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. 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. In addition, properties of the MNW allocations for binary valuations are known. In this paper, we show that when all agents' valuations are binary and linear for each good, an MNW allocation for mixed goods satisfies the envy-freeness up to any good for mixed goods. 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. Moreover, our result holds for a general fairness notion based on minimizing a symmetric strictly convex function. For the general additive valuations, we also provide a formal proof that an MNW allocation satisfies a weaker notion than EFM.