We study a dynamic allocation problem in which $T$ sequentially arriving divisible resources are to be allocated to a number of agents with linear utilities. The marginal utilities of each resource to the agents are drawn stochastically from a known joint distribution, independently and identically across time, and the central planner makes immediate and irrevocable allocation decisions. Most works on dynamic resource allocation aim to maximize the utilitarian welfare, i.e., the efficiency of the allocation, which may result in unfair concentration of resources on certain high-utility agents while leaving others' demands under-fulfilled. In this paper, aiming at balancing efficiency and fairness, we instead consider a broad collection of welfare metrics, the H\"older means, which includes the Nash social welfare and the egalitarian welfare. To this end, we first study a fluid-based policy derived from a deterministic surrogate to the underlying problem and show that for all smooth H\"older mean welfare metrics it attains an $O(1)$ regret over the time horizon length $T$ against the hindsight optimum, i.e., the optimal welfare if all utilities were known in advance of deciding on allocations. However, when evaluated under the non-smooth egalitarian welfare, the fluid-based policy attains a regret of order $\Theta(\sqrt{T})$. We then propose a new policy built thereupon, called Backward Infrequent Re-solving with Thresholding ($\mathsf{BIRT}$), which consists of re-solving the deterministic surrogate problem at most $O(\log\log T)$ times. We prove the $\mathsf{BIRT}$ policy attains an $O(1)$ regret against the hindsight optimal egalitarian welfare, independently of the time horizon length $T$. We conclude by presenting numerical experiments to corroborate our theoretical claims and to illustrate the significant performance improvement against several benchmark policies.

翻译：我们研究一个动态分配问题，其中$T$个顺序到达的可分割资源需分配给多个具有线性效用的智能体。每个资源对智能体的边际效用独立同分布地从一个已知联合分布中随机抽取，中央规划者需做出即时且不可撤销的分配决策。大多数动态资源分配研究旨在最大化功利主义福利，即分配效率，但这可能导致资源不公平地集中于某些高效用智能体，而其他智能体的需求得不到满足。本文旨在平衡效率与公平，转而考虑一系列广泛的福利指标——Hölder均值，其中包括纳什社会福利和平等主义福利。为此，我们首先研究一种基于流体策略的方法，该方法源自底层问题的确定性近似，并证明对于所有光滑的Hölder均值福利指标，该策略在时间跨度$T$上相对于后见最优（即若所有效用事先已知时的最优分配福利）取得$O(1)$的遗憾。然而，在非光滑的平等主义福利指标评估下，基于流体策略的遗憾达到$\Theta(\sqrt{T})$。随后，我们提出一种基于此的新策略，称为带阈值的后向不频繁重新求解（$\mathsf{BIRT}$），该策略最多$O(\log\log T)$次重新求解确定性近似问题。我们证明$\mathsf{BIRT}$策略在后见最优平等主义福利上取得$O(1)$的遗憾，且与时间跨度$T$无关。最后，我们通过数值实验验证理论结果，并展示该策略相对于若干基准策略的显著性能提升。