Decision Processes with Total-cost Criteria'

By a decision process is meant a pair (X, r), where X is an arbitrary set (the state space), and r associates to each point x in X an arbitrary nonempty collection of discrete probability measures (actions) on X. In a decision process with nonnegative costs depending on the current state, the action taken, and the following state, there is always available a Markov strategy which uniformly (nearly) minimizes the expected total cost. If the costs are strictly positive and depend only on the current state, there is even a stationary strategy with the same property. In a decision process with a fixed goal g in X, there is always a stationary strategy which uniformly (nearly) minimizes the expected time to the goal, and, if X is countable, such a stationary strategy exists which also (nearly) maximizes the probability of reaching the goal.