# Almost Envy-Freeness in Group Resource Allocation

## TL;DR

This paper investigates fair allocation of indivisible goods among groups of agents, exploring relaxations of envy-freeness under various valuation assumptions and introducing a flexible group partitioning model.

## Contribution

It provides new existence results for EF1 and EFX allocations under different valuation models and introduces a novel model allowing dynamic group partitioning.

## Key findings

- Existence of EF1 and EFX allocations for various valuation types.
- Full characterization of group sizes for binary valuations.
- A new model enabling flexible group partitioning with guaranteed fairness.

## Abstract

We study the problem of fairly allocating indivisible goods between groups of agents using the recently introduced relaxations of envy-freeness. We consider the existence of fair allocations under different assumptions on the valuations of the agents. In particular, our results cover cases of arbitrary monotonic, responsive, and additive valuations, while for the case of binary valuations we fully characterize the cardinalities of two groups of agents for which a fair allocation can be guaranteed with respect to both envy-freeness up to one good (EF1) and envy-freeness up to any good (EFX). Moreover, we introduce a new model where the agents are not partitioned into groups in advance, but instead the partition can be chosen in conjunction with the allocation of the goods. In this model, we show that for agents with arbitrary monotonic valuations, there is always a partition of the agents into two groups of any given sizes along with an EF1 allocation of the goods. We also provide an extension of this result to any number of groups.