The Existence And Stability Of Some Equilibriums

This paper mainly studies the existence of the extended Nash equilibrium of non-monetized noncooperative games and the stability of the equilibriums of coalition games.Noncooperative games in which both of collection of strategies and the ranges of thepayof functions for the players are posets are said to be nonmonetized. The generalizedNash equilibrium of nonmonetized noncooperative games, because of its huge potentialapplication, has caused widespread attention. many authors studied the existence of thegeneralized Nash equilibrium under diferent conditions. In this paper, we will considerthe existence of the extended Nash equilibrium, which is more general than the general-ized Nash equilibrium. Under normal circumstances, the strategies sets and the payofspaces are either complete spaces equipped with some metric topological and algebra-ic structure or order compact spaces equipped with certain ordered sequence when wediscuss the existence of extended Nash equilibrium. But it is not an easy thing, whichrequires very strong assumptions. Therefore, in the third chapter, we study the existenceof extended Nash equilibrium without the consideration of a topological structure or analgebraic structure. Moreover, we also relax assumptions of the concept of the inductiveset so as to facilitate the practical application.Coalition games mainly research how to form coalitions to make as large as possiblecommon interests and how to distribute the interests of the coalitions. The developmentof the coalition game theory is very slow and imperfect, especially the stability of the e-quilibriums of the coalition games. Recently, the stability problem of the equilibriums forcoalition games is converted into the stability problem of the vector function optimizationby many researchers. In the fourth chapter, we also use the stability theory of the vec-tor function optimization to discuss the stability of the quasi-equilibrium of the coalitiongames, where the stability is that when the initial coefcients of the linear payof func-tions are subject to certain perturbations, the equilibrium remains the same, namely theinitial equilibrium. We mainly from the two aspects to consider this problem: First, theinitial coefcients within the scope of what changes can guarantee the stability of the ini-tial equilibrium; Second, the maximum perturbation of the initial coefcient is calculatedby the stability radius and the accurate radius. The full text is divided into five chapters:First chapter is an introduction. Firstly, we introduce the development of the equilib-rium of games, especially the Nash equilibrium of non-cooperative games is very detailedintroduced; Then we mainly analyze the research status of the game theory and point outthe significance and the latest trends of the research in detail; Finally, we point out theproblems will be solved in this paper, namely, the existence of the extended Nash equi-librium of nonmonetized noncooperative games and the stability of the equilibrium of thecoalition games.Second chapter is preliminaries. We mainly introduce the definitions of the nonco-operative games and the coalition games. Then the definitions and related lemma of theorder poset are recalled, which will be the foundation of the following research work.In the third chapter, we mainly discuss the existence problems of the extended Nashequilibrium of the n person nonmonetized noncooperative games. In the first part, we ex-tend and improve the fixed point theorem in [36] for set-valued functions in posets withoutthe consideration of a topological structure or an algebraic structure. Then we extend aseries of related de definitions of inductive poset. Lastly, some related properties of in-ductive poset are given, for example, an inductive set has maximal element and minimalelement. In the second part, by applying the fixed point theorem and the order preservingproperty of the set-valued mapping, we prove an existence theorem of the extended Nashequilibrium of nonmonetized noncooperative games and its corollary. In the third part,some examples are given here to illustrative the applicability of our results.The fourth chapter mainly focuses on the stability of the quasi-equilibrium of thecoalition games. Firstly, we introduce the concept of the quasi-equilibrium of coalitiongame. Then, by using the stability of vector optimization theory, some stability theoremsof the quasi-equilibrium is obtained. Finally, we discuss the stability problems of s equi-librium with the same method. It is worth mentioning that the quasi-equilibrium is moregeneral than the J equilibrium, therefore the stability theorem of the quasi-equilibriumis more universal and practical.Finally, the fifth chapter makes a brief summary and presents some prospects forfurther studies.