Let <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">{X_{n}}</tex> be a Markov process with finite state space and transition probabilities <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p_{ij}(u_{i}, v_{i})</tex> depending on u <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</inf> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">v_{i}.</tex> State 0 is the capture state (where the game ends; <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p_{oi} \equiv \delta_{oi})</tex> ; <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u = {u_{i}}</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">v = {v_{i}}</tex> are the pursuer and evader strategies, respectively, and are to be chosen so that capture is advanced or delayed and the cost <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C_{i^{u,v}} = E[\Sum_{0}^{\infty} k (u(X_{n}), v(X_{n}), X_{n}) | X_{0} = i]</tex> is minimaxed (or maximined), where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k(\alpha, \beta, 0) \equiv 0</tex> . The existence of a saddle point and optimal strategy pair or e-optimal strategy pair is considered under several conditions. Recursive schemes for computing the optimal or ε-optimal pairs are given.