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

2026-04-20

AI Summary

This study addresses the central open problem concerning the existence of EFX (envy-free up to any good) allocations in discrete fair division. By encoding the EFX condition as a Boolean satisfiability (SAT) instance and leveraging the SPASS-SAT solver together with the DRAT-trim proof checker and the Lean formal verification system, the authors achieve both efficient search and rigorous validation. Their main contributions include the first proof that EFX allocations always exist for three agents and seven goods, as well as the construction of a counterexample for three agents and eight goods. This counterexample is further generalized to all instances with $n \geq 3$ agents and $m \geq n + 5$ goods, thereby refuting the long-standing conjecture that EFX allocations universally exist and resolving a pivotal question in the field.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• Constraint Satisfaction and Optimization: Satisfiability

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

SAT solving has recently been proven effective in tackling open combinatorial problems. We contribute two additional results in the context of fair distribution of indivisible goods. Specifically, we demonstrate that EFX (envy-freeness up to any good) allocations always exist for three agents and seven goods, while we provide a counterexample for the case of $n \ge 3$ agents and $m \ge n + 5$ goods. An allocation is EFX if no agent would envy the allocation of any other agent if any single item were to be removed from the other agent's bundle of goods. Each agent's preferences are modeled by a monotone valuation function on all potential bundles. After analyzing theoretical aspects of the problem, we encode the negation of the EFX instances into SAT. Satisfiability of the respective SAT formula constitute a counter-example to EFX, unsatisfiability of the respective SAT formula implies that EFX holds. The theoretical foundations of the encoding are proven correct in LEAN. For the three agents and seven goods case, we obtained a proof of unsatisfiability using SPASS-SAT of size about 30 GB in about 30 hours. It was shown to be correct by DRAT-trim. In the case of three agents and eight goods, SPASS-SAT computed satisfiability indicating a counterexample in the form of three specific agent valuations in about 20 hours. It was verified by probing all possible bundle assignments; the verification takes seconds. The extension of the counterexample to $n \ge 4$ agents and $m \ge n + 5$ goods does not involve SAT-solving. This counterexample resolves, in the negative, one of the central questions in the theory of discrete fair division.

Problem

• EFX
• fair division
• indivisible goods
• monotone valuations
• counterexample

Innovation

• EFX
• SAT solving
• fair division
• monotone valuations
• formal verification