Envy-freeness up to one item: Shall we add or remove resources?

TL;DR

This paper introduces new fairness axioms for dividing indivisible items, showing that EFX+- and PO allocations exist where previous fairness notions do not, and establishes related impossibility results.

Contribution

It proposes EF1+- and EFX+- axioms, demonstrating their existence and compatibility with Pareto optimality in cases where traditional fairness allocations fail.

Findings

EF1+- and EFX+- allocations often exist where EF1 and EFX do not

EFX+- and PO allocations always exist when EFX and PO do not

Several new impossibility and incompatibility results are proved

Abstract

We consider a fair division model in which agents have general valuations for bundles of indivisible items. We propose two new axiomatic properties for allocations in this model: EF1+- and EFX+-. We compare these with the existing EF1 and EFX. Although EF1 and EF1+- allocations often exist, our results assert eloquently that EFX+- and PO allocations exist in each case where EFX and PO allocations do not exist. Additionally, we prove several new impossibility and incompatibility results.