Existence and stability of equilibrium for the uncertainty abstract economy in fuzzy environment

Document Type : Research Paper

Authors

College of Mathematics and statistics, Guizhou University, Guizhou, Guiyang , China

Abstract

For a class of uncertainty abstract economies in fuzzy environment (UASIFEs), we establish for each UASIFE, the existence
of fuzzy strong Berge equilibrium by the Kakutani-Fan Glicksberg fixed point theorem. Moreover, we investigate
the stability of equilibrium, and prove that most UASIFEs are essential and each UASIFE has at least an essential
component by Fort theorem and reduction to absurdity, respectively.

Keywords

Main Subjects

1] K. Y. Abalo, M. M. Kostreva, Equi-well-posed games, Journal of Optimization Theory and Applications, 89(1)
(1996), 89-99. http://doi.org/10.1007/BF02192642

[2] K. Y. Abalo, M. M. Kostreva, Some existence theorems of Nash and Berge equilibria, Applied Mathematics Letters,
17(5) (2004), 569-573. https://doi.org/10.1016/S0893-9659(04)90127-9

[3] C. D. Aliprantis, K. C. Border, Infinite dimensional analysis, New York: Springer-Verlag, 2006.
https://doi.org/10.1006/jfan.1996.0067

[4] R. J. Aumann, Subjectivity and correlation in randomized strategies, Journal of Mathematical Economics, 1(1)
(1974), 67-96. https://ideas.repec.org/r/cor/louvrp/167.html

[5] C. Berge, Theorie generale des jeux an personnes, Paris: Gauthier-Villars, 1957.

[6] S. Chang, Y. Zhu, On variational inequalities for fuzzy mappings, Fuzzy Sets and Systems, 32(3) (1989), 359-367.
https://doi.org/10.1016/0165-0114(89)90268-6

[7] X. C. Deng, S. W. Xiang, Existence of strong Berge equilibrium in generalized games under uncertainty, Acta
Mathematicae Applicatae Sinica, 38(2) (2015), 200-211. https://doi.org/CNKI:SUN:YYSU.0.2015-02-002

[8] X. C. Deng, S. W. Xiang, Y. Zuo, Existence of strong Berge equilibrium under uncertainty, Operations Research
Transactions, 17(3) (2013), 101-107. https://doi.org/10.3969/j.issn.1007-6093.2013.03.011

[9] K. Fan, Fixed point and minimax theorems in locally convex topological linear spaces, Proceedings of the National
Academy of Sciences, 38(2) (1952), 121-126. https://doi.org/10.1073/pnas.38.2.121

[10] M. K. Fort, Essential and non-essential fixed points, American Journal of Mathematics, 72(2) (1950), 315-322.
https://doi.org/10.2307/2372035

[11] M. K. Fort, Points of continuity of semicontinuous functions, Publicationes Mathematicae Debrecen, 2 (1951),
100-102. https://doi.org/10.5486/PMD.1951.2.2.03

[12] J. Gao, D. H. Wu, G. Zhang, Generic stability of equilibrium for n-person non-cooperative games
under uncertainty, Communication on Applied Mathematics and Computation, 28(3) (2014), 36-342.
https://doi.org/10.3969/j.issn.1006-6330.2014.03.010

[13] I. L. Glicksberg, A further generalization of the Kakutani fixed point theorem with application to Nash equilibrium
points, Proceedings of the American Mathematical Society, 3(1) (1952), 170-174. https://doi.org/10.2307/2032478

[14] V. N. Hung, V. M. Tam, D. O’Regan, et al, A new class of generalized multiobjective games in bounded rationality
with fuzzy mappings: Structural (λ, ε)−stability and (λ, ε)−robustness to ε−equilibria, Journal of Computational
and Applied Mathematics, 372 (2020), 112-735. https://doi.org/10.1016/j.cam.2020.112735

[15] T. Ichiishi, A social coalitional equilibrium existence lemma, Econometrica: Journal of the Econometric Society,
(1981), 369-377. https://doi.org/10.2307/1913316

[16] W. K. Kim, On a generalized Berge strong equilibrium, Rairo Operations Research, 29(2) (2014), 367-377.
http://doi.org/10.4134/CKMS.2014.29.2.367

[17] W. K. Kim, K. H. Lee, Generalized fuzzy games and fuzzy equilibria, Fuzzy Sets and Systems, 122(2) (2001),
293-301. https://doi.org/10.1016/S0165-0114(00)00073-7

[18] M. Larbani, H. Lebbah, A concept of equilibrium for a game under uncertainty, European Journal of Operational
Research, 117(1) (1999), 145-156. https://doi.org/10.1016/S0377-2217(98)00079-4

[19] M. Larbani, R. Nessah, Sur l’´equilibre fort selon Berge, Rairo Operations Research, 35(4) (2001), 439-451.
http://doi.org/10.1051/ro:2001124

[20] J. Q. Liu, G. H. Yu, Fuzzy Kakutani-Fan-Glicksberg fixed point theorem and existence of Nash equilibria for fuzzy
games, Fuzzy Sets and Systems, 447 (2022), 100-112. https://doi.org/10.1016/j.fss.2022.02.002

[21] D. T. Luc, Theory of vector optimization, 319, Berlin: Springer-Verlag, (2003), 1-70.

[22] J. F. Nash, Non-cooperative games, Annals of Mathematics, 54 (1951), 286-295. https://doi.org/10.2307/1969529

[23] R. Nessah, M. Larbani, T. Tazdait, A note on Berge equilibrium, Applied Mathematics Letters, 20(8) (2007),
926-932. https://doi.org/10.1016/j.aml.2006.09.005

[24] R. Nessah, T. Tazdait, M. Larbani, Strong Berge and Pareto equilibrium existence for a non-cooperative game,
Working Paper, 2008.

[25] V. Scalzo, On the existence of maximal elements, fixed points and equilibria of generalized games in a fuzzy environment,
Fuzzy Sets and Systems, 272 (2015), 126-133. https://doi.org/10.1016/j.fss.2015.02.006

[26] V. Scalzo, Remarks on the existence and stability of some relaxed Nash equilibrium in strategic form games, Economic
Theory, 61(3) (2016), 571-586. https://doi.org/10.1007/s00199-015-0917-4

[27] T. Schelling, The strategy of conflict, Harvard University Press, 1960. https://doi.org/10.1177/002200275800200

[28] R. Selten, Reexamination of the perfectness concept for equilibrium points in extensive games, International Journal
of Game Theory, 4(1) (1975), 25-55. https://doi.org/10.1007/978-94-015-7774-8-1

[29] N. Singla, P. Kaur, U. C. Gupta, A new approach to solve intuitionistic fuzzy bi-matrix games involving multiple
opinions, Iranian Journal of Fuzzy Systems, 20(1) (2023), 185-197. https://doi.org/10.22111/IJFS.2023.7354

[30] R. Verma, A. Aggarwal, On matrix games with 2-tuple intuitionistic fuzzy linguistic payoffs, Iranian Journal of
Fuzzy Systems, 18(4) (2021), 149-167. https://doi.org/10.22111/ijfs.2021.6182

[31] X. Wang, K. L. Teo, Generalized Nash equilibrium problem over a fuzzy strategy set, Fuzzy Sets and Systems, 434
(2022), 172-184. https://doi.org/10.1016/j.fss.2021.06.006

[32] W. T. Wu, J. H. Jiang, Essential equilibrium points of n-person noncooperative games, Scientia Sinica, 11 (1962),
1307-1322. https://doi.org/10.1360/ya1962-11-10-1307

[33] Z. Yang, Existence of Berge-NS equilibrium in noncooperative games under generalized uncertainty, Journal of
Systems Science and Mathematical Sciences, 9 (2015), 8. https://doi.org/CNKI:SUN:STYS.0.2015-09-008

[34] Z. Yang, A coalitional extension of generalized fuzzy games, Fuzzy Sets and Systems, 383 (2020), 68-79.
https://doi.org/10.1016/j.fss.2019.06.010

[35] Z. Yang, Y. J. Pu, Existence of NS equilibrium in generalized games under generalized uncertainty, Chinese Journal
of Management Science, 21(5) (2013), 165-171. https://doi.org/CNKI:SUN:ZGGK.0.2013-05-021

[36] Z. Yang, A. Wang, Existence and stability of the α−core for fuzzy games, Fuzzy Sets and Systems, 341 (2018),
59-68. https://doi.org/10.1016/j.fss.2015.02.006

[37] J. Yu, Essential equilibria of n-person noncooperative games, Journal of Mathematical Economics, 31(3) (1999),
361-372. https://doi.org/10.1016/S0304-4068(97)00060-8

[38] J. Yu, The existence and stability of nash equilibria, Journal of Systems Science and Mathematical Sciences, 22(3)
(2002), 296-311. https://doi.org/10.3969/j.issn.1000-0577.2002.03.006

[39] J. Yu, Game theory and nonlinear analysis, Beijing: Science Press, 2008.

[40] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353. https://doi.org/10.1016/S0019-9958(65)90241-
X
[41] G. Zhang, D. H. Wu, J. X. Tang, One leader-follower game under uncertainty and its application
ε− Equilibrium stability analysis, Operations Research and Management Science, 27(1) (2018), 23-30.
https://doi.org/10.12005/orms.2018.0004

[42] H. J. Zhang, Q. Zhang, Existence of simple Berge equilibrium in non cooperative games under uncertainty, System
Engineering-Theory and Practice, 30(9) (2010), 1630-1635. https://doi.org/CNKI:SUN:XTLL.0.2010-09-015

[43] H. J. Zhang, Q. Zhang, Existence of strong Nash equilibrium in non cooperative games under uncertainty, Control
and Decision, 8 (2010), 1251-1254. https://doi.org/10.3724/SP.J.1087.2010.02828

[44] W. Zhao, H. Yang, X. Y.Wu, Existence and general stability of group game equilibrium under uncertain parameters,
Acta Mathematicae Applicatae Sinica, 43(4) (2020), 627-638. https://doi.org/10.12387/C2020048

[45] L. Zhou, W. S. Jia, L. P. Liu, Essential stability of fuzzy equilibria for generalized multiobjective games with fuzzy
constraint mappings, Fuzzy Sets and Systems, 447 (2022), 113-122. https://doi.org/10.1016/j.fss.2021.11.012

[46] V. I. Zhukovskii, A. A. Chikrii, Linear quadratic differential games, Naoukova Doumka: Kiev, 1994.
https://doi.org/10.1142/9789814596237-0006

References

[1] K. Y. Abalo, M. M. Kostreva, Equi-well-posed games, Journal of Optimization Theory and Applications, 89(1)
(1996), 89-99. http://doi.org/10.1007/BF02192642
[2] K. Y. Abalo, M. M. Kostreva, Some existence theorems of Nash and Berge equilibria, Applied Mathematics Letters,
17(5) (2004), 569-573. https://doi.org/10.1016/S0893-9659(04)90127-9
[3] C. D. Aliprantis, K. C. Border, Infinite dimensional analysis, New York: Springer-Verlag, 2006.
https://doi.org/10.1006/jfan.1996.0067
[4] R. J. Aumann, Subjectivity and correlation in randomized strategies, Journal of Mathematical Economics, 1(1)
(1974), 67-96. https://ideas.repec.org/r/cor/louvrp/167.html
[5] C. Berge, Theorie generale des jeux an personnes, Paris: Gauthier-Villars, 1957.
[6] S. Chang, Y. Zhu, On variational inequalities for fuzzy mappings, Fuzzy Sets and Systems, 32(3) (1989), 359-367.
https://doi.org/10.1016/0165-0114(89)90268-6
[7] X. C. Deng, S. W. Xiang, Existence of strong Berge equilibrium in generalized games under uncertainty, Acta
Mathematicae Applicatae Sinica, 38(2) (2015), 200-211. https://doi.org/CNKI:SUN:YYSU.0.2015-02-002
[8] X. C. Deng, S. W. Xiang, Y. Zuo, Existence of strong Berge equilibrium under uncertainty, Operations Research
Transactions, 17(3) (2013), 101-107. https://doi.org/10.3969/j.issn.1007-6093.2013.03.011
[9] K. Fan, Fixed point and minimax theorems in locally convex topological linear spaces, Proceedings of the National
Academy of Sciences, 38(2) (1952), 121-126. https://doi.org/10.1073/pnas.38.2.121
[10] M. K. Fort, Essential and non-essential fixed points, American Journal of Mathematics, 72(2) (1950), 315-322.
https://doi.org/10.2307/2372035
[11] M. K. Fort, Points of continuity of semicontinuous functions, Publicationes Mathematicae Debrecen, 2 (1951),
100-102. https://doi.org/10.5486/PMD.1951.2.2.03
[12] J. Gao, D. H. Wu, G. Zhang, Generic stability of equilibrium for n-person non-cooperative games
under uncertainty, Communication on Applied Mathematics and Computation, 28(3) (2014), 36-342.
https://doi.org/10.3969/j.issn.1006-6330.2014.03.010
[13] I. L. Glicksberg, A further generalization of the Kakutani fixed point theorem with application to Nash equilibrium
points, Proceedings of the American Mathematical Society, 3(1) (1952), 170-174. https://doi.org/10.2307/2032478
[14] V. N. Hung, V. M. Tam, D. O’Regan, et al, A new class of generalized multiobjective games in bounded rationality
with fuzzy mappings: Structural (λ, ε)−stability and (λ, ε)−robustness to ε−equilibria, Journal of Computational
and Applied Mathematics, 372 (2020), 112-735. https://doi.org/10.1016/j.cam.2020.112735
[15] T. Ichiishi, A social coalitional equilibrium existence lemma, Econometrica: Journal of the Econometric Society,
(1981), 369-377. https://doi.org/10.2307/1913316
[16] W. K. Kim, On a generalized Berge strong equilibrium, Rairo Operations Research, 29(2) (2014), 367-377.
http://doi.org/10.4134/CKMS.2014.29.2.367
[17] W. K. Kim, K. H. Lee, Generalized fuzzy games and fuzzy equilibria, Fuzzy Sets and Systems, 122(2) (2001),
293-301. https://doi.org/10.1016/S0165-0114(00)00073-7
[18] M. Larbani, H. Lebbah, A concept of equilibrium for a game under uncertainty, European Journal of Operational
Research, 117(1) (1999), 145-156. https://doi.org/10.1016/S0377-2217(98)00079-4
[19] M. Larbani, R. Nessah, Sur l’´equilibre fort selon Berge, Rairo Operations Research, 35(4) (2001), 439-451.
http://doi.org/10.1051/ro:2001124
[20] J. Q. Liu, G. H. Yu, Fuzzy Kakutani-Fan-Glicksberg fixed point theorem and existence of Nash equilibria for fuzzy
games, Fuzzy Sets and Systems, 447 (2022), 100-112. https://doi.org/10.1016/j.fss.2022.02.002
[21] D. T. Luc, Theory of vector optimization, 319, Berlin: Springer-Verlag, (2003), 1-70.
[22] J. F. Nash, Non-cooperative games, Annals of Mathematics, 54 (1951), 286-295. https://doi.org/10.2307/1969529
[23] R. Nessah, M. Larbani, T. Tazdait, A note on Berge equilibrium, Applied Mathematics Letters, 20(8) (2007),
926-932. https://doi.org/10.1016/j.aml.2006.09.005
[24] R. Nessah, T. Tazdait, M. Larbani, Strong Berge and Pareto equilibrium existence for a non-cooperative game,
Working Paper, 2008.
[25] V. Scalzo, On the existence of maximal elements, fixed points and equilibria of generalized games in a fuzzy environment,  Fuzzy Sets and Systems, 272 (2015), 126-133. https://doi.org/10.1016/j.fss.2015.02.006
[26] V. Scalzo, Remarks on the existence and stability of some relaxed Nash equilibrium in strategic form games, Economic
Theory, 61(3) (2016), 571-586. https://doi.org/10.1007/s00199-015-0917-4
[27] T. Schelling, The strategy of conflict, Harvard University Press, 1960. https://doi.org/10.1177/002200275800200
[28] R. Selten, Reexamination of the perfectness concept for equilibrium points in extensive games, International Journal
of Game Theory, 4(1) (1975), 25-55. https://doi.org/10.1007/978-94-015-7774-8-1
[29] N. Singla, P. Kaur, U. C. Gupta, A new approach to solve intuitionistic fuzzy bi-matrix games involving multiple
opinions, Iranian Journal of Fuzzy Systems, 20(1) (2023), 185-197. https://doi.org/10.22111/IJFS.2023.7354
[30] R. Verma, A. Aggarwal, On matrix games with 2-tuple intuitionistic fuzzy linguistic payoffs, Iranian Journal of
Fuzzy Systems, 18(4) (2021), 149-167. https://doi.org/10.22111/ijfs.2021.6182
[31] X. Wang, K. L. Teo, Generalized Nash equilibrium problem over a fuzzy strategy set, Fuzzy Sets and Systems, 434
(2022), 172-184. https://doi.org/10.1016/j.fss.2021.06.006
[32] W. T. Wu, J. H. Jiang, Essential equilibrium points of n-person noncooperative games, Scientia Sinica, 11 (1962),
1307-1322. https://doi.org/10.1360/ya1962-11-10-1307
[33] Z. Yang, Existence of Berge-NS equilibrium in noncooperative games under generalized uncertainty, Journal of
Systems Science and Mathematical Sciences, 9 (2015), 8. https://doi.org/CNKI:SUN:STYS.0.2015-09-008
[34] Z. Yang, A coalitional extension of generalized fuzzy games, Fuzzy Sets and Systems, 383 (2020), 68-79.
https://doi.org/10.1016/j.fss.2019.06.010
[35] Z. Yang, Y. J. Pu, Existence of NS equilibrium in generalized games under generalized uncertainty, Chinese Journal
of Management Science, 21(5) (2013), 165-171. https://doi.org/CNKI:SUN:ZGGK.0.2013-05-021
[36] Z. Yang, A. Wang, Existence and stability of the α−core for fuzzy games, Fuzzy Sets and Systems, 341 (2018),
59-68. https://doi.org/10.1016/j.fss.2015.02.006
[37] J. Yu, Essential equilibria of n-person noncooperative games, Journal of Mathematical Economics, 31(3) (1999),
361-372. https://doi.org/10.1016/S0304-4068(97)00060-8
[38] J. Yu, The existence and stability of nash equilibria, Journal of Systems Science and Mathematical Sciences, 22(3)
(2002), 296-311. https://doi.org/10.3969/j.issn.1000-0577.2002.03.006
[39] J. Yu, Game theory and nonlinear analysis, Beijing: Science Press, 2008.
[40] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353. https://doi.org/10.1016/S0019-9958(65)90241-
X
[41] G. Zhang, D. H. Wu, J. X. Tang, One leader-follower game under uncertainty and its application
ε− Equilibrium stability analysis, Operations Research and Management Science, 27(1) (2018), 23-30.
https://doi.org/10.12005/orms.2018.0004
[42] H. J. Zhang, Q. Zhang, Existence of simple Berge equilibrium in non cooperative games under uncertainty, System
Engineering-Theory and Practice, 30(9) (2010), 1630-1635. https://doi.org/CNKI:SUN:XTLL.0.2010-09-015
[43] H. J. Zhang, Q. Zhang, Existence of strong Nash equilibrium in non cooperative games under uncertainty, Control
and Decision, 8 (2010), 1251-1254. https://doi.org/10.3724/SP.J.1087.2010.02828
[44] W. Zhao, H. Yang, X. Y.Wu, Existence and general stability of group game equilibrium under uncertain parameters,
Acta Mathematicae Applicatae Sinica, 43(4) (2020), 627-638. https://doi.org/10.12387/C2020048
[45] L. Zhou, W. S. Jia, L. P. Liu, Essential stability of fuzzy equilibria for generalized multiobjective games with fuzzy
constraint mappings, Fuzzy Sets and Systems, 447 (2022), 113-122. https://doi.org/10.1016/j.fss.2021.11.012
[46] V. I. Zhukovskii, A. A. Chikrii, Linear quadratic differential games, Naoukova Doumka: Kiev, 1994.
https://doi.org/10.1142/9789814596237-0006
Iranian Journal of Fuzzy Systems
Volume 21, Issue 2
March and April 2024
Pages 129-140

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