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We express Gittins indices for multi-armed bandit problems as Laurent expansions around discount factor 1. The coefficients of these expansions are then used to characterize stationary optimal policies when the Ž optimality criteria are sensitive-discount optimality otherwise known as . Blackwell optimality , average-reward optimality and average-overtaking optimality. We also obtain bounds and derive optimality conditions for policies of a type that continue playing the same bandit as long as the state of that bandit remains in prescribed sets.

Finite state multi-armed bandit problems: sensitive-discount, average-reward and average-overtaking optimality