Tian, Guoqiang and Zhou, Jianxin (1990): The Maximum Theorem and the Existence of Nash Equilibrium of (Generalized) Games without Lower Semicontinuities. Published in: Journal of Mathematical Analysis and Applications , Vol. 166, (1992): pp. 351364.

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Abstract
In this paper we generalize Berge's Maximum Theorem to the case where the payoff (utility) functions and the feasible action correspondences are not lowersemicontinuous. The condition we introduced is called the Feasible Path Transfer Lower Semicontinuity (in short, FPT l.s.c.). By applying our Maximum Theorem to game theory and economics, we are able to prove the existence of equilibrium for the generalized games (the socalled abstract economics) and Nash equilibrium for games where the payoff functions and the feasible strategy correspondences are not lowersemicontinuous. Thus the existence theorems given in this paper generalize many existence theorems on Nash equilibrium and equilibrium for the generalized games in the literature.
Item Type:  MPRA Paper 

Original Title:  The Maximum Theorem and the Existence of Nash Equilibrium of (Generalized) Games without Lower Semicontinuities 
Language:  English 
Keywords:  Maximum Theorem; Existence; Nash Equilibrium; Lower Semicontinuities 
Subjects:  D  Microeconomics > D5  General Equilibrium and Disequilibrium > D50  General 
Item ID:  41300 
Depositing User:  Guoqiang Tian 
Date Deposited:  19 Sep 2012 11:40 
Last Modified:  22 Apr 2016 09:12 
References:  J. P. Aubin, “Mathematical Methods of Games and Economic Theory,” NorthHolland, Amsterdam, 1979. M. Baye, G. Tian and J. Zhou, “The Existence of Pure Nash Equilibrium in Games with Nonquasiconcave Payoffs,” Texas A&M University, mimeo, 1990. C. Berge, “Escapes Topologiques et Fonctions Multivoques,” Dunod, Paris, 1959. C. Berge, “Topological Spaces” (Translated by E.M. Patterson), Macmillan Co., New York, 1963. J. Bertrand, EdgeworthBertrand Duopoly revisited, in “Operations ResearchVerfahren, III” (R. Henn, Ed.), pp. 5568, Sonderdruck, Verlag, Anton Hein, Meisenhein, 1965. A. Borglin and H. Keiding, Existence of equilibrium actions and of equilibrium: A note on the ‘new’ existence theorem, J. Math Econom. 3 (1976), 313316. P. Dasgupta and E. Maskin, The existence of equilibrium in discontinuous economic games. I. Theory, Rev. Econom. Stud. 53 (1986), 126. G. Debreu, A social equilibrium existence theorem, Proc. Nat. Acad. Sci. U.S.A. 38 (1952). F.M. Edgeworth, “Papers Relating to Political Economy,” Vol. I, Macmillan & Co., London, 1953. K. Fan, Fixedpoint and minimax theorems in locally convex topological linear spaces, Proc. Nat. Acad. Sci. U.S.A. 38 (1952), 131136. K. Fan, Minimax theorem, Proc. Nat. Acad. Sci. U.S.A. 39 (1953), 4247. I. L. Glicksberg, A further generalization of the Kakutani fixed theorem with applications to Nash equilibrium points, Proc. Amer. Math. Soc. 38 (1952), 170174. M. Ali. Khan, Equilibrium points of nonatomic games over a Banach space, Trans. Amer. Math. Soc. 29 (1986), 737749. M. Ali. Khan and R. Vohra, Equilibrium in abstract economics without ordered preferences and with a measure of agents, J. Math. Econom. 13 (1984), 133142. M. Ali. Khan and P. Papageorgiou, “on Cournot Nash Equilibria in Generalized Quantitative Games with a Continuum of Players,” University of Illinois, Urbana, mimeo, 1985. 
URI:  https://mpra.ub.unimuenchen.de/id/eprint/41300 