On Cournot-Nash equilibrium distributions for games with a nonmetrizable action space and upper semi-continuous payoffs

Khan, M. Ali

doi:10.1090/S0002-9947-1989-0970267-X

On Cournot-Nash equilibrium distributions for games with a nonmetrizable action space and upper semi-continuous payoffs
HTML articles powered by AMS MathViewer

by M. Ali Khan

Trans. Amer. Math. Soc. 315 (1989), 127-146

DOI: https://doi.org/10.1090/S0002-9947-1989-0970267-X

PDF | Request permission

Abstract:

We report results on the existence of a Cournot-Nash equilibrium distribution for games in which the action space is not necessarily metrizable and separable and the payoff functions are not necessarily continuous. Our work relies on the theory of Radon measures as developed by Schwartz-Topsoe and on the epitopology as developed by Dolecki-Salinetti-Wets

References

Sterling K. Berberian, Measure and integration, The Macmillan Company, New York; Collier Macmillan Ltd., London, 1965. MR 0183839

Topological spaces

Patrick Billingsley, Convergence of probability measures, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0233396
C. Castaing and M. Valadier, Convex analysis and measurable multifunctions, Lecture Notes in Mathematics, Vol. 580, Springer-Verlag, Berlin-New York, 1977. MR 0467310, DOI 10.1007/BFb0087685
Partha Dasgupta and Eric Maskin, The existence of equilibrium in discontinuous economic games. I. Theory, Rev. Econom. Stud. 53 (1986), no. 1, 1–26. MR 823958, DOI 10.2307/2297588
Szymon Dolecki, Gabriella Salinetti, and Roger J.-B. Wets, Convergence of functions: equi-semicontinuity, Trans. Amer. Math. Soc. 276 (1983), no. 1, 409–430. MR 684518, DOI 10.1090/S0002-9947-1983-0684518-7
James Dugundji, Topology, Allyn and Bacon, Inc., Boston, Mass., 1966. MR 0193606
G. A. Edgar, Disintegration of measures and the vector-valued Radon-Nikodým theorem, Duke Math. J. 42 (1975), no. 3, 447–450. MR 412381, DOI 10.1215/S0012-7094-75-04242-8
Sergiu Hart, Werner Hildenbrand, and Elon Kohlberg, On equilibrium allocations as distributions on the commodity space, J. Math. Econom. 1 (1974), no. 2, 159–166. MR 436929, DOI 10.1016/0304-4068(74)90006-8
Werner Hildenbrand, Core and equilibria of a large economy, Princeton Studies in Mathematical Economics, No. 5, Princeton University Press, Princeton, N.J., 1974. With an appendix to Chapter 2 by K. Hildenbrand. MR 0389160

Probablity in

spaces

Erwin Klein and Anthony C. Thompson, Theory of correspondences, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1984. Including applications to mathematical economics; A Wiley-Interscience Publication. MR 752692
M. Ali Khan, Equilibrium points of nonatomic games over a Banach space, Trans. Amer. Math. Soc. 293 (1986), no. 2, 737–749. MR 816322, DOI 10.1090/S0002-9947-1986-0816322-3
M. Ali Khan and Nikolaos S. Papageorgiou, On Cournot-Nash equilibria in generalized qualitative games with a continuum of players, Nonlinear Anal. 11 (1987), no. 6, 741–755. MR 893778, DOI 10.1016/0362-546X(87)90040-X
M. Ali Khan and Nikolaos S. Papageorgiou, On Cournot-Nash equilibria in generalized qualitative games with a continuum of players, Nonlinear Anal. 11 (1987), no. 6, 741–755. MR 893778, DOI 10.1016/0362-546X(87)90040-X
Andreu Mas-Colell, On a theorem of Schmeidler, J. Math. Econom. 13 (1984), no. 3, 201–206. MR 777118, DOI 10.1016/0304-4068(84)90029-6
Kiiti Morita, Products of normal spaces with metric spaces, Math. Ann. 154 (1964), 365–382. MR 165491, DOI 10.1007/BF01362570

Modern general topology

K. R. Parthasarathy, Probability measures on metric spaces, Probability and Mathematical Statistics, No. 3, Academic Press, Inc., New York-London, 1967. MR 0226684

On the hypo-convergence of probability measures

David Schmeidler, Equilibrium points of nonatomic games, J. Statist. Phys. 7 (1973), 295–300. MR 465225, DOI 10.1007/BF01014905
Laurent Schwartz, Radon measures on arbitrary topological spaces and cylindrical measures, Tata Institute of Fundamental Research Studies in Mathematics, No. 6, Published for the Tata Institute of Fundamental Research, Bombay by Oxford University Press, London, 1973. MR 0426084
Flemming Topsøe, Topology and measure, Lecture Notes in Mathematics, Vol. 133, Springer-Verlag, Berlin-New York, 1970. MR 0422560, DOI 10.1007/BFb0069481

On a compactness theorem for epi-convergent sequences of functions

Stephen Willard, General topology, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1970. MR 0264581