Fair Resource Allocation in Weakly Coupled Markov Decision Processes

We consider fair resource allocation in sequential decision-making environments modeled as weakly coupled Markov decision processes, where resource constraints couple the action spaces of $N$ sub-Markov decision processes (sub-MDPs) that would otherwise operate independently. We adopt a fairness definition using the generalized Gini function instead of the traditional utilitarian (total-sum) objective. After introducing a general but computationally prohibitive solution scheme based on linear programming, we focus on the homogeneous case where all sub-MDPs are identical. For this case, we show for the first time that the problem reduces to optimizing the utilitarian objective over the class of "permutation invariant" policies. This result is particularly useful as we can exploit efficient algorithms that optimizes the utilitarian objective such as Whittle index policies in restless bandits to solve the problem with this fairness objective. For more general settings, we introduce a count-proportion-based deep reinforcement learning approach. Finally, we validate our theoretical findings with comprehensive experiments, confirming the effectiveness of our proposed method in achieving fairness.