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Equilibrium points in nperson games

Communicated by S. Lefschetz, November 16, 1949
One may define a concept of an nperson game in which each player has a finite set of pure strategies and in which a definite set of payments to the n players corresponds to each ntuple of pure strategies, one strategy being taken for each player. For mixed strategies, which are probability distributions over the pure strategies, the payoff 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 ntuple 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 ntuple counters another if the strategy of each player in the countering ntuple yields the highest obtainable expectation for its player against the n − 1 strategies of the other players in the countered ntuple. A selfcountering ntuple is called an equilibrium point.
The correspondence of each ntuple with its set of countering ntuples gives a onetomany 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 payoff functions we see that the graph of the mapping is closed. The closedness is equivalent to saying: if P_{1}, P_{2}, … and Q_{1}, Q_{2}, …, Q_{n}, … are sequences of points in the product space where Q_{n} → Q, P_{n} → P and Q_{n} counters P_{n} 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^{1} that the mapping has a fixed point (i.e., point contained in its image). Hence there is an equilibrium point.
In the twoperson zerosum 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.
Footnotes

↵*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.