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The robustness of generalized abstract fuzzy economies in generalized convex spaces. (English) Zbl 1235.91123

Summary: We study the economic model proposed by Anderlini and Canning, a parameterized class of generalized abstract fuzzy economies together with an associated abstract rationality function, and show that the structural stability of this model implies its robustness to \(\varepsilon \)-equilibria.

MSC:

91B54 Special types of economic markets (including Cournot, Bertrand)
91B52 Special types of economic equilibria
Full Text: DOI

References:

[1] Aubin, J. P., Cooperative fuzzy games, Math. Oper. Res., 6, 1-13 (1981) · Zbl 0496.90092
[2] Aubin, J. P.; Ekeland, I., Applied Nonlinear Analysis (1984), Wiley: Wiley New York · Zbl 0641.47066
[3] Anderini, L.; Canning, D., Structural stability implies robustness to bounded rationality, J. Econ. Theory, 101, 395-442 (2001) · Zbl 0996.91078
[4] Arrow, K. J.; Debreu, G., Existence of an equilibrium for a competitive economy, Econometrica, 22, 265-290 (1952) · Zbl 0055.38007
[5] Azrieli, Y.; Lehrer, E., On some families of cooperative fuzzy games, Int. J. Game Theory, 36, 1-15 (2007) · Zbl 1128.91005
[6] Bellman, R. E.; Zadeh, L. A., Decision-making in a fuzzy environment, Manage. Sci., 17, 141-164 (1970) · Zbl 0224.90032
[7] Billot, A., Economic Theory of Fuzzy Equilibria (1992), Springer-Verlag: Springer-Verlag New York · Zbl 0758.90008
[8] Borglin, A.; Keiding, H., Existence of equilibrium action and equilibrium: a note on the ‘new’ existence theorems, J. Math. Econ., 3, 313-316 (1976) · Zbl 0349.90157
[9] Borkotokey, S., Cooperative games with fuzzy coalitions and fuzzy characteristic functions, Fuzzy Sets Syst., 159, 138-151 (2008) · Zbl 1168.91311
[10] Briec, W.; Horvath, C., Nash points, Ky Fan inequality and equilibria of abstract economies in Max-Plus and B-convexity, J. Math. Anal. Appl., 341, 188-199 (2008) · Zbl 1151.91017
[11] Butnariu, D., Fuzzy games: a description of the concept, Fuzzy Sets Syst., 1, 181-192 (1978) · Zbl 0389.90100
[12] Campos, F. A.; Villar, J.; Barqu, J.; Ruipez, J., Robust mixed strategies in fuzzy non-cooperative Nash games, Eng. Optim., 40, 459-474 (2008)
[13] Chang, S. S.; Tan, K. K., Equilibria and maximal elements of abstract fuzzy economies and qualitative fuzzy games, Fuzzy Sets Syst., 125, 389-399 (2002) · Zbl 1011.91063
[14] Debreu, G., A social equilibrium existence theorem, Proc. Natl. Acad. Sci. USA, 38, 886-895 (1952) · Zbl 0047.38804
[15] Ding, X. P.; Feng, H. L., Fixed point theorems and existence of equilibrium points of noncompact abstract economies for \(L_F^\ast \)-majorized mappings in \(\operatorname{FC} \)-spaces, Nonlinear Anal. TMA, 72, 65-76 (2010) · Zbl 1216.54011
[16] Ding, X. P.; Wang, L., Fixed points, minimax inequalities and equilibria of noncompact abstract economies in FC-spaces, Nonlinear Anal. TMA, 69, 730-746 (2008) · Zbl 1157.47037
[17] Fodor, J. C.; Roubens, M. R., Fuzzy Preference Modelling and Multicriteria Decision Support (1994), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 0827.90002
[18] Gale, D.; Mas-Colell, A., An equilibrium existence theorem for a general model without ordered preferences, J. Math. Econ., 2, 9-15 (1975) · Zbl 0324.90010
[19] Horvath, C., Contractibility and general convexity, J. Math. Anal. Appl., 156, 341-357 (1991) · Zbl 0733.54011
[20] Horvath, C., Nonlinear and Convex Analysis (1987), Dekker: Dekker New York
[21] Huang, N. J., Some new equilibrium theorems for abstract economies, Appl. Math. Lett., 11, 1, 41-45 (1998) · Zbl 1075.91580
[22] Huang, N. J., A new equilibrium existence theorem for abstract fuzzy economies, Appl. Math. Lett., 12, 5, 1-5 (1999) · Zbl 0979.91062
[23] Huang, N. J., Existence of equilibrium for generalized abstract fuzzy economies, Fuzzy Sets Syst., 117, 151-156 (2001) · Zbl 0967.91038
[24] Hwang, Y. A.; Liao, Y. H., The consistent value of fuzzy games, Fuzzy Sets Syst., 160, 644-656 (2009) · Zbl 1180.91060
[25] Hwang, Y. A.; Liao, Y. H., Max-consistency, complement-consistency and the core of fuzzy games, Fuzzy Sets Syst, 159, 152-163 (2008) · Zbl 1168.91312
[26] Kim, W. K.; Kum, S. H.; Lee, K. H., On general best proximity pairs and equilibrium pairs in free abstract economies, Nonlinear Anal. TMA, 68, 2216-2227 (2008) · Zbl 1136.91309
[27] Kim, W. K.; Lee, K. H., Fuzzy fixed point and existence of equilibria of fuzzy games, J. Fuzzy Math., 6, 193-202 (1998) · Zbl 0903.90181
[28] Kim, W. K.; Lee, K. H., Generalized fuzzy games and fuzzy equilibria, Fuzzy Sets Syst., 122, 293-301 (2001) · Zbl 1022.91002
[29] Kim, W. K.; Tan, K. K., New existence theorems of equilibria and applications, Nonlinear Anal. TMA, 47, 531-542 (2001) · Zbl 1042.47534
[30] Li, S. J.; Zhang, Q., A simplified expression of the Shapley function for fuzzy game, Eur. J. Oper. Res., 196, 234-245 (2009) · Zbl 1159.91315
[31] Lin, L. J.; Chen, L. F.; Ansari, Q. H., Generalized abstract economy and systems of generalized vector quasi-equilibrium problems, J. Comput. Appl. Math., 208, 341-353 (2007) · Zbl 1124.91046
[32] Lin, L. J.; Liu, Y. H., The study of abstract economies with two constraint correspondences, J. Optim. Theory Appl., 137, 41-52 (2008) · Zbl 1141.91034
[33] Mas-Colell, A., An equilibrium existence theorem without complete or transitive preferences, J. Math. Econ., 1, 237-246 (1974) · Zbl 0348.90033
[34] Nash, J., Equilibrium states in N-person games, Proc. Natl. Acad. Sci. USA, 36, 48-49 (1950) · Zbl 0036.01104
[35] Park, S., Fixed points of better admissible maps on generalized convex spaces, J. Korea Math. Soc., 37, 6, 885-899 (2000) · Zbl 0967.47039
[36] Park, S.; Kim, H., Coincidence theorems for admissible multifunctions on generalized convex spaces, J. Math. Anal. Appl., 197, 173-187 (1996) · Zbl 0851.54039
[37] Park, S.; Kim, H., Foundations of the KKM theory on generalized convex spaces, J. Math. Anal. Appl., 209, 551-571 (1997) · Zbl 0873.54048
[38] Patriche, M., Equilibria of free abstract fuzzy economies, An. Stiint. Univ. Ovidius Constanta, 17, 143-154 (2009) · Zbl 1199.91135
[39] Tan, K. K.; Yu, J.; Z Yuan, X., The stability of coincident points for multivalued mappings, Nonlinear Anal. TMA, 25, 163-198 (1995) · Zbl 0856.54045
[40] Tan, K. K.; Zhang, X. L., Fixed point theorem on G-convex spaces and applications, Nonlinear Funct. Anal. Appl., 1, 1-19 (1996)
[41] Wang, L.; Huang, N. J.; Lee, C. S., Some new existence theorems of generalized abstract fuzzy economies with applications, Taiwanese J. Math., 14, 47-57 (2010) · Zbl 1192.91140
[42] Yu, C.; Yu, J., On structural stability and robustness to bounded rationality, Nonlinear Anal. TMA, 65, 583-592 (2006) · Zbl 1186.91053
[43] Yuan, X. Z., KKM Theory and Applications in Nonlinear Analysis (1999), Marcel Dekker, Inc: Marcel Dekker, Inc New York · Zbl 0936.47034
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