在强扰动下广义最大元的稳定性
Stability of Generalized Maximal Elements under Strong Perturbations
DOI: 10.12677/PM.2023.1312377, PDF,    国家自然科学基金支持
作者: 周应辉, 杨彦龙*：贵州大学数学与统计学院，贵州 贵阳
关键词: 稳定性；Hausdorff半度量；广义最大元；Stability； Hausdorff Semi-Metric； Generalized Maximal Element
摘要: 本文通过引入由半度量定义的一种更强的扰动，即Hausdorff半度量，代入该扰动后证明广义最大元中强本质集的存在性，进一步得出广义最大元点集稳定性的一些结果。作为应用，该方法也证明了广义最大元策略的Nash均衡的稳定性，对支付函数扰动具有较强的鲁棒性。
Abstract: In this paper, we further derive some results on the stability of generalized maximal elements by introducing a stronger perturbation defined by the semi-metric measure, i.e. the Hausdorff semi-metric measure, and proving the existence of a strong essential set in the generalized maximal element by substituting this perturbation. As an application, the method also proves the stability of the Nash equilibrium of the generalized maximal element strategy, which is robust to payment function perturbations.
文章引用：周应辉, 杨彦龙. 在强扰动下广义最大元的稳定性[J]. 理论数学, 2023, 13(12): 3638-3645. https://doi.org/10.12677/PM.2023.1312377

参考文献

[1] Knaster, B., Kuratowski, K. and Mazurkiewicz, S. (1929) Ein beweis des fixpunksates furn-dimensional simplexe. Fundamenta Mathematicae, 14, 132-137. [Google Scholar] [CrossRef] 
[2] Fang, M. and Ding, X.P. (2003) Generalized L-R-KKM Theorem in and Applications in L-Convex Space. Journal of Sichuan Normal University, 26, 461-463.
[3] Ansari, Q.H., Chan, W.K. and Yang, X.Q. (2004) The System of Vector Quasi-Equilibrium Problems with Applications. Journal of Global Optimization, 29, 45-57. [Google Scholar] [CrossRef] 
[4] Ansari, Q.H., Oettli, W. and Schläger, D. (1997) A Generalization of Vector Equilibria. Mathematical Methods of Operations Research, 46, 147-152. [Google Scholar] [CrossRef] 
[5] Ansari, Q.H., Schaible, S. and Yao, J.C. (2000) System of Vector Equi-librium Problems and Their Applications. Journal of Optimization Theory and Applications, 107, 547-557. [Google Scholar] [CrossRef] 
[6] Blum, E. and Oettli, W. (1994) From Optimiztion and Variational Inequalities to Equilibirum Problems. Mathematics Student-India, 63, 123-145.
[7] Chen, G.Y. and Craven, B.D. (1989) Approximate Dual and Approximate Vector Variational Inequality for Multi-Objective Optimization. Journal of the Australian Mathematical Society, 47, 418-423. [Google Scholar] [CrossRef] 
[8] Chadli, O., Chiang, Y. and Yao, T.C. (2002) Equilibrium Problems with Lower and Upper Bounds. Applied Mathematics Letters, 15, 327-331. [Google Scholar] [CrossRef] 
[9] Gale, D. and Mas-Colell, A. (1975) An Equilibrium Exist-ence Theorem for a General Model without Ordered Preferences. Journal of Mathematical Economics, 2, 9-15. [Google Scholar] [CrossRef] 
[10] Yu, J. (1999) Essential Equilibria of N-Person Non-Cooperative Game. Math Economics, 31, 361-372. [Google Scholar] [CrossRef] 
[11] Kohlberg, E. and Mertens, J.F. (1986) On the Strategic Sta-bility of Equilibria. Econometrica, 54, 1003-1037. [Google Scholar] [CrossRef] 
[12] Hillas, J. (1990) On the Definition of the Strategic Stability of Equilibria. Econometrica, 58, 1365-1390. [Google Scholar] [CrossRef] 
[13] Tan, K.K., Yu, J. and Yuan, X.Z. (1995) The Stability of Ky Fan’s Points. Proceedings of the American Mathematical Society, 123, 1511-1519. [Google Scholar] [CrossRef] 
[14] Jiang, J.H. (1962) Essential Fixed Points of the Multi-valued Mappings. Scientia Sinica, 11, 293-298.
[15] Tan, K.K., Yu, J. and Yuan, X.Z. (1995) The Stability of Coinci-dent Points for Multivalued Mappings. Nonlinear Analysis: Theory, Methods & Applications, 25, 163-168. [Google Scholar] [CrossRef] 
[16] Yu, J. and Yang, H. (2004) The Essential Components of the Set of Equilibrium Points for Set-Valued Maps. Journal of Mathematical Analysis and Applications, 300, 334-342. [Google Scholar] [CrossRef] 
[17] Yu, J. and Luo, Q. (1999) On Essential Components of the Solu-tion Set of Generalized Games. Journal of Mathematical Analysis and Applications, 230, 303-310. [Google Scholar] [CrossRef] 
[18] Yu, J. and Xiang, S.W. (1999) On Essential Component of the Set of Nash Equilibrium Points. Nonlinear Analysis: Theory, Methods & Applications, 38, 259-264. [Google Scholar] [CrossRef] 
[19] Yu, J. and Xiang, S.W. (2003) The Stability of the Set of KKM Points. Nonlinear Analysis: Theory, Methods & Applications, 54, 839-844. [Google Scholar] [CrossRef] 
[20] Xiang, S.W., He, J.H., et al. (2017) Some Further Results on the Stability of Ky Fan’s Points. Journal of Inequalities and Applications, 2017, Article No. 289. [Google Scholar] [CrossRef] [PubMed]
[21] 俞建. 博弈论与非线性分析[M]. 北京: 科学出版社, 2008.
[22] 杨光惠, 向淑文. 广义最大元的通有稳定性[J]. 广西师范大学学报(自然科学版), 2010, 28(2): 50-52.
[23] 陈治友, 夏顺友. 抽象凸空间中广义最大元的稳定性[J]. 西南大学学报(自然科学版), 2012, 34(8): 116-118.

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