Weighted Envy-Freeness for Submodular Valuations

TL;DR

This paper extends envy-freeness concepts to agents with submodular valuations, proposing new fairness notions and allocation rules that improve fairness guarantees for indivisible goods.

Contribution

It introduces two families of envy-based fairness notions for submodular valuations and demonstrates their satisfaction through generalized allocation rules, improving fairness guarantees.

Findings

New envy-based fairness notions for submodular valuations

Generalized allocation rules satisfying these notions

Stronger fairness guarantees with harmonic welfare measures

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.