  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.
