The Power of Share-Based Notions in Proving Envy-Based Fairness Guarantees

Abstract

We study the problem of fairly allocating indivisible goods among agents with monotone valuations. We introduce a new share-based fairness notion, the residual maximin share (RMMS), and show that it provides a unified framework for several existing lone-divider style techniques in fair division. RMMS satisfies two key properties: feasibility and self-maximization. Using RMMS, we give simple proofs of the existence of partial allocations that are both RMMS and envy-free up to any good (EFX), and complete allocations that are both RMMS and envy-free up to one good (EF1), in fact satisfying the stronger notion of EFL. This unifies and strengthens several previously known results. We further demonstrate the power of the share-based approach by studying the compatibility of fairness notions related to the long-standing EFX problem. While allocations satisfying either epistemic EFX (EEFX) or EF1 are known to exist for general monotone valuations, whether they can always be achieved simultaneously has remained open in every setting where EFX existence itself is unresolved. For additive valuations, we resolve this question affirmatively by proving the existence of allocations that satisfy both EEFX and EFL. Our proof introduces the strong EEFX share, a new share notion implying EEFX feasibility of bundles. We show that the strong EEFX share is upper bounded by RMMS, enabling us to derive EEFX+EFL allocations via the RMMS framework. This answers the main open question of Akrami and Rathi (2025). Finally, although our algorithm for computing EEFX and EF1 allocations may take exponential time in general, we develop a polynomial-time algorithm for restricted additive valuations. Unlike the lone-divider approach, our algorithm exploits the structural properties of restricted additive valuations to compute allocations satisfying both EEFX and EF1.