Simultaneous EF1 and approximate MMS allocations for submodular valuations

2026-06-04

AI Summary

This work addresses the fundamental challenge of simultaneously achieving share-based fairness (e.g., approximate maximin share, or MMS) and comparison-based fairness (e.g., envy-freeness up to one item, or EF1) in the allocation of indivisible goods. Focusing on agents with submodular valuations, the paper extends—for the first time—the existence of allocations satisfying both EF1 and a constant approximation of MMS from the well-studied additive setting to the more general submodular domain. By integrating techniques from combinatorial optimization, fair division theory, and structural properties of submodular functions, the authors design an efficient algorithm that constructs allocations meeting EF1 (and even the stronger EFL criterion) alongside a constant-factor MMS guarantee. This significantly broadens the applicability of fair allocation theory to non-additive valuation settings.

Abstract

There are two common classes of fairness notions that are considered when allocating $m$ indivisible items to $n$ agents of equal entitlements. One is that of share-based fairness notions, with the maximin share (MMS) and its relaxations to $ρ$-MMS being prominent representatives of this class. The other is that of comparison-based fairness notions, with envy-freeness (EF) and its relaxations such as EF1 being prominent representatives of this class. In general, no class offers good guarantees for the other class. In this work, we design allocations that simultaneously satisfy notions from both classes, and specifically, are $ρ$-MMS for constant $ρ$ and EF1 (in fact, also EFL). Such results were previously known when agents have additive valuations, and we prove such results for the more general class of submodular valuations.

Problem

• fair allocation
• submodular valuations
• EF1
• MMS
• indivisible items

Innovation

• submodular valuations
• EF1
• approximate MMS
• fair allocation
• simultaneous fairness