Stability Analysis Of Asymmetric Snowdrift Games

Evolutionary game combining the basic theory of game theory and dynamics is a powerful tool for stability analysis.Evolutionary game is a research direction of game theory and an important branch of dynamical systems.Snowdrift game is one of classical models,which can profoundly reflect the cooperative and competitive relationship between individuals.In this paper,the snowdrift game is reasonably extended.Two types of asymmetric snowdrift games are proposed,and the influence of spatial structure on them is considered.Firstly,an asymmetric dynamic snowdrift game model based on penalty mechanism is constructed,and the stability of the model is analyzed.Evolutionary stability under replicator equation is obtained for infinite population.It is found that penalty mechanism and discount factor can promote cooperation for finite population.Fixed probabilities for the selection-mutation Moran process are calculated for finite population.What's more,the conditions for stability and strong stability for the stochastic process are obtained.Secondly,the influence of spatial structure on the stability of asymmetric snowdrift game is analyzed.The results show that the rule diagrams under BD,DB,IM and PC updating rules can change the dynamic behavior of asymmetric snowdrift game,that is,bifurcation occurs at the internal equilibrium point.Furthermore,the stochastic stable equilibrium points on asymmetric heterogeneous graphs under PC updating rules are obtained.On this basis,it is deduced that degree variance can change the value of stochastic stable equilibrium points and promote cooperation.Finally,a role asymmetric snowdrift game is proposed,and the Nash equilibrium under infinite group and the invariant distribution of fixed probability under finite group are obtained.