The Existence And Stochastic Stability Of Nash Equilibrium In Limited Group Game

We mainly study the existence and stochastic stability of Nash equilibrium in finite population games.First of all,a counterexample shows that the Nash equilibrium of finite population games does not necessarily exist.Secondly,the existence of the Nash equilibrium for finite population games generated by two strategies symmetric normal form games is proved.We show that the two-strategy games is a potential game and Nash equilibrium of finite population potential games is equivalent to the local maximum point of potential function.Finally,a simple two-strategy symmetric coordination games model is introduced,which forms a Markov chain under certain perturbations.As the disturbance fades away,the stochastic stability of the process is analyzed by two examples.We find the stochastic stable state of the general two-strategy games according to the relationship between the maximum point of the potential function and stochastic stable state in finite population potential games.The innovations in this article are:(1)The existence conditions of Nash equilibrium for finite population games are considered based on the fact that finite population have practical significance.(2)We analyze the stochastic stability of Nash equilibrium in finite population games by two methods.This thesis is organized as follows:Chapter 1,the background is introduced which contains the history of development and research status of game theory,population games,finite population games,especially the results of existence and stability of the Nash equilibrium points.Chapter 2,we briefly introduce the model of population games and the definition and properties of Nash equilibrium points.We introduce full potential games and potential games and the relationship between them in continuous population games.Chapter 3,the basic model and definition of Nash equilibrium points are brieflyintroduced in finite population games,we introduce the finite population full potential games and the potential games and the relation between them.It is proved that the finite population congestion games is a potential game.Chapter 4,firstly,we give a counterexample to the non-existence of Nash equilibrium in finite population games.Secondly,it is proved that the existence of the Nash equilibrium point of the two-strategy symmetric formal games.It is verified that the two-strategy games is potential games.Finally,we prove the existence of Nash equilibrium in potential games.Chapter 5,we introduce a simple two-strategy symmetric coordination games model,which forms a Markov chain under certain perturbations.The stochastic stability of Nash equilibrium is analyzed by example.We find the stochastic stable state of the general two-strategy games according to the relationship between the maximum point of the potential function and stochastic stable state in finite population potential games.