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

Envy-freeness up to one good (EF1) and envy-freeness any (EFX) are two well-known extensions of for the case indivisible items. It is shown that EF1 can always be guaranteed agents with subadditive valuations. In sharp contrast, it unknown whether or not an EFX allocation exists, even four additive addition, best approximation guarantee (φ − 1) ≃ 0.61 by Amanitidis et al.. order find a middle ground bridge this gap, in paper we suggest another fairness criterion, namely random EFR, which weaker than EFX, yet stronger EF1. For notion, provide polynomial-time 0.73-approximation algorithm. our algorithm introduce Nash Social Welfare Matching makes connection between envy freeness.