHOR]

• Nonlinear payoff based on synergy and discounting is considered in SPGG model. • Two parameters to describe the synergy and discounting effect are introduced. • Phase transition process and stable state of the evolutionary system are explored. • Effects of synergy and discounting on cooperation are investigated. Hauert et al. [1] first observed the effects of discounting and synergy in social dilemma games and systematically studied these effects on cooperation in an unstructured population. This paper extends the effects of discounting and synergy into spatial public goods games (PGG) with a population structure. Specifically, an economy-scale threshold and an amplitude parameter to describe conditions and degrees of discounting and synergy are introduced, respectively. Simulation results show that nonlinear payoff based on synergy and discounting effects can promote cooperation more significantly than linear payoff structure. Moreover, the smaller the threshold value, the easier to trigger the synergy effect and remove the discounting effect; the larger the amplitude parameter, the stronger the effect of synergy and discounting to play a role. In both situations, it is more favorable for the emergence and prosperity of cooperation. These results are helpful to understand the evolution of cooperation in nonlinear PGG. [ABSTRACT FROM AUTHOR]