Nash-type equilibrium theorems and competitive Nash-type equilibrium theorems. (English) Zbl 1103.91308

Lin, Lai-Jiu; Cheng, Shih Feng

Comput. Math. Appl. 44, No. 10-11, 1369-1378 (2002).

Summary: We consider two types of equilibrium problems. We study the constrained Nash-type equilibrium problems with multivalued payoff functions. We also study the competitive Nash-type equilibrium problems with multivalued payoff functions. In these two equilibrium problems, we want to find a strategy combination such that each player wishes to find a minimal loss from his multivalued payoff function. We use a fixed-point theorem of Park to prove the existence results of these two types of equilibrium problems.

MSC:

91A10	Noncooperative games
91A44	Games involving topology, set theory, or logic

Keywords:

Nash equilibrium problem; Closed multimap; \(C\)-quasiconvex; Continuous; selection

References:

[1]	Ding, X.P., Existence of Pareto equilibria for constrained multiobjective games in H-space, Computers math. applic., 39, 9/10, 125-134, (2000) · Zbl 0960.91007
[2]	Wang, S.Y., Existence of a Pareto equilibrium, J. optim. theory and appl., 79, 373-384, (1993) · Zbl 0797.90124
[3]	Wang, S.Y., An existence theorem of a Pareto equilibrium, Appl. math. lett., 4, 3, 61-63, (1991) · Zbl 0744.90112
[4]	Yu, J.; Yuan, G.X.-z., The study of Pareto equilibria for multiobjective games by fixed point and Ky Fan minimax inequality methods, Computers math. applic., 35, 9, 17-24, (1998) · Zbl 1005.91008
[5]	Nash, J., Noncooperative games, Ann. math., 54, 286-295, (1951) · Zbl 0045.08202
[6]	Debreu, G., A social equilibrium theorem, (), 386-393
[7]	Park, S., (), 248-277
[8]	Luc, D.T.; Vargas, C., A saddle point theorem for set-valued maps, Nonlinear anal., theory, methods & applications, 18, 1-7, (1992)
[9]	Lin, L.J.; Yu, Z.T., On generalized vector quasi-equilibrium problems for multimaps, J. computational and applied math., 129, 171-183, (2001)
[10]	Aubin, J.P.; Frankowska, H., Set-valued analysis, (1990), Springer-Verlag Birkhauser, Boston, MA
[11]	Lassonde, M., On the use of KKM multifunctions in fixed point theory and related topics, J. math. anal. appl., 97, 151-201, (1983) · Zbl 0527.47037
[12]	Horvath, C.D., Extensions and selection theorems in topological vector space with a generalized convexity structure, Ann. fac. sci. Toulouse, 2, 253-269, (1993) · Zbl 0799.54013
[13]	Lee, B.S.; Lee, G.M.; Chang, S.S., Generalized vector variational inequalities for multifunctions, (), 193-202 · Zbl 1012.47050
[14]	Luc, D.T., ()
[15]	Massey, W.S., Singular homology theory, (1980), Springer Verlag New York · Zbl 0377.55005
[16]	Fan, K., Fixed point and minimax theorems in locally convex topological linear spaces, Pro. nat. acad. sci. U.S.A., 38, 121-126, (1952) · Zbl 0047.35103