Fair Policy Optimization in Major-Minor Weakly Coupled Markov Decision Processes

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.

Fairness is an important aspect of decision-making in multi-objective reinforcement learning (MORL), where policies must ensure both optimality and equity across multiple, potentially conflicting objectives. While single-policy MORL methods can learn fair policies for fixed user preferences using welfare functions such as the generalized Gini welfare function (GGF), they fail to provide the diverse set of policies necessary for dynamic or unknown user preferences. To address this limitation, we formalize the fair optimization problem in multi-policy MORL, where the goal is to learn a set of Pareto-optimal policies that ensure fairness across all possible user preferences. Our key technical contributions are threefold: (1) We show that for concave, piecewise-linear welfare functions (e.g., GGF), fair policies remain in the convex coverage set (CCS), which is an approximated Pareto front for linear scalarization. (2) We demonstrate that non-stationary policies, augmented with accrued reward histories, and stochastic policies improve fairness by dynamically adapting to historical inequities. (3) We propose three novel algorithms, which include integrating GGF with multi-policy multi-objective Q-Learning (MOQL), state-augmented multi-policy MOQL for learning non-statoinary policies, and its novel extension for learning stochastic policies. We evaluate our algorithms across various domains and compare our methods against the state-of-the-art MORL baselines. The empirical results show that our methods learn a set of fair policies that accommodate different user preferences.

Dynamic multi-resource allocation is a central problem in shared computing environments, where users' demands arrive sequentially and resources must be distributed fairly without knowledge of future demands. Existing methods emphasize fairness guarantees such as Sharing Incentive, Envy Freeness, and Dynamic Pareto Optimality, but often overlook system utility. Moreover, these fairness criteria are mutually incompatible, preventing strict enforcement of them at the same time. We propose a neural allocation mechanism that reconciles fairness with utility through multi-objective optimization during sequential rollout. We first formalize fairness in the dynamic setting via stepwise loss functions for Sharing Incentive, Envy Freeness, and Dynamic Pareto Optimality, enabling differentiable training. Leveraging non-wastefulness, we parameterized the solutions by constraining allocations to the subspace of demand while allowing elastic over-allocation when resources remain available. Empirical results demonstrate that our learned allocator achieves substantially higher utility at comparable levels of fairness, uncovering clear Pareto-frontier-like tradeoffs across metrics.