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

Abstract

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

CLC Number:

O225

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.

Figures/Tables 4

References 29

1	Nash J. Non-cooperative games[D]. Princeton$: $ Princeton University, 1950.
2	SmithM.Evolution and the Theory of Games[M].Cambridge:Cambridge University Press,1982.
3	Bjornerstedt J, Weibull J W. Nash equilibrium and evolution by imitation[EB/OL]. [2022-10-21]. https://www.econstor.eu/bitstream/10419/94705/1/wp407.pdf.
4	HaroldW K,JohnC H,ReinhardS,et al.The work of John Nash in game theory: Nobel seminar, December 8, 1994[J].Journal of Economic Theory,1996,69(1):153-185. doi: 10.1006/jeth.1996.0042
5	Weibull J W, Jorgen W. The mass-action interpretation of Nash equilibrium[EB/OL]. [2022-10-21]. https://www.econstor.eu/bitstream/10419/95056/1/wp427.pdf.
6	HofbauerJ.From Nash and Brown to Maynard Smith: Equilibria, dynamics and ESS[J].Selection,2005,1(1):81-88.
7	BlumeL E.Population games[J].Game Theory & Information,1998,11(3):211-217.
8	HauerJ F,TrudnowskiD J,RogersG,et al.Evolutionary games and population dynamics[J].IEEE Computer Applications in Power,1998,10(4):50-54.
9	SandholmW H.Population Games and Evolutionary Dynamics[M].Massachusetts:MIT Press,2011.
10	YangG H,YangH.Stability of weakly Pareto-Nash equilibria and Pareto-Nash equilibria for multiobjective population games[J].Set-Valued and Variational Analysis,2017,25(2):427-439. doi: 10.1007/s11228-016-0391-6
11	YangG H,YangH,SongQ Q.Stability of weighted Nash equilibrium for multiobjective population games[J].The Journal of Nonlinear Sciences and Applications,2016,9(6):4167-4176. doi: 10.22436/jnsa.009.06.59
12	YangZ,ZhangH Q.Essential stability of cooperative equilibria for population games[J].Optimization Letters,2019,13(7):1573-1582. doi: 10.1007/s11590-018-1303-5
13	陈华鑫,贾文生.群体博弈的逼近定理及通有收敛性[J].数学物理学报,2021,41(5):1566-1573.
14	SmithJ M,PriceG R.The logic of animal conflict[J].Nature,1973,246,15-18. doi: 10.1038/246015a0
15	TaylorP D,JonkerL B.Evolutionarily stable strategies and game dynamics[J].Mathematical Biosciences,1978,40(1-2):145-156. doi: 10.1016/0025-5564(78)90077-9
16	HofbauerJ,SchusterP,SigmundK.A note on evolutionary stable strategies and game dynamics[J].Journal of Theoretical Biology,1979,81(3):609-612. doi: 10.1016/0022-5193(79)90058-4
17	Zeeman E C. Population dynamics from game theory[M]//Nitecki Z, Robinson C. (eds. ) Global Theory of Dynamical Systems. Berlin: Springer, 1979, 819: 471-497.
18	HofbauerJ,SigmundK.The Theory of Evolution and Dynamical Systems[M].Cambridge:Cambridge University Press,1988.
19	NowakM A,SigmundK.Evolutionary dynamics of biological games[J].Science,2004,303(5659):793-799. doi: 10.1126/science.1093411
20	SasakiT,UnemiT.Replicator dynamics in public goods games with reward funds[J].Journal of Theoretical Biology,2011,287,109-114. doi: 10.1016/j.jtbi.2011.07.026
21	CressmanR,TaoY.The replicator equation and other game dynamics[J].Proceedings of the National Academy of Sciences of the United States of America,2014,111,10810-10817.
22	ThomasB.On evolutionarily stable sets[J].Journal of Mathematical Biology,1985,22,105-115.
23	ThomasB.Evolutionary stable sets in mixed-strategist models[J].Theoretical Population Biology,1985,28(3):332-341. doi: 10.1016/0040-5809(85)90033-4
24	张维迎.博弈论与信息经济学[M].上海:上海人民出版社,2004.
25	蔡阳洋,向淑文.$n$人非合作博弈弱Nash均衡点的存在性[J].重庆工商大学学报(自然科学版),2020,37(1):54-58.
26	王明婷,杨光惠.群体博弈弱Nash平衡的存在性和通有稳定性[J].数学的实践与认识,2021,51(15):187-193.
27	CooperR,DeJongD,ForsytheR,et al.Communication in the battle of the sexes game: Some experimental results[J].RAND Journal of Economics,1989,20(4):568-587. doi: 10.2307/2555734
28	SugdenR.The Economics of Rights, Co-Operation and Welfare[M].Oxford:Basil Blackwell,1986.
29	AxelrodR.Effective choice in the prisoner's dilemma[J].The Journal of Conflict Resolution,1980,24(3):3-25.

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] $	–

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Weak Nash equilibrium and evolution analysis of population games

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