Envy-freeness and maximum Nash welfare for mixed divisible and indivisible goods

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.