Counterexamples to EFX for Submodular and Subadditive Valuations
Three agents, eight goods, and no fair allocation exists even with identical preferences.
A fundamental question in fair division asks whether it is always possible to allocate goods so that no agent envies another's bundle after removing any single good—known as EFX (envy-free up to any good). While EFX allocations have been shown to exist for many settings, the general case has remained open. Now, Simon Mackenzie and Mashbat Suzuki have constructed concrete counterexamples showing EFX can fail even for simple, symmetric valuations.
Their three-agent, eight-good instance uses submodular valuations (specifically weighted coverage functions) for which no EFX allocation exists. For the broader class of monotone subadditive valuations, they prove that no allocation can satisfy α-EFX for any α > 1/∛√2 ≈ 0.89. A key insight is symmetry: agents' valuations are identical up to relabeling of goods, meaning the impossibility arises purely from how items are named. This yields simple, human-verifiable combinatorial obstructions and marks a significant advance in understanding the limits of fairness in resource allocation.
- First counterexample to EFX for submodular valuations with 3 agents and 8 goods
- For subadditive valuations, α-EFX cannot exceed ~0.89, disproving perfect fairness
- Construction is symmetric (identical preferences up to relabeling) and fully human-verifiable
Why It Matters
Settles a long-standing open problem in fair division, proving fundamental limits to envy-free allocations.