Stability of fixed points of set-valued mappings and strategic stability of Nash equilibria
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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.
Share and Cite
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
Keywords
- Stability
- Nash equilibria
- fixed point
- strong \(\delta\)-perturbation
- stable set.
MSC
References
-
[1]
O. Carbonell-Nicolau, On strategic stability in discontinuous games, Econom. Lett., 113 (2011), 120–123.
-
[2]
O. Carbonell-Nicolau, Further results on essential Nash equilibria in normal-form games, Econom. Theory, 59 (2015), 277–300.
-
[3]
M. K. Fort, Jr., Points of continuity of semi-continuous functions, Publ. Math. Debrecen, 2 (1951), 100–102.
-
[4]
S. Govindan, R. Wilson, Essential equilibria, Proc. Natl. Acad. Sci. USA, 102 (2005), 15706–15711.
-
[5]
J. Hillas, On the definition of the strategic stability of equilibria, Econometrica, 58 (1990), 1365–1390.
-
[6]
J. Hillas, M. Jansen, J. Potters, D. Vermeulen, On the relation among some definitions of strategic stability, Math. Oper. Res., 26 (2001), 611–635.
-
[7]
J. Hillas, M. Jansen, J. Potters, D. Vermeulen, Independence of inadmissible strategies and best reply stability: a direct proof, Special issue on stable equilibria, Internat. J. Game Theory, 32 (2004), 371–377.
-
[8]
W.-S. Jia, S.-W. Xiang, J.-H. He, Y.-L. Yang, Existence and stability of weakly Pareto-Nash equilibrium for generalized multiobjective multi-leaderfollower games, J. Global Optim., 61 (2015), 397–405.
-
[9]
E. Kalai, D. Samet, Persistent equilibria in strategic games, Internat. J. Game Theory, 13 (1984), 129–144.
-
[10]
E. Klein, A. C. Thompson, Theory of correspondences, Including applications to mathematical economics, Canadian Mathematical Society Series of Monographs and Advanced Texts, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York (1984)
-
[11]
E. Kohlberg, J. F. Mertens, On the strategic stability of equilibria, Econometrica, 54 (1986), 1003–1037.
-
[12]
D. M. Kreps, R. Wilson, Sequential equilibria, Econometrica, 50 (1982), 863–894.
-
[13]
A. McLennan, Fixed points of contractible valued correspondences, Internat. J. Game Theory, 18 (1989), 175–184.
-
[14]
J. F. Mertens, Stable equilibria–a reformulation, I, Definition and basic properties, Math. Oper. Res., 14 (1989), 575– 625.
-
[15]
R. B. Myerson, Refinements of the Nash equilibrium concept, Internat. J. Game Theory, 7 (1978), 73–80.
-
[16]
A. B. Sadanand, V. Sadanand, Equilibria in non-cooperative games, I, Perturbations based refinements of Nash equilibrium, Bull. Econ. Res., 43 (1994), 197–224.
-
[17]
V. Scalzo, Essential equilibria of discontinuous games, Econom. Theory, 54 (2013), 27–44.
-
[18]
V. Scalzo, On the existence of essential and trembling-hand perfect equilibria in discontinuous games, Econ. Theory Bull., 1 (2014), 1–12.
-
[19]
R. Selten, Reexamination of the perfectness concept for equilibrium points in extensive games, Internat. J. Game Theory, 4 (1975), 25–55.
-
[20]
E. van Damme, Strategic equilibrium, Handbook of Game Theory with Economic Applications, 3 (2002), 1521–1596.
-
[21]
A. J. Vermeulen, J. A. M. Potters, M. J. M. Jansen, On quasi-stable sets, Internat. J. Game Theory, 25 (1996), 43–49.
-
[22]
A. J. Vermeulen, J. A. M. Potter, M. J. M. Jansen, On stable sets of equilibria, Game theoretical applications to economics and operations research, Bangalore, (1996), Theory Decis. Lib. Ser. C Game Theory Math. Program. Oper. Res., Kluwer Acad. Publ., Boston, MA, 18 (1997), 133–148.
-
[23]
W.-T. Wu, J.-H. Jiang, Essential equilibrium points of n-person non-cooperative games, Sci. Sinica, 11 (1962), 1307–1322.
-
[24]
S.-W. Xiang, G.-D. Liu, Y.-H. Zhou, On the strongly essential components of Nash equilibria of infinite n-person games with quasiconcave payoffs, Nonlinear Anal., 63 (2005), e2637–e2647.
-
[25]
J. Yu, Q. Luo, On essential components of the solution set of generalized games, J. Math. Anal. Appl., 230 (1999), 303–310.
-
[26]
J. Yu, S.-W. Xiang, On essential components of the set of Nash equilibrium points, Nonlinear Anal., 38 (1999), 259–264.
-
[27]
Y.-H. Zhou, J. Yu, S.-W. Xiang, Essential stability in games with infinitely many pure strategies, Internat. J. Game Theory, 35 (2007), 493–503.