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

Abstract 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.

[1]  M. Nowak.  Evolutionary Dynamics: Exploring the Equations of Life , 2006 .

[2]  Ichiro Nishizaki,et al.  Equilibrium solutions in multiobjective bimatrix games with fuzzy payoffs and fuzzy goals , 2000, Fuzzy Sets Syst..

[3]  G. Szabó,et al.  Evolutionary prisoner's dilemma game on a square lattice , 1997, cond-mat/9710096.

[4]  T. Maeda,et al.  Characterization of the Equilibrium Strategy of the Bimatrix Game with Fuzzy Payoff , 2000 .

[5]  Jun Tanimoto,et al.  Various error settings bring different noise-driven effects on network reciprocity in spatial prisoner's dilemma , 2018, Chaos, Solitons & Fractals.

[6]  M. Nowak,et al.  Evolutionary game dynamics in a Wright-Fisher process , 2006, Journal of mathematical biology.

[7]  M. Nowak.  Five Rules for the Evolution of Cooperation , 2006, Science.

[8]  Martin A. Nowak,et al.  Analytical Results for Individual and Group Selection of Any Intensity , 2008, Bulletin of mathematical biology.

[9]  T. Antal,et al.  Fixation of Strategies for an Evolutionary Game in Finite Populations , 2005, Bulletin of mathematical biology.

[10]  Alireza Arshadi Khamseh,et al.  An optimization model of three-player payoff based on fuzzy game theory in green supply chain , 2019, Comput. Ind. Eng..

[11]  Ran Xu,et al.  Evolutionary game analysis for third-party governance of environmental pollution , 2018, J. Ambient Intell. Humaniz. Comput..

[12]  Xianjia Wang,et al.  The roles of particle swarm intelligence in the prisoner's dilemma based on continuous and mixed strategy systems on scale-free networks , 2019, Appl. Math. Comput..

[13]  Drew Fudenberg,et al.  Evolutionary game dynamics in finite populations , 2004, Bulletin of mathematical biology.

[14]  Didier Dubois,et al.  Fuzzy sets and systems ' . Theory and applications , 2007 .

[15]  Long Wang,et al.  Aspiration dynamics of multi-player games in finite populations , 2014, Journal of The Royal Society Interface.

[16]  D. Fudenberg,et al.  Emergence of cooperation and evolutionary stability in finite populations , 2004, Nature.

[17]  A. Cabrales.  Stochastic replicator dynamics , 2000 .

[18]  Josef Hofbauer,et al.  Evolutionary Games and Population Dynamics , 1998 .

[19]  Jun Tanimoto,et al.  Scaling the phase-planes of social dilemma strengths shows game-class changes in the five rules governing the evolution of cooperation , 2018, Royal Society Open Science.

[20]  M. Sakawa,et al.  Max-min solutions for fuzzy multiobjective matrix games , 1994 .

[21]  Yibin Kang,et al.  Fixation times in evolutionary games with the Moran and Fermi processes. , 2015, Journal of theoretical biology.

[22]  Zhi Xiong,et al.  Research on Multi-Players Evolutionary Game of Environmental Pollution in System Dynamics Model , 2016 .

[23]  James J. Buckley,et al.  Fuzzy Two-Person Zero-Sum Games , 2007 .

[24]  Xianjia Wang,et al.  Evolutionary game dynamics of combining the Moran and imitation processes , 2019, Chinese Physics B.

[25]  Lourdes Campos.  Fuzzy linear programming models to solve fuzzy matrix games , 1989 .

[26]  M. Nowak,et al.  A symmetry of fixation times in evoultionary dynamics. , 2006, Journal of theoretical biology.

[27]  M. Nowak,et al.  Non-commercial Research and Educational Use including without Limitation Use in Instruction at Your Institution, Sending It to Specific Colleagues That You Know, and Providing a Copy to Your Institution's Administrator. All Other Uses, Reproduction and Distribution, including without Limitation Comm , 2022 .

[28]  P. Moran,et al.  The statistical processes of evolutionary theory. , 1963 .

[29]  Hsuan-Shih Lee,et al.  The revised method of ranking fuzzy numbers with an area between the centroid and original points , 2008, Comput. Math. Appl..

[30]  Yibin Kang,et al.  Fixation probabilities in evolutionary games with the Moran and Fermi processes. , 2015, Journal of theoretical biology.

[31]  Jun Tanimoto,et al.  Relationship between dilemma occurrence and the existence of a weakly dominant strategy in a two-player symmetric game , 2007, Biosyst..

[32]  J M Smith,et al.  Evolution and the theory of games , 1976 .

[33]  Weixin Yang,et al.  Does Whistleblowing Work for Air Pollution Control in China? A Study Based on Three-party Evolutionary Game Model under Incomplete Information , 2019, Sustainability.

[34]  A. Traulsen,et al.  Fixation times in evolutionary games under weak selection , 2008, 0812.0851.

[35]  S. Kokubo,et al.  Universal scaling for the dilemma strength in evolutionary games. , 2015, Physics of life reviews.

[36]  Ming Zhang,et al.  New evolutionary game model of the regional governance of haze pollution in China , 2018, Applied Mathematical Modelling.

[37]  Jun Tanimoto,et al.  Promotion of cooperation by payoff noise in a 2x2 game. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[38]  Danyang Jia,et al.  Expectation driven by update willingness promotes cooperation in the spatial prisoner's dilemma game , 2019, Appl. Math. Comput..

[39]  M. Perc.  Coherence resonance in a spatial prisoner's dilemma game , 2006 .

[40]  C. Hauert,et al.  Coevolutionary dynamics: from finite to infinite populations. , 2004, Physical review letters.

[41]  C. R. Bector,et al.  Matrix games with fuzzy goals and fuzzy payoffs , 2005 .

[42]  Francisco R. Fernández,et al.  Set-valued cooperative games with fuzzy payoffs. The fuzzy assignment game , 2013, Eur. J. Oper. Res..

[43]  Jun Tanimoto,et al.  Fundamentals of Evolutionary Game Theory and its Applications , 2015 .

[44]  C. C. Li.  The Statistical Processes of Evolutionary Theory , 1962 .

[45]  Takashi Maeda,et al.  On characterization of equilibrium strategy of two-person zero-sum games with fuzzy payoffs , 2003, Fuzzy Sets Syst..

[46]  J. M. Smith,et al.  The Logic of Animal Conflict , 1973, Nature.

[47]  P. Taylor,et al.  Evolutionarily Stable Strategies and Game Dynamics , 1978 .

[48]  Sankar Kumar Roy,et al.  Bi-level Programming for Stackelberg Game with Intuitionistic Fuzzy Number: a Ranking Approach , 2019, Journal of the Operations Research Society of China.

[49]  Yibin Kang,et al.  Fixation of strategies with the Moran and Fermi processes in evolutionary games , 2017 .

[50]  Ji Quan,et al.  Evolutionary game dynamics of the Wright-Fisher process with different selection intensities. , 2019, Journal of theoretical biology.

[51]  Suresh Chandra,et al.  Optimal solutions for group matrix games involving interval-valued Fuzzy Numbers , 2019, Fuzzy Sets Syst..