# Asymptotic Existence of Fair Divisions for Groups

## TL;DR

This paper studies the conditions under which fair, envy-free resource divisions exist among groups with multiple players, showing that the number of items relative to players influences existence and proposing a mechanism for approximate fairness.

## Contribution

It establishes asymptotic conditions for envy-free divisions in group settings and analyzes the effectiveness of a simple truthful mechanism for approximate fairness.

## Key findings

- Envy-free divisions are likely when items exceed total players by a logarithmic factor.
- Envy-free divisions are unlikely when items are fewer than total players.
- The random assignment mechanism achieves approximate envy-freeness with high probability.

## Abstract

The problem of dividing resources fairly occurs in many practical situations and is therefore an important topic of study in economics. In this paper, we investigate envy-free divisions in the setting where there are multiple players in each interested party. While all players in a party share the same set of resources, each player has her own preferences. Under additive valuations drawn randomly from probability distributions, we show that when all groups contain an equal number of players, a welfare-maximizing allocation is likely to be envy-free if the number of items exceeds the total number of players by a logarithmic factor. On the other hand, an envy-free allocation is unlikely to exist if the number of items is less than the total number of players. In addition, we show that a simple truthful mechanism, namely the random assignment mechanism, yields an allocation that satisfies the weaker notion of approximate envy-freeness with high probability.