On the Number of Almost Envy-Free Allocations

TL;DR

This paper investigates the minimum number of allocations satisfying fairness criteria EF1 and EFX for two agents with indivisible items, revealing exponential and minimal bounds respectively, and explains the robustness gap between these notions.

Contribution

It establishes tight lower bounds on the number of EF1 and EFX allocations for two agents, using hypercube isoperimetric inequalities, clarifying the robustness gap.

Findings

EFX allocations can be as few as two regardless of item count

EF1 allocations grow exponentially with the number of items

The results explain the robustness gap between EF1 and EFX

Abstract

Envy-freeness is a standard benchmark of fairness in resource allocation. Since it cannot always be satisfied when the resource consists of indivisible items even when there are two agents, the relaxations envy-freeness up to one item (EF1) and envy-freeness up to any item (EFX) are often considered. We establish tight lower bounds on the number of allocations satisfying each of these benchmarks in the case of two agents. In particular, while there can be as few as two EFX allocations for any number of items, the number of EF1 allocations is always exponential in the number of items. Our results apply a version of the vertex isoperimetric inequality on the hypercube and help explain the large gap in terms of robustness between the two notions.