PROPm Allocations of Indivisible Goods to Multiple Agents

TL;DR

This paper proves the universal existence of PROPm allocations for indivisible goods among any number of agents, providing a polynomial-time algorithm for computing such fair allocations.

Contribution

It extends previous results by showing PROPm allocations exist for all instances and introduces a polynomial-time algorithm to find them.

Findings

PROP m allocations exist for any number of agents and goods.

The proposed algorithm computes allocations in polynomial time.

This work generalizes previous results limited to five agents.

Abstract

We study the classic problem of fairly allocating a set of indivisible goods among a group of agents, and focus on the notion of approximate proportionality known as PROPm. Prior work showed that there exists an allocation that satisfies this notion of fairness for instances involving up to five agents, but fell short of proving that this is true in general. We extend this result to show that a PROPm allocation is guaranteed to exist for all instances, independent of the number of agents or goods. Our proof is constructive, providing an algorithm that computes such an allocation and, unlike prior work, the running time of this algorithm is polynomial in both the number of agents and the number of goods.