Summarization Of Game Theory's Developments And Functional Analysis' Applications In Game Theory

It is made up of the developments of functional analysis in game theory in the Chapter 1. The research about games in which each players has a number of pure strategies and mixed strategies have come into being grown-up theories and methods. People have got many results with the mixed game with infinite pure strategy sets. For instance, It is well- known that there exists optimal mixed strategies in games which each player has a payoff function with a continual pure strategy set on the unit square (consult[10][11]).G.owen has also discussed the existence of optimal mixed strategies in which each player has a payoff function with a discontinuous pure strategies. However, compared with finite pure strategy set, there remains a number of unsolvable questions of games in which each players has an infinite pure strategy set and fixed strategies. In the Chapter 1, a max-min theorem and a max theorem are proved on local convex topological linear spaces, as an application, it is obtained that there exists optimal mixed strategies in a game with an infinite pure strategy set on a compact Hausdorff space on the basis of analytic methods.We concern the developments of game theory in the chapter 2. At first, it introduces the general situation, the historical stages and main achievements of game theory's developments. In the second part, it goes on with static game of perfect information and brings in Nash Equilibrium. We analyze Nash Equilibrium of the classical theory game of Prisoner's Dilemma and point out the solution unpractical although it has the chance to optimize theoretically. As an application, we explain the exist necessity of many environment problems by the case of "sheep eat grass". During the third part we introduce sub game perfect Nash Equilibrium, which is the rational equilibrium solution of dynamic game of perfect information. We use the backward induction to work out the optimal equilibrium solution of optimal tariff game, at the same time, explain the importance of tariff among the international competition and trade punishment. In the fourth part, it gives the definition of Bayesian Nash Equilibrium in the static game of incomplete information and discusses the game of sealed auction and draws some conclusions according to it. During the fifth part, we introduce perfect Bayesian Nash Equilibrium, which is the optimal solution of dynamic game of incomplete information, meanwhile, analyze the reason of an abnormal phenomenon in Chinese Stock Markets. Finally, combined with the reform of economic construction and the building of market economics, it opens up the applying regions of game theory in China.