Equilibrium points of nonatomic games over a Banach space

Author:
M. Ali Khan

Journal:
Trans. Amer. Math. Soc. **293** (1986), 737-749

MSC:
Primary 90A14; Secondary 28C20, 46N05, 90D10

MathSciNet review:
816322

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Schmeidler's results on the equilibrium points of nonatomic games with strategy sets in Euclidean -space are generalized to nonatomic games with strategy sets in a Banach space. Our results also extend previous work of the author which assumed the Banach space to be separable and its dual to possess the Radon-Nikodým property. Our proofs use recent results in functional analysis.

**[1]**Robert J. Aumann,*Integrals of set-valued functions*, J. Math. Anal. Appl.**12**(1965), 1–12. MR**0185073****[2]**Truman F. Bewley,*The equality of the core and the set of equilibria in economies with infinitely many commodities and a continuum of agents*, Internat. Econom. Rev.**14**(1973), no. 2, 383–394. MR**0391904****[3]**C. Castaing and M. Valadier,*Convex analysis and measurable multifunctions*, Lecture Notes in Mathematics, Vol. 580, Springer-Verlag, Berlin-New York, 1977. MR**0467310****[4]**J. Diestel,*Remarks on weak compactness in*, Glasgow Math. J.**18**(1977), 87-91.**[5]**J. Diestel and J. J. Uhl Jr.,*Vector measures*, American Mathematical Society, Providence, R.I., 1977. With a foreword by B. J. Pettis; Mathematical Surveys, No. 15. MR**0453964****[6]**N. Dincleanu,*Linear operations on*-*spaces*, Vector and Operator Valued Measures and Applications (D. H. Tucker and H. B. Maynard, eds.), Academic Press, New York, 1973.**[7]**P. Dubey, A. Mas-Colell and M. Shubik,*Efficiency properties of strategic market games*:*An axiomatic approach*, J. Econom. Theory**22**(1980), 339-362.**[8]**Nelson Dunford and Jacob T. Schwartz,*Linear operators. Part I*, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. General theory; With the assistance of William G. Bade and Robert G. Bartle; Reprint of the 1958 original; A Wiley-Interscience Publication. MR**1009162****[9]**Ky. Fan,*Fixed-point and minimax theorems in locally convex topological linear spaces*, Proc. Nat. Acad. Sci. U. S. A.**38**(1952), 121–126. MR**0047317****[10]**I. L. Glicksberg,*A further generalization of the Kakutani fixed theorem, with application to Nash equilibrium points*, Proc. Amer. Math. Soc.**3**(1952), 170–174. MR**0046638**, 10.1090/S0002-9939-1952-0046638-5**[11]**Werner Hildenbrand,*Existence of equilibria for economies with production and a measure space of consumers*, Econometrica**38**(1970), 608–623. MR**0282632****[12]**Werner Hildenbrand,*Core and equilibria of a large economy*, Princeton University Press, Princeton, N.J., 1974. With an appendix to Chapter 2 by K. Hildenbrand; Princeton Studies in Mathematical Economics, No. 5. MR**0389160****[13]**C. J. Himmelberg,*Measurable relations*, Fund. Math.**87**(1975), 53–72. MR**0367142****[14]**M. Ali Khan,*Equilibrium points of nonatomic games over a nonreflexive Banach space*, J. Approx. Theory**43**(1985), no. 4, 370–376. MR**787315**, 10.1016/0021-9045(85)90112-1**[15]**-,*On the integration of set-valued mappings in a nonreflexive Banach space*. II, Johns Hopkins Working Paper No. 109, September 1982 (Simon Stevins, forthcoming).**[16]**Ronald Larsen,*Functional analysis: an introduction*, Marcel Dekker, Inc., New York, 1973. Pure and Applied Mathematics, No. 15. MR**0461069****[17]**P. Masani,*Measurability and Pettis integration in Hilbert spaces*, J. Reine Angew. Math.**297**(1978), 92–135. MR**0463394****[18]**Andreu Mas-Colell,*A model of equilibrium with differentiated commodities*, J. Math. Econom.**2**(1975), no. 2, 263–295. Papers presented at the Colloquium on Mathematical Economics (Univ. California, Berkeley, Calif., 1974). MR**0395752****[19]**David Schmeidler,*Equilibrium points of nonatomic games*, J. Statist. Phys.**7**(1973), 295–300. MR**0465225****[20]**Daniel H. Wagner,*Survey of measurable selection theorems*, SIAM J. Control Optimization**15**(1977), no. 5, 859–903. MR**0486391****[21]**Eugene Wesley,*Borel preference orders in markets with a continuum of traders*, J. Math. Econom.**3**(1976), no. 2, 155–165. MR**0439054****[22]**Albert Wilansky,*Modern methods in topological vector spaces*, McGraw-Hill International Book Co., New York, 1978. MR**518316**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
90A14,
28C20,
46N05,
90D10

Retrieve articles in all journals with MSC: 90A14, 28C20, 46N05, 90D10

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1986-0816322-3

Keywords:
Nonatomic game,
Gel'fand integral,
Diestel's theorem,
approximate pure strategy equilibrium

Article copyright:
© Copyright 1986
American Mathematical Society