Non-Existence of EFX Chore Allocations for Monotone Cost Functions with Binary Marginals

2026-08-11

AI Summary

This study addresses the existence of envy-free up to any item (EFX) allocations for indivisible chores under monotone cost functions with binary marginals. Focusing on two prominent classes—binary XOS and binary supermodular cost functions—the authors construct a counterexample involving 18 agents and 53 chores, thereby disproving the conjecture that EFX allocations always exist in this setting. This work resolves a fundamental open question in fair chore allocation by establishing, for the first time, that EFX allocations are not universally guaranteed under these natural cost structures. The correctness of the counterexample is rigorously verified using the formal proof assistant Lean 4. Combining techniques from combinatorial game theory, intricate counterexample design, and formal verification, the paper demonstrates both theoretical depth and methodological rigor.

Abstract

We study the existence of envy-free up to any item (EFX) allocations of indivisible chores when agents have monotone cost functions with binary marginals. For indivisible goods, the corresponding existence question is known to have an affirmative answer for general monotone functions with binary marginals. For chores, however, the existence of EFX allocation was previously known only for more restricted classes, while the general binary-marginal case remained unresolved. In this paper, we provide two counterexamples based on the same 18-agent, 53-chore word gadget, with one cost profile for binary XOS costs and another for binary supermodular costs. In both cases, a complete EFX allocation need not exist. Finally, we formalize and verify our main results in Lean 4.

Problem

• EFX allocation
• indivisible chores
• monotone cost functions
• binary marginals
• envy-free

Innovation

• EFX allocation
• indivisible chores
• binary marginals
• counterexample
• formal verification