| 摘要: |
| 针对以往集值映射Nash均衡点无约束的问题，提出了有约束条件下的广义集值映射Nash均衡点的概念，它以通常的Nash均衡点及Loose Nash均衡点为特例，首先，使用KKM定理的等价形式，得到了广义集值映射Nash均衡点的存在定理；其次，针对广义集值映射Nash均衡点的稳定性，通过定义Levitin-Polyak近似解序列，证明了Levitin-Polyak良定性的充分和必要条件，在此基础上，得到了广义集值映射Nash均衡点的Levitin-Polyak良定性结果；此外，通过给出实际例子，验证了广义集值映射Nash均衡点的存在性和Levitin-Polyak良定性结果，说明了大多数的广义集值映射Nash均衡点具有稳定的性质,同样，当其支付或可行约束对应映射退化为单值函数时，其存在结果和Levitin-Polyak良定性结果依然成立。 |
| 关键词: 集值映射 Nash均衡点 Levitin-Polyak良定性 |
| DOI： |
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| Existence and Levitin-Polyak Well-posedness of Generalized Nash Equilibria for Set-valued Mappings |
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LIU Lei, LIN Zhi, PENG Zai-yun, WANG Yan-cheng
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School of Mathematics and Statistics, Chongqing Jiaotong University, Chongqing 400074, China
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| Abstract: |
| As the previous Nash equilibrium for set-valued mappings is unconstrained, in order to solve this problem, a concept of Generalized Nash equilibrium with constraints for set-valued mappings is proposed, which includes usual Nash equilibrium and Loose Nash equilibrium as special cases. Firstly, by using the equivalent form of KKM Theorem, the existence theorem of generalized Nash equilibria for set-valued mappings is obtained. Secondly, in order to discuss the stability of generalized Nash equilibria for set-valued mappings, the sufficient and necessary conditions of Levitin-Polyak well-posedness are proved by defining the Levitin-Polyak approximating sequence. On this basis, the results of Levitin-Polyak well-posedness of generalized Nash equilibria for set-valued mappings are obtained. In addition, the results of existence and Levitin-polyak well-posedness of generalized Nash equilibria for set-valued mappings are verified by practical examples. It is shown that the most of generalized Nash equilibria for set-valued mappings are stable. Similarly, when the mappings of corresponding degenerates to a singleton valued function, the results of existence and Levitin-Polyak well-posedness still hold. |
| Key words: set-valued mapping Nash equilibrium point Levitin-Polyak well-posedness |
