Indexability of Finite State Restless Multi-Armed Bandit and Rollout Policy

TL;DR

This paper investigates the indexability of finite state restless multi-armed bandits, analyzing structural properties, proposing algorithms for verification, and comparing index and rollout policies with myopic approaches.

Contribution

It introduces a structural analysis of single-armed restless bandits, proposes an algorithm for indexability verification, and compares the performance of index and rollout policies.

Findings

Indexability is established in certain cases.

The proposed algorithm effectively verifies indexability.

Index and rollout policies outperform myopic policy in simulations.

Abstract

We consider finite state restless multi-armed bandit problem. The decision maker can act on M bandits out of N bandits in each time step. The play of arm (active arm) yields state dependent rewards based on action and when the arm is not played, it also provides rewards based on the state and action. The objective of the decision maker is to maximize the infinite horizon discounted reward. The classical approach to restless bandits is Whittle index policy. In such policy, the M arms with highest indices are played at each time step. Here, one decouples the restless bandits problem by analyzing relaxed constrained restless bandits problem. Then by Lagrangian relaxation problem, one decouples restless bandits problem into N single-armed restless bandit problems. We analyze the single-armed restless bandit. In order to study the Whittle index policy, we show structural results on the single armed bandit model. We define indexability and show indexability in special cases. We propose an alternative approach to verify the indexable criteria for a single armed bandit model using value iteration algorithm. We demonstrate the performance of our algorithm with different examples. We provide insight on condition of indexability of restless bandits using different structural assumptions on transition probability and reward matrices. We also study online rollout policy and discuss the computation complexity of algorithm and compare that with complexity of index computation. Numerical examples illustrate that index policy and rollout policy performs better than myopic policy.