有限理性下广义多目标多主多从博弈受控系统解的稳定性

摘要/Abstract

摘要：

该文在有限理性条件下, 构建了一类新的广义多目标多主多从博弈受控系统, 并给出了该博弈受控系统解的存在性条件. 进一步, 利用非线性标量化方法构造了适当的理性函数, 并证明了该模型是结构稳定的, 同时对 $\varepsilon$-平衡也是鲁棒的, 即在 Baire 分类意义下, 大多数广义多目标多主多从博弈受控系统问题都是稳定的, 表明在一定条件下该模型的精确解可用有限理性条件下的近似解逼近.

关键词: 广义多目标多主多从博弈受控系统, 有限理性, 结构稳定性, 鲁棒性

Abstract:

In this paper, the stability of solutions for controlled systems of generalized multiobjective multi-leader-follower games is studied under the framework of bounded rationality. We construct an appropriate rational function by the nonlinear scalarization method, and prove that the model is structurally stable and robust to $\varepsilon$-equilibrium. This means that most of the problems for controlled systems of generalized multiobjective multi-leader-follower games are stable on the meaning of Baire category, and also shows that, under certain conditions, the problem model can be approximated to complete rationality by bounded rationality.

Key words: Controlled systems of generalized multiobjective multi-leader-follower games, Bounded rationality, Structural stability, Robustness

中图分类号:

O225

张咏雪, 贾文生. 有限理性下广义多目标多主多从博弈受控系统解的稳定性[J]. 数学物理学报, 2024, 44(6): 1689-1702.

Zhang Yongxue, Jia Wensheng. Stability of Solutions for Controlled Systems of Generalized Multiobjective Multi-Leader-Follower Games Under Bounded Rationality[J]. Acta mathematica scientia,Series A, 2024, 44(6): 1689-1702.

参考文献 39

[1]	Stackelberg H V. Marktform und Gleichgewicht. Berlin:Springer, 1934
[2]	Yu J, Wang H L. An existence theorem for equilibrium points for multi-leader-follower games. Nonlinear Analysis: Theory, Methods and Applications, 2008, 69(5/6): 1775-1777
[3]	Leyffer S, Munson T. Solving multi-leader-common-follower games. Optimisation Methods and Software, 2010, 25(4): 601-623
[4]	Ding X P. Equilibrium existence theorems for multi-leader-follower generalized multiobjective games in $ FC$-spaces. Journal of Global Optimization, 2012, 53: 381-390
[5]	Jia W S, Xiang S W, He J H, Yang Y L. Existence and stability of weakly Pareto-Nash equilibrium for generalized multiobjective multi-leader-follower games. Journal of Global Optimization, 2015, 61: 397-405
[6]	Zeng J, Zhang W Y. Existence and Levitin-Polyak well-posedness for a class of generalized multiobjective multi-leader-follower games. Pacific Journal of Optimization, 2016, 12(4): 717-726
[7]	Julien L A. On noncooperative oligopoly equilibrium in the multiple leader-follower game. European Journal of Operational Research, 2017, 256(2): 650-662
[8]	Yang Z, Gong Q B. Existence of weakly cooperative equilibria for infinite-leader-infinite-follower games. Journal of the Operations Research Society of China, 2019, 7: 643-654
[9]	Aussel D, Svensson A. A short state of the art on multi-leader-follower games. Bilevel Optimization: Advances and Next Challenges, 2020: 53-76
[10]	Herty M, Steffensen S, Thünen A. Solving quadratic multi-leader-follower games by smoothing the follower's best response. Optimization Methods and Software, 2022, 37(2): 772-799
[11]	帅轩越, 马志程, 王秀丽, 等. 基于主从博奔理论的共享储能与综合能源微网优化运行研究. 电网技术, 2023, 47(2): 679-687
	Shuai X Y, Ma Z C, Wang X L, et al. Optimal operation of shared energy storage and integrated energy microgrid based on leader-follower game theory. Power System Technology, 2023, 47(2): 679-687
[12]	Liu Z L, Wang G L, Yang G H. Existence of equilibrium solution for leader-follower games with fuzzy goals and parameters. Fuzzy Sets and Systems, 2023, 473: 108731
[13]	Aussel D, Lepaul S, von Niederhäusern L. A multi-leader-follower game for energy demand-side management. Optimization, 2023, 72(2): 351-381
[14]	南江霞, 李明月, 吴小勇, 张茂军. 基于区块链技术下两条竞争供应链的策略选择. 系统科学与数学, 2023, 43(2): 452-477 doi: 10.12341/jssms22436
	Nan J X, Li M Y, Wu X Y, Zhang M J. Strategy selection of two competing supply chains based on blockchain technology. Journal of System Science and Mathematical Science Chinese Series, 2023, 43(2): 452-477 doi: 10.12341/jssms22436
[15]	Chen T, Chen K, Zhang Y. Existence and continuity theorems of $\alpha$-core of multi-leader-follower games with set payoffs. Journal of Computational and Applied Mathematics, 2024, 439: 115610
[16]	Park J, Jeong J, Kang Y. Optimal control of parabolic variational inequalities with delays and state constraint. Nonlinear Analysis: Theory, Methods and Applications, 2009, 71(12): 329-339
[17]	Liu Z, Migórski S, Zeng B. Existence results and optimal control for a class of quasi mixed equilibrium problems involving the $(f, g, h)$-quasimonotonicity. Applied Mathematics and Optimization, 2019, 79: 257-277
[18]	Chadli O, Ansari Q H, Al-Homidan S, Alshahrani M. Optimal control of problems governed by mixed quasi-equilibrium problems under monotonicity-type conditions with applications. Applied Mathematics and Optimization, 2021, 83(3): 2185-2209
[19]	Hung N V, Keller A A. Painlevé-Kuratowski convergence of the solution sets for controlled systems of fuzzy vector quasi-optimization problems with application to controlling traffic networks under uncertainty. Computational and Applied Mathematics, 2021, 40(1): Article 28
[20]	Hung N V. Generalized Levitin-Polyak well-posedness for controlled systems of FMQHI-fuzzy mixed quasi-hemivariational inequalities of Minty type. Journal of Computational and Applied Mathematics, 2021, 386: 113263
[21]	Hung N V. LP well-posed controlled systems for bounded quasi-equilibrium problems and their application to traffic networks. Journal of Computational and Applied Mathematics, 2022, 401: 113792
[22]	Hung N V, Keller A A. Existence and generic stability conditions of equilibrium points to controlled systems for $n$-player multiobjective generalized games using the Kakutani-Fan-Glicksberg fixed-point theorem. Optimization Letters, 2022, 16(5): 1477-1493
[23]	Hung N V, Keller A A. Optimal control of generalized multiobjective games with application to traffic networks modeling. Mathematische Nachrichten, 2023, 296(8): 3676-3698
[24]	Arrow K J, Debreu G. Existence of an equilibrium for a competitive economy. Econometrica: Journal of the Econometric Society, 1954: 265-290
[25]	Anderlini L, Canning D. Structural stability implies robustness to bounded rationality. Journal of Economic Theory, 2001, 101(2): 395-422
[26]	Yu C, Yu J. On structural stability and robustness to bounded rationality. Nonlinear Analysis: Theory, Methods and Applications, 2006, 65(3): 583-592
[27]	Yu C, Yu J. Bounded rationality in multiobjective games. Nonlinear Analysis: Theory, Methods and Applications, 2007, 67(3): 930-937
[28]	俞建. 几类考虑有限理性平衡问题解的稳定性. 系统科学与数学, 2009, 29(7): 999-1008 doi: 10.12341/jssms08438
	Yu J. Bounded rationality and stability of solutions of some equilibrium problems. Journal of Systems Science and Mathematical Sciences, 2009, 29(7): 999-1008 doi: 10.12341/jssms08438
[29]	Wang L, Cho Y J, Huang N. The robustness of generalized abstract fuzzy economies in generalized convex spaces. Fuzzy Sets and Systems, 2011, 176(1): 56-63
[30]	Miyazaki Y, Azuma H. $(\lambda, \varepsilon )$-stable model and essential equilibria. Mathematical Social Sciences, 2013, 65(2): 85-91
[31]	Loi A, Matta S. Increasing complexity in structurally stable models: An application to a pure exchange economy. Journal of Mathematical Economics, 2015, 57: 20-24
[32]	Yu J, Yang Z, Wang N F. Further results on structural stability and robustness to bounded rationality. Journal of Mathematical Economics, 2016, 67: 49-53
[33]	俞建. 有限理性与博弈论中平衡点集的稳定性. 北京: 科学出版社, 2017
	Yu J. Bounded Rationality and the Stability of Equilibrium Point Set in Game Theory. Beijing: Science Press, 2017
[34]	Hung N V, Tam V M, O'Regan D, Cho Y J. A new class of generalized multiobjective games in bounded rationality with fuzzy mappings: Structural $(\lambda, \varepsilon )$-stability and $(\lambda, \varepsilon )$-robustness to $\varepsilon $-equilibria. Journal of Computational and Applied Mathematics, 2020, 372: 112735
[35]	Aubin J P, Ekeland I. Applied Nonlinear Analysis. New York: John Wiley and Sons, 1984
[36]	Klein E, Thompson A. Theory of Correspondences. New York: John Wiley and Sons, 1984
[37]	Gerstewitz C. Nichtkonvexe dualit$\ddot{\text{a}}$t in der vektoroptimierung. Wissenschaftliche Zeitschrift der Technischen Hochschule f$\ddot{\text{u}}$r Chemie Leuna-Merseburg, 1983, 25(3): 357-364
[38]	Deguire P, Tan K K, Yuan G X Z. The study of maximal elements, fixed points for $L_{s}$-majorized mappings and their applications to minimax and variational inequalities in product topological spaces. Nonlinear Analysis: Theory, Methods and Applications, 1999, 37(7): 933-951
[39]	刘德海, 王维国. 多寡头市场下领先者、追随者和竞争者的市场绩效比较分析. 数学的实践与认识, 2010, 40(11): 20-28
	Liu D H, Wang W G. Multiple oligopolies market performance analysis among leader, competitors and followers. Mathematics in Practice and Theory, 2010, 40(11): 20-28