  We consider the infinite-horizon, average-reward restless bandit problem in
discrete time. We propose a new class of policies that are designed to drive a
progressively larger subset of arms toward the optimal distribution. We show
that our policies are asymptotically optimal with an $O(1/\sqrt{N})$ optimality
gap for an $N$-armed problem, provided that the single-armed MDP is unichain
and aperiodic under the optimal single-armed policy. Our approach departs from
most existing work that focuses on index or priority policies, which rely on
the Uniform Global Attractor Property (UGAP) to guarantee convergence to the
optimum, or a recently developed simulation-based policy, which requires a
Synchronization Assumption (SA).
