2023/02/26 by Koichi Nishimura, Nishimura, Koichi, Hanna Sumita +1 · 1 citation
Economics, Econometrics and Finance · Social Sciences · #Computer Science and Game Theory (cs.GT) #Experimental Behavioral Economics Studies #FOS: Computer and information sciences #Law, Economics, and Judicial Systems
paper · pdf · doi:10.48550/arxiv.2302.13342
openalex publication_date 2023/02/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.