Brouwer 固定點定理與 Nash 均衡之研究

Abstract

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.