凸合成随机合作对策解的研究
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摘要
对策论中，合作对策得到了广泛的关注和研究，随机合作对策作为对策论的一个分支，自Suijs等人在1995年引入后，主要研究在合作对策形成过程中对局中人的吸引力的大小为基础的合作对策，吸引了不少学者的研究，并取得了很好的成果。
论文主要是在随机合作对策模型下加入凸合成对策理论，建立了凸合成随机合作对策。在随机合作对策条件下，介绍了它的凸性和超可加性，并给出了随机合作对策的核、核仁和Shapley值的概念，讨论了它们的性质。将凸合成对策与随机合作对策结合，给出了凸合成随机合作对策的核仁、核心和稳定集结构，并通过对优超定义的推广，给出了凸合成随机合作对策的弱核心与弱稳定集，对它们的性质和关系进行了研究。
首先介绍对策论的发展史，论文产生的背景以及论文研究的实用价值。
其次介绍了与论文相关的随机合作对策的基础知识。研究随机合作对策的凸性和超可加性，给出了随机合作对策的核、核仁与Shapley值的概念及性质，为下文研究凸合成随机合作对策奠定了理论基础。
最后建立了凸合成随机合作对策的模型，给出了核心、稳定集和强ε核心的概念，并给出了它的几种不同解的结构以及它们的性质与关系；将优超的概念推广，定义了弱优超，并给出了它的弱核心和弱稳定集的概念和它们之间的关系。
In game theory, cooperative game has received a concern and become a researchfocus. Stochastic cooperative game, as a branch of game theory, it has attracted manyscholars'interest, and some significant results are obtained.
The main paper is the stochastic cooperative game model by adding convexcompound game theory, set up convex compound stochastic cooperative game, under thecondition in the stochastic cooperative game, introduce their convexity and superadditive,we present the new concepts of kernel，nucleolus and Shapley value, and discuss theproperties of them. Then we combine the convex compound game and stochasticcooperative game, give their structure of nucleolus, core and stable set, and by expandingthe definition of dominance, we present the new concepts of weak core and weak stableset, study their properies and the relations of the core and stable set.
First, providing histories of classical games theory, background and practical value ofthe paper.
Second, introduce the stochastic cooperative game which related to the thesis,introduce the convexity and superadditive of the stochastic cooperative game, focusing ontheir concepts and characteristics of kernel and nucleolus for the following.
In the last part, we define the convex compound stochastic cooperative game model,define the concept of core, strong ε-core and stable set, study the properties of them, thengive the relations between the core and stable set. Then we define the weak dominance,give the concepts of weak core and weak stable set, and the relations between them.
论文主要是在随机合作对策模型下加入凸合成对策理论，建立了凸合成随机合作对策。在随机合作对策条件下，介绍了它的凸性和超可加性，并给出了随机合作对策的核、核仁和Shapley值的概念，讨论了它们的性质。将凸合成对策与随机合作对策结合，给出了凸合成随机合作对策的核仁、核心和稳定集结构，并通过对优超定义的推广，给出了凸合成随机合作对策的弱核心与弱稳定集，对它们的性质和关系进行了研究。
首先介绍对策论的发展史，论文产生的背景以及论文研究的实用价值。
其次介绍了与论文相关的随机合作对策的基础知识。研究随机合作对策的凸性和超可加性，给出了随机合作对策的核、核仁与Shapley值的概念及性质，为下文研究凸合成随机合作对策奠定了理论基础。
最后建立了凸合成随机合作对策的模型，给出了核心、稳定集和强ε核心的概念，并给出了它的几种不同解的结构以及它们的性质与关系；将优超的概念推广，定义了弱优超，并给出了它的弱核心和弱稳定集的概念和它们之间的关系。
In game theory, cooperative game has received a concern and become a researchfocus. Stochastic cooperative game, as a branch of game theory, it has attracted manyscholars'interest, and some significant results are obtained.
The main paper is the stochastic cooperative game model by adding convexcompound game theory, set up convex compound stochastic cooperative game, under thecondition in the stochastic cooperative game, introduce their convexity and superadditive,we present the new concepts of kernel，nucleolus and Shapley value, and discuss theproperties of them. Then we combine the convex compound game and stochasticcooperative game, give their structure of nucleolus, core and stable set, and by expandingthe definition of dominance, we present the new concepts of weak core and weak stableset, study their properies and the relations of the core and stable set.
First, providing histories of classical games theory, background and practical value ofthe paper.
Second, introduce the stochastic cooperative game which related to the thesis,introduce the convexity and superadditive of the stochastic cooperative game, focusing ontheir concepts and characteristics of kernel and nucleolus for the following.
In the last part, we define the convex compound stochastic cooperative game model,define the concept of core, strong ε-core and stable set, study the properties of them, thengive the relations between the core and stable set. Then we define the weak dominance,give the concepts of weak core and weak stable set, and the relations between them.
引文
[1] Nash J. Equilibrium Points in n-Person Game. Proc[J]. Nat. Acad. Sci. U.S.A.,1950,36:48-49.
[2]王仲英,李小申. NASH博弈问题的模拟算法[J].河南科技大学学报（自然科学版）2009,30(5):70-73.
[3]杨哲,蒲勇建.大博弈中NASH均衡的存在性[J].系统科学与数学.2010,30(12):1606-1612.
[4] Neumann V, Morgenstern. Theory of games and economic behavior[M]. Princeton UniversityPress,1944,7-17.
[5]雷建平,丰淑云.对策论在工程施工中的应用[J].水科学与工程技术.2007,4,43-44.
[6]杨顺文,彭定涛,黄茂胜等. n人非合作对策Nash平衡点稳定性的进一步推广[J].贵州大学学报（自然科学版）.2006,23(1):26-30.
[7]刘微,李靖,封梅.凸合作对策核心的的一扩展性质[J].石家庄铁道大学学报（自然科学版），2010,19(2):41-43.
[8]王振吉.连续凸对策的一个特殊解[J].河北工程大学学报(自然科学版).2007，24(3):104-106.
[9]王振吉,刘微,李靖等.具有可数个局中人的连续凸对策的核心[J].河北师范大学学报,2008,32(2).162-164.
[10]孙红霞,张强.具有联盟结构博弈的联盟核心的公理化[J].北京理工大学学报.2010,30(10):1256-1260.
[11] Shapley L S. A value for N-Person Games[J]. Ann of Math. Studies,1953,28:307-295.
[12] Aumann R J, Maschler M. The Bargaining Set for Cooperative Games[J]. In Dresher M, ShapleyLS and Tucker A W.(ed) Advances in Game Theory, Princeton University Press,1964:443-476.
[13] Schmeidler D. The Nucleolus of a Characteristic Function Game[J]. Slam J. Appl. Math,1969,17:1163-1170.
[14] Charnes A, Granot D. Prior Solutions: Extensions of Convex Nucleolus Solutions toChance-Constrained games[J]. Proceedings of the Computer Science and Statistics. SeventhSymposium at Lowa State University,1973,33:323-332.
[15] Charnes A, Granot D. Coalitional and Chance-Constrained Solutions to N-Person Game[J]. SiamAppl Math,1976,31:358-367.
[16] Granot D. Cooperrative Games in Stochastic Characteristic Function Form[J]. Management,1977,23:621-630.
[17] Suijs J, Borm P. Stochastic Cooperative Games: Superadditivity, Convexity, and CertaintyEquivalents[J]. Games and Economic Behavior,2000,27(2):331-345.
[18] Maslov V P. Phase Transitions in a Stochastic Games[J]. Mathematical Notes,2003,73:598-601.
[19] Jacek Mi'ekisz. Stochastic Stability in Spatial Three-player Games[J]. Physica A,2004,343:175-184.
[20] Horst U. Stationary Equilibria in Discounted Stochastic Games with Weakly Interacting Players[J].Games and Economic Behavior,2005,51:83–108.
[21] Thuijsman F. Perfect Information Stochastic Games and Related Classes[J]. International Journalof Game Theory,1997,26:403-408.
[22] Yeung D W K, Petrosyan L A. Subgame Consistent Cooperative Solutions in StochasticDifferential Games[J]. Journal of Optimization Theory and Application,2004,120:651-666.
[23] Julien G. Note Equilibrium Payoffs in Stochastic Games of Incomplete Information: the GeneralSymmetric Case[J]. Game Theory,2001,30:449-452.
[24] Solan E. Correlated Equilibrium in Stochastic Games[J]. Games and Economic Behavior,2002,38(1):362-399.
[25] Somal R. New Algorithms for Solving Simple Stochastic Games[J]. Electronic Notes inTheoretical Computer Science,2005,19:51-65.
[26] Ronen I, Brafman A, Tennenholtz M. A Near-optimal Polynomial Time Algorithm forLearning inCertain Classes of Stochastic Games[J]. Artificial Intelligence,2000,121:31-47.
[27] Vieille N. Small Perturbations and Stochastic Games[J]. Math,2002,119B:127-144.
[28] Somla R. New Algorithms for Solving Simple Stochastic Games[J]. Electronic Notes inTheoretical Computer Science,2005,119:51-65.
[29] Marita L. Cooperative and Non-cooperative Harvesting in a Stochastic Sequential Fishery[J].Journal of Environmental Economics and Management,2003,45:454-473.
[30] Milman E. Approachable Sets of Vector Payoffs in Stochastic Games[J]. Games and EconomicBehavior,2006,56:135-147.
[31] Mrinal K, Ghosh K, Kumar S. A Nonzero-sum Stochastic Differential Game in the Orthant[J].Math. Anal. Appl,2005,3(5):158-174.
[32]白红信.重复n人随机合作对策的序列相容支付[J].保定学院学报.2011,24(3):31-33.
[33] Somla R. New Algorithms for Solving Simple Stochastic Games[J]. Electronic Notes inTheoretical Computer Science,2005,119:51-65.
[34] Subir K, Chakrabarti. Pure Strategy Markov Equilibrium in Stochastic Games with a Continuumof Players[J]. Journal of Mathematical Economics,2003,39:693-724.
[35] Subir K, Chakrabarti. Markov Equilibria in Discounted Stochastic Games[J]. Journal of EconomicTheory,1999,85:294-327.
[36]刘喜华，王双成.保险风险分配的随机合作博弈模型[J].运筹与管理，2007,16:69-73.
[37]王天虹，宋业新.随机矩阵对策及其在舰艇作战方案中的应用[J].兵工自动化,2010,29(6):22-24.
[38]高作峰,王友,王国成.对策理论与经济管理决策[M].北京:中国林业出版社,2006,60-104.
[39]张盛开,张亚东.对策论与决策方法[M].大连:东北财经大学出版社,2000,152-396.
[40] Sharkey W. Convex Games without Side Payments[J]. Int.j Game Theory,1981,39:101-106.
[41] Vieille N. Equilibrium in2-Person Stochastic Games[J]. A Reduction Math,2002,119A:55–91.
[42] Suijs J, Borm P. Stochastic Cooperative Games: Superadditivity, Convexity, and CertaintyEquivalents[J]. Games and Economic Behavior,2000,27(2):331-345.
[43] Suijs J, Borm P, Tijs S. Cooperative Games with Stochastic Payoffs[J]. European Journal ofOperational Research,1999,133:193-205.
[44]刘广智. Shapley值的扩充表达式[J].大连轻工业学院学报.2002,21(2):149-151.
[45] Rockafellar R T. Convex Analysis[M]. Princeton:Press Princeton NJ,1973.
[46] Broom M. Bounds on the Number of Esss of A Matrix Game[J]. Math,2000,167:163–175.
[47] Suijs J, Borm P.Stochastic Cooperative Gzmes:Superadditiviety, Covexity, and CertaintyEquivalents[J]. Game and Economic Behavior.2000,27(2):331-345.
[48]张丽超,孔亮,刘岩.覆盖对策模型的均衡性[J].河北科技师范学院学报.2009,23(3):44-46.
[49]许敏，王清,白红信等.随机合作对策的核仁[J].科学技术与工程.2008,23:6196-6198.
[50]刘广智，王菲，张盛开.凸对策核仁的非空性及合成凸对策核仁的单调性[J].应用数学,2001,14(增):86-90.
[51]谢政.对策论[M].武汉:国防科技大学出版社,2004,183-241.
[52]刘薇，高作峰,张晓玲等.凸随机合作对策的核心[J].运筹与管理，2005,14(5):59-62.
[53]白红信，高作峰,于永波等.重复n人随机合作对策的强ε核心[J].统计与决策,2008,12:30-32.
[54]高作峰,鄂成国,徐东方等.重复n人随机合作对策的核心[J].高校应用数学学报A辑,2007,22(1):1-7.
[55]张素婷，高作峰，白红信等.重复模糊合作对策的弱核心与弱稳定集[J].山东理工大学学报（自然科学版），2008,4:53-56.
[56]高红伟.结盟对策中弱稳定集的引入与探讨[J].青岛大学学报,1998,11(4):1-4.
[2]王仲英,李小申. NASH博弈问题的模拟算法[J].河南科技大学学报（自然科学版）2009,30(5):70-73.
[3]杨哲,蒲勇建.大博弈中NASH均衡的存在性[J].系统科学与数学.2010,30(12):1606-1612.
[4] Neumann V, Morgenstern. Theory of games and economic behavior[M]. Princeton UniversityPress,1944,7-17.
[5]雷建平,丰淑云.对策论在工程施工中的应用[J].水科学与工程技术.2007,4,43-44.
[6]杨顺文,彭定涛,黄茂胜等. n人非合作对策Nash平衡点稳定性的进一步推广[J].贵州大学学报（自然科学版）.2006,23(1):26-30.
[7]刘微,李靖,封梅.凸合作对策核心的的一扩展性质[J].石家庄铁道大学学报（自然科学版），2010,19(2):41-43.
[8]王振吉.连续凸对策的一个特殊解[J].河北工程大学学报(自然科学版).2007，24(3):104-106.
[9]王振吉,刘微,李靖等.具有可数个局中人的连续凸对策的核心[J].河北师范大学学报,2008,32(2).162-164.
[10]孙红霞,张强.具有联盟结构博弈的联盟核心的公理化[J].北京理工大学学报.2010,30(10):1256-1260.
[11] Shapley L S. A value for N-Person Games[J]. Ann of Math. Studies,1953,28:307-295.
[12] Aumann R J, Maschler M. The Bargaining Set for Cooperative Games[J]. In Dresher M, ShapleyLS and Tucker A W.(ed) Advances in Game Theory, Princeton University Press,1964:443-476.
[13] Schmeidler D. The Nucleolus of a Characteristic Function Game[J]. Slam J. Appl. Math,1969,17:1163-1170.
[14] Charnes A, Granot D. Prior Solutions: Extensions of Convex Nucleolus Solutions toChance-Constrained games[J]. Proceedings of the Computer Science and Statistics. SeventhSymposium at Lowa State University,1973,33:323-332.
[15] Charnes A, Granot D. Coalitional and Chance-Constrained Solutions to N-Person Game[J]. SiamAppl Math,1976,31:358-367.
[16] Granot D. Cooperrative Games in Stochastic Characteristic Function Form[J]. Management,1977,23:621-630.
[17] Suijs J, Borm P. Stochastic Cooperative Games: Superadditivity, Convexity, and CertaintyEquivalents[J]. Games and Economic Behavior,2000,27(2):331-345.
[18] Maslov V P. Phase Transitions in a Stochastic Games[J]. Mathematical Notes,2003,73:598-601.
[19] Jacek Mi'ekisz. Stochastic Stability in Spatial Three-player Games[J]. Physica A,2004,343:175-184.
[20] Horst U. Stationary Equilibria in Discounted Stochastic Games with Weakly Interacting Players[J].Games and Economic Behavior,2005,51:83–108.
[21] Thuijsman F. Perfect Information Stochastic Games and Related Classes[J]. International Journalof Game Theory,1997,26:403-408.
[22] Yeung D W K, Petrosyan L A. Subgame Consistent Cooperative Solutions in StochasticDifferential Games[J]. Journal of Optimization Theory and Application,2004,120:651-666.
[23] Julien G. Note Equilibrium Payoffs in Stochastic Games of Incomplete Information: the GeneralSymmetric Case[J]. Game Theory,2001,30:449-452.
[24] Solan E. Correlated Equilibrium in Stochastic Games[J]. Games and Economic Behavior,2002,38(1):362-399.
[25] Somal R. New Algorithms for Solving Simple Stochastic Games[J]. Electronic Notes inTheoretical Computer Science,2005,19:51-65.
[26] Ronen I, Brafman A, Tennenholtz M. A Near-optimal Polynomial Time Algorithm forLearning inCertain Classes of Stochastic Games[J]. Artificial Intelligence,2000,121:31-47.
[27] Vieille N. Small Perturbations and Stochastic Games[J]. Math,2002,119B:127-144.
[28] Somla R. New Algorithms for Solving Simple Stochastic Games[J]. Electronic Notes inTheoretical Computer Science,2005,119:51-65.
[29] Marita L. Cooperative and Non-cooperative Harvesting in a Stochastic Sequential Fishery[J].Journal of Environmental Economics and Management,2003,45:454-473.
[30] Milman E. Approachable Sets of Vector Payoffs in Stochastic Games[J]. Games and EconomicBehavior,2006,56:135-147.
[31] Mrinal K, Ghosh K, Kumar S. A Nonzero-sum Stochastic Differential Game in the Orthant[J].Math. Anal. Appl,2005,3(5):158-174.
[32]白红信.重复n人随机合作对策的序列相容支付[J].保定学院学报.2011,24(3):31-33.
[33] Somla R. New Algorithms for Solving Simple Stochastic Games[J]. Electronic Notes inTheoretical Computer Science,2005,119:51-65.
[34] Subir K, Chakrabarti. Pure Strategy Markov Equilibrium in Stochastic Games with a Continuumof Players[J]. Journal of Mathematical Economics,2003,39:693-724.
[35] Subir K, Chakrabarti. Markov Equilibria in Discounted Stochastic Games[J]. Journal of EconomicTheory,1999,85:294-327.
[36]刘喜华，王双成.保险风险分配的随机合作博弈模型[J].运筹与管理，2007,16:69-73.
[37]王天虹，宋业新.随机矩阵对策及其在舰艇作战方案中的应用[J].兵工自动化,2010,29(6):22-24.
[38]高作峰,王友,王国成.对策理论与经济管理决策[M].北京:中国林业出版社,2006,60-104.
[39]张盛开,张亚东.对策论与决策方法[M].大连:东北财经大学出版社,2000,152-396.
[40] Sharkey W. Convex Games without Side Payments[J]. Int.j Game Theory,1981,39:101-106.
[41] Vieille N. Equilibrium in2-Person Stochastic Games[J]. A Reduction Math,2002,119A:55–91.
[42] Suijs J, Borm P. Stochastic Cooperative Games: Superadditivity, Convexity, and CertaintyEquivalents[J]. Games and Economic Behavior,2000,27(2):331-345.
[43] Suijs J, Borm P, Tijs S. Cooperative Games with Stochastic Payoffs[J]. European Journal ofOperational Research,1999,133:193-205.
[44]刘广智. Shapley值的扩充表达式[J].大连轻工业学院学报.2002,21(2):149-151.
[45] Rockafellar R T. Convex Analysis[M]. Princeton:Press Princeton NJ,1973.
[46] Broom M. Bounds on the Number of Esss of A Matrix Game[J]. Math,2000,167:163–175.
[47] Suijs J, Borm P.Stochastic Cooperative Gzmes:Superadditiviety, Covexity, and CertaintyEquivalents[J]. Game and Economic Behavior.2000,27(2):331-345.
[48]张丽超,孔亮,刘岩.覆盖对策模型的均衡性[J].河北科技师范学院学报.2009,23(3):44-46.
[49]许敏，王清,白红信等.随机合作对策的核仁[J].科学技术与工程.2008,23:6196-6198.
[50]刘广智，王菲，张盛开.凸对策核仁的非空性及合成凸对策核仁的单调性[J].应用数学,2001,14(增):86-90.
[51]谢政.对策论[M].武汉:国防科技大学出版社,2004,183-241.
[52]刘薇，高作峰,张晓玲等.凸随机合作对策的核心[J].运筹与管理，2005,14(5):59-62.
[53]白红信，高作峰,于永波等.重复n人随机合作对策的强ε核心[J].统计与决策,2008,12:30-32.
[54]高作峰,鄂成国,徐东方等.重复n人随机合作对策的核心[J].高校应用数学学报A辑,2007,22(1):1-7.
[55]张素婷，高作峰，白红信等.重复模糊合作对策的弱核心与弱稳定集[J].山东理工大学学报（自然科学版），2008,4:53-56.
[56]高红伟.结盟对策中弱稳定集的引入与探讨[J].青岛大学学报,1998,11(4):1-4.