Fair Allocation of Indivisible Goods with Variable Groups
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911255821811712 |
|---|---|
| author | Gölz, Paul Igarashi, Ayumi Manurangsi, Pasin Suksompong, Warut |
| author_facet | Gölz, Paul Igarashi, Ayumi Manurangsi, Pasin Suksompong, Warut |
| contents | 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 $m = ω(\log n)$, 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 $m = o(\sqrt{n})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_06218 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fair Allocation of Indivisible Goods with Variable Groups Gölz, Paul Igarashi, Ayumi Manurangsi, Pasin Suksompong, Warut Computer Science and Game Theory Discrete Mathematics Probability 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 $m = ω(\log n)$, 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 $m = o(\sqrt{n})$. |
| title | Fair Allocation of Indivisible Goods with Variable Groups |
| topic | Computer Science and Game Theory Discrete Mathematics Probability |
| url | https://arxiv.org/abs/2511.06218 |