On the application of dynamic programming to the determination of optimal play in chess and checkers

One of the fundamental concepts in mathematics is that of transformation. The study of the unfolding over time of a physical process leads naturally to investigations of the effects of the repetition of a transformation, which is to say to the study of multistage processes. Much of classical and contemporary analysis stems from this source: iteration, ergodic theory, the theory of semigroups [1], the theory of branching processes [2], random transformations at fixed times and deterministic transformations at stochastic times [3, 4]. We wish to indicate still another direction of research, that of multistage decision processes. What happens when we allow a choice of the transformation to be employed at each time?