单种群多人博弈中的ESS、NIS和GIS
摘要
演化博弈理论最重要的基本概念是演化稳定策略(ESS)。ESS能够成功抵御其他变异策略的入侵。另一方面,一个策略是否是种群长期演化选择的结果,也在于其能否在种群演化过程中成功进入。进入者策略是能够成功进入种群的策略,所以长期演化过程中也可能被种群中的个体所采用。本文旨在从抵御入侵与可以进入两个角度分析策略的演化稳定性。
本文基于单种群多人矩阵博弈与一类非矩阵博弈模型,讨论演化稳定策略(ESS)、邻域进入者策略(NIS)与全局进入者策略(GIS)三个主要概念,并着重研究了ESS、NIS、GIS的性质及关系,获得了相应的结论。
在多人矩阵博弈模型中,GIS一定是ESS;GIS、ESS不能共存,除非策略本身是ESS;GIS具有唯一性;若存在多个ESS则无GIS。二人矩阵博弈中,ESS与NIS等价;GIS是全局优超的,从而在动力系统中是全局渐近稳定的。多人矩阵博弈与二人矩阵博弈有着不同的性质与结论。
在非矩阵博弈模型中,具有一定线性性时ESS与NIS等价,并且具有与矩阵博弈相同的性质:GIS一定是ESS;GIS、ESS不能共存,除非策略本身是ESS;GIS具有唯一性;若存在多个ESS则无GIS。
最后讨论了策略的可入侵性与不可入侵性,并分别得到可入侵性及不可入侵性在连续、离散动力系统中的等价条件。
Evolutionary stable strategy (ESS) is the basic concept of Evolutionary Game Theory. ESS can successfully resist the invasion of other strategy. On the other hand, it is important for a strategy to enter the population occupied by other strategy if it can be the evolutionary stable choice by the population in the long run. Invader strategy is a strategy that be able to invade all established communities, so it also can be adopted by the individuals of population in the long run. We analyze the evolutionary stability from two aspects: resisting the invasion of other strategy and successfully invading the population.
In this paper, we discuss the concepts of evolutionary stable strategy (ESS), neighborhood invader strategy (NIS) and global invader strategy (GIS) in matrix game and a kind of non-matrix game. The main content is the properties of ESS, NIS,
GIS and relationship among them, hence some correspond conclusions.
In multi-player matrix game, we show that: a GIS is always an ESS; a GIS cannot coexist with an ESS unless it is itself an ESS; If a GIS exists, then it is unique; If there is more than one ESS, then there are no GIS. We also get the conclusion that in a pair wise game MS is equivalent to ESS, GIS is globally superior and it is globally asymptotically stable in the dynamics of duplicator. Also, there are some results in multi-player games different from those in pair wise games.
In non-matrix game, NIS is equivalent to ESS on the assumption that the payoff is linear. And we get the same properties as multi-player matrix game: a GIS is always an ESS; a GIS cannot coexist with an ESS unless it is itself an ESS; if a GIS exists, then it is unique; if there is more than one ESS, then there are no GIS.
Finally, we discuss the invasion and non-invasion of the strategy, and get the equivalent conditions of the invasion and non-invasion in continuous-time dynamic and discrete-time dynamic.
本文基于单种群多人矩阵博弈与一类非矩阵博弈模型,讨论演化稳定策略(ESS)、邻域进入者策略(NIS)与全局进入者策略(GIS)三个主要概念,并着重研究了ESS、NIS、GIS的性质及关系,获得了相应的结论。
在多人矩阵博弈模型中,GIS一定是ESS;GIS、ESS不能共存,除非策略本身是ESS;GIS具有唯一性;若存在多个ESS则无GIS。二人矩阵博弈中,ESS与NIS等价;GIS是全局优超的,从而在动力系统中是全局渐近稳定的。多人矩阵博弈与二人矩阵博弈有着不同的性质与结论。
在非矩阵博弈模型中,具有一定线性性时ESS与NIS等价,并且具有与矩阵博弈相同的性质:GIS一定是ESS;GIS、ESS不能共存,除非策略本身是ESS;GIS具有唯一性;若存在多个ESS则无GIS。
最后讨论了策略的可入侵性与不可入侵性,并分别得到可入侵性及不可入侵性在连续、离散动力系统中的等价条件。
Evolutionary stable strategy (ESS) is the basic concept of Evolutionary Game Theory. ESS can successfully resist the invasion of other strategy. On the other hand, it is important for a strategy to enter the population occupied by other strategy if it can be the evolutionary stable choice by the population in the long run. Invader strategy is a strategy that be able to invade all established communities, so it also can be adopted by the individuals of population in the long run. We analyze the evolutionary stability from two aspects: resisting the invasion of other strategy and successfully invading the population.
In this paper, we discuss the concepts of evolutionary stable strategy (ESS), neighborhood invader strategy (NIS) and global invader strategy (GIS) in matrix game and a kind of non-matrix game. The main content is the properties of ESS, NIS,
GIS and relationship among them, hence some correspond conclusions.
In multi-player matrix game, we show that: a GIS is always an ESS; a GIS cannot coexist with an ESS unless it is itself an ESS; If a GIS exists, then it is unique; If there is more than one ESS, then there are no GIS. We also get the conclusion that in a pair wise game MS is equivalent to ESS, GIS is globally superior and it is globally asymptotically stable in the dynamics of duplicator. Also, there are some results in multi-player games different from those in pair wise games.
In non-matrix game, NIS is equivalent to ESS on the assumption that the payoff is linear. And we get the same properties as multi-player matrix game: a GIS is always an ESS; a GIS cannot coexist with an ESS unless it is itself an ESS; if a GIS exists, then it is unique; if there is more than one ESS, then there are no GIS.
Finally, we discuss the invasion and non-invasion of the strategy, and get the equivalent conditions of the invasion and non-invasion in continuous-time dynamic and discrete-time dynamic.
引文
[1]丁二维生.博弈论与经济[M].北京:高等教育出版社,2007:429-483.
[2]肖条军.博弈论及其应用[M].上海:上海三联书店,2004:429-483.
[3]Lewontin,R.C.Evolution and the theory of games.Journal of Theoretieal Biology,1960,1:382-403.
[4]Maynard Smith,J.,Price G.R.The logic of animal conflict.Nature,1973,246:15-18.
[5]Apaloo,J.Revisiting matrix games:The concept of neighborhood invader strategies.Theoretic Population Biology.2006,69:235-242.
[6]Apaloo.J.Revisiting Strategic Models of Evolution:The Concept of Neighborhood Invader Strategies.Theoretic Population Biology.1997a,52:71-77.
[7]McKelvey,R.,Apaloo,J.The structure and evolution of competition-organized ecological communities.Rocky Mount.J.Math.1995,25(1):417-436.MathSciNet.
[8]Kisidi E.,Meszena G.Density dependent life history evolution in fluctuating environment,in "Adaptation in a Stochastic Environment"(J.Yoshimura,and C.Clark,Eds.).Lecture Notes in Biomathematics.Springer,Berlin.1993,99:26-62.
[9]Kisidi E.,Meszena G.Life Histories with Lottery Competition in a Stochastic Enviornment:ESSs which do not prevail.Theor.Popul.Biol.1995,47:191-211.
[10]Doanld Ludwig,Simon A.Levin.Evolutionary stability of plant communities and the maintenance of multiple dispersal types.Theor.Popul.Biol.1991,40:285-307.
[11]Lessard.Evolutionary stability:one concept,several meanings,Theoretic Population Biology.1990,37:159-170.
[12]Apaloo.J.Ecological species coevolution,J.Biol.Syst 1997b,5:17-34.
[13]Apaloo.J.Single species evolutionary dynamics,Evolutionary Ecology.2003,5:33-49.
[14]Apaloo.J.Multi-species evolutionary dynamics,Evolutionary Ecology.2005a,18:55-77.
[15]Apaloo.J.Inaccessible continuously stable strategies.Nat.Resour.Modeling.2005b,19:521-535.
[16]Apaloo.J.Evolutionary matrix games and optimization theory Journal of Theoretical Biology.2009,257:84-89.
[17]Ding et al.ESS,NIS and GIS for multi-player matrix game in single population, Mathematical Methods in the Applied Sciences.2009,DOI:10.1002/mma.1173
[18]Maynard Smith.J.Evolution and the Theory of Games.Cambridge:Cambridge University Press.1982.
[19]Hofbauer J.,Sigmund K.Evolutionary Games and Population Dynamics.Cambridge:Cambridge University Press.1998.
[20]Bishop.D.T.,Cannings.C.A generalized war of attrition.J.Theor.Popul.Biol.1978,70:85-124.
[21]Maynard Smith.J.Evolution and the Theory of Games.Cambridge:Cambridge University Press.1982.
[22]Cressman,R.The Stability Concept of Evolutionary Game Theory:a dynamic approach.Lecture Notes in Biomathematics,Springer,Berlin.1992,94.
[23]Maciej Bukowski,Jacek Miekisz.Evolutionary and asymptotic stability in symmetric multi-player games.Int J Game Theory.2004.33:41-54.
[24]Editorial.ESS theory now.Theoretic Population Biology.2006,69:231-233.
[25]Weibull.J.W.Evolutionary Game Theory,The MIT Press,London,1995,ISBN 0-262-23181-6.
[26]Bukowski.M.,Miekisz.J.Evolutionary and asymptotic stability in symmetric multi-player games,International Journal of Game Theory,2004,33:41-54.
[27]陆征一,罗勇.进化对策与种群动力学[M].成都:四川科学技术出版社.2002.9
[28]Cressman.R.Uninvadability in N-species frequency models for resident-mutant systems with discrete or continuous time.Theoretic Population Biology.2006,69:253-262.
[29]Cressman.R.Evolutionary Dynamics and Extensive Form Games.MIT Press,Cambridge,MA.2003.
[30]Bomze,I.M.,Potscher,B.M.Game Theoretical Foundations of Evolutionary Stability.Lecture Notes in Biomathematics,Springer,Berlin.1989:324.
[2]肖条军.博弈论及其应用[M].上海:上海三联书店,2004:429-483.
[3]Lewontin,R.C.Evolution and the theory of games.Journal of Theoretieal Biology,1960,1:382-403.
[4]Maynard Smith,J.,Price G.R.The logic of animal conflict.Nature,1973,246:15-18.
[5]Apaloo,J.Revisiting matrix games:The concept of neighborhood invader strategies.Theoretic Population Biology.2006,69:235-242.
[6]Apaloo.J.Revisiting Strategic Models of Evolution:The Concept of Neighborhood Invader Strategies.Theoretic Population Biology.1997a,52:71-77.
[7]McKelvey,R.,Apaloo,J.The structure and evolution of competition-organized ecological communities.Rocky Mount.J.Math.1995,25(1):417-436.MathSciNet.
[8]Kisidi E.,Meszena G.Density dependent life history evolution in fluctuating environment,in "Adaptation in a Stochastic Environment"(J.Yoshimura,and C.Clark,Eds.).Lecture Notes in Biomathematics.Springer,Berlin.1993,99:26-62.
[9]Kisidi E.,Meszena G.Life Histories with Lottery Competition in a Stochastic Enviornment:ESSs which do not prevail.Theor.Popul.Biol.1995,47:191-211.
[10]Doanld Ludwig,Simon A.Levin.Evolutionary stability of plant communities and the maintenance of multiple dispersal types.Theor.Popul.Biol.1991,40:285-307.
[11]Lessard.Evolutionary stability:one concept,several meanings,Theoretic Population Biology.1990,37:159-170.
[12]Apaloo.J.Ecological species coevolution,J.Biol.Syst 1997b,5:17-34.
[13]Apaloo.J.Single species evolutionary dynamics,Evolutionary Ecology.2003,5:33-49.
[14]Apaloo.J.Multi-species evolutionary dynamics,Evolutionary Ecology.2005a,18:55-77.
[15]Apaloo.J.Inaccessible continuously stable strategies.Nat.Resour.Modeling.2005b,19:521-535.
[16]Apaloo.J.Evolutionary matrix games and optimization theory Journal of Theoretical Biology.2009,257:84-89.
[17]Ding et al.ESS,NIS and GIS for multi-player matrix game in single population, Mathematical Methods in the Applied Sciences.2009,DOI:10.1002/mma.1173
[18]Maynard Smith.J.Evolution and the Theory of Games.Cambridge:Cambridge University Press.1982.
[19]Hofbauer J.,Sigmund K.Evolutionary Games and Population Dynamics.Cambridge:Cambridge University Press.1998.
[20]Bishop.D.T.,Cannings.C.A generalized war of attrition.J.Theor.Popul.Biol.1978,70:85-124.
[21]Maynard Smith.J.Evolution and the Theory of Games.Cambridge:Cambridge University Press.1982.
[22]Cressman,R.The Stability Concept of Evolutionary Game Theory:a dynamic approach.Lecture Notes in Biomathematics,Springer,Berlin.1992,94.
[23]Maciej Bukowski,Jacek Miekisz.Evolutionary and asymptotic stability in symmetric multi-player games.Int J Game Theory.2004.33:41-54.
[24]Editorial.ESS theory now.Theoretic Population Biology.2006,69:231-233.
[25]Weibull.J.W.Evolutionary Game Theory,The MIT Press,London,1995,ISBN 0-262-23181-6.
[26]Bukowski.M.,Miekisz.J.Evolutionary and asymptotic stability in symmetric multi-player games,International Journal of Game Theory,2004,33:41-54.
[27]陆征一,罗勇.进化对策与种群动力学[M].成都:四川科学技术出版社.2002.9
[28]Cressman.R.Uninvadability in N-species frequency models for resident-mutant systems with discrete or continuous time.Theoretic Population Biology.2006,69:253-262.
[29]Cressman.R.Evolutionary Dynamics and Extensive Form Games.MIT Press,Cambridge,MA.2003.
[30]Bomze,I.M.,Potscher,B.M.Game Theoretical Foundations of Evolutionary Stability.Lecture Notes in Biomathematics,Springer,Berlin.1989:324.