Dynamical Behaviors Of Direct Sum Matrix Game

For biological science, evolutionary game theory (EGT) provides an effective math-ematical framework for Darwin’s theory of natural selection. The essential idea of the theory of natural selection can be summarized as follows:excessive multiplication leads to survival competition, and individual differences (genetic variations) make different fit-ness, and the fittest survive. Competition among individuals is an instinct of organisms. Viewing from heredity, there is a notion of "selfish" gene, which means that the activity of genes in an organism always tend to be beneficial to the spread of the genes itself. At first sight, it seems to be impossible for the cooperative behaviors, which result in the harm of their self-interest, to be exist. But cooperation is a widespread phenomenon existing in various levels from the microscopic level of molecular to the macro level of population and ecosystem, and social system. For this reason, the origin and evolution of cooperation is a question to which evolutionary game theorists pay very close attentions.The most commonly used game theoretical models for cooperation are prisoner’s dilemma(PD) and snowdrift(SD), and their variants. The reason is that both models rep-resent the essence of cooperation, which is the conflicts between the interests of individuals and the collective, in the simplest forms. In this thesis, a multiple roles game(MRG) model has been costructed with these two models in direct sum interaction matrix form. The dynamical behaviors of the model were studied in details. The principal contributions in this thesis can be summarized as follows:First, based on the2×2matrix games of prisoner’s dilemma and snowdrift, a direct sum matrix multiple roles game is constructed. A parameter p is introduced to indicate the probability that individuals choose PD game in a pairwise interplay. The direct sum interaction matrix is changed into4×4payoff matrix with the parameter p. Eight equilibrium points of the replicator equation are calculated out. Amang them, there are only two points related to the parameter p with some existence conditions.Second, the dynamical behaviors on square lattice of the multiple roles game are investigated by numerical simulation. It is found that, comparing with p=1which indicate the population structure with pure PD game, mixing SD activities to pure PD population inhibits the proportion of cooperators in PD, and omparing with p=0which indicate the population structure with pure SD game, mixing PD activities to pure SD population stimulates the proportion of cooperators in SD. Third, based on the numerical simulation, our results show that there is roles reciproc-ities besides spatial reciprocities between different types of individuals. Roles reciprocity provides a possible approach for the evolution of cooperation. Compared with general reci-procity, the distinguishing feature of roles reciprocity is that individuals in MRG benefit from themselves, rather than from other individuals.More specifically, in a specific range of parameters, an individual with defection strategy(D) in PD and cooperative strategy(c) in SD or with C in PD and d in SD would have a better chance of survival than that with Cc or Dd.The fact that the range of parameters with roles reciprocity is too small reflects the frangibility of that toy ecosystem with Dc and Cd.