Fair Dynamic Rationing

We study the allocative challenges that governmental and nonprofit organizations face when tasked with equitable and efficient rationing of a social good among agents whose needs (demands) realize sequentially and are possibly correlated. As one example, early in the COVID-19 pandemic, the Federal Emergency Management Agency faced overwhelming, temporally scattered, a priori uncertain, and correlated demands for medical supplies from different states. In such contexts, social planners aim to maximize the minimum fill rate across sequentially arriving agents, where each agent's fill rate is determined by an irrevocable, one-time allocation. For an arbitrarily correlated sequence of demands, we establish upper bounds on the expected minimum fill rate (ex-post fairness) and the minimum expected fill rate (ex-ante fairness) achievable by any policy. Our upper bounds are parameterized by the number of agents and the expected demand-to-supply ratio, yet we design a simple adaptive policy called projected proportional allocation (PPA) that simultaneously achieves matching lower bounds for both objectives (ex-post and ex-ante fairness), for any set of parameters. Our PPA policy is transparent and easy to implement, as it does not rely on distributional information beyond the first conditional moments. Despite its simplicity, we demonstrate that the PPA policy provides significant improvement over the canonical class of non-adaptive target-fill-rate policies. We complement our theoretical developments with a numerical study motivated by the rationing of COVID-19 medical supplies based on a standard SEIR modeling approach that is commonly used to forecast pandemic trajectories. In such a setting, our PPA policy significantly outperforms its theoretical guarantee as well as the optimal target-fill-rate policy.