In denumerable state, denumerable action sequential decision problems in which the reward function has uniformly bounded second moment, the optimal reward for the decisionmaker who restricts himself to the countable set of stationary policies consisting of those that choose some arbitrary action at all but a finite number of states will be the same as the optimal reward for the decisionmaker who optimizes overall stationary policies. Under some further restriction, he can do almost as well simply by solving a large finite state truncation of the orginal problem. 13 pp. Ref.
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