Uniformly Bounded Regret in Dynamic Fair Allocation

We study a dynamic allocation problem in which $T$ sequentially arriving divisible resources need to be allocated to $n$ fixed agents with additive utilities. Agents' utilities are drawn stochastically from a known distribution, and decisions are made immediately and irrevocably. Most works on dynamic resource allocation aim to maximize the utilitarian welfare of the agents, which may result in unfair concentration of resources at select agents while leaving others' demands under-fulfilled. In this paper, we consider the egalitarian welfare objective instead, which aims at balancing the efficiency and fairness of the allocation. To this end, we first study a fluid-based policy derived from a deterministic approximation to the underlying problem and show that it attains a regret of order $\Theta(\sqrt{T})$ against the hindsight optimum, i.e., the optimal egalitarian allocation when all utilities are known in advance. We then propose a new policy, called Backward Infrequent Re-solving with Thresholding ($\mathsf{BIRT}$), which consists of re-solving the fluid problem at most $O(\log\log T)$ times. We prove the $\mathsf{BIRT}$ policy attains $O(1)$ regret against the hindsight optimum, independently of the time horizon length $T$ and initial welfare. We also present numerical experiments to illustrate the significant performance improvement against several benchmark policies.