群体博弈弱Nash平衡的存在性和通有稳定性  

The Existence and Generic Stability of Weak Nash Equilibria for Population Games

作　　者：王明婷 杨光惠 WANG Ming-ting;YANG Guang-hui(School of Mathematics and Statistics,Guizhou University,Guiyang 550025,China)

机构地区：[1]贵州大学数学与统计学院,贵州贵阳550025

出　　处：《数学的实践与认识》2021年第15期187-193,共7页Mathematics in Practice and Theory

基　　金：国家自然科学基金项目(11271098);贵州省科技计划项目(黔科合基础[2019]1067号);贵州大学引进人才科研项目([2017]59);贵州省教学改革项目(2019008)。

摘　　要：Nash平衡是群体博弈的一个重要概念,即当某一策略被博弈代理人选用时,代理人不能通过单方面改变策略而获得比当前更高的收益.但由于博弈中策略改变的同时也会产生成本,通过举例说明成本对经典的Nash平衡具有一定的影响.因此,对经典群体博弈引入了博弈代理人的成本函数,建立了新的群体博弈模型,并提出一种新的平衡概念--弱Nash平衡,其思想是当博弈代理人改变策略时所产生的成本大于或等于所增加收益时,会使得代理人没有改变策略的动机因而达到弱Nash平衡.进一步,运用Brouwer不动点定理证明了群体博弈弱Nash平衡的存在性;最后运用Fort定理证明了当纯收益函数发生扰动时群体博弈弱Nash平衡的通有稳定性.A Nash equilibrium is a significant notion in population games,that is,when a strategy is chosen by a game player,the player cannot unilaterally change the strategy to obtain higher profits than the incumbent one.However,the switches for strategies in real decision-problems also cause the resulting costs.And an example is illustrated the effects of costs on Nash equilibria of classical population games.This paper introduces the cost function of game agents to classical population games and establishes a new model for population games.Then,a new concept,a weak Nash equilibrium,is proposed.The underlying idea is that when the resulting cost of changing strategies is greater than or equal to the increase in profits,the game agent has no incentive to switch the current strategy and thus the undergoing population games reach a weak Nash equilibrium.Furthermore,the existence of weak Nash equilibria is proved by Brouwer’s fixed point theorem.Finally,the generic stability of weak Nash equilibria for population games is proved by Fort theorem when the net profit function is perturbed.

关 键 词：群体博弈 弱Nash平衡 BROUWER不动点定理 存在性 通有稳定性 

分 类 号：O225[理学—运筹学与控制论]