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• In this paper, 2 × 2 symmetric games in which the game payoff matrix is represented by fuzzy numbers were studied. • We introduce the fuzzy dilemma strength to classify fuzzy games. • The fuzzy fixation probability of the strategy is calculated when the game payoff matrix is represented by the normal fuzzy numbers. • The proposed fuzzy Moran model was applied to solve the problem of strategy selection in the interaction between pollution-producing enterprises. Most of the previous studies on the evolutionary dynamics of the Moran process assumed that the payoff obtained by the participant from the games is deterministic and expressed in precise numbers. However, because of the influence of various uncertain factors of the environment, the individual's payoff is not an accurate number but needs to be expressed by a fuzzy number. In this paper, 2 × 2 symmetric games in which the game payoff matrix is represented by fuzzy numbers were studied. Firstly, we introduce the fuzzy dilemma strength to classify fuzzy games. Then, the evolution dynamics of a fuzzy Moran process of a finite population are analyzed by using the operation of a fuzzy number. Under the condition of weak selection, the fuzzy fixation probability of the strategy is calculated when the game payoff matrix is represented by the normal fuzzy numbers. Furthermore, the conditions under which natural selection favors one strategy to be fixed in population and a strategy to become a fuzzy evolutionary stability strategy are analyzed. Lastly, the proposed fuzzy Moran model was applied to solve the problem of strategy selection in the interaction between pollution-producing enterprises. By numerical analysis, the effect of fuzzy dilemma strength on the fuzzy fixation probability was illustrated and then the feasibility and effectiveness of the method were verified. The extension of classical game dynamics to a fuzzy environment enriches the theory of evolutionary games and is more realistic. [ABSTRACT FROM AUTHOR]

• A group-level selection dynamic induced by the implementation of an individual-level selection is proposed. • The existence of an independent cooperator group is beneficial to the rise of cooperation in the population with multiple groups. • Mutation and migration will inhibit the prevalence of cooperation by destroying the independence and purity of cooperator groups. • A special kind of multi-person migration, group division, will inhibit the anti-cooperation effect of the mutation. It is a challenge to understand why cooperation is prevalent in human society, especially without a mechanism that benefits cooperators such as direct, indirect and strong reciprocity. We study an evolutionary process of public goods game among individuals in multiple groups in a population. We find that the existence of an independent cooperator group is beneficial to the rise of cooperation in the population, which is applicable to both Moran process and a so-called disaster update process proposed in this paper. However, the introduction of mutation can completely overturn the emerging trend of cooperation, and even let the defectors occupy the whole population for a short period of time, by continuously reducing the number of cooperator groups. In addition, the migration of individuals between groups can also inhibit cooperation. More specifically, a higher frequency of migration can reduce the probability of cooperators occupying the population, and extend the time it takes for the population to reach the stationary state. Furthermore, we find that a special kind of multi-individual migration, namely group division, can resist the anti-cooperation effect of mutation to some extent, mainly by maintaining a certain number of cooperator groups in the population. The aforementioned results suggest that cooperation can rise without strengthening the ability of cooperators to directly confront the defectors in a population composed of multiple groups. [ABSTRACT FROM AUTHOR]

• Climate change dilemma is formulated as a public goods game with environmental feedback. • A dynamic model based on Particle Swarm Optimization algorithm is proposed with restricted exchange of information. • Theoretical analysis and simulation results both show that environmental feedback promotes cooperation. • Continuous contribution is more conducive to cooperation. • Population composed of individuals that make decisions replying more on historical information than social information can promote cooperation. This paper formulates the climate change dilemma as an adaption of public goods game. The Nash equilibrium of the climate change dilemma is analyzed in the cases of discrete contribution and continuous contribution. Analytic results show that environmental feedback promotes cooperation to a certain extent, but as the number of players increases, zero contribution becomes the only Nash equilibrium in most cases. A dynamic model based on Particle Swarm Optimization algorithm is then proposed for the climate change dilemma, where the information exchange is restricted by a network. Simulation results show that the proposed dynamic model can effectively promote cooperation, especially in the case of continuous contribution, and making decisions based on historical information can largely boost the average contribution. [ABSTRACT FROM AUTHOR]