We study fair allocation of resources consisting of both divisible and indivisible goods to agents with additive valuations. Recently, a fairness notion called envy-freeness for mixed goods (EFM) has been introduced for this setting. The EFM is a natural combination of classic fairness notions called envy-freeness for divisible goods and envy-freeness up to one good for indivisible goods. When either divisible or indivisible goods exist, it is known that an allocation that achieves the maximum Nash welfare (MNW) satisfies the classic fairness notions. On the other hand, for mixed goods, an MNW allocation does not necessarily entail EFM. In this paper, we formally prove that an MNW allocation for mixed goods is envy-free up to one (indivisible) good for mixed goods.

翻译：我们研究向具有可加性估值的代理人分配包含可分与不可分物品的资源时的公平分配问题。针对这一设定，近期引入了一种名为"混合物品免嫉妒性"（EFM）的公平性概念。EFM是经典公平性概念——可分物品的免嫉妒性与不可分物品的"至多一物品免嫉妒性"的自然结合。已知当仅存在可分或不可分物品时，实现最大纳什福利（MNW）的分配满足经典公平性概念。然而对于混合物品，MNW分配未必满足EFM。本文正式证明：混合物品的MNW分配满足"至多一（不可分）物品的免嫉妒性"。