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

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.