On The Pursuit of EFX for Chores: Non-Existence and Approximations

TL;DR

This paper investigates the existence and approximation of envy-free up to any item (EFX) allocations for chores, proving non-existence in certain cases, NP-completeness of decision, and positive results under specific conditions.

Contribution

It provides the first non-existence proof for EFX in chores, establishes NP-completeness of deciding EFX existence, and offers improved approximation guarantees for specific cases.

Findings

No EFX solutions for six chores among three agents with superadditive costs.

Deciding EFX existence is NP-complete.

Existence of EFX for up to n+2 chores with monotone costs.

Abstract

We study the problem of fairly allocating a set of chores to a group of agents. The existence of envy-free up to any item (EFX) allocations is a long-standing open question for both goods and chores. We resolve this question by providing a negative answer for the latter, presenting a simple construction that admits no EFX solutions for allocating six items to three agents equipped with superadditive cost functions, thus proving a separation result between goods and bads. In fact, we uncover a deeper insight, showing that the instance has unbounded approximation ratio. Moreover, we show that deciding whether an EFX allocation exists is NP-complete. On the positive side, we establish the existence of EFX allocations under general monotone cost functions when the number of items is at most $n+2$. We then shift our attention to additive cost functions. We employ a general framework in order to improve the approximation guarantees in the well-studied case of three additive agents, and provide several conditional approximation bounds that leverage ordinal information.