Indexability and Rollout Policy for Multi-State Partially Observable Restless Bandits

Restless multi-armed bandits with partially observable states has applications in communication systems, age of information and recommendation systems. In this paper, we study multi-state partially observable restless bandit models. We consider three different models based on information observable to decision maker -- 1) no information is observable from actions of a bandit 2) perfect information from bandit is observable only for one action on bandit, there is a fixed restart state, i.e., transition occurs from all other states to that state 3) perfect state information is available to decision maker for both actions on a bandit and there are two restart state for two actions. We develop the structural properties. We also show a threshold type policy and indexability for model 2 and 3. We present Monte Carlo (MC) rollout policy. We use it for whittle index computation in case of model 2. We obtain the concentration bound on value function in terms of horizon length and number of trajectories for MC rollout policy. We derive explicit index formula for model 3. We finally describe Monte Carlo rollout policy for model 1 when it is difficult to show indexability. We demonstrate the numerical examples using myopic policy, Monte Carlo rollout policy and Whittle index policy. We observe that Monte Carlo rollout policy is good competitive policy to myopic.