Existence of EFX allocations for additive valuations

Establish whether an envy-free-up-to-any-good (EFX) allocation exists for every fair-division instance with indivisible goods and additive valuations.

Background

The paper defines EFX as the property that, for every pair of agents, one agent’s own bundle is at least as valuable as the other agent’s bundle after removing any one good from that bundle. The paper notes that PMMS is stronger than EFX under mild positivity conditions on the valuations.

Although the paper resolves the existence question for PMMS by exhibiting a three-agent counterexample, it does not resolve the broader EFX existence problem for additive valuations. The problem is included because the authors explicitly identify it as a major unresolved question in fair division.

References

One reason to be interested in this axiom is that it is stronger (under mild conditions, for example that all agent--good values are positive) than envy-freeness up to any good (EFX), whose existence for all additive valuations is arguably the largest open question in fair division:

— PMMS Allocations Need Not Exist for 3 Agents with Additive Valuations  (2609.08954 - Gölz, 8 Sep 2026) in Section 1, Model