Almost and Approximate EFX for Few Types of Agents

TL;DR

This paper advances fair division by proving the existence of approximate EFX allocations for any number of agents with up to four valuation types and extends EFX with charity results to multiple valuation types.

Contribution

It establishes new existence results for approximate EFX allocations with limited valuation types and extends EFX with charity to multiple valuation classes.

Findings

A 2/3-approximate EFX allocation exists for any number of agents with up to four valuation types.

A (1-ε)-EFX allocation with bounded charity exists for any number of agents with up to k valuation types.

The charity size bound depends on the number of valuation types, not agents.

Abstract

We study the problem of fair allocation of a set of indivisible goods among $n$ agents with $k$ distinct additive valuations, with the goal of achieving approximate envy-freeness up to any good ($\alpha-\mathrm{EFX}$). It is known that EFX allocations exist for $n$ agents when there are at most three distinct valuations due to HV et al. Furthermore, Amanatidis et al. showed that a $\frac{2}{3}-\mathrm{EFX}$ allocation is guaranteed to exist when number of agents is at most seven. In this paper, we show that a $\frac{2}{3}-\mathrm{EFX}$ allocation exists for any number of agents when there are at most four distinct valuations. Secondly, we consider a relaxation called $\mathrm{EFX}$ with charity, where some goods remain unallocated such that no agent envies the set of unallocated goods. Akrami et al. showed that for $n$ agents and any $\varepsilon \in \left(0, \frac{1}{2}\right]$, there exists a $(1-\varepsilon)-\mathrm{EFX}$ allocation with at most $\tilde{\mathcal{O}}((n/\varepsilon)^{\frac{1}{2}})$ goods to charity. In this paper, we show that a $(1-\varepsilon)-\mathrm{EFX}$ allocation with a $\tilde{\mathcal{O}}(k/\varepsilon)^{\frac{1}{2}}$ charity exists for any number of agents when there are at most $k$ distinct valuations.