We study the problem of allocating indivisible items to budget-constrained agents, aiming to provide fairness and efficiency guarantees. Specifically, our goal is to ensure that the resulting allocation is envy-free up to any item (EFx) while minimizing the amount of inefficiency that this needs to introduce. We first show that there exist two-agent problem instances for which no EFx allocation is Pareto efficient. We, therefore, turn to approximation and use the Nash social welfare maximizing allocation as a benchmark. For two-agent instances, we provide a procedure that always returns an EFx allocation while achieving the best possible approximation of the optimal Nash social welfare that EFx allocations can achieve. For the more complicated case of three-agent instances, we provide a procedure that guarantees EFx, while achieving a constant approximation of the optimal Nash social welfare for any number of items.

翻译：我们研究在预算约束下将不可分割物品分配给多个主体的问题，旨在兼顾公平性与效率保障。具体而言，目标是确保最终分配满足任意物品无嫉妒性（EFx），同时最小化由此引入的效率损失。首先证明，存在两个主体的任务实例，其中任何EFx分配均无法实现帕累托最优。因此，我们转向近似方法，并以最大化纳什社会福利的分配作为基准。针对两个主体的实例，我们提出一种程序，该程序始终返回EFx分配，同时实现EFx分配所能达到的最优纳什社会福利的最佳可能近似。对于更为复杂的三个主体实例，我们给出一种程序，该程序保证EFx性质，并在任意物品数量下实现对最优纳什社会福利的常数近似。