Epistemic EFX Allocations Exist for Monotone Valuations

TL;DR

This paper proves that epistemic EFX allocations, a relaxed fairness concept, exist for any number of agents with general monotone valuations, and discusses the computational complexity of finding such allocations.

Contribution

It establishes the existence of EEFX allocations for agents with general monotone valuations, extending previous results beyond additive valuations.

Findings

Existence of EEFX allocations for arbitrary agents with monotone valuations.

Proves PLS-hardness of computing EEFX allocations even with identical submodular valuations.

Shows exponential query complexity for computing EEFX allocations.

Abstract

We study the fundamental problem of fairly dividing a set of indivisible items among agents with (general) monotone valuations. The notion of envy-freeness up to any item (EFX) is considered to be one of the most fascinating fairness concepts in this line of work. Unfortunately, despite significant efforts, existence of EFX allocations is a major open problem in fair division, thereby making the study of approximations and relaxations of EFX a natural line of research. Recently, Caragiannis et al. introduced a promising relaxation of EFX, called epistemic EFX (EEFX). We say an allocation to be EEFX if, for every agent, it is possible to shuffle the items in the remaining bundles so that she becomes "EFX-satisfied". Caragiannis et al. prove existence and polynomial-time computability of EEFX allocations for additive valuations. A natural question asks what happens when we consider valuations more general than additive? We address this important open question and answer it affirmatively by establishing the existence of EEFX allocations for an arbitrary number of agents with general monotone valuations. To the best of our knowledge, EEFX is the only known relaxation of EFX (beside EF1) to have such strong existential guarantees. Furthermore, we complement our existential result by proving computational and information-theoretic lower bounds. We prove that even for an arbitrary number of (more than one) agents with identical submodular valuations, it is PLS-hard to compute EEFX allocations and it requires exponentially-many value queries to do so.