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. 351-364.
Download (603kB) | Preview
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 so-called 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|
|Keywords:||Maximum Theorem; Existence; Nash Equilibrium; Lower Semicontinuities|
|Subjects:||D - Microeconomics > D5 - General Equilibrium and Disequilibrium > D50 - General|
|Depositing User:||Guoqiang Tian|
|Date Deposited:||19. Sep 2012 11:40|
|Last Modified:||17. Feb 2013 17:00|
J. P. Aubin, “Mathematical Methods of Games and Economic Theory,” North-Holland, 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, Edgeworth-Bertrand Duopoly revisited, in “Operations Research-Verfahren, III” (R. Henn, Ed.), pp. 55-68, 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), 313-316.
P. Dasgupta and E. Maskin, The existence of equilibrium in discontinuous economic games. I. Theory, Rev. Econom. Stud. 53 (1986), 1-26.
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, Fixed-point and minimax theorems in locally convex topological linear spaces, Proc. Nat. Acad. Sci. U.S.A. 38 (1952), 131-136.
K. Fan, Minimax theorem, Proc. Nat. Acad. Sci. U.S.A. 39 (1953), 42-47.
I. L. Glicksberg, A further generalization of the Kakutani fixed theorem with applications to Nash equilibrium points, Proc. Amer. Math. Soc. 38 (1952), 170-174.
M. Ali. Khan, Equilibrium points of nonatomic games over a Banach space, Trans. Amer. Math. Soc. 29 (1986), 737-749.
M. Ali. Khan and R. Vohra, Equilibrium in abstract economics without ordered preferences and with a measure of agents, J. Math. Econom. 13 (1984), 133-142.
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.