On maximizing the average time at a goal

In a decision process (gambling or dynamic programming problem) with finite state space and arbitrary decision sets (gambles or actions), there is always available a Markov strategy which uniformly (nearly) maximizes the average time spent at a goal. If the decision sets are closed, there is even a stationary strategy with the same property.Examples are given to show that approximations by discounted or finite horizon payoffs are not useful for the general average reward problem.