Fuzzy games for a general Bayesian abstract fuzzy economy model of product measurable spaces. (English) Zbl 1348.91203
Summary: In this paper, we introduce a general Bayesian abstract fuzzy economy model of product measurable spaces, and we prove the existence of Bayesian fuzzy equilibrium for this model. Our results extend and improve the corresponding recent results announced by Patriche and many authors from the literature. It captures the idea that the uncertainties characterize the individual feature of the decisions of the agents involved in different economic activities. In this paper, the uncertainties can be described by using random fuzzy mappings. Further attention is needed for the study of applications of the established result in the game theory and the fuzzy economic field.
Keywords:
Bayesian abstract fuzzy economy; Bayesian fuzzy equilibrium; incomplete information; random fuzzy mappingReferences:
| [1] | NashJ. Equilibrium points in n‐person games. Proceedings of the National Academy of Sciences, U.S.A.1950; 36(1): 48-49. · Zbl 0036.01104 |
| [2] | DebreuG. A socail equilibrium existence theorem. Proceedings of the National Academy of Sciences, U.S.A.1952; 38: 886-903. · Zbl 0047.38804 |
| [3] | ArrowKJ, DebreuG. Existence of an equilibrium for a competitive economy. Econometrica. 1952; 22: 265-290. · Zbl 0055.38007 |
| [4] | ShaferW, SonnenscheinH. Equilibrium in abstract economies without ordered preferences. Journal of Mathematical Economics. 1975; 2: 345-348. · Zbl 0312.90062 |
| [5] | BorglinA, KeidingH. Existence of equilibrium actions and of equilibrium: a note on the “new” existence theorem. Journal of Mathematical Economics. 1976; 3: 313-316. · Zbl 0349.90157 |
| [6] | HuangNJ. Some new equilibrium theorems for abstract economies. Applied Mathematics Letters. 1998; 11(1): 41-45. · Zbl 1075.91580 |
| [7] | KimWK, TanKK. New existence theorems of equilibria and applications. Nonlinear Analysis: Theory, Methods and Applications. 2001; 47: 531-542. · Zbl 1042.47534 |
| [8] | LinLJ, ChenLF, AnsariQH. Generalized abstract economy and systems of generalized vector quasi‐equilibrium problems. Journal of Computational and Applied Mathematics. 2007; 208: 341-353. · Zbl 1124.91046 |
| [9] | BriecW, HorvathC. C: Nash points, Ky Fan inequality and equilibria of abstract economies in Max‐Plus and B‐convexity. Journal of Mathematical Analysis and Applications. 2008; 341: 188-199. · Zbl 1151.91017 |
| [10] | DingXP, WangL. Fixed points, minimax inequalities and equilibria of noncompact abstract economies in FC‐spaces. Nonlinear Analysis: Theory, Methods and Applications. 2008; 69: 730-746. · Zbl 1157.47037 |
| [11] | KimWK, KumSH, LeeKH. On general best proximity pairs and equilibrium pairs in free abstract economies. Nonlinear Analysis: Theory, Methods and Applications. 2008; 68: 2216-2227. · Zbl 1136.91309 |
| [12] | LinLJ, LiuYH. The study of abstract economies with two constraint correspondences. Journal of Optimization Theory and Applications. 2008; 137: 41-52. · Zbl 1141.91034 |
| [13] | DingXP, FengHL. Fixed point theorems and existence of equilibrium points of noncompact abstract economies for \(L_F^\ast \)‐majorized mappings in FC‐spaces. Nonlinear Analysis: Theory, Methods and Applications. 2010; 72: 65-76. · Zbl 1216.54011 |
| [14] | WangL, ChoYJ, HuangNJ. The robustness of generalized abstract fuzzy economies in generalized convex spaces. Fuzzy Sets and Systems. 2011; 176: 56-63. · Zbl 1235.91123 |
| [15] | ZadehLA. Fuzzy sets. Information and Control. 1965; 8: 338-353. · Zbl 0139.24606 |
| [16] | KimT, PrikryK, YannelisNC. On Carateheodory‐type selection theolem. Journal of Mathematical Analysis and Applications. 1988; 135: 664-670. · Zbl 0676.28006 |
| [17] | PatricheM. Bayesian abstract economy with a measure space of agents. Abstract and Applied Analysis. 2009; 2009: 11. Article ID 523619, DOI: 10.1155/2009/52361. · Zbl 1176.91079 |
| [18] | PatricheM. Equilibrium of Bayesian fuzzy economies and quasi‐variational inequalities with random fuzzy mappings. Journal of Inequalities and Applications. 2013; 2013: 374, Article ID 58E35. · Zbl 1285.91081 |
| [19] | PatricheM. Existence of equilibrium for an abstract economy with private information and a countable space of actions. Mathematical Reports. 2013; 15(65)(3): 233-242. · Zbl 1389.91062 |
| [20] | PatricheM. Fuzzy games with a countable space of actions and applications to systems of generalized quasi‐variational inequalities. Fixed Point Theory and Applications. 2014; 2014: 124. · Zbl 1345.54066 |
| [21] | PatricheM. Equilibrium in games and competitive economies. The Publishing House of the Romanian Academy: Bucharest, 2011. |
| [22] | ChangSS, ZhuYG. On variational inequalities for fuzzy mappings. Fuzzy Sets and Systems. 1989; 32: 359-367. · Zbl 0677.47037 |
| [23] | NoorMA. Variational inequalities for fuzzy mappings III. Fuzzy Sets and Systems. 2000; 110: 101-108. · Zbl 0940.49011 |
| [24] | ParkJY, LeeSY, JeongJU. Completely generalized strongly quasivariational inequalities for fuzzy mapping. Fuzzy Sets and Systems. 2000; 110: 91-99. · Zbl 0940.49012 |
| [25] | DingXP, ParkJY. A new class of generalized nonlinear implicit quasi‐variational inclusions with fuzzy mappings. Journal of Computational and Applied Mathematics. 2002; 138: 243-257. · Zbl 0996.65067 |
| [26] | NoorMA, ElsanousiSA. Iterative algorithms for random variational inequalities. Pan‐American Mathematical Journal. 1993; 3: 39-50. · Zbl 0845.49007 |
| [27] | AubinJP, FrankowskaH. Set‐valued Analysis. Basel: Birkhäuser, 1990. · Zbl 0713.49021 |
| [28] | DebreuG. 1966. Integration of correspondences. In Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, vol. 2 University of California Press: Berkeley; 351-372. · Zbl 0211.52803 |
| [29] | YannelisNC. 1998. On the existence of Bayesian Nash equilibrium. In Functional Analysis and Economic Theory, AgramovichY (ed.), AvgerinosE (ed.), YannelisNC (ed.) (eds).Springer: Berlin; 291-296. · Zbl 0914.90037 |
| [30] | YannelisNC. 1991. Integration of Banach‐valued correspondences. In Equilibrium Theory in Infinite Dimensional Spaces, KhanMA (ed.), YannelisNC (ed.) (eds)., Studies in Economic TheorySpringer: Berlin; 2-35. · Zbl 0747.28005 |
| [31] | FanK. Fixed‐point and minimax theorems in locally convex topological linear spaces. Proceedings of the National Academy of Sciences, U.S.A.1952; 38: 121-126. · Zbl 0047.35103 |
| [32] | CastaingC, ValadierM. Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics, vol. 580. Springer: New York, 1977. · Zbl 0346.46038 |
| [33] | DiestelJ, UhlJ. Vector measures. In Mathematical Surveys, Vol. 15: Am. Math. Soc., Providence, 1977, 15: 101-105. · Zbl 0369.46039 |
| [34] | ChangSS. Coincidence theorems and variational inequalities for fuzzy mappings. Fuzzy Sets and Systems. 1994; 61: 359-368. · Zbl 0830.47052 |
| [35] | Onjai‐UeaN, KumamP. A generalized nonlinear random equations with random fuzzy mappings in uniformly smooth Banach spaces. Journal of Inequalities and Applications. 2010; 2010: 15. Article ID 728452, DOI: 10.1155/2010/728452. · Zbl 1210.47102 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
