In this talk, I will discuss a generalized Moran process from the evolutionary game theory. The generalization incorporates arrangement of by graphs and games among individuals. For these additional features, there has been consistent interest in using general spatial structure as a way to explain the ubiquitous game behavior in biological evolutions; the introduction of games leads to technical complications as basic as nonlinearity and asymmetry in the model. The talk will be centered around a seminal finding in the evolutionary game theory that was obtained more than a decade ago. By an advanced mean-field method, it reduces the infinite-dimensional problem of solving for the game fixation probabilities to a one-dimensional diffusion problem in the limit of a large population. The recent mathematical results and some related mathematical methods will be explained.