双矩阵博弈完美平衡的稳定性与灵敏度分析
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摘要
本文研究了双矩阵博弈完美平衡的稳定性，运用线性规划的方法对模型进行灵敏度分析，然后通过双矩阵博弈的完美平衡与线性规划的联系对双矩阵博弈的完美平衡做稳定性的分析。
第一章为背景知识，介绍了博弈论的出现，发展，以及现况，描述了博弈论的类型以及理论结构。叙述了非合作博弈论的一些成果，并通过经典例子来认识非合作博弈。
第二章是预备知识，主要介绍本文用到的一些相关知识，说扰动博弈，完美平衡的概念以及相关的引理。
第三章是应用线性规划的方法如何来验证双矩阵博弈的完美平衡点。
第四章用线性规划的方法对双矩阵博弈的完美平衡做了稳定性的研究。本章主要是对线性规划的模型进行灵敏度分析，然后通过线性规划的模型和双矩阵博弈的联系分析双矩阵博弈完美平衡的稳定性。
In this paper,we research the stability of perfect equilibrium of bimatrixgame by applying linear programming. Specifically, we research the stability andsensitivity analysis of perfect equilibrium of bimatrix game through the relation ofperfect equilibrium and linear programming.
The first chapter is the background knowledge. We introduce the game theoryappearance, development, present situation and it's type as well as the theory structure.In addition, we introduce some achievements and classical examples of non-cooperation game, through which we can obtain more understanding to the non-cooperation game.
The second chapter is preliminary knowledge, which mainly contains somerelated knowledge, such as perturbation game, perfect equilibrium concept and somelemma.
In the third chapter, we confirm the perfect equilibrium of bimatrix game byapplying linear programming.
In the fourth chapter, we analyse the stability of perfect equilibrium of bimatrixgame by applying linear programming. This chapter mainly carries on the sensitivityanalysis to the linear programming model. Then we analyze the stability of perfectequilibrium of bimatrix game through the relation of the linear programming modeland bimatrix game.
第一章为背景知识，介绍了博弈论的出现，发展，以及现况，描述了博弈论的类型以及理论结构。叙述了非合作博弈论的一些成果，并通过经典例子来认识非合作博弈。
第二章是预备知识，主要介绍本文用到的一些相关知识，说扰动博弈，完美平衡的概念以及相关的引理。
第三章是应用线性规划的方法如何来验证双矩阵博弈的完美平衡点。
第四章用线性规划的方法对双矩阵博弈的完美平衡做了稳定性的研究。本章主要是对线性规划的模型进行灵敏度分析，然后通过线性规划的模型和双矩阵博弈的联系分析双矩阵博弈完美平衡的稳定性。
In this paper,we research the stability of perfect equilibrium of bimatrixgame by applying linear programming. Specifically, we research the stability andsensitivity analysis of perfect equilibrium of bimatrix game through the relation ofperfect equilibrium and linear programming.
The first chapter is the background knowledge. We introduce the game theoryappearance, development, present situation and it's type as well as the theory structure.In addition, we introduce some achievements and classical examples of non-cooperation game, through which we can obtain more understanding to the non-cooperation game.
The second chapter is preliminary knowledge, which mainly contains somerelated knowledge, such as perturbation game, perfect equilibrium concept and somelemma.
In the third chapter, we confirm the perfect equilibrium of bimatrix game byapplying linear programming.
In the fourth chapter, we analyse the stability of perfect equilibrium of bimatrixgame by applying linear programming. This chapter mainly carries on the sensitivityanalysis to the linear programming model. Then we analyze the stability of perfectequilibrium of bimatrix game through the relation of the linear programming modeland bimatrix game.
引文
[1] 《运筹学》教材编写组。运筹学[M]。北京：清华大学出版社，2005。
[2] 李小军。线性规划中的灵敏度分析[J]。广州师院学报。2000，21(3)：15-20。
[3] 张维迎。博弈论与信息经济学[M]。上海：上海人民出版社，1996。
[4] 庞留勇，黄伟亮。线性规划多变量系数变化的灵敏度分析[J]。天中学刊。2005，20(5)：8-9。
[5] 胡运权。运筹学教程[M]。北京：清华大学出版社，1998。
[6] 俞建。对策论与非线性分析[J]。贵州科学。2003，21(1-2)：1-4。
[7] 谢政。对策论[M]。长沙：国防科技大学。2004。
[8] 解心江。线性规划模型减少约束时的灵敏度分析[J]。农业系统科学与综合研究，2002，18(3)：178-179。
[9] Aumann, R. Subjectivity and Correlation in Randomized Strategies[J]. Journal of Mathematical Economics1974, 1: 67-96.
[10] B¨orgers, T. Perfect Equilibrium Histories of Finite and Infinite Horizon Games[J]. Journal of Economic Theory. 1989, 47: 218-227.
[11] Debreu, D., A Social Equilibrium Existence Theorem[J]. Proceedings of the National Academy of Science 1952, 38: 886-893.
[12] Fan, K., Fixed point and minimax theorems in locally convex topological linear spaces[J]. Proceedings of the national Academy of Sciences 1952, 38: 121-126.
[13] Fan, K. Minimax theorems[J]. Proceedings of the National Academy of Sciences 1953, 39: 42-47.
[14] Fudunberg, D.,and J. Tirole. A theory of exit in duopoly[J]. Econometrica 1986, 54: 943-960.
[15] Fudenberg, Drew and Jean Tirole. Game Theory[M]. MIT Press. 1991.
[16] Glicksberg, I.L, A Further Generalization of the Kakutani Fixed Point Theorem with Application to Nash Equilibrium Points[J]. Proceedings of the National Academy of Science 1952, 38:170-174.
[17] Harsanyi, J., Games with Incomplete Information Played by Bayesian Players Parts Ⅰ Ⅱ and Ⅲ [J]. Management Science 1967, 14: 159-82, 320-34, 486-502.
[18]Harsanyi, J. Games with Randomly Distributed Payoffs :A New Rationale for Mixed Strategy Equilibrium Points[J] International Journal of Game Theory. 1973, 2:1-23.
[19]Harsanyi, J. Oddness of the number of equilibrium points: A new proof[J]. International Journal of Game Theory .1973, 2:235-250.
[20]Kreps. D., and R. Wilson..Sequential equilibrium[J]. Econometrica 1982, 50:863-894.
[21]Kreps. D., and R. Wilson. Reputation and imperfect information[J]. Journal of Economic Theory .1982, 27:253-279.
[22]Maynard Smith, J. Evolution and the Theory of Games[M]. Cambridge: Cambridge University Press. 1982.
[23]Myerson, R.B. Refinements of the Nash equilibrium concept[J]. International Journal of Game Theory .1978,15:73-80.
[24]Nash,J. The bargaining problem[J]. Econometrica. 1950,18:155-162.
[25]Nash,J. Equilibrium Points in n-Person Games[J]. Proceedings of the National Academy of Sciences. 1950, 36:48-49.
[26]Nash,J. Noncooprative Games[J]. Annals of Mathematics. 1951, 54:286-295.
[27]Rosen, J.B, Existence and Uniqueness of Equilibrium Points for Concave n-person Games [J]. Econometria. 1965, 33:520-534.
[28]Selten,R. Spieltheoretische Behandlung eines Oligopolmodells mit Nachfagetragheit[J] . Zeitschrift fur die gesamte Staatswissenschaft. 1965, 12:301-324.
[29]Selten,R. Reexamination of the perfectness concept for equilibrium points in extensive games[J]. International Journal of Game Theory. 1975, 4:25-55.
[30]Shapley, L. Some Topics in Two-person Games[J], in Contributions to the Theory of Games (Princeton Annals of Mathematical Studies,on.52) 1964.
[31 ]Tucker, A.W and. Luce, R.D. Contributions to the Theory of Games[M], Princeton University Press, 1959.
[32]Van Damme, .E.,Stability and Prefection of Nash Equilibria[M], Springer —Verlag, Berlin, 1987.
[33] Van Damme, .E., Stable equilibria and forward induction[J].Journal of Economic Theory. 1989, 48: 476-496.
[34] Von Neumann, J. Zur Theorie der Gesellschaftsspiele[J]. Math. Annalen 1928, 100: 295-320.
[35] Von Neumann, J. and Morgenstern, O. Theory of Games and Economic Behaviour[M]. New York: Wiley, 1944.
[2] 李小军。线性规划中的灵敏度分析[J]。广州师院学报。2000，21(3)：15-20。
[3] 张维迎。博弈论与信息经济学[M]。上海：上海人民出版社，1996。
[4] 庞留勇，黄伟亮。线性规划多变量系数变化的灵敏度分析[J]。天中学刊。2005，20(5)：8-9。
[5] 胡运权。运筹学教程[M]。北京：清华大学出版社，1998。
[6] 俞建。对策论与非线性分析[J]。贵州科学。2003，21(1-2)：1-4。
[7] 谢政。对策论[M]。长沙：国防科技大学。2004。
[8] 解心江。线性规划模型减少约束时的灵敏度分析[J]。农业系统科学与综合研究，2002，18(3)：178-179。
[9] Aumann, R. Subjectivity and Correlation in Randomized Strategies[J]. Journal of Mathematical Economics1974, 1: 67-96.
[10] B¨orgers, T. Perfect Equilibrium Histories of Finite and Infinite Horizon Games[J]. Journal of Economic Theory. 1989, 47: 218-227.
[11] Debreu, D., A Social Equilibrium Existence Theorem[J]. Proceedings of the National Academy of Science 1952, 38: 886-893.
[12] Fan, K., Fixed point and minimax theorems in locally convex topological linear spaces[J]. Proceedings of the national Academy of Sciences 1952, 38: 121-126.
[13] Fan, K. Minimax theorems[J]. Proceedings of the National Academy of Sciences 1953, 39: 42-47.
[14] Fudunberg, D.,and J. Tirole. A theory of exit in duopoly[J]. Econometrica 1986, 54: 943-960.
[15] Fudenberg, Drew and Jean Tirole. Game Theory[M]. MIT Press. 1991.
[16] Glicksberg, I.L, A Further Generalization of the Kakutani Fixed Point Theorem with Application to Nash Equilibrium Points[J]. Proceedings of the National Academy of Science 1952, 38:170-174.
[17] Harsanyi, J., Games with Incomplete Information Played by Bayesian Players Parts Ⅰ Ⅱ and Ⅲ [J]. Management Science 1967, 14: 159-82, 320-34, 486-502.
[18]Harsanyi, J. Games with Randomly Distributed Payoffs :A New Rationale for Mixed Strategy Equilibrium Points[J] International Journal of Game Theory. 1973, 2:1-23.
[19]Harsanyi, J. Oddness of the number of equilibrium points: A new proof[J]. International Journal of Game Theory .1973, 2:235-250.
[20]Kreps. D., and R. Wilson..Sequential equilibrium[J]. Econometrica 1982, 50:863-894.
[21]Kreps. D., and R. Wilson. Reputation and imperfect information[J]. Journal of Economic Theory .1982, 27:253-279.
[22]Maynard Smith, J. Evolution and the Theory of Games[M]. Cambridge: Cambridge University Press. 1982.
[23]Myerson, R.B. Refinements of the Nash equilibrium concept[J]. International Journal of Game Theory .1978,15:73-80.
[24]Nash,J. The bargaining problem[J]. Econometrica. 1950,18:155-162.
[25]Nash,J. Equilibrium Points in n-Person Games[J]. Proceedings of the National Academy of Sciences. 1950, 36:48-49.
[26]Nash,J. Noncooprative Games[J]. Annals of Mathematics. 1951, 54:286-295.
[27]Rosen, J.B, Existence and Uniqueness of Equilibrium Points for Concave n-person Games [J]. Econometria. 1965, 33:520-534.
[28]Selten,R. Spieltheoretische Behandlung eines Oligopolmodells mit Nachfagetragheit[J] . Zeitschrift fur die gesamte Staatswissenschaft. 1965, 12:301-324.
[29]Selten,R. Reexamination of the perfectness concept for equilibrium points in extensive games[J]. International Journal of Game Theory. 1975, 4:25-55.
[30]Shapley, L. Some Topics in Two-person Games[J], in Contributions to the Theory of Games (Princeton Annals of Mathematical Studies,on.52) 1964.
[31 ]Tucker, A.W and. Luce, R.D. Contributions to the Theory of Games[M], Princeton University Press, 1959.
[32]Van Damme, .E.,Stability and Prefection of Nash Equilibria[M], Springer —Verlag, Berlin, 1987.
[33] Van Damme, .E., Stable equilibria and forward induction[J].Journal of Economic Theory. 1989, 48: 476-496.
[34] Von Neumann, J. Zur Theorie der Gesellschaftsspiele[J]. Math. Annalen 1928, 100: 295-320.
[35] Von Neumann, J. and Morgenstern, O. Theory of Games and Economic Behaviour[M]. New York: Wiley, 1944.
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