The Properties And Equilibrium In Mixed Strategies Of Continuous Game Theory

Game theory is the research of mathematical models about cooperating and conflicting among the players who are gaming. Game theory became most important economics and took a fair place in economics theory in the 1950's. In 1994, it is Nash, Harsany and Selten who won the Nobel Economic Prize that given the most authoritative to made Game theory became one subject of economics. In the later, J.A.Mirrlees and N. Vickrey who is the expert of Game theory and economic information won the Nobel Economic Prize for the basic research of power on asymmetry information in 1996. As well as, the Noble prize award to GAkerlof, A.M Spence and J.Stiglitz for their contribute of found Game theory in asymmetry information, and strengthened the development of the direction of Game theory. In 2005, R. Aumann and T.C.Sehelling attained the Noble prize together, due to one made progress in the felicitousness concept of Game theory, and another gained importance achievements in deep study on Game theory and contribution of applications on philosophic theory. It show Game theory possess extraordinary consequence and infection again. Now, Game theory has come into being a largish system information. "Game theory become the basic law of demonstration of economy and society academic", it's power must be the importance application of many way and many directions expanding.In the progress of form and development of Game theory, introducing mixed strategy (random strategy) and application have great effect in Game theory. In case of uncertainty or have risk, player of assumed is to seek a maximum of expectation payment on distribution always. Thus expectation is determined by random variable distribution completely. Therefore ,there is not strict determinate equilibrium, and neither player hoped else predominated his choice, the best method is assuming each player shall choose distribution in his strategies space ashis mixed strategies, so player seek a maximum of expectation payment. And interprets the choices who are gaming though the corresponding model, and research payment according player's mutual effect. In the development of Game theory, it is mean method with mixed strategy to analysis Game, introducing the strategy guarantees the existence of Nash equilibrium. It is the embodiment of essence of predicting non-cooperative game. The mixed strategy Nash equilibrium is the most fundamental concept in economic activities.In the economic practicality, the strategy space of player is immensity, for example, there are measureless strategies whether the player look the quantity or price of products. Therefore, the research of immensity Game theory and solution concept also is the equilibrium became one of the focus and masterstroke of economy, Many continuous Game models have much effect in simulation and interpretation of the player who play in economy activity. In the research of continuous Game in infinite Game, the existence of equilibrium effect in theory and result, too. The research on existence of equilibrium of continuous Game now can sum up as follow: The theory of existence of mixed strategies of continuous Game in the theory that existence of Nash equilibrium(Glicksgerg, 1952), and theory of pure strategy in continuous Game(see also I. Glicksberg, 1952; G. Debreu, 1952; Ky. Fan, 1952. Cournot brought forward the concept of equilibrium first in 1838, and were named Cournot equilibrium by people , and also named Nash equilibrium now; The latter is particular case as the former.The following is the mainly work and content of this thesis.On the base of corresponding work Fudenberg and Tirole, this paper mostly work is discussing that the properties of mixed strategy and proving of theory of existence of mixed Nash equilibrium of continuous Game. The method of continuous mixed strategies is presented by examples. Further, represent a method that prove the existence theorem of Nash equilibrium by relation which compact metric space infinite set can sufficient approximation by finite set, so we may constitute continuous with finite Game.Thesis includes the following specific elements.The first is prelude which is summarization. The second is provision knowledge which presented some basic concept and mark. Repeating some contentmainly comparison the properties and equilibrium of mixed strategies in perfect information static Game(see also Liu-zongqian,2004,2006). In the third potion, represent infinite Game, continuous Game, smooth game and response corresponding and corresponding function, solving Nash equilibrium of the continuous Game by discussing and examples. In the fourth part, giving and proving some properties which is basic and importance, this properties in mixed continuous is resemble to finite Game, proving the existence of mixed Nash equilibrium utilizing the method in the prelude. About 2-person constant sum Game could be solved by response functions equations and can to seek the mixed equilibrium by maxmin method of theorem of a saddle point. And presented and proved some general proposition. In the fifth part, erecting relation between continuous and finite Game, giving and proving concerned theorem, especially approximation theorem, in order to stress application which is infinite mathematics knot which compact metric space is approximate sufficient by finite set. In the sixth part of this paper, there are represented another method that proved the existence theorem of mixed Nash equilibrium in continuous Game. In the end, the part of epilogue, showing and proving what is pure strategy equilibrium in perfect information strategy Game.