We study fair allocation of resources consisting of both divisible and indivisible goods to agents with additive valuations. When only divisible or indivisible goods exist, it is known that an allocation that achieves the maximum Nash welfare (MNW) satisfies the classic fairness notions based on envy. Moreover, the literature shows the structures and characterizations of MNW allocations when valuations are binary and linear (i.e., divisible goods are homogeneous). In this paper, we show that when all agents' valuations are binary linear, an MNW allocation for mixed goods satisfies the envy-freeness up to any good for mixed goods (EFXM). This notion is stronger than an existing one called envy-freeness for mixed goods (EFM), and our result generalizes the existing results for the case when only divisible or indivisible goods exist. When all agents' valuations are binary over indivisible goods and identical over divisible goods (e.g., the divisible good is money), we extend the known characterization of an MNW allocation for indivisible goods to mixed goods, and also show that an MNW allocation satisfies EFXM. For the general additive valuations, we also provide a formal proof that an MNW allocation satisfies a weaker notion than EFM.

翻译：我们研究具有加性估值的主体间对包含可分与不可分物品的资源的公平分配问题。当仅存在可分或不可分物品时，已知实现最大纳什福利的分配满足基于嫉妒的经典公平性概念。此外，现有文献在估值为二元线性（即可分物品为同质）时，已揭示了最大纳什福利分配的结构与特征。本文证明，当所有主体的估值均为二元线性时，混合物品的最大纳什福利分配满足针对混合物品的任意物品嫉妒自由性。该概念强于现有混合物品嫉妒自由性概念，且我们的结果推广了仅存在可分或不可分物品情形的现有结论。当所有主体对不可分物品的估值均为二元性、对可分物品的估值具有同一性（例如可分物品为货币）时，我们将不可分物品最大纳什福利分配的已知特征推广至混合物品情形，并证明最大纳什福利分配满足任意物品嫉妒自由性。对于一般加性估值情形，我们也给出了最大纳什福利分配满足弱于混合物品嫉妒自由性概念的正式证明。