Evolutionary Stability Problems In The Game

It is well know that the important shortcoming of classical non-cooperative game theory is the rationalistic foundation. Not only is it required that agents be optimizers, but it also presumes a large degree of coordination of different agents' expectations. With the development of evolutionary game theory, the puzzle of classical non-cooperative game theory has been well solved. The demanding of evolutionary game theory is not complete ration, but is bound ration. The first developed model is the single-population model, there are a few of good conclusion that have been shown.. for example : evolutionary stable strategy is asymptotically stable in replicator dynamics; neutral stable strategy is lyapunov stable in replicator dynamics. However, evolutionary stable strategy or neutral stable strategy is not always existence, Thus it is a worthy of studying to disscusion the stable of Nash equilibrium while abandon evolutionary stable strategy and neutral stable strategy.On the other hand, there are a lot of multi-population model in real living, especially in economic application. It has been shown that the strategy combination must be strict Nash equilibrium if it is asymptotically stable in SPS dynamics. However, there are not strict Nash equilibrium in many games. Especially, there are only Nash equilibrium sets in extensive games. So it is not make sense to search an asymptotically stable Nash equilibrium in SPS dynamic. Thus we can only quit asymptotically stable Nash equilibrium and search new method. The one way is we can search the Nash equilibrium of lyapunov stable. The other way is we can turn to study the stable of a set while abandon the single Nash equilibrium.In this paper, Firstly, we study the stable of Nash equilibrium in a few of dynamics, this study is builded in single-population. We get two sufficiency condition that a Nash equilibrium is lyapunov stable in SPS dynamics and We get a sufficiency and necessary condition which a Nash equilibrium is asymptotically stable in replicator dynamics.Secondly, the stable of the single Nash equilibrium has been studied ,A sufficiency condition has been get that a pure Nash equilibrium is lyapunov stable in SPS dynamics ,and we get a new sufficiency and necessary condition which the face is asymptotically stable, whereas these model is multi-population model.Lastly the evolutionary stable of the strict equilibrium set is studied in SPS dynamics.we get some relation about the strict equilibrium set and strategy stable.