- 學位論文
Brouwer 固定點定理與 Nash 均衡之研究
A Study of Brouwer's Fixed Point Theorem and Nash's Equilibrium
摘要
本論文旨在對著名的Brouwer固定點定理與Nash均衡定理做一仔細研究。第一節為導論。第二節中我們簡介單體的三角剖分,相關的定義如仿射獨立、仿射組合、仿射包、凸包 、重心坐標均有列出,與之相關的一些性質也均有陳述。第三節中我們介紹單體的重心與上升單體序列,並詳細討論單體重心細分的存在性,它保證單體可以被三角剖分成一些任意小的單體。在第四節中,我們對凸集與凸體仔細探究,可 知某些特殊的同胚映射將凸體映成適當維度的單體。第五節中,以典型的方式研究Sperner引理,Knaster-Kuratowski-Mazurkiewicz定理及Brouwer固定點定理。在第六節中,介紹了標準型的有限非合作賽局,相關的概念如參賽者、純策、混策、局面、報償、均衡點均有介紹及例子。從證明中可以看到非合作賽局中的均衡點是與之相關Nash映射的固定點。
並列摘要
The purpose of this thesis is to give a detail study of the well known Brouwer's fixed point theorem and Nash's equilibrium theorem. Section 1 is an introduction. Notions about triangulations of simplexes are recalled in section 2, the definitions of the affine independence, affne combination, affine hull, convex hull, barycentric coordinate, simplex, face, and triangulation are given and some basic properties of them are listed. In section 3, the definitions of barycenters of simplexes and sequences of ascending simplexes are introduced. After some discussions, the existence of a triangulation, the k-th barycentric subdivision, of a simplex is established, in case k is large enough, the maximum diameter of its members can be arbitrarily small. In section 4, we study some topological properties of convex sets and convex bodies, we see that some special homeomorphisms map convex bodies onto simplexes of suitable dimensions. In section 5, we study the celebrate theorems of Sperner, Knaster-Kuratowski-Mazurkiewicz, and Brouwer in a typical way. In section 6, the definition of noncooperative finite games in normal form is given, notions of the player, pure or mixed strategy, situation, payoff and equilibrium point are introduced with some examples. We see that the fixed points of Nash mappings are exactly the equilibrium points in mixed strategies of the corresponding games.