# Best-of-Both-Worlds Fair Allocation of Indivisible and Mixed Goods

arXiv: https://arxiv.org/abs/2410.06877 · HTML: https://mlsanity.com/paper/2410.06877/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Oct 2024.

## Built on by (most cited first, 2 of 2)

- [Probing EFX via PMMS: (Non-)Existence Results in Discrete Fair Division](https://mlsanity.com/paper/2507.14957/) (2025)
- [Simultaneous Envy and Equitability Guarantees](https://mlsanity.com/paper/2608.26410/) (2026)

## Abstract (arXiv)

We study the problem of fairly allocating either a set of indivisible goods or a set of mixed divisible and indivisible goods (i.e., mixed goods) to agents with additive utilities, taking the best-of-both-worlds perspective of guaranteeing fairness properties both ex ante and ex post. The ex-post fairness notions considered in this paper are relaxations of envy-freeness, specifically, EFX for indivisible-goods allocation, and EFM for mixed-goods allocation. For two agents, we show that there is a polynomial-time randomized algorithm that achieves ex-ante envy-freeness and ex-post EFX / EFM simultaneously. For n agents with bi-valued utilities, we show there exist randomized allocations that are (i) ex-ante proportional and ex-post EFM, and (ii) ex-ante envy-free, ex-post EFX, and ex-post fractionally Pareto optimal.