集值映射不动点的本质集与相关问题研究

发布时间：2018-03-25 19:39

本文选题：集值映射不动点　切入点：Nash平衡　出处：《贵州大学》2016年硕士论文

【摘要】：Brouwer不动点定理和Kakutani不动点定理被成功的运用于博弈论、数理经济等领域,尤其是Nash用Brouwer不动点定理建立了非合作博弈理论,Arrow和Debreu用Kakutani不动点定理证明了一般均衡的存在性后,不动点理论引起了更加广泛的关注.博弈论是运筹学的一个分支,非合作博弈在博弈论中处于基础和核心的地位,Nash平衡是非合作博弈理论中最主要、最核心的概念.Nash平衡与非线性问题的解有着某种等价关系,例如与集值映射不动点、变分不等式解的等价性.Nash平衡的存在性与稳定性可以转化为相关非线性问题解的存在性与稳定性.本文正是从这些等价性出发,通过集值映射不动点、Ky-Fan引理及变分不等式的解的稳定性来研究Nash平衡的稳定性.主要是通过改进集值映射和函数的扰动,即引入几种更强的扰动来研究这些问题解的稳定性.本文研究了强?-扰动意义下集值映射不动点的强本质性,并由此给出了Nash平衡的强稳定集,研究了Ky-Fan截口问题、Ky-Fan不等式的解集在强?-扰动下本质连通区的存在性,研究了非合作向量博弈Pareto-Nash平衡的稳定性.第一章:基础知识.主要有:集值映射的上、下半连续性、Fan-Glicksberg不动点定理、Fort引理、稠密剩余集、本质不动点、不动点的本质集、不动点的本质连通区、Hausdorff距离等.第二章:研究了强?-扰动下不动点的稳定性和Nash平衡策略集的稳定性.强?-扰动下得到的不动点稳定性最强,对应Nash平衡策略的稳定集——SBR-稳定集,具有关于映射扰动、映射和定义域扰动的稳定性.SBR-稳定集对应的强?-扰动是用Hausdorff距离来定义的,没有附加其他任何距离上的条件,所以这种方法可以用来研究任何非合作有限博弈.第三章:运用强?-扰动的方法,本节给出了Ky-Fan截口问题解的更强的连续性和稳定性,给出了在强?-扰动扰动下的本质连通区和在图像?-扰动下的本质连通区.并由Ky-Fan截口问题解的强本质性,得到了Ky-Fan不等式解集的强本质连通区.再由Ky-Fan点集的强本质连通区,得到了Nash平衡的强本质连通区.第3.5节针对Ky-Fan点和Nash平衡点的稳定性,考虑不等式函数或者支付函数有可能满足Ky-Fan点或Nash平衡点条件的点处所产生的扰动,进一步拓展Ky-Fan点集和Nash平衡点集稳定性的扰动范围.第四章:从优化的角度,研究了非合作向量博弈Pareto-Nash平衡的稳定性.向量博弈的理想-Nash平衡点、满意-Nash平衡点和分层次Pareto-Nash平衡点,都是Pareto有效点.它们都具有Pareto-Nash平衡点的连续性(上半连续性需要条件)、通有稳定性和本质连通区性质.
[Abstract]:Brouwer fixed point theorem and Kakutani fixed point theorem have been successfully applied in the fields of game theory, mathematical economics and so on, especially after Nash established the non-cooperative game theory by Brouwer fixed point theorem and Debreu proved the existence of general equilibrium by Kakutani fixed point theorem. Game theory is a branch of operational research, and non-cooperative game plays an important role in non-cooperative game theory. The most important concept, the Nash equilibrium, has some equivalent relationship with the solution of nonlinear problems, such as fixed points with set-valued mappings, Equivalence of Solutions of variational inequalities. The existence and stability of Nash equilibrium can be transformed into the existence and stability of solutions for related nonlinear problems. The stability of Nash equilibrium is studied by means of Ky-Fan Lemma of set-valued mapping and stability of solutions of variational inequalities, mainly by improving set-valued mappings and perturbation of functions. In this paper, we study the stability of the solutions of these problems by introducing several stronger perturbations. The strong property of fixed point of set-valued mapping in perturbed sense is given, and the strong stable set of Nash equilibrium is obtained. The solution set of Ky-Fan inequality for Ky-Fan cut problem is studied. -existence of essential connected region under perturbation, the stability of Pareto-Nash equilibrium in non-cooperative vector game is studied. Chapter 1: basic knowledge. The main contents are as follows: upper, lower semi-continuous Fan-Glicksberg fixed point theorem Fort Lemma, dense residual set. Essential fixed points, essential set of fixed points, essential connected regions of fixed points, Hausdorff distance, etc. -the stability of fixed points under perturbation and the stability of the set of Nash equilibrium strategies. The stable set corresponding to the Nash equilibrium strategy is the stable set of SBR-stable set, which has the strong stability of the map perturbation, mapping and domain perturbation. SBR-stable set corresponding to the Nash equilibrium strategy, and the stable set corresponding to the Nash equilibrium strategy, are strongly correlated with the stability of the map perturbation, the mapping and the perturbation of the defined domain. -perturbation is defined by Hausdorff distance without any other conditions on the distance, so this method can be used to study any non-cooperative finite game. In this section, we give the stronger continuity and stability of the solution of the Ky-Fan section problem, and give the stronger continuity and stability of the solution. -intrinsically connected regions and images under perturbed perturbations? By using the strong nature of the solution of the Ky-Fan cut problem, the strongly essential connected region of the solution set of the Ky-Fan inequality is obtained, and then the strongly essential connected region of the Ky-Fan point set is obtained. The strongly essential connected region of Nash equilibrium is obtained. For the stability of Ky-Fan point and Nash equilibrium point, the perturbation at the point where the inequality function or the payment function may satisfy the Ky-Fan point or Nash equilibrium condition is considered. Chapter 4: from the point of view of optimization, we study the stability of Pareto-Nash equilibrium in non-cooperative vector game. Both satisfied -Nash equilibrium points and hierarchical Pareto-Nash equilibrium points are Pareto efficient points. They all have the continuity of Pareto-Nash equilibrium points (the upper semicontinence needs the condition), and they all have the properties of stability and essential connected region.
【学位授予单位】：贵州大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：O225;O177.91

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