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. To better achieve their dual aims of equity and efficiency, social planners intend to maximize the minimum fill rate across agents, where each agent's fill rate is determined by a one-time allocation that must be irrevocably decided upon its arrival. For an arbitrarily correlated sequence of demands, we establish upper bounds on both the expected minimum fill rate (ex-post fairness) and the minimum expected fill rate (ex-ante fairness) achievable by any policy. Our bounds are parameterized by the number of agents and the expected demand-to-supply ratio. Further, we show that for any set of parameters, a simple adaptive policy of projected proportional allocation achieves the best possible fairness guarantee, ex post as well as ex ante. We obtain the performance guarantees of our proposed adaptive policy by inductively designing lower-bound functions on its corresponding value-to-go. Our policy is transparent and easy to implement, as it does not rely on distributional information beyond the first conditional moments. Despite our policy's simplicity, we demonstrate that it provides significant improvement over the class of non-adaptive target-fill-rate policies by characterizing the performance of the optimal such policy. We complement our theoretical developments with a numerical study motivated by the rationing of COVID-19 medical supplies based on a standard SEIR model approach that is commonly used to forecast pandemic trajectories. In such a setting, our simple adaptive policy significantly outperforms its theoretical guarantee as well as the optimal target-fill-rate policy.