Outcome-Fair Restless Multi-Armed Bandits for Stochastic Deadline Scheduling

Whittle index policies offer a scalable method for restless multi-armed bandits, but under partial observability even determining the indifference subsidy at a single belief requires solving an infinite-horizon belief-state problem with no closed-form value function. Liu [10] addresses this difficulty by linearizing the unknown decision boundary, leading to a linear system and a closed-form approximate Whittle index. However, the resulting threshold uses only a one-step active--passive comparison and does not account for longer-horizon continuation values. We extend this framework to a \emph{$t$-step lookahead threshold policy}. For each subsidy $m$, the threshold is defined by the active-minus-passive advantage under $t$-step finite-horizon value iteration. At $t=1$, the threshold is $m$-independent and recovers the linear threshold of Liu [10]; for $t>1$, it becomes subsidy-dependent through the induced first-crossing structure and tracks the exact decision boundary more closely. The proposed algorithm does not require indexability as an input and includes an indexability verification. Under the original Whittle indexability, we prove that the $t$-step approximate Whittle index converges geometrically to the exact Whittle index, $$|\widehat W_t(ω)-W(ω)|=O(β^t).$$ Numerically, all 2,715 tested three-state instances are verified as indexable according to the proposed criterion. The P95 index error decreases from $2.18\times10^{-2}$ at $t=1$ to $8.93\times10^{-4}$ at $t=8$. In an exact-comparable instance with $β=0.9999$, $t=2$ already recovers the exact Whittle-index ordering. Moderate-depth threshold policies also outperform the one-step baseline and remain close to the optimal dynamic-programming benchmark, while runtime grows mildly with $t$.

Minimum-exposure constraints arise in recommendation, content curation, and regulated allocation when each provider, arm, or group must receive guaranteed exposure inside a period rather than only in aggregate. We study stochastic bandits with exact exposure floors and show that the right object is a rounding problem: a fractional fair schedule is realized as integral pulls, and the exposure error is exactly a discrepancy vector. The main contribution is a blockwise model with time-varying floors. BDQ-UCB satisfies every block floor deterministically and has fair regret governed by the nonmandatory budget $R$, not the horizon $T$, with high-probability regret $O(\sqrt{KR\log(KT)})$. A MOSS residual variant attains $O(\sqrt{KR})$, and a matching lower bound gives the minimax rate $Θ(\sqrt{KR})$, even with positive mandatory exposure; a kl-UCB$^{++}$ residual rule adds instance-dependent optimality. The formulation becomes essential for overlapping group floors: per-arm rounding can violate a group constraint by $Ω(s)$ in the group size, whereas Beck--Fiala null-space rounding meets every group floor within the block budget with violation below the arm degree $t$, and composes with UCB at the same $R$-parametrized regret. For learned group plans, we close disjoint systems at $\widetildeΘ(\sqrt{KT})$, give a dual-ledger decomposition explaining why naive index rules fail under overlap, and prove a plan-sampling rule that is pathwise feasible under an initial cover-slack condition and attains a conditional $\widetilde O(\sqrt{KT})$ guarantee, leaving the condition-free overlap rate open. Experiments on synthetic floors, MovieLens-100k genre exposure, and deployment stress tests show exact feasibility without penalty tuning and regret competitive with tuned Lagrangian baselines.

We study the problem of fair online resource allocation, motivated by applications such as refugee resettlement and airline scheduling, where agents arrive sequentially and must be assigned to facilities with limited capacities. We introduce a model that maximizes the overall welfare subject to resource constraints and a Lipschitz fairness requirement, which ensures that similar agents arriving in the same batch receive similar expected outcomes. We first analyze the offline problem, proving that the value of the optimal fair allocation is at least an $Ω(1/γ)$ fraction of the optimal unfair allocation, where $γ$ is the fairness coefficient, thereby bounding the price of fairness. For the online setting, we propose an algorithm based on dual mirror descent that enforces fairness constraints within batches while estimating optimal dual variables. We prove that this algorithm achieves sublinear regret relative to the optimal offline fluid benchmark. Finally, we validate our theoretical results using real-world data from the Refugee Economies Programme, demonstrating the algorithm's performance and examining the trade-offs between welfare maximization and fairness enforcement.