A Verification Theorem for Threshold-Indexability of Real-State Discounted Restless Bandits

Abstract

The Whittle index, which characterizes optimal policies for controlling certain single restless bandit projects (a Markov decision process with two actions: active and passive) is the basis for a widely used heuristic index policy for the intractable restless multiarmed bandit problem. Yet two roadblocks need to be overcome to apply such a policy: the individual projects in the model at hand must be shown to be indexable, so that they possess a Whittle index; and the index must be evaluated. Such roadblocks can be especially vexing when project state spaces are real intervals, as in recent sensor scheduling applications. This paper presents sufficient conditions for indexability (relative to a generalized Whittle index) of general real-state discrete-time restless bandits under the discounted criterion, which are not based on elucidating properties of the optimal value function and do not require proving beforehand optimality of threshold policies as in prevailing approaches. The main contribution is a verification theorem establishing that, if project performance metrics under threshold policies and an explicitly defined marginal productivity (MP) index satisfy three conditions, then the project is indexable with its generalized Whittle index being given by the MP index, and threshold policies are optimal for dynamic project control.