  Punishment, especially selfish punishment, has recently been identified as a
potent promoter in sustaining or even enhancing the cooperation among unrelated
individuals. However, without other key mechanisms, the first-order social
dilemma and second-order social dilemma are still two enduring conundrums in
biology and the social sciences even with the presence of punishment. In the
present study, we investigate a spatial evolutionary four-strategy prisoner's
dilemma game model with avoiding mechanism, where the four strategies are
cooperation, defection, altruistic and selfish punishment. By introducing the
low level of random mutation of strategies, we demonstrate that the presence of
selfish punishment with avoiding mechanism can alleviate the two kinds of
social dilemmas for various parametrizations. In addition, we propose an
extended pair approximation method, whose solutions can essentially estimate
the dynamical behaviors and final evolutionary frequencies of the four
strategies. At last, considering the analogy between our model and the
classical Lotka-Volterra system, we introduce interaction webs based on the
spatial replicator dynamics and the transformed payoff matrix to qualitatively
characterize the emergent co-exist strategy phases, and its validity are
supported by extensive simulations.
