A Mathematical Analysis For The Evolution Of Public Goods Game Under The Social Structure Of Global Learning And Interaction

The evolution and generation of autonomous cooperation is always a great important research area for social sciences. Reciprocity theory and kin selection theory are great efficient for the explanation of some instances of cooperation behavior, but they are inefficient for the case with a multi-agent game consisting of a group of strange agents.For the problem above, many researchers studied the evolution of cooperation behavior in the perspective of the group selection theory. For instance, Jung kyoo choi (2003) and Xi Xie (2009) researched the affection of social structure on the cooperation evolution of groups by using agent-based modeling approach, where the public goods game was taken as an example. Unfortunately, their results are fully different under the same conditions, such as model parameters and calculation approach. The result is that the cooperation frequencies are 1.71% and 83.34% for Jung kyoo choi and Xi Xie, respectively, under the social structure of global interaction and learning.For the contradiction above, in this research, we adopt a mathematical analysis approach rather than a computer simulation approach to investigate the evolution of public goods game under the social structure of global interaction and learning. First, the evolution of public goods game based on genetic algorithms is analyzed, and a model of public goods game is developed. Second, a mathematic analysis approach with Markov chains which is greatly efficient for the calculation of the long-run behavior of the evolution is discussed in the light of the model of public goods game. But there is a disadvantage for the mathematic approach requiring a large amount of computer memory. Therefore, a memory-efficient approach is proposed by taking into account the sparsity of matrices. Finally, the proposed mathematic approach is implemented by the MATLAB mathematic tool, and the long-run limit distribution of the public goods game is calculated.By comparing the results derived from the proposed mathematic approach and the computer simulation approach of Xi Xie, a conclusion is proved that the result of Xi Xie is more convincing than the one of Jung kyoo choi. Furthermore, three other critical conclusions are also obtained: (1) there are only a few strategies are used during the evolution of the public goods game, (2) there is no a optimal strategy during the evolution of the public goods game, (3) the strategies which are helpful to enhance the evolution of the public goods game have a instinct of cooperation, and this instinct are maintained only under the environment with a large number of cooperative agents.Finally, the proposed mathematic analysis approach in the research is also great helpful for the other studies of social sciences.