Study On The Generalized Set-valued Nash Equilibrium Theory

Aiming at the problem that the Nash equilibrium of the set-valued mappings has no constraints in the past,we propose the concept of the Nash equilibrium of the generalized set-valued mappings with constraints.It contains a wide range of contents.such as common Nash equilibrium,multi-objective Nash equilibrium,generalized Nash equilibrium,Loose Nash equilibrium and so on,which are special cases of Nash equilibrium of generalized set-valued mappings.In this paper,we mainly study the existence and stability of Nash equilibrium for generalized setvalued mappings.The main work and innovations include the following aspects:1.The classical game systems and the corresponding Nash equilibrium model are introduced,and their existence is discussed by using Fan-Glicksberg fixed point theorem.Furthermore,we introduce the Nash equilibrium model of set-valued mapping,and obtain the existence property of Nash equilibrium of set-valued mapping by using the equivalent form of generalized KKM.2.In this paper,a game system model of generalized set-valued mapping is proposed.Under the condition of compactness,the existence of Nash equilibrium of generalized set-valued mapping is obtained by using the equivalent form of generalized KKM.In addition,under the noncompact condition,using the generalized Browder fixed point theorem,the existence of Nash equilibrium for generalized setvalued mappings is discussed,and the positive result is obtained,which weakens the existence condition from compactness to noncompact.3.From the perspective of the essential equilibrium point,the stability of the Nash equilibria for the generalized set-valued mappings is discussed,and it is proved that when the payment mapping of the countermeasure system changes slightly,the Nash equilibria will not change accordingly.Therefore,The generic stability of the Nash equilibria for generalized set-valued mappings is discussed,and it is proved that the most of Nash equilibria for generalized set-valued mappings(in the sense of Baire topology)is stable.4.In addition,we discuss the well-posedness of Nash equilibrium problems for generalized set-valued mappings.By defining the sequence of approximate solutions of Levitin-polyak,we prove the sufficient and necessary conditions for the well-posedness of Levitin-polyak.On this basis,we obtain the results of the well-posedness of Nash equilibrium problems for generalized set-valued mappings,and obtain that the Nash equilibrium problems for generalized set-valued mappings are well-posedness in most cases.