Rufur I r a a c r l
The RAND Corporation
The general problem of a marksman versus a mobile target, with a time lag in the gunner's information as to the target's position, appears in many guises in many situations. It is a classic military problem. Formulated in t e r m s of game theory, the desiderata are: How should the target best maneuver to confound prediction of his position7 How and when should the marksman make this prediction? What hit probability is to be expected when both participants behave optimally?
This paper discusses this general c l a s s of problems and then settles on one which s e e m s to be the simplest possible example that is not trivial. Nevertheless it is difficult. In two previous papers devoted to it, the evader's best strategy and value of the game were given. Here the emphasis is on the marksman. He h a s no optimal strategy, but does have a n ideal strategy with the property that every near optimal strategy is close to it. He also has a class of passive โฌ-strategies such that if and only if he obeys their dictates will he either come within E of the best hit probabilityor else always remain in a position where i t is possible t o do so.
2For peripheral positions we should make a correction if we take any inaccuracy of bomber into account. We will not do so here.
8. THE TRUNCATED VERSIONS
We can apply a technique here that proved useful in [2]; we amend the rules by requiring that P can fire only on the first n moves. The max min functions will be replaced by f n ( r ) and hn (r). The functional equations (6) and (7) become recurrence relations; we affix to each f or h the subscript n + 1 when it appears on the left side and n when it appears on the right. We may take fo(r) and ho(r) as 0; all the functions are then determined recurrently.