Stability of weighted Nash equilibrium for multiobjective population games

Volume 9, Issue 6, pp 4167--4176 http://dx.doi.org/10.22436/jnsa.009.06.59

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 1667 Downloads
• 2834 Views

Authors

Guanghui Yang - School of Mathematics and Statistics, Guizhou University, Guiyang, 550025, P. R. China. Hui Yang - School of Mathematics and Statistics, Guizhou University, Guiyang, 550025, P. R. China. Qiqing Song - School of Science, Guilin University of Technology, Guilin, 541004, P. R. China. - School of Mathematics and Statistics, Guizhou University, Guiyang, 550025, P. R. China.

Abstract

This paper studies the existence and stability of weighted Nash equilibria for multiobjective population games. By constructing a Nash's mapping, the existence of weighted Nash equilibria is established. Furthermore, via the generic continuity method, each weighted Nash equilibrium is shown to be stable for most of multiobjective population games when weight combinations and payoff functions are simultaneously perturbed. Besides, this leads to the stability of Nash equilibria for classical population games with the perturbed payoff functions. These results play cornerstone role in the research concerning multiobjective population games.

Share and Cite

Keywords

• Multiobjective population game
• weighted Nash equilibrium
• stability.

MSC

•  90C29
•  90C31
•  91A10

References

• [1] Q. H. Ansari, Z. Khan , On existence of Pareto equilibria for constrained multiobjective games, Southeast Asian B. Math., 27 (2004), 973-982.


• [2] I. Arieli, H. P. Young, Fast convergence in population games, Department of economics Discussion Paper Series, University of Oxford (2011)


• [3] J. P. Aubin, I. Ekeland, Applied nonlinear analisis, North-Holland, Amsterdam (1984)


• [4] K. C. Border, Fixed point theorems with applications to economics and game theory, Cambridge University Press, Cambridge (1985)


• [5] J. Canals, F. Vega-Redondo , Multi-level evolution in population games, Int. J. Game theory, 27 (1998), 21-35.


• [6] G. Y. Chen, X. X. Huang, X. Q. Yang, Vector Optimization: set-valued and variational analysis, Spring-Verlag, Berlin (2005)


• [7] I. Eshel, E. Sansone, Evolutionary and dynamic stability in continuous population games, J. Math. Biol., 46 (2003), 445-459.


• [8] M. K. Fort , Points of Continuity of Semi-continuous Functions, Publ. Math. Deb-recen, 2 (1951), 100-102.


• [9] P. Q. Khanh, N. H. Quan, Generic stability and essential components of generalized KKM points and applications, J. Optim. Theory Appl., 148 (2011), 488-504.


• [10] E. Klein, A. Thompson, Theory of correspondences, Wiley, New York (1984)


• [11] A. Lanni , Learning Correlated Equilibria in population games, Math. Social Sci., 42 (2001), 271-294.


• [12] G. M. Lee, N. D. Yen, A result on vector variational inequalities with polyhedral constraint sets, J. Optim. Theory Appl., 109 (2001), 193-197.


• [13] Z. Lin, Essential components of the set of weakly Pareto-Nash equilibrium points for multiobjective generalized games in two different topological spaces, J. Optimiz. Theory Appl., 124 (2005), 387-405.


• [14] D. T. Luc , Theorey of Vector Optimization, Springer-Verlag, Berlin (1989)


• [15] J. Nash , Equilibrium points in N-person games, Proc. Nat. Ac. Sci., 36 (1950), 48-49.


• [16] J. Nash, Noncooperative games, Dept. Mathematics, Dissertation, Princeton University (1950)


• [17] J. Nash, Noncooperative games, Ann. Math., 54 (1951), 286-295.


• [18] D. Oyama, W. H. Sandholm, O. Tercieux, Sampling best response dynamics and deterministic equilibrium selection, Theor. Econ., 10 (2015), 243-281.


• [19] W. H. Sandholm, Population Games and Evolutionary Dynamics, MIT Press, London (2011)


• [20] Y. Sawaragi, H. Nakayama, T. Tanino, Theory of Multiobjective Optimization, Academic Press, Inc, London (1985)


• [21] Q. Q. Song, L. S. Wang, On the stability of the solution for multiobjective generalized games with the payoffs perturbed, Nonlinear Anal., 73 (2010), 2680-2685.


• [22] S. Y. Wang, Existence of a Pareto equilibrium, J. Optim. Theory Appl., 79 (1993), 373-384


• [23] S. W. Xiang, W. S. Jia, J. He, S. Y. Xia, Z. Y. Chen, Some results concerning the generic continuity of set-valued mappings, Nonlinear Anal., 75 (2012), 3591-3597.


• [24] S. W. Xiang, S. H. Xiang , Generic stability on weight factors in multiobjective optimization problems, Panamer. Math. J., 7 (1997), 79-84.


• [25] S. W. Xiang, W. S. Yin , Stability Results for Efficient Solutions of Vector Optimization Problems, J. Optim. Theory Appl., 134 (2007), 385-398.


• [26] Z. Yang, Y. J. Pu, On existence and essential components for solution set for system of strong vector quasi- equilibrium problems, J. Global Optim., 55 (2013), 253-259.


• [27] J. Yu, Essential equilibria of n-person noncooperative games, J. Math. Econom., 31 (1999), 361-372.


• [28] J. Yu, G. X.-Z. Yuan, The study of Pareto equilibria for multiobjective games by fixed point and Ky Fan minimax inequality methods, Comput. Math. Appl., 35 (1998), 17-24.