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

TL;DR

This paper introduces a new fairness criterion called envy-freeness up to a random good (EFR), provides a polynomial-time approximation algorithm for it, and connects it with Nash Social Welfare Matching to bridge gaps in existing fairness notions.

Contribution

It proposes EFR as a middle ground between EF1 and EFX, and introduces Nash Social Welfare Matching, a novel approach linking social welfare and envy-freeness.

Findings

Provides a 0.73-approximation algorithm for EFR.

Establishes Nash Social Welfare Matching as a new concept.

Bridges the gap between EF1 and EFX fairness notions.

Abstract

Envy-free up to one good (EF1) and envy-free up to any good (EFX) are two well-known extensions of envy-freeness for the case of indivisible items. It is shown that EF1 can always be guaranteed for agents with subadditive valuations. 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 $(\phi -1) \simeq 0.61$ by Amanitidis et al.. 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 introduce Nash Social Welfare Matching which makes a connection between Nash Social Welfare and envy freeness. We believe Nash Social Welfare Matching will find its applications in future work.