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• We introduce multi-game populations and explore the evolutionary dynamics in this kind of populations. • We consider the diversity of perceptions among all agents. • We consider the perceptions of agents obeys a norm distribution. This paper introduces a multi-game population to describe the fact that people have heterogeneous perceptions on their interactions. Besides, we also consider the diversity of perceptions among all agents in a structured population. Then, we established the theoretical model to investigate the evolutionary dynamics of cooperation in a spatial multi-game population. By means of the Monte Carlo method, we get some insightful observations regarding to the promotion of cooperation. Firstly, simulation results show that agents are more likely to take cooperation action in a multi-game population than in a homogeneous population. Moreover, we also find out that agents playing prisoner's dilemma game are possible to adopt cooperation strategy when they interact with agents playing snowdrift game. [ABSTRACT FROM AUTHOR]

• We introduce the multigame environment and prove that the evolutionary dynamics is different from that in previous work. • We combine the evolutionary dynamics with the Particle Swarm Optimization. • We consider a continuous strategy version. This paper studies the evolution of cooperation in a so-called multigame environment based on the Particle Swarm Optimization (PSO) algorithm. In a multigame environment, players use different game payoff matrices and acquire their utilities from their interactions with their neighbors. According to the PSO algorithm, each player updates its strategy according to both the strategy adopted by the player with the highest utility in its neighborhood and the most profitable strategy in their own past actions. Simulation results show that the multigame environment is conducive to the promotion of cooperation. Besides, within the multigame environment, for any player, imitating the most profitable strategy in its past actions promotes cooperation more effectively than imitating the strategy adopted by the player with the highest utility in its neighborhood. [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]