Characterizing maximum Nash welfare: Envy−freeness and proportionality

Abstract

When allocating indivisible goods, the maximum Nash welfare rule is known to be the unique welfarist rule satisfying envy-freeness up to one good. We strengthen this characterization by showing that envy-freeness up to one good can be replaced by substantially weaker fairness requirements: envy-freeness up to k goods or proportionality up to k goods, for any fixed k. Our characterization holds even for profiles with identical goods.