EQUILIBRIUM POINTS IN N-PERSON GAMES

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

strategies.

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.

source:

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