Fair Allocation of Indivisible Goods with Variable Groups

TL;DR

This paper investigates fair allocation of indivisible goods among variable groups, proving the existence of envy-free up to one good (EF1) outcomes under various conditions, and providing algorithms and probabilistic analyses for such allocations.

Contribution

It generalizes EF1 existence results from individuals to groups, introduces efficient algorithms, and analyzes probabilistic models for fair allocations.

Findings

EF1 outcomes always exist for any number of groups and sizes.

An EF1 outcome exists even with connected bundles on a path.

High probability of EF1 existence when goods are divisible by groups and utilities are random.

Abstract

We study the fair allocation of indivisible goods with variable groups. In this model, the goal is to partition the agents into groups of given sizes and allocate the goods to the groups in a fair manner. We show that for any number of groups and corresponding sizes, there always exists an envy-free up to one good (EF1) outcome, thereby generalizing an important result from the individual setting. Our result holds for arbitrary monotonic utilities and comes with an efficient algorithm. We also prove that an EF1 outcome is guaranteed to exist even when the goods lie on a path and each group must receive a connected bundle. In addition, we consider a probabilistic model where the utilities are additive and drawn randomly from a distribution. We show that if there are $n$ agents, the number of goods $m$ is divisible by the number of groups $k$, and all groups have the same size, then an envy-free outcome exists with high probability if $m = \omega(\log n)$, and this bound is tight. On the other hand, if $m$ is not divisible by $k$, then an envy-free outcome is unlikely to exist as long as $m = o(\sqrt{n})$.