27 Sep 2015 sye   » (Apprentice)

By John F. Nash, Jr. , Princeton Univ.
Communicated by S. Lefschetz, Nov. 16, 1949

One may define a concept of an n-person game in which each player
has a finite set of pure strategies and in which a definite set of payments
to n players corresponds to each n-tuple of pure strategies, one strategy
being taken for each player. For mixed strategies, which are probability
distributions over the pure strategies, the pay-off functions are the
expectations of the players, thus becoming polylinear forms in the
probabilities with which the various players play their various pure

Any n-tuple of strategies, one for each player, may be regarded as a
point in the product space obtained by multiplying the- n strategy spaces
of the players. One:-such n-tuple counters another if the strategy of each
player in the countering n-tuple yields the highest obtainable expectation
for its player against, the n - 1 strategies of the other players in the
countered n-tuple. A self-countering n-tuple is called an equilibrium point.

The correspondence of each n-tuple with its set of countering n-tuples
gives a one-to-many mapping of the product space into itself. From the
definition of countering we-see that the set of countering points of a point
is convex. By using the continuity of the pay-off functions we see that the
graph of the mapping is closed. The closedness is equivalent to saying:
if Pi, P2, ... and Qi, Q2, .... Qn, ... are sequences of points in the product
space where Q. -n Q, P n P and Q,, counters P,, then Q counters P.

Since the graph is closed and since the-image of each point under the
mapping is convex, we infer from Kakutani's theorem' that the mapping
has a fixed point (i.e., point contained in its image). Hence there is an
equilibrium point.

In the two-person zero-sum case the "main theorem"2 and the existence
of, an equilibrium point are equivalent. In this case any two equilibrium
points lead to the-same expectations for the players, but this need not occur
in general.

* The author is indebted to Dr. David Gale for suggesting
the use of Kakutani's theorem to simplify the proof and to the A. E. C.
for financial support.
1. 'Kakutani, S., Duke Math. J., 8, 457-459 (1941).
2 Von Neumann, J., and Morgenstern, O., The Theory of Games and Economic Behaviour,
Chap. 3, Princeton University Press, Princeton, 1947.


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