群体博弈的弱Nash平衡与演化分析

doi:10.15960/j.cnki.issn.1007-6093.2025.04.010

运筹学学报（中英文） ›› 2025, Vol. 29 ›› Issue (4): 112-120.doi: 10.15960/j.cnki.issn.1007-6093.2025.04.010

群体博弈的弱Nash平衡与演化分析

汤卫¹^,², 王春²^,*(), 杨光惠², 皮进修²

1. 贵州开放大学 (贵州职业技术学院) 信息工程学院, 贵州贵阳 550023
2. 贵州大学数学与统计学院, 贵州贵阳 550025

收稿日期:2022-11-20 出版日期:2025-12-15 发布日期:2025-12-11
通讯作者: 王春 E-mail:751164229@qq.com
基金资助:
国家自然科学基金(11271098);贵州省科技计划项目(黔科合基础-ZK[2022]般168)

Weak Nash equilibrium and evolution analysis of population games

Wei TANG¹^,², Chun WANG²^,*(), Guanghui YANG², Jinxiu PI²

1. School of Information Engineering, Guizhou Open University (Guizhou Vocational Technology Institute), Guiyang 550023, Guizhou, China
2. School of Mathematics and Statistics, Guizhou University, Guiyang 550025, Guizhou, China

Received:2022-11-20 Online:2025-12-15 Published:2025-12-11
Contact: Chun WANG E-mail:751164229@qq.com

摘要/Abstract

摘要：

在博弈的策略转移过程中, 如果局中人会获得策略转移成本, 那么一些局中人的最优回应策略可能没有最大化支付。对此, 本文在群体博弈模型下引入更一般的策略转移成本函数, 对弱Nash平衡概念进行改进, 新的弱Nash平衡包含Nash平衡集。此外, 我们在2×2对称矩阵博弈诱导的单群体博弈下, 通过复制动力学得到了弱Nash平衡稳定性, 并进行了仿真实验。结果表明, 相比未考虑策略转移成本的情形, 演化稳定状态有所增加。

关键词: 策略转移成本, 群体博弈, 弱Nash平衡, 复制动力学, 演化稳定状态

Abstract:

If the cost is generated in switching strategies of games, the best reply strategy of some players can be not maximize their payoffs. In order to improve weak Nash equilibrium, this paper introduces a more general cost function with switching strategies under population games model, and the new weak Nash equilibria contain Nash equilibria. In addition, according to replication dynamics, we obtain the stability of weak Nash equilibrium state in single-population game induced by the \times2$ symmetric matrix game with switching strategy cost. By simulation, it is easy to show that evolutionary stable state also is increased compared without switching strategies cost.

Key words: transfer cost, group game, weak Nash equilibrium, replication kinetics, evolutionary steady state

中图分类号:

O225

汤卫, 王春, 杨光惠, 皮进修. 群体博弈的弱Nash平衡与演化分析[J]. 运筹学学报（中英文）, 2025, 29(4): 112-120.

Wei TANG, Chun WANG, Guanghui YANG, Jinxiu PI. Weak Nash equilibrium and evolution analysis of population games[J]. Operations Research Transactions, 2025, 29(4): 112-120.

图/表 4

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参与人1	参与人2
参与人1	策略1	策略2
策略1	$ (a,a) $	$ (b,c) $
策略2	$ (c,b) $	$ (d,d) $

序号	参数条件	Nash平衡	弱Nash平衡	复制动力学静态点	ESS	NSS
(1)	$a+d-b-c>0,$ $d-b\geqslant 0$	$\{0\}\cup \{1\} \cup$ $\left\{\frac{d-b}{a+d-b-c}\right\}$	$\{0\}\cup \{1\} \cup$ $\left[\max\left(\frac{d-b-v}{a+d-b-c},0\right),\right.$ $\left.\min\left(\frac{d-b+w}{a+d-b-c}, 1\right)\right]$	$\{0\}\cup \{1\} \cup$ $\left[\max\left(\frac{d-b-v}{a+d-b-c},0\right),\right.$ $\left.\min\left(\frac{d-b+w}{a+d-b-c}, 1\right)\right]$	$ \{0\}\cup \{1\} $	$\left(\max\left(\frac{d-b-v}{a+d-b-c},0\right),\right.$ $\left.\min\left(\frac{d-b+w}{a+d-b-c}, 1\right)\right)$
(2)	$a+d-b-c <0,$ $d-b\leqslant 0$	$ \left\{\frac{d-b}{a+d-b-c}\right\} $	$\left[\max\left(\frac{d-b+w}{a+d-b-c},0\right)\right.,$ $\left.\min\left(\frac{d-b-v}{a+d-b-c}, 1\right)\right]$	$\{0\}\cup \{1\} \cup$ $\left[\max\left(\frac{d-b+w}{a+d-b-c},0\right)\right.,$ $\left.\min\left(\frac{d-b-v}{a+d-b-c}, 1\right)\right]$	$\left[\max\left(\frac{d-b+w}{a+d-b-c},0\right)\right.,$ $\left.\min\left(\frac{d-b-v}{a+d-b-c}, 1\right)\right]$	–
(3)	$a+d-b-c\geqslant 0,$ $d-b <0$	{1}	$\{1\}\cup$ $\left[0, \max\left(\frac{d-b+w}{a+d-b-c},0\right)\right]$	$\{0\}\cup \{1\} \cup$ $\left[0, \max\left(\frac{d-b+w}{a+d-b-c},0\right)\right]$	{1}	$ \left(0, \max\left(\frac{d-b+w}{a+d-b-c}, 0\right)\right) $
(4)	$a+d-b-c\leqslant 0,$ $d-b>0$	{0}	$ \left[0, \max\left(\frac{d-b-v}{a+d-b-c}, 0\right)\right] $	$\{0\}\cup \{1\} \cup$ $\left[0, \max\left(\frac{d-b-v}{a+d-b-c}, 0\right)\right]$	$ \left[0, \max\left(\frac{d-b-v}{a+d-b-c}, 0\right)\right] $	–

群体博弈的弱Nash平衡与演化分析

Weak Nash equilibrium and evolution analysis of population games

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