Equilibrium points in n-person games

Equilibrium Points in n-Person Games Author(s): John F. Nash Source: Proceedings of the National Academy of Sciences of the United States of America, Vol. 36, No. 1 (Jan. 15, 1950), pp. 48-49 Published by: National Academy of Sciences Stable URL: http://www.jstor.org/stable/88031 Accessed: 24/06/2009 02:35
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MATHEMATICS:

J. F. NASH, JR.

PROC.N. A. S.

This follows from the arguments used in a forthcoming paper.13 It is proved by constructing an "abstract" mapping cylinder of X and transcribing into algebraic terms the proof of the analogous theorem on CWcomplexes.
* This note arose from consultations during the tenure of a JohnSimon Guggenheim Memorial Fellowship by MacLane. 2 Whitehead, J. H. C., "Combinatorial Homotopy I and II," Bull. A.M.S., 55, 214-245 and 453-496 (1949). We refer to these papers as CH I and CH II, respectively. 3 By a complex we shall mean a connected CW complex, as defined in ?5 of CH I. We do not restrict ourselves to finite complexes. A fixed 0-cell e? e K? will be the base point for all thehomotopy groups in K. 4 MacLane, S., "Cohomology Theory in Abstract Groups III," Ann. Math., 50, 736-761 (1949), referred to as CT III. 5 An (unpublished) result like Theorem 1 for the homotopy type was obtained prior to these results by J. A. Zilber. 6 CT III uses in place of equation (2.4) the stronger hypothesis that XBcontains the center of A, but all the relevant developments there apply under theweaker assumption (2.4). 7 Eilenberg, S., and MacLane, S., "Cohomology Theory in Abstract Groups II," Ann. Math., 48, 326-341 (1947). 8 Eilenberg, S., and MacLane, S., "Determination of the Second Homology ... by
Means

9 Blakers, A. L., "Some Relations Between Homology and Homotopy Groups," Ann. Math., 49, 428-461 .(1948), ?12. 10The hypothesis of Theorem C, requiring that v-1 (1) not becyclic, can be readily realized by suitable choice of the free group X, but this hypothesis is not needed here
(cf. 6).
11 Eilenberg, S., and MacLane, S., "Homology of Spaces with Operators II," Trans. A.M.S., 65, 49-99 (1949); referred to as HSO II. 12 C(K) here is the C(K)'of CH II. Note that K exists and is a CW complex by (N) of p. 231 of CH I and that p-lKn = Kn, where p is the projection p:K --K. 13Whitehead, J. H. C., "Simple Homotopy Types." If W = 1, Theorem 5 follows from (17: 3) on p. 155 of S. Lefschetz, Algebraic Topology, (New York, 1942) and arguments in ?6 of J. H. C. Whitehead, "On Simply Connected 4-Dimensional Polyhedra" (Comm. Math. Helv., 22, 48-92 (1949)). However this proof cannot be generalized to 1. the case W

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