A game over spaces of probability distributions

This paper analyzes a two person zero‐sum game in which the strategies on the two sides are probability distributions. The solutions always turn out to contain jumps. In most cases the distributions are combinations of delta functions and density functions. The problem is as follows: a submarine chooses a range r, within a declared war zone, at which to fire his missile. If he is detected at a larger range he attempts to fire at that larger range with the effectiveness at that range decreased by defense measures such as attempts to kill the submarine, shoot down the missile, or protect the target. If this defense effectiveness is denoted by ω, with ω = 0 referring to perfect defensive reaction measures, and ω = 1 referring to poor defensive reaction measures, the following is true. If ω = 0, the problem is analogous to a problem (“The Two Machine‐gun Duel”) solved by L. Gillman and the author in 1949 (Ref. 2) and is not difficult. The defenses in this case are in close to the coast. If ω = 1 the problem is different but not difficult and the defenses are well out towards (and in some cases at) maximum missile range. There are for ω = 1 no defenses near the coast; this is referred to as an “initial gap.”. The first problem that arises is to find for what values of ω is there an “initial gap.” A necessary and sufficient condition for such gaps was found. For general ω there may be instead an “interior gap,” i.e., defense contiguous to the coast and out near or at maximum missile range, but none at intermediate ranges. Necessary and sufficient conditions for such interior gaps were found. In a typical case the solution for the defense follows one function out to a certain target, then is zero, then follows another function to maximum missile range, then has a delta function at maximum missile range. It was necessary to study in detail the behavior of expressions involving discontinuous functions by breaking into continuous parts and discontinuous parts. A great deal of the complexity of the paper arises from this fact. A second cause is that it is essential to admit completely arbitrary distribution functions on the two sides as strategies. There is no a priori reason for instance, why the attacking player could not choose his plays from the Cantor distribution, a function increasing on (0, 1) from 0 to 1, but having almost everywhere a zero derivative. These possibilities emphasize the need for great care in the analysis and for the somewhat tedious existence proof of a later section of this paper.