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Envy-free up to one good (EF1) and envy-free up to any good (EFX) are twowell-known extensions of envy-freeness for the case of indivisible items. It isshown that EF1 can always be guaranteed for agents with subadditive valuations.In sharp contrast, it is unknown whether or not an EFX allocation alwaysexists, even for four agents and additive valuations. In addition, the bestapproximation guarantee for EFX is $(\phi -1) \simeq 0.61$ by Amanitidis etal..  In order to find a middle ground to bridge this gap, in this paper we suggestanother fairness criterion, namely envy-freeness up to a random good or EFR,which is weaker than EFX, yet stronger than EF1. For this notion, we provide apolynomial-time $0.73$-approximation allocation algorithm. For our algorithm,we introduce Nash Social Welfare Matching which makes a connection between NashSocial Welfare and envy freeness. We believe Nash Social Welfare Matching willfind its applications in future work.

Almost Envy-freeness, Envy-rank, and Nash Social Welfare Matchings