## Abstract

We consider the problem of fair division, where a set of indivisible goods should be distributed fairly among a set of agents with combinatorial valuations. To capture fairness, we adopt the notion of shares, where each agent is entitled to a fair share, based on some fairness criterion, and an allocation is considered fair if the value of every agent (weakly) exceeds her fair share. A share-based notion is considered universally feasible if it admits a fair allocation for every profile of monotone valuations. A major question arises: is there a non-trivial share-based notion that is universally feasible? The most well-known share-based notions, namely the proportional share and the maximin share, are not universally feasible, nor are any constant approximations of them. We propose a novel share notion, where an agent assesses the fairness of a bundle by comparing it to her valuation in a random allocation. In this framework, a bundle is considered q-quantile fair, for q∈[0,1], if it is at least as good as a bundle obtained in a uniformly random allocation with probability at least q. Our main question is whether there exists a constant value of q for which the q-quantile share is universally feasible. Our main result establishes a strong connection between the feasibility of quantile shares and the classical Erdos Matching Conjecture. Specifically, we show that if a version of this conjecture is true, then the 1/2e-quantile share is universally feasible. Furthermore, we provide unconditional feasibility results for additive, unit-demand and matroid-rank valuations for constant values of q. Finally, we discuss the implications of our results for other share notions.

Original language | English |
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Title of host publication | STOC 2024 - Proceedings of the 56th Annual ACM Symposium on Theory of Computing |

Editors | Bojan Mohar, Igor Shinkar, Ryan O�Donnell |

Publisher | Association for Computing Machinery |

Pages | 1235-1246 |

Number of pages | 12 |

ISBN (Electronic) | 9798400703836 |

DOIs | |

State | Published - 10 Jun 2024 |

Event | 56th Annual ACM Symposium on Theory of Computing, STOC 2024 - Vancouver, Canada Duration: 24 Jun 2024 → 28 Jun 2024 |

### Publication series

Name | Proceedings of the Annual ACM Symposium on Theory of Computing |
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ISSN (Print) | 0737-8017 |

### Conference

Conference | 56th Annual ACM Symposium on Theory of Computing, STOC 2024 |
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Country/Territory | Canada |

City | Vancouver |

Period | 24/06/24 → 28/06/24 |

### Funding

Funders | Funder number |
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European Research Council | |

Horizon 2020 | 866132 |

Horizon 2020 | |

NSF-BSF | 2020788 |

## Keywords

- Erdos Matching Conjecture
- Fair Division
- Quantile Share