Evolutionary game dynamics of Moran process with fuzzy payoffs and its application

•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.