# Multiple Birds with One Stone: Beating 1/2 for EFX and GMMS via Envy Cycle Elimination

arXiv: https://arxiv.org/abs/1909.07650 · HTML: https://mlsanity.com/paper/1909.07650/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Sep 2019.

## Built on by (most cited first, 8 of 8)

- [Online Fair Division with Additional Information](https://mlsanity.com/paper/2505.24503/) (2025)
- [Pushing the Frontier on Approximate EFX Allocations](https://mlsanity.com/paper/2406.12413/) (2024)
- [Allocating Indivisible Goods to Strategic Agents: Pure Nash Equilibria and Fairness](https://mlsanity.com/paper/2109.08644/) (2021)
- [Understanding EFX Allocations: Counting and Variants](https://mlsanity.com/paper/2504.03951/) (2025)
- [PROPm Allocations of Indivisible Goods to Multiple Agents](https://mlsanity.com/paper/2105.11348/) (2021)
- [On the existence of EFX allocations in multigraphs](https://mlsanity.com/paper/2502.09777/) (2025)
- [The Cost of EFX: Generalized-Mean Welfare and Complexity Dichotomies with Few Surplus Items](https://mlsanity.com/paper/2601.12849/) (2026)
- [Probing EFX via PMMS: (Non-)Existence Results in Discrete Fair Division](https://mlsanity.com/paper/2507.14957/) (2025)

## Abstract (arXiv)

Several relaxations of envy-freeness, tailored to fair division in settings with indivisible goods, have been introduced within the last decade. Due to the lack of general existence results for most of these concepts, great attention has been paid to establishing approximation guarantees. In this work, we propose a simple algorithm that is universally fair in the sense that it returns allocations that have good approximation guarantees with respect to four such fairness notions at once. In particular, this is the first algorithm achieving a (φ-1)-approximation of envy-freeness up to any good (EFX) and a frac2φ +2-approximation of groupwise maximin share fairness (GMMS), where φ is the golden ratio (φ ≈ 1.618). The best known approximation factor for either one of these fairness notions prior to this work was 1/2. Moreover, the returned allocation achieves envy-freeness up to one good (EF1) and a 2/3-approximation of pairwise maximin share fairness (PMMS). While EFX is our primary focus, we also exhibit how to fine-tune our algorithm and improve the guarantees for GMMS or PMMS. Finally, we show that GMMS -- and thus PMMS and EFX -- allocations always exist when the number of goods does not exceed the number of agents by more than two.