广义不确定性下广义博弈中NS均衡的存在性

中国管理科学 ›› 2013, Vol. 21 ›› Issue (5): 165-171.

广义不确定性下广义博弈中NS均衡的存在性

杨哲¹, 蒲勇健²

1. 上海财经大学经济学院, 上海 200433;
2. 重庆大学经济与工商管理学院, 重庆 400044

收稿日期:2011-07-04 修回日期:2012-08-28 出版日期:2013-10-30 发布日期:2013-10-15

Existence of NS Equilibrium Points in Generalized Games Under Generalized Uncertainty

YANG Zhe¹, PU Yong-jian²

1. School of Economics, Shanghai University of Finance and Economics, Shanghai 200433, China;
2. College of Economics and Business Administration, Chongqing University, Chongqing 400044, China

Received:2011-07-04 Revised:2012-08-28 Online:2013-10-30 Published:2013-10-15

摘要/Abstract

摘要： 把不确定性引入广义博弈的研究之中,在此博弈中,局中人策略之间存在相互影响,局中人的策略可以改变不确定参数的变化范围,而且局中人的支付函数和策略可行反应映射都受到不确定参数的作用,此类型博弈定义为广义不确定性下的广义博弈问题。进一步定义出此类型博弈中的NS均衡,并且凭借Fan-Glicksberg不动点定理,证明此均衡点的存在性。最后给出算例验证其可行性。

关键词: 广义博弈, 广义不确定性, NS均衡, 存在性

Abstract: Uncertainty is introduced in generalized games is this paper. In the game, people's strategies are chteracted on each other, and can change the uncertain parameter variation range. What's more, payoff functions and feasible mappings of strategy are affected by uncertain parameters. This kindof game is defined by generalized games under generalized uncertainty. Further, NS equilibrium points in generalized games under generalized uncertainty are defined, and the existence theorem of NS equilibrium points is proved by mean of Fan-Glicksberg fixed point theorem. Finally, a numeric example is given to demonstrate the feasibility.

Key words: generalized games, generalized uncertainty, NS equilibrium, existence

中图分类号:

杨哲, 蒲勇健. 广义不确定性下广义博弈中NS均衡的存在性[J]. 中国管理科学, 2013, 21(5): 165-171.

YANG Zhe, PU Yong-jian. Existence of NS Equilibrium Points in Generalized Games Under Generalized Uncertainty[J]. Chinese Journal of Management Science, 2013, 21(5): 165-171.

参考文献

[1] Neumann J V, Morgenstern O. Theory of game and economic behavior[M]. Princeton:Princeton University Press, 1994.
[2] Nash J. Equilibrium point in N-person games[J]. Proceedings of the National Academy of Sciences, 1950(36): 48-49.
[3] Nash J. Noncooperative games[J]. The Annals of mathematics, 1951(54): 286-295.
[4] Aubin J P. Mathematical methods of games and economic theory [M]. Amsterdam:North-Holland, 1982.
[5] Yu Jian. Essential equilibria of n-person noncooperative Games[J]. Journal of Mathematical Economics, 1999 (31): 361-372.
[6] Yu Jian, Yuan Xianzhi.. The study of Pareto equilibria for multiobjective games by fixed point and Ky Fan minimax inequality methods[J]. Computers Mathematics with Applications, 1998, 35(9): 17-24.
[7] Harker P T. Generalized Nash games and quasi-variational inequalities [J]. European Journal of Operational Research, 1991, 54(1): 81-94.
[8] Facchinei F, Fischer A, Piccial-li V. On generalized Nash games and variational inequalities [J]. Operations Research Letters, 2007, 35(2): 159-164.
[9] Kim W K, Yuan Xianzhi. Existence of equilibria for generalized games and generalized social systems with coordination[J]. Nonlinear Functional Analysis and Applications,1998,3:77-102.
[10] Kim W K, Lee K H. Generalized fuzzy games and fuzzy equilibria[J]. Fuzzy Sets and Systems, 2001, 122(2): 293-301.
[11] Yuan Xianzhi. The existence of equilibria for noncompact generali-zed games[J]. Applied Mathematics Letters, 2000, 13(1): 57-63.
[12] Hou Jicheng. Existence of equilibria for generalized games without parcompactness[J]. Nonlinear Analysis, 2004, 56(4): 625-632.
[13] Llinares J V. Existence of equilibrium in generalized games with abstract convexity structure [J]. Journal of Optimization Theory and Applications, 2000, 105(1): 149-160.
[14] Hou Jicheng. An existence theorem of equilibrium for generalized games in H-spaces [J]. Applied Mathematics Letters, 2003, 16(1): 97-103.
[15] Cubiotti P. Existence of nash equilibria for generalized games without upper semicontinuity[J]. International Journal of Game Theory, 1997, 26(2): 267-273.
[16] Cubiotti P,Yao J C. Nash equilibria of generalized games in normed spaces without upper semicontinuity[J]. Journal of Global Optimization, 2010, 46(4): 509-519.
[17] Ding Xieping, Yuan Xianzhi. The study of existence of equilibria for generalized games without lower semicontinuity in locally topologi/cal vector spaces[J]. Journal of Mathematical Analysis and Applica- tions, 1998, 227(2): 420-438.
[18] Facchinei F, Fischer A , Piccialli V. Generalized Nash equilibrium problems and Newton methods[J]. Mathematical Programm ing, 2009, 117(1-2): 163-194.
[19] Lin Zhi, Yu Jian. On well-posedness of the multiobjective generalized game[J]. Applied Mathematics-A Journal of Chinese Universities, 2004, 19(3): 327-334.
[20] Lin Zhi. Essential components of the set of weakly pareto-Nash equilibrium points for multiobjective generalized games in two different topological spaces[J]. Journal of Optimization Theory and Applications, 2005, 124(2): 387-405.
[21] Song Qiqing, Wang Laisheng. On the stability of the solution for multiobjective generalized games with the payoffs perturbed[J]. Nonlinear Analysis, 2010, 73(8): 2680-2685.
[22] Zhukovskii V I. Linear quadratic differential games[M]. Naoukova Doumka: Kiev, 1994.
[23] Larbani M, Lebbah H. A concept of equilibrium for a game under uncertainty [J]. European J of Operational Research, 1999, 117(1): 145-156.
[24] 张会娟, 张强.不确定性下非合作博弈强Nash均衡的存在性[J]. 控制与决策, 2010, 25(8): 1251-1254.
[25] 张会娟, 张强. 不确定性下非合作博弈简单Berge均衡的存在性[J].系统工程理论与实践, 2010, 30(9): 1630-1635.
[26] 杨哲,蒲勇健. 不确定性下多主从博弈中均衡的存在性[J].控制与决策,2012, 27(5):736-740.
[27] 俞建. 博弈论与非线性分析[M]. 北京:科学出版社,2008.
[28] Aubin J P, Ekeland I. Applied nonlinear analysis[M]. New York:John Wiley and Sons Inc, 1984.