A Counterexample to EFX $n \ge 3$ Agents, $m \ge n + 5$ Items, Submodular Valuations via SAT-Solving

TL;DR

This paper provides the first counterexample showing EFX allocations do not always exist for three or more agents with certain numbers of goods, using SAT solving and formal verification.

Contribution

It introduces the first counterexample to EFX existence for agents with monotone valuations and proves positive results for three-agent cases with relaxations.

Findings

Counterexample for EFX non-existence with n ≥ 3 agents and m ≥ n+5 goods.

Every three-agent instance with seven goods admits an EFX allocation.

SAT solving and formal verification in Lean were used to establish these results.

Abstract

The existence of EFX allocations is a central open problem in discrete fair division. An allocation is EFX (envy-free up to any good) if no agent envies another agent after the removal of any single good from the other agent's bundle. We resolve this longstanding question by providing the \textbf{first-ever counterexample} to the existence of EFX allocations for agents with monotone valuations, which in turn immediately implies a counterexample for submodular valuations. Specifically, we show that EFX allocations need not exist for instances with $n \ge 3$ agents and $m \ge n+5$ goods. In contrast, we prove that every instance with three agents and seven goods admits an EFX allocation. Both results are obtained via SAT solving. We encode the negation of EFX existence as a SAT instance: satisfiability yields a counterexample, while unsatisfiability establishes universal existence. The correctness of the encoding is formally verified in Lean. Finally, we establish positive guarantees for fair allocations with three agents and an arbitrary number of goods. Although EFX allocations may fail to exist, we prove that every instance with three agents and monotone valuations admits at least one of two natural relaxations of EFX: tEFX, or EF1 and EEFX.