# Best-of-Both-Worlds Fair-Share Allocations

arXiv: https://arxiv.org/abs/2102.04909 · HTML: https://mlsanity.com/paper/2102.04909/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Feb 2021.

## Built on by (most cited first, 8 of 8)

- [Fair Shares: Feasibility, Domination and Incentives](https://mlsanity.com/paper/2205.07519/) (2022)
- [Stable Matching with Ties: Approximation Ratios and Learning](https://mlsanity.com/paper/2411.03270/) (2024)
- [Fair Allocation in Dynamic Mechanism Design](https://mlsanity.com/paper/2406.00147/) (2024)
- [Individual Fairness under Varied Notions of Group Fairness in Bipartite Matching - One Framework to Approximate Them All](https://mlsanity.com/paper/2208.09951/) (2022)
- [Fair Grading Algorithms for Randomized Exams](https://mlsanity.com/paper/2304.06254/) (2023)
- [First-Choice Maximality Meets Ex-ante and Ex-post Fairness](https://mlsanity.com/paper/2305.04589/) (2023)
- [Reachability Poorman Discrete-Bidding Games](https://mlsanity.com/paper/2307.15218/) (2023)
- [Simultaneous Envy and Equitability Guarantees](https://mlsanity.com/paper/2608.26410/) (2026)

## Abstract (arXiv)

We consider the problem of fair allocation of indivisible items among n agents with additive valuations, when agents have equal entitlements to the goods, and there are no transfers. Best-of-Both-Worlds (BoBW) fairness mechanisms aim to give all agents both an ex-ante guarantee (such as getting the proportional share in expectation) and an ex-post guarantee. Prior BoBW results have focused on ex-post guarantees that are based on the "up to one item" paradigm, such as envy-free up to one item (EF1). In this work we attempt to give every agent a high value ex-post, and specifically, a constant fraction of his maximin share (MMS). The up to one item paradigm fails to give such a guarantee, and it is not difficult to present examples in which previous BoBW mechanisms give agents only a frac1n fraction of their MMS. Our main result is a deterministic polynomial time algorithm that computes a distribution over allocations that is ex-ante proportional, and ex-post, every allocation gives every agent at least his proportional share up to one item, and more importantly, at least half of his MMS. Moreover, this last ex-post guarantee holds even with respect to a more demanding notion of a share, introduced in this paper, that we refer to as the truncated proportional share (TPS). Our guarantees are nearly best possible, in the sense that one cannot guarantee agents more than their proportional share ex-ante, and one cannot guarantee agents more than a fracn2n-1 fraction of their TPS ex-post.