An analog of the minimax theorem for vector payoffs

TLDR

In a two‑person zero‑sum game, von Neumann’s theorem guarantees that each player can secure an expected payoff of at least v (or at most v for the opponent), and these results describe how players can control the long‑run center of gravity of actual payoffs. This paper investigates whether, for matrices whose entries are points in ΛΓ‑space, a player can force the long‑run center of gravity of payoffs to lie within or arbitrarily close to a given set S in iV‑space with probability approaching one. The authors formulate the problem precisely and provide a complete solution in two cases: when JV = 1 and when the set S is convex.

Abstract

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