Punishment, especially selfish punishment, has recently been identified as apotent promoter in sustaining or even enhancing the cooperation among unrelatedindividuals. However, without other key mechanisms, the first-order socialdilemma and second-order social dilemma are still two enduring conundrums inbiology and the social sciences even with the presence of punishment. In thepresent study, we investigate a spatial evolutionary four-strategy prisoner'sdilemma game model with avoiding mechanism, where the four strategies arecooperation, defection, altruistic and selfish punishment. By introducing thelow level of random mutation of strategies, we demonstrate that the presence ofselfish punishment with avoiding mechanism can alleviate the two kinds ofsocial dilemmas for various parametrizations. In addition, we propose anextended pair approximation method, whose solutions can essentially estimatethe dynamical behaviors and final evolutionary frequencies of the fourstrategies. At last, considering the analogy between our model and theclassical Lotka-Volterra system, we introduce interaction webs based on thespatial replicator dynamics and the transformed payoff matrix to qualitativelycharacterize the emergent co-exist strategy phases, and its validity aresupported by extensive simulations.
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