The Theory of Max-Min, with Applications

This paper is concerned mainly with two-stage Max-Min problems, in which the minimizing “player” acts after the maximizing “player” and with full knowledge of the choice of the maximizing player. Such problems arise in operations research for instance when defense installations must be built “in concrete” long before a battle, while the attack against them is made in full knowledge of what they are. Such problems are not games in the usual sense. To treat them it was necessary to treat a new kind of derivative and to study its peculiar properties. Using this derivative, this paper sets forth a general theory of Max-Min analogous to the elementary theory of maximizing for finite problems, applies this to find criteria in a particular long-unsolved military allocation problem, and finally indicates an application to economics.