EFX for Additive Chores: Nonexistence, Pareto Incompatibility, and Bi-Valued Existence

2026-06-07

AI Summary

This study investigates the existence of envy-free up to any chore (EFX) allocations under additive cost functions in the context of indivisible chore division. Through combinatorial constructions and counterexample analysis, it establishes for the first time that EFX allocations are not guaranteed to exist under three-valued additive costs. The work further reveals an incompatibility between EFX and Pareto optimality in the two-valued setting, without relying on items of zero cost. Additionally, it proves that EFX allocations always exist when there are four agents. These results delineate the limits of EFX’s universal existence—showing its failure in three-valued and higher-cost settings—characterize the boundary of its conflict with Pareto optimality, and provide existence guarantees for small-scale instances.

Abstract

We consider the fair division problem of indivisible chores and resolve the long-standing open problem for the existence of EFX allocations with additive cost functions. We show that, even for tri-valued additive cost functions, for every $n\geq 4$, there exists an instance with $n$ agents where no EFX allocation exists. Our counterexample only uses three types of chores, which is also tight on the number of types, as an EFX allocation is known to exist for two types of chores. We then consider bi-valued instances. We show that, for every $n\geq 4$, there exists an instance with $n$ agents where every EFX allocation is not Pareto-optimal. This is also the first example showing the incompatibility of EFX and Pareto-optimality when the costs of items are positive: existing examples showing the incompatibility of EFX and Pareto-optimal exploit items with $0$ costs. Our result shows such an example exists even for bi-valued instances. The number of agents $n$ is also tight: for $n\leq 3$, it is known that EFX is compatible with Pareto-optimality. Finally, we also show that an EFX allocation is guaranteed to exist for $n=4$.

Problem

• EFX
• fair division
• additive chores
• Pareto-optimality
• indivisible chores

Innovation

• EFX allocation
• indivisible chores
• additive cost functions
• Pareto-optimality
• bi-valued instances