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代理人，旨在提供公平性与效率保障。具体而言，我们的目标是确保最终分配满足对任意物品无嫉妒（EFx），同时最小化由此引入的效率损失。我们首先证明了存在两个代理人的问题实例，其中没有任何EFx分配是帕累托最优的。因此，我们转向近似方法，并以纳什社会福利最大化分配作为基准。对于两个代理人的实例，我们提出了一种始终返回EFx分配的程序，同时达到EFx分配所能实现的最优纳什社会福利的最佳可能近似。对于更复杂的三个代理人实例，我们提供了一种能保证EFx的程序，同时针对任意数量的物品实现了最优纳什社会福利的常数近似。