Almost Envy-Freeness, Envy-Rank, And Nash Social Welfare Matchings

Envy-freeness up to one good (EF1) and envy-freeness up to any good (EFX) are two well-known extensions of envyfreeness for the case of indivisible items. It is shown that EF1 can always be guaranteed for agents with subadditive valuations (Lipton et al. 2004). In sharp contrast, it is unknown whether or not an EFX allocation always exists, even for four agents and additive valuations. In addition, the best approximation guarantee for EFX is (phi - 1) similar or equal to 0.61 by Amanatidis et al. (Amanatidis, Markakis, and Ntokos 2020).In order to find a middle ground to bridge this gap, in this paper we suggest another 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 a polynomial-time 0.73-approximation allocation algorithm. For our algorithm we use Nash Social Welfare Matching which makes a new connection between Nash Social Welfare and envy freeness.