We study the problem of allocating a set of indivisible items among agents whose preferences include externalities. Unlike the standard fair division model, agents may derive positive or negative utility not only from items allocated directly to them, but also from items allocated to other agents. Since exact envy-freeness cannot be guaranteed, prior work has focused on its relaxations. However, two central questions remained open: does there always exist an allocation that is envy-free up to one item (EF1), and if not, what is the optimal relaxation EF-$k$ that can always be attained? We settle both questions by deriving tight asymptotic bounds on the number of items sufficient to eliminate envy. We show that for any instance with $n$ agents, an allocation that is envy-free up to $O(\sqrt{n})$ items always exists and can be found in polynomial time, and we prove a matching $Ω(\sqrt{n})$ lower bound showing that this result is tight even for binary valuations, which rules out the existence of EF1 allocations when agents have externalities.

翻译：本文研究在偏好包含外部性的智能体之间分配不可分物品的问题。与标准公平分配模型不同，智能体不仅从直接分配给自己的物品中获得正效用或负效用，也可能从分配给其他智能体的物品中获得效用。由于无法保证精确的无嫉妒性，先前研究主要关注其松弛形式。然而，两个核心问题始终悬而未决：是否存在总是满足"至多一件物品差异的无嫉妒性"（EF1）的分配方案？若不存在，可始终实现的最优松弛形式EF-$k$中的$k$值是多少？我们通过推导消除嫉妒所需物品数量的紧渐近界，同时解决了这两个问题。我们证明：对于任意包含$n$个智能体的实例，始终存在满足"至多$O(\sqrt{n})$件物品差异的无嫉妒性"的分配方案，且该方案可在多项式时间内找到；同时我们给出了匹配的$Ω(\sqrt{n})$下界，证明该结果即使在二值估值下也是紧的，这排除了存在外部性时EF1分配方案存在的可能性。