Planning to Fairly Allocate: Probabilistic Fairness in the Restless Bandit Setting

Restless and collapsing bandits are commonly used to model constrained resource allocation in settings featuring arms with action-dependent transition probabilities, such as the allocation of health interventions among patients [Whittle, 1988; Mate et al., 2020]. However, state-of-the-art Whittle-index-based approaches to this planning problem either do not consider fairness among arms or incentivize fairness without guaranteeing it [Mate et al., 2021]. Additionally, their optimality guarantees only apply when arms are indexable and threshold-optimal. We demonstrate that the incorporation of hard fairness constraints necessitates the coupling of arms, which undermines the tractability, and by extension, indexability of the problem. We then introduce ProbFair, a probabilistically fair stationary policy that maximizes total expected reward and satisfies the budget constraint, while ensuring a strictly positive lower bound on the probability of being pulled at each timestep. We evaluate our algorithm on a real-world application, where interventions support continuous positive airway pressure (CPAP) therapy adherence among obstructive sleep apnea (OSA) patients, as well as on a broader class of synthetic transition matrices.