Restless and collapsing bandits are commonly used to model constrained resource allocation in settings featuring arms with action-dependent transition probabilities, such as allocating 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 simulations on a broader class of synthetic transition matrices.