Optimality of myopic policy for a class of monotone affine restless multi-armed bandits

We formulate a general class of restless multi-armed bandits with n independent and stochastically identical arms. Each arm is in a real-valued state s ∈ [s₀, s_max]. Selecting an arm with state s yields an immediate reward with expectation R(s). The state of the arm that is selected stochastically jumps from its current value s to either s_max or s₀ with probability p(s) or 1 - p(s) respectively. The state of the arms that are not selected evolve according to a function τ (s). We assume that τ (s), p(s), and R(s) are all monotonically increasing affine functions, and τ (s) is a contraction mapping. We then derive a condition on τ (s) under which the simple myopic policy, which selects at each time the arm with the highest immediate reward, is optimal. This extends and generalizes recent results in the literature pertaining to arms evolving as two-state Markov chains.