Do EFX allocations always exist for additive valuations?

When indivisible goods are shared, full envy-freeness can be impossible, so fair division asks for envy-freeness up to any good (EFX): no agent should envy another once any single good is removed from the other's bundle. Whether such an allocation always exists when agents value bundles additively is open even for four agents. Ariel Procaccia has called it the successor of envy-free cake cutting as fair division's biggest problem [4].

Problem

There are $n$ agents and $m$ indivisible goods. Agent $i$ values good $g$ at $v_i(g) \ge 0$ and a bundle $S$ at $v_i(S) = \sum_{g \in S} v_i(g)$ (additive valuations). An allocation is a partition $(A_1, \dots, A_n)$ of the goods; bundles may be empty. It is EFX if for all agents $i \ne j$ and every good $g \in A_j$ with $v_i(g) > 0$,

$$v_i(A_i) \ge v_i(A_j \setminus \{g\}).$$

Question. Does every instance with additive valuations have an EFX allocation? A counterexample needs at least four agents, since EFX allocations always exist for three [2].

Background

Envy-free allocations of indivisible goods need not exist (two agents, one good both want), so relaxations are studied. EFX, introduced by Caragiannis, Kurokawa, Moulin, Procaccia, Shah and Wang in 2016 (doi:10.1145/3355902), is among the strongest that might always be achievable. Plaut and Roughgarden proved the first general results about it [1], and Chaudhury, Garg and Mehlhorn proved constructively that an EFX allocation always exists for three agents with additive valuations [2]. In 2026 the analogous question was answered negatively for chores [3] and for goods with submodular valuations [5, 6]; Epoch reads the chores result as some evidence that additive goods could fail too.

What is known

• EFX allocations exist for two agents with arbitrary monotone valuations, and for any number of agents with identical valuations (Plaut and Roughgarden) [1].
• They exist for three agents with additive valuations (Chaudhury, Garg and Mehlhorn, 2020; journal version 2024) [2].
• For four additive agents, Alkassar, Fouz and Mehlhorn (August 2026, arXiv:2608.08590) report a computer-assisted proof that EFX allocations exist whenever there are at most nine goods, up from the earlier bound $m \le n + 3$. This is an unrefereed preprint.
• Beyond additive valuations the answer is negative. Akrami, Mayorov, Mehlhorn, Srinivas and Weidenbach found, by SAT solving with a Lean-verified encoding, submodular instances with no EFX allocation for every $n \ge 3$ agents and $m \ge n + 5$ goods [5]. Mackenzie and Suzuki gave a compact instance with three agents, eight goods and submodular (weighted coverage) valuations [6]. Both are unrefereed 2026 preprints, and neither applies to additive valuations.
• For additive chores, He and Tao (2026) built, for every $n \ge 4$, an instance with $n$ agents and no EFX allocation, using only three types of chores and two types of agents [3].
• For goods with additive valuations, existence is open for every $n \ge 4$.

What would count as progress

• A valuation matrix with $n \ge 4$ rows of non-negative integers such that none of the $n^m$ allocations is EFX. For small $n$ and $m$ this is checked by brute force; Epoch's verifier enumerates all allocations, so $n^m$ must stay small.
• An existence proof for four additive agents with more than nine goods, or with any number of goods.

References

1. B. Plaut and T. Roughgarden, Almost envy-freeness with general valuations, SIAM J. Discrete Math. 34 (2020), 1039–1068. doi:10.1137/19M124397X
2. B. R. Chaudhury, J. Garg and K. Mehlhorn, EFX exists for three agents, J. ACM 71 (2024), 1–27. doi:10.1145/3616009 (arXiv:2002.05119)
3. W. He and B. Tao, EFX for additive chores: nonexistence, Pareto incompatibility, and bi-valued existence, 2026. arXiv:2606.08872
4. A. D. Procaccia, Technical perspective: an answer to fair division's most enigmatic question, Comm. ACM 63 (2020), 118. doi:10.1145/3382131
5. H. Akrami, A. Mayorov, K. Mehlhorn, S. Srinivas and C. Weidenbach, A counterexample to EFX for $n \ge 3$ agents, $m \ge n + 5$ items, submodular valuations via SAT-solving, 2026. arXiv:2604.18216
6. S. Mackenzie and M. Suzuki, Counterexamples to EFX for submodular and subadditive valuations, 2026. arXiv:2605.06451

Source

Imported from FrontierMath: Open Problems, efx, curated by Epoch AI. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.