The Strategies' Evolution In The Ultimatum Game

The ultimatum game (UG) is a non-zero-sum and dynamic game with two players. Repeating the game is similar to the process of bargain. It has a significant meaning to investigating the evolution of behaviors, strategies and fairness in this game. The evolutionary game dynamics is the application of the population evolutionary dynamics method to the game theory, which is introduced by evolutionary biologists, and is predicted by the classical game theory. This paper studied the evolution of individuals'behavior with special strategies under evolutionary game dynamics.In the first and second chapter, we introduced the deterministic evolutionary dynamics for infinite populations, the theory of stochastic evolutionary dynamics for finite populations, and the general model and the equilibrium of the UG, as well as the definition of the biased behavior and the cooperation during the ultimatum game's evolution.The third chapter gave the classical model of the UG described by dynamic evolution.For infinite population, the dynamics is described by determining the replicator equation and adaptive equation. A model which is described by replicator equation shows that under certain conditions, the rational strategy will eventually dominate the fairness strategy. The single parameter model described by the adaptive dynamic equation shows that the natural selection feature is fair, and the two parameter model shows that the natural selection is not fair, and finally the rational solution is obtained. For finite population, two typical stochastic dynamics based on the Moran process and Fermi process are introduced. The choice of the Moran process shows that although there is no repeated interactions or reputation mechanism, an ultimatum altruistic strategy in a suitable degree has more advantages than the greedy strategy.A new strategy of the UG in finite population is raised innovatively in the fourth chapter. By introducing the mechanism of the new strategy adjustment, we defined the greedy preference and altruism preference of individual's behaviors. We simulated respectively in the single strategy model and multi strategy model, discussing the relationship between cooperation and wealth accumulation. The results show that, in the single strategy model, the altruistic groups will be more likely to evolve cooperation, thus the indirect reciprocity can obtain a higher degree of fitness. The greedy groups will tend to do not cooperate, and the overall fitness of the group is relatively lower. When there is no deviation in the whole behavior, there is a diving phenomenon. At the same time, the groups of all single strategies will evolve five different social wealth structures, including the Pyramid structure, the inverted Pyramid structure, the dumbbell structure, the Double-peak structure and the olive structure. In the multi strategy without learning model, altruistic individuals still prone to cooperate than greedy ones, and the higher the degree of altruism, the possibility of cooperation is higher, when the behaviors are unbiased, diving phenomenon disappeared; but the group with higher greedy degree will get higher fitness. Namely, in the multi strategy model, greedy individuals dominate the altruistic ones. When the Fermi learning is introduced in the multi strategy model, the strategy invading appears in sufficient time. Fermi learning process is a common mechanism to select the best strategy. In the process of learning, there are altruistic strategy retained. To some extent, the results echoes those by Moran process in the UG.