# Approximation Algorithms for Maximin Fair Division

## TL;DR

This paper develops efficient algorithms for approximately fair division of indivisible goods among agents with additive and submodular valuations, achieving guarantees close to the maximin share and introducing new methods for submodular cases.

## Contribution

It provides a simple, efficient algorithm matching the 2/3 approximation for additive valuations and initiates the study of approximate maximin fairness under submodular valuations with a 0.21 guarantee.

## Key findings

- Achieves a 2/3-approximate maximin fair allocation for additive valuations.
- Introduces a 0.21-approximate solution for submodular valuations.
- Uses multilinear extensions to analyze the round-robin algorithm.

## Abstract

We consider the problem of allocating indivisible goods fairly among n agents who have additive and submodular valuations for the goods. Our fairness guarantees are in terms of the maximin share, that is defined to be the maximum value that an agent can ensure for herself, if she were to partition the goods into n bundles, and then receive a minimum valued bundle. Since maximin fair allocations (i.e., allocations in which each agent gets at least her maximin share) do not always exist, prior work has focused on approximation results that aim to find allocations in which the value of the bundle allocated to each agent is (multiplicatively) as close to her maximin share as possible. In particular, Procaccia and Wang (2014) along with Amanatidis et al. (2015) have shown that under additive valuations a 2/3-approximate maximin fair allocation always exists and can be found in polynomial time. We complement these results by developing a simple and efficient algorithm that achieves the same approximation guarantee. Furthermore, we initiate the study of approximate maximin fair division under submodular valuations. Specifically, we show that when the valuations of the agents are nonnegative, monotone, and submodular, then a 0.21-approximate maximin fair allocation is guaranteed to exist. In fact, we show that such an allocation can be efficiently found by using a simple round-robin algorithm. A technical contribution of the paper is to analyze the performance of this combinatorial algorithm by employing the concept of multilinear extensions.