The Random Evolutional Dynamics For The Strategies Of Direct Reciprocity Under Kin Selection

Direct reciprocity is an important part of evolutionary game theory. Economists and socialscientists have applied this theory in their fields respectively and achieved excellent results. Thestudy about direct reciprocity and the evolution of cooperation is becoming a mature field.However, we have not found much work on the evolutionary dynamics about the strategies ofdirect reciprocity such as TFT (tit-for-tat), ALLD (always-defect), WSLS (win-stay, lose-shift)under kin selection, which deserves further research. In addition, the selection intensity plays avital role in the evolutionary dynamics of the finite populations. It influences directly the relationof the payoff and the fitness. Nevertheless, the emergence of mutation breaks the existing steadystate, promotes the biodiversity and changes the evolutionary dynamics of the populations. Thusanalyzing the limit distribution of multi-strategies with the weak selection and small mutation aremeaningful.In chapter1we introduce the background and the preliminary remarks of direct reciprocity intwo person games. In chapter2, we build relevant models based on the payoff matrices of theprisoner’s dilemma and the snowdrift game, and obtain the payoff matrix for the TFT and ALLDstrategies. We then discuss the evolutionary game dynamics of TFT and ALLD in both the finiteand infinite populations. In chapter3, we first study the evolutionary dynamics of WSLS under thenetwork reciprocity. The results show that network reciprocity favors the evolutionary stability ofWSLS under the DB and IM update mechanisms on regular graph. Then we present the models ofthe game between the WSLS and ALLD strategies based on the payoff matrix of the prisoner’sdilemma. We analyze the evolutional conditions for WSLS on regular graph. Findings suggest thatthe network reciprocity promotes the evolution of cooperation under kin selection. In chapter4,based on Moran process we first compute the specific expressions for the limit distribution ofstrategies in the22symmetric game with the weak selection and small mutation. Then wefurther derive the expressions for the corresponding limit distribution of each strategy in33symmetric game based on pairwise comparison process.