An analog of the minimax theorem for vector payoffs.

for all i, j . Thus in the (two-person, zero-sum) game with matrix Λf, player I has a strategy insuring an expected gain of at least v, and player II has a strategy insuring an expected loss of at most v. An alternative statement, which follows from the von Neumann theorem and an appropriate law of large numbers is that, for any e>0, I can, in a long series of plays of the game with matrix M, guarantee, with probability approaching 1 as the number of plays becomes infinite, that his average actual gain per play exceeds v — e and that II can similarly restrict his average actual loss to v-he. These facts are assertions about the extent to which each player can control the center of gravity of the actual payoffs in a long series of plays. In this paper we investigate the extent to which this center of gravity can be controlled by the players for the case of matrices M whose elements m(i9 j) are points of ΛΓ-space. Roughly, we seek to answer the following question. Given a matrix M and a set S in iV-space, can I guarantee that the center of gravity of the payoffs in a long series of plays is in or arbitrarily near St with probability approaching 1 as the number of plays becomes infinite ? The question is formulated more precisely below, and a complete solution is given in two cases: the case JV=1 and the case of convex S. Let