Equilibrium points in N-person games

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 the 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.