Characterization and computation of restless bandit marginal productivity indices

The Whittle index [P. Whittle (1988). Restless bandits: Activity allocation in a changing world. J. Appl. Probab. 25A, 287-298] yields a practical scheduling rule for the versatile yet intractable multi-armed restless bandit problem, involving the optimal dynamic priority allocation to multiple stochastic projects, modeled as restless bandits, i.e., binary-action (active/passive) (semi-) Markov decision processes. A growing body of evidence shows that such a rule is nearly optimal in a wide variety of applications, which raises the need to efficiently compute the Whittle index and more general marginal productivity index (MPI) extensions in large-scale models. For such a purpose, this paper extends to restless bandits the parametric linear programming (LP) approach deployed in [J. Niño-Mora. A (2/3)$n^{3}$ fast-pivoting algorithm for the Gittins index and optimal stopping of a Markov chain, INFORMS J. Comp., in press], which yielded a fast Gittins-index algorithm. Yet the extension is not straightforward, as the MPI is only defined for the limited range of socalled indexable bandits, which motivates the quest for methods to establish indexability. This paper furnishes algorithmic and analytical tools to realize the potential of MPI policies in largescale applications, presenting the following contributions: (i) a complete algorithmic characterization of indexability, for which two block implementations are given; and (ii) more importantly, new analytical conditions for indexability — termed LP-indexability — that leverage knowledge on the structure of optimal policies in particular models, under which the MPI is computed faster by the adaptive-greedy algorithm previously introduced by the author under the more stringent PCL-indexability conditions, for which a new fast-pivoting block implementation is given. The paper further reports on a computational study, measuring the runtime performance of the algorithms, and assessing by a simulation study the high prevalence of indexability and PCL-indexability.

Restless bandits, partial conservation laws and indexability

We show that if performance measures in a stochastic scheduling problem satisfy a set of so-called partial conservation laws (PCL), which extend previously studied generalized conservation laws (GCL), then the problem is solved optimally by a priority-index policy for an appropriate range of linear performance objectives, where the optimal indices are computed by a one-pass adaptive-greedy algorithm, based on Klimov's. We further apply this framework to investigate the indexability property of restless bandits introduced by Whittle, obtaining the following results: (1) we identify a class of restless bandits (PCL-indexable) which are indexable; membership in this class is tested through a single run of the adaptive-greedy algorithm, which also computes the Whittle indices when the test is positive; this provides a tractable sufficient condition for indexability; (2) we further indentify the class of GCL-indexable bandits, which includes classical bandits, having the property that they are indexable under any linear reward objective. The analysis is based on the so-called achievable region method, as the results follow from new linear programming formulations for the problems investigated.Stochastic scheduling, Markov decision chains, bandit problems, achievable region

Risk, Unexpected Uncertainty, and Estimation Uncertainty: Bayesian Learning in Unstable Settings

Recently, evidence has emerged that humans approach learning using Bayesian updating rather than (model-free) reinforcement algorithms in a six-arm restless bandit problem. Here, we investigate what this implies for human appreciation of uncertainty. In our task, a Bayesian learner distinguishes three equally salient levels of uncertainty. First, the Bayesian perceives irreducible uncertainty or risk: even knowing the payoff probabilities of a given arm, the outcome remains uncertain. Second, there is (parameter) estimation uncertainty or ambiguity: payoff probabilities are unknown and need to be estimated. Third, the outcome probabilities of the arms change: the sudden jumps are referred to as unexpected uncertainty. We document how the three levels of uncertainty evolved during the course of our experiment and how it affected the learning rate. We then zoom in on estimation uncertainty, which has been suggested to be a driving force in exploration, in spite of evidence of widespread aversion to ambiguity. Our data corroborate the latter. We discuss neural evidence that foreshadowed the ability of humans to distinguish between the three levels of uncertainty. Finally, we investigate the boundaries of human capacity to implement Bayesian learning. We repeat the experiment with different instructions, reflecting varying levels of structural uncertainty. Under this fourth notion of uncertainty, choices were no better explained by Bayesian updating than by (model-free) reinforcement learning. Exit questionnaires revealed that participants remained unaware of the presence of unexpected uncertainty and failed to acquire the right model with which to implement Bayesian updating

On the asymptotic optimality of greedy index heuristics for multi-action restless bandits

The class of restless bandits as proposed by Whittle (1988) have long been known to be intractable. This paper presents an optimality result which extends that of Weber and Weiss (1990) for restless bandits to a more general setting in which individual bandits have multiple levels of activation but are subject to an overall resource constraint. The contribution is motivated by the recent works of Glazebrook et al. (2011a), (2011b) who discussed the performance of index heuristics for resource allocation in such systems. Hitherto, index heuristics have been shown, under a condition of full indexability, to be optimal for a natural Lagrangian relaxation of such problems in which a resource is purchased rather than constrained. We find that under key assumptions about the nature of solutions to a deterministic differential equation that the index heuristics above are asymptotically optimal in a sense described by Whittle. We then demonstrate that these assumptions always hold for three-state bandits