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We study fair allocation of resources consisting of both divisible andindivisible goods to agents with additive valuations. When only divisible orindivisible goods exist, it is known that an allocation that achieves themaximum Nash welfare (MNW) satisfies the classic fairness notions based onenvy. In addition, properties of the MNW allocations for binary valuations areknown. In this paper, we show that when all agents' valuations are binary andlinear for each good, an MNW allocation for mixed goods satisfies theenvy-freeness up to any good for mixed goods. This notion is stronger than anexisting one called envy-freeness for mixed goods (EFM), and our resultgeneralizes the existing results for the case when only divisible orindivisible goods exist. Moreover, our result holds for a general fairnessnotion based on minimizing a symmetric strictly convex function. For thegeneral additive valuations, we also provide a formal proof that an MNWallocation satisfies a weaker notion than EFM.

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