Weighted Envy-Freeness for Submodular Valuations
Luisa Montanari, Ulrike Schmidt-Kraepelin, Warut Suksompong, Nicholas Teh
TL;DR
This work advances fair division for indivisible goods by addressing weighted entitlements under submodular and matroid-rank valuations. It introduces two envy-based notions, TWEF$(x,1-x)$ and WMEF$(x,1-x)$, and provides algorithmic frameworks—adjusted picking sequences, MWNW, a weighted transfer algorithm, and harmonic-welfare notions—to satisfy them. The results show that MWNW cannot guarantee strong envy-freeness in general, but the proposed harmonic-welfare approaches yield stronger fairness guarantees under matroid-rank valuations, with Pareto-optimality preserved. Overall, the paper broadens the toolkit for fair allocation with non-additive valuations, offering practical procedures for weighted settings and insights into when stronger fairness can be achieved.
Abstract
We investigate the fair allocation of indivisible goods to agents with possibly different entitlements represented by weights. Previous work has shown that guarantees for additive valuations with existing envy-based notions cannot be extended to the case where agents have matroid-rank (i.e., binary submodular) valuations. We propose two families of envy-based notions for matroid-rank and general submodular valuations, one based on the idea of transferability and the other on marginal values. We show that our notions can be satisfied via generalizations of rules such as picking sequences and maximum weighted Nash welfare. In addition, we introduce welfare measures based on harmonic numbers, and show that variants of maximum weighted harmonic welfare offer stronger fairness guarantees than maximum weighted Nash welfare under matroid-rank valuations.