On the Fairness of Additive Welfarist Rules

TL;DR

This paper investigates additive welfarist rules in fair division, establishing the uniqueness of the maximum Nash welfare rule in guaranteeing EF1 across various instances, and exploring alternative rules under integer utilities.

Contribution

It strengthens the characterization of MNW as the unique EF1-guaranteeing additive welfarist rule and explores other rules providing EF1 with integer utilities.

Findings

MNW is uniquely EF1-guaranteeing in identical-good, two-value, and normalized instances.

Other rules can guarantee EF1 when agents' utilities are integers.

Characterizations of these alternative rules are provided for different instance classes.

Abstract

Allocating indivisible goods is a ubiquitous task in fair division. We study additive welfarist rules, an important class of rules which choose an allocation that maximizes the sum of some function of the agents' utilities. Prior work has shown that the maximum Nash welfare (MNW) rule is the unique additive welfarist rule that guarantees envy-freeness up to one good (EF1). We strengthen this result by showing that MNW remains the only additive welfarist rule that ensures EF1 for identical-good instances, two-value instances, as well as normalized instances with three or more agents. On the other hand, if the agents' utilities are integers, we demonstrate that several other rules offer the EF1 guarantee, and provide characterizations of these rules for various classes of instances.