“Recursive” games were first defined and studied by Everett. Related results can be found in Gillette, Milnor and Shapley, and Blackwell and Ferguson. In this paper we introduce the notion of a recursive matrix game, which we believe eliminates the vagueness but none of the useful generality of the earlier definition. We then give an inductive proof (different from the proof in [3]) that these games have a value, with ∊-optimal stationary strategies available to each player. We also apply the result and show how a class of games studied in a different framework are games of this type and thus have a value.