A Class Of Stochastic Evolution Models Based On The Classic Games

Evolutionary game theory is a new research field combining the evolutionary theory in biology and the game theory. After decades of development, evolutionary game theory has play an important role in studying the cooperation and competition between individuals. For example, some scholars study the phenomena of competition and cooperation in human society and the animal world based on the Hawk-Dove game. However, the cooperative and competitive behavior is influenced by many factors in the real world. So it is meaningful to do in-depth research for the classic games.In chapter 1 we simply introduce the background and the preliminary knowledge about two person games and the main work of this paper. In chapter 2 two different maximum demand limits(for Hawk and Dove respectively) are introduced in the traditional Hawk-Dove game to obtain our new models. Based on the new models we study the evolutionary game dynamics of cooperation and competition in both infinite and finite populations and the effects of all parameters on the evolutionary game dynamics. In chapter 3 we establish new models based on the traveler’s dilemma game. Firstly we consider the traveler’s dilemma game with two strategies, that is, the airline may allow travelers to choose from two values in the real situations, and the airline will punish the travelers who choose the high value. We assume that the punishment variable is random(i.e., a random variable), and then we get a two-dimensional random payoff matrix. Thus we derive the expressions for the expectations of the fixation probabilities under weak selection. Secondly, we assume that the values for punishment variable vary in a reasonable interval in the traditional traveler’s dilemma game. We then discuss the stochastic evolutionary dynamics in finite populations. In chapter 4 we establish several new selection mechanisms in finite populations based on the Moran process and study the limit dynamics of stochastic evolution in large populations. We prove that corresponding stochastic evolutionary dynamics approach the adjusted replicator dynamics, and the results are similar based on the Fermi process. We finally generalize these results from two-strategy games to multi-strategy games.