The Existence Of Nash Equilibriums Of Discontinuous Games With Ordered Strategy Spaces

Finding equilibrium points is one of the important contents of game theory,and the theory of fixed points is an important tool for studying equilibrium problems.Specifically,people often transform the mathematical model of games to some operator,then give appropriate assumptions to establish the existence of fixed points for such operator which guarantees the existence of equilibrium point.A large number of references have given sufficient conditions and necessary conditions for the existence of pure strategy Nash equilibrium.However,in terms of theory and application,these conditions are relatively strong,inconveniently to apply and difficult to verify.Therefore,it is interest and necessary to weaken and improve some conditions in some existing results.Meanwhile,in this article we will present several existence results to pure strategy Nash equilibriums which are more profound and more practical,also lay a theoretical foundation for future research.In this article,we will do a few aspects of the work: Firstly,we introduce the concept of preorder in the strategy spaces instead to consider the topological structure and algebraic structure.Secondly,we get rid of the constraint of the(weak)continuity assumption of the payoffs and explore the existence of Nash equilibriums of discontinuous games.In other words,we will provide sufficient and necessary conditions of the existence for Nash equilibriums of games with discontinuous payoffs and the strategy spaces neither a topological structure nor an algebraic structure.Instead,we introduce a preorder via payoffs.Our results generalize and improve many well-known results.Moreover,we present some examples to show that our results can be used to obtain the existence of Nash equilibrium points for the discontinuous games due to real economic activities which con not obtained by corresponding results in the existing literature.This article is divided into five chapters:The first chapter is the introduction.Firstly,we introduce the historical research background and the current development of game theory;then,we analyze the research meaning and the current situation of the existence of Nash equilibriums;finally,we put forward the improvement based on the existing achievements: that is to get rid of the constraint of the continuity assumption of the payoffs and the requirement of topological structure,we consider the existence of Nash equilibriums of discontinuous games with ordered strategy spaces.The second chapter is the preliminaries.We firstly introduce the definitions of the preordered set and inductive set.Then we review some relevant existing results,which establish the theoretical foundation for the later writing.The third chapter is the improvement of this article.Firstly,we give an existence theorem of the pure strategy Nash equilibrium which can get rid of the constraint of the continuity assumption of the payoffs and the requirement of topological structure and then prove it.Secondly,we give some examples to compare our conclusion with the existing research results and illustrate its practicability.The fourth chapter is the practical application of the results.We firstly prove that our results can be used in a wide range of common continuous games,for which we verify the existence of Nash equilibrium of Bertrand duopoly competition model.Then we introduce a discontinuous game called diagonal game and use our results to prove the existence of Nash equilibrium in the game.Finally we introduce a special case in diagonal games which called second price auction and verify the economic significance and practical value of the results obtained in this article.The fifth chapter makes a brief summary of this paper and presents some prospects for further research.