Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2602.11732

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

Hannaneh Akrami·Uriel Feige·Ryoga Mahara·Kurt Mehlhorn·Nidhi Rathi
View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

cs.GT

Keep this discovery

BibTeXRIS

Hannaneh Akrami, Uriel Feige, Ryoga Mahara, Kurt Mehlhorn, Nidhi Rathi. 2026-02-12. The Power of Share-Based Notions in Proving Envy-Based Fairness Guarantees. https://arxiv.org/abs/2602.11732

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

MMS Allocation for Chores with Online Agent Arrivals

We study the fair allocation of $m$ indivisible chores to $n$ agents with subadditive cost functions arriving online in an arbitrary order. Upon an agent's arrival, we are informed of her cost function and must irrevocably assign her a set of chores. We focus on the Maximin Share (MMS) fairness notion and aim to compute an allocation in which all items are assigned, and no agent incurs a cost more than $\alpha$ times her MMS. Without any prior information about the instance (other than $n$ and $m$), we design an algorithm with a competitive ratio of $O(\min\{n, k\log^{1+\epsilon}k, \log m\})$ for any constant $\epsilon > 0$, where $k$ denotes the number of cost function types. Our bound matches the best known offline approximation guarantees for MMS under subadditive costs and is nearly optimal with respect to all three parameters: we show that even for binary additive cost functions, no online algorithm can achieve a competitive ratio of $o(\min\{n, k\log k, \log m\})$. We then consider the setting in which the $k$ cost function types are known in advance (though the realized types of arriving agents are not). For additive cost functions, we provide an algorithm with a competitive ratio of $O(\min\{\log k, \log(kn)/\log\log(kn)\})$, and show that constant-competitive algorithms do not exist for general $k$, even for the binary additive setting. For binary additive functions when $k \le n$, we propose a $3$-competitive algorithm and establish a lower bound of $2$.

cs.GT↗
arXiv2026

Truncated Noisy Best-Response Algorithms: Toward Game Theoretic Learning with Safety Guarantees

We consider a game theoretic approach to solve multi-agent coordination problems with submodular maximization objectives. It is known for such problems that the Nash equilibria for the corresponding game are always within 50% of the optimal, but that the equilibria which achieve this worst-case bound are not stable. To exploit this instability, we propose a family of algorithms which we call Truncated Noisy Best-Response (TNBR) Algorithms. These algorithms are flexibly characterized by agents asynchronously and stochastically selecting actions from a neighbourhood of their best response payoffs. We compute bounds on the recurrent classes of TNBR algorithms' associated Markov chains. Our bounds fall into two categories: first, "Performance" bounds ensure that TNBR algorithms always have a high-value recurrent state; second, "Safety" bounds ensure that TNBR algorithms never have arbitrarily-bad recurrent states. Furthermore, these two types of bounds are linked by a waterbed-like effect: every game with a poor Safety guarantee necessarily has a favorable Performance guarantee.

cs.GT↗
arXiv2026

Existence of the Core in Approval-Based Committee Elections

We settle the main open question in the theory of approval-based multi-winner elections: we show that there always exists a committee in the core. The core is a stability and group fairness concept. The proof introduces a new voting rule that optimizes an entropy-like objective function over committees and payment systems. All local optima of this objective function lie in the core, which implies that a core committee can be found in polynomial time.

cs.GT↗