Berge’s maximum theorem to vector-valued functions with some applications. (English) Zbl 1412.47027
Summary: In this paper, we introduce pseudocontinuity for Berge’s maximum theorem for vector-valued functions which is weaker than semicontinuity. We prove the Berge’s maximum theorem for vector-valued functions with pseudocontinuity and obtain the set-valued mapping of the solutions is upper semicontinuous with nonempty and compact values. As applications, we derive some existence results for weakly Pareto-Nash equilibrium for multiobjective games and generalized multiobjective games both with pseudocontinuous vector-valued payoffs. Moreover, we obtain the existence of essential components of the set of weakly Pareto-Nash equilibrium for these discontinuous games in the uniform topological space of best-reply correspondences. Some examples are given to investigate our results.
MSC:
| 49J53 | Set-valued and variational analysis |
| 91A44 | Games involving topology, set theory, or logic |
| 90C29 | Multi-objective and goal programming |
| 47H04 | Set-valued operators |
| 90C31 | Sensitivity, stability, parametric optimization |
| 91A10 | Noncooperative games |
Keywords:
maximum theorem; vector-valued functions; set-valued mapping; pseudocontinuity; weakly Pareto-Nash equilibrium; essential componentsReferences:
| [1] | Aliprantis, C. D.; Border, K. C., Infinite dimensional analysis, A hitchhiker’s guide, Third edition, Springer, Berlin (2006) · Zbl 1156.46001 |
| [2] | Anh, N. L. H.; Khanh, P. Q., Calculus and applications of Studniarski’s derivatives to sensitivity and implicit function theorems, Control Cybernet., 43, 33-57 (2014) · Zbl 1318.49028 |
| [3] | Berge, C., Topological spaces, Including a treatment of multi-valued functions, vector spaces and convexity, Translated from the French original by E. M. Patterson, Reprint of the 1963 translation, Dover Publications, Inc., Mineola, NY (1997) · Zbl 0114.38602 |
| [4] | Carbonell-Nicolau, O., Essential equilibria in normal-form games, J. Econom. Theory, 145, 421-431 (2010) · Zbl 1202.91009 · doi:10.1016/j.jet.2009.06.002 |
| [5] | Dutta, P. K.; Mitra, T., Maximum theorems for convex structures with an application to the theory of optimal intertemporal allocation, J. Math. Econom., 18, 77-86 (1989) · Zbl 0679.90016 · doi:10.1016/0304-4068(89)90006-2 |
| [6] | Fan, K., Fixed-point and minimax theorems in locally convex topological linear spaces, Proc. Nat. Acad. Sci. U. S. A., 38, 121-126 (1952) · Zbl 0047.35103 · doi:10.1073/pnas.38.2.121 |
| [7] | Feinberg, E. A.; Kasyanov, P. O., Continuity of minima: local results, Set-Valued Var. Anal., 23, 485-499 (2015) · Zbl 1321.49030 · doi:10.1007/s11228-015-0318-7 |
| [8] | Feinberg, E. A.; Kasyanov, P. O.; Voorneveld, M., Berge’s maximum theorem for noncompact image sets, J. Math. Anal. Appl., 413, 1040-1046 (2014) · Zbl 1327.49031 · doi:10.1016/j.jmaa.2013.12.011 |
| [9] | Glicksberg, I. L., A further generalization of the Kakutani fixed theorem, with application to Nash equilibrium points, Proc. Amer. Math. Soc., 3, 170-174 (1952) · Zbl 0046.12103 · doi:10.2307/2032478 |
| [10] | Han, Y.; Huang, N.-J., Some characterizations of the approximate solutions to generalized vector equilibrium problems, J. Ind. Manag. Optim., 12, 1135-1151 (2016) · Zbl 1328.90147 · doi:10.3934/jimo.2016.12.1135 |
| [11] | Hillas, J., On the definition of the strategic stability of equilibria, Econometrica, 58, 1365-1390 (1990) · Zbl 0734.90125 · doi:10.2307/2938320 |
| [12] | Jiang, J.-H., Essential component of the set of fixed points of the multivalued mappings and its application to the theory of games, Sci. Sinica, 12, 951-964 (1963) · Zbl 0126.38604 · doi:10.1360/ya1963-12-7-951 |
| [13] | Kohlberg, E.; Mertens, J. F., On the strategic stability of equilibria, Econometrica, 54, 1003-1037 (1986) · Zbl 0616.90103 |
| [14] | Lin, Z., Essential components of the set of weakly Pareto-Nash equilibrium points for multiobjective generalized games in two different topological spaces, J. Optim. Theory Appl., 124, 387-405 (2005) · Zbl 1102.91024 · doi:10.1007/s10957-004-0942-0 |
| [15] | Lin, Z.; Yu, J., The existence of solutions for the system of generalized vector quasi-equilibrium problems, Appl. Math. Lett., 18, 415-422 (2005) · Zbl 1074.90041 · doi:10.1016/j.aml.2004.07.023 |
| [16] | Liu, C.-P.; Yang, X.-M., Characterizations of the approximate solution sets of nonsmooth optimization problems and its applications, Optim. Lett., 9, 755-768 (2015) · Zbl 1342.90186 · doi:10.1007/s11590-014-0780-4 |
| [17] | Luc, D. T., Theory of vector optimization, Lecture Notes in Economics and Mathematical Systems, Springer-Verlag, Berlin (1989) · Zbl 0695.90002 |
| [18] | Morgan, J.; Scalzo, V., Pseudocontinuous functions and existence of Nash equilibria, J. Math. Econom., 43, 174-183 (2007) · Zbl 1280.91010 · doi:10.1016/j.jmateco.2006.10.004 |
| [19] | Scalzo, V., Hadamard well-posedness in discontinuous non-cooperative games, J. Math. Anal. Appl., 360, 697-703 (2009) · Zbl 1175.91020 · doi:10.1016/j.jmaa.2009.07.007 |
| [20] | Scalzo, V., On the uniform convergence in some classes of non-necessarily continuous functions, Int. J. Contemp. Math. Sci., 4, 617-624 (2009) · Zbl 1191.54013 |
| [21] | Song, Q. Q., On essential sets of fixed points for functions, Numer. Funct. Anal. Optim., 36, 942-950 (2015) · Zbl 1356.54048 · doi:10.1080/01630563.2015.1043370 |
| [22] | Song, Q. Q.; Wang, L. S., On the stability of the solution for multiobjective generalized games with the payoffs perturbed, Nonlinear Anal., 73, 2680-2685 (2010) · Zbl 1193.91031 · doi:10.1016/j.na.2010.06.048 |
| [23] | Terazono, Y.; Matani, A., Continuity of optimal solution functions and their conditions on objective functions, SIAM J. Optim., 25, 2050-2060 (2015) · Zbl 1327.90224 · doi:10.1137/110850189 |
| [24] | Tian, G. Q.; Zhou, J.-X., The maximum theorem and the existence of Nash equilibrium of (generalized) games without lower semicontinuities, J. Math. Anal. Appl., 166, 351-364 (1992) · Zbl 0761.90110 · doi:10.1016/0022-247X(92)90302-T |
| [25] | Yang, Z., On existence and essential stability of solutions of symmetric variational relation problems, J. Inequal. Appl., 2014, 1-13 (2014) · Zbl 1372.49023 · doi:10.1186/1029-242X-2014-5 |
| [26] | Yang, H.; Yu, J., Essential components of the set of weakly Pareto-Nash equilibrium points, Appl. Math. Lett., 15, 553-560 (2002) · Zbl 1016.91008 · doi:10.1016/S0893-9659(02)80006-4 |
| [27] | Yu, J., Essential equilibria of n-person noncooperative games, J. Math. Econom., 31, 361-372 (1999) · Zbl 0941.91006 · doi:10.1016/S0304-4068(97)00060-8 |
| [28] | Yu, J., Game theory and nonlinear analysis (Continued), Science Press, Beijing (2011) |
| [29] | Yu, J.; Luo, Q., On essential components of the solution set of generalized games, J. Math. Anal. Appl., 230, 303-310 (1999) · Zbl 0916.90278 · doi:10.1006/jmaa.1998.6202 |
| [30] | Yu, J.; Xiang, S.-W., On essential components of the set of Nash equilibrium points, Nonlinear Anal., 38, 259-264 (1999) · Zbl 0967.91004 · doi:10.1016/S0362-546X(98)00193-X |
| [31] | Yu, J.; Yang, H.; Xiang, S.-W., Unified approach to existence and stability of essential components, Nonlinear Anal., 63, 1-2415 (2005) · Zbl 1224.90199 · doi:10.1016/j.na.2005.03.048 |
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