We discuss a random walk in a convex set of Euclidean space ruled by two opponents. They may as usual independently choose a row and a column of the matrix of given random vectors. The surface of this set absorbs a moving point, and the payoff is defined in absorbation points. The determinateness of such games is proved with uniqueness theorems for Bellman-type functional equations under a somewhat artificial condition (cf. (66)). For the one-dimensional case (which is a generalization of Bellman–Milnor–Shapley’s “games of survival”) a more explicit analysis is given.Absorbation time is also considered as a payoff function both in one and multi-dimensional cases.