Stability of fixed points of set-valued mappings and strategic stability of Nash equilibria


Authors

Shuwen Xiang - School of Mathematics and Statistics, Guizhou University, Guiyang, 550025, P. R. China. Shunyou Xia - School of Mathematics and Statistics, Guizhou University, Guiyang, 550025, P. R. China. - School of Mathematics and Computer Science, Guizhou Education University, Guiyang, 550018, P. R. China. Jihao He - School of Mathematics and Statistics, Guizhou University, Guiyang, 550025, P. R. China. Yanlong Yang - School of Mathematics and Statistics, Guizhou University, Guiyang, 550025, P. R. China. Chenwei Liu - School of Mathematics and Statistics, Guizhou University, Guiyang, 550025, P. R. China.


Abstract

In this paper, we mainly focus on the stability of Nash equilibria to any perturbation of strategy sets. A larger perturbation, strong \(\delta\)-perturbation, will be proposed for set-valued mapping. The class of perturbed games considered in the definition of strong \(\delta\)-perturbation is richer than those considered in many other definitions of stability of Nash equilibria. The strong \(\delta\)-perturbation of the best reply correspondence will be used to define an appropriate stable set for Nash equilibria, called SBR-stable set. As an SBR-stable set is stable to any strong \(\delta\)-perturbation and, various perturbations of strategy sets are not beyond the range of strong \(\delta\)-perturbation, it has the stability that various stable sets possess, such as fully stable set, stable set, quasistable set, and essential set. An SBR-stable set is stable to any perturbation of strategy sets, so it will provide convenience for study in strategic stability, which is even used to study any noncooperative game.


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ISRP Style

Shuwen Xiang, Shunyou Xia, Jihao He, Yanlong Yang, Chenwei Liu, Stability of fixed points of set-valued mappings and strategic stability of Nash equilibria, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 7, 3599--3611

AMA Style

Xiang Shuwen, Xia Shunyou, He Jihao, Yang Yanlong, Liu Chenwei, Stability of fixed points of set-valued mappings and strategic stability of Nash equilibria. J. Nonlinear Sci. Appl. (2017); 10(7):3599--3611

Chicago/Turabian Style

Xiang, Shuwen, Xia, Shunyou, He, Jihao, Yang, Yanlong, Liu, Chenwei. "Stability of fixed points of set-valued mappings and strategic stability of Nash equilibria." Journal of Nonlinear Sciences and Applications, 10, no. 7 (2017): 3599--3611


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