On the Existence of Fair Allocations for Goods and Chores under Dissimilar Preferences

TL;DR

This paper establishes explicit bounds for the existence of envy-free allocations in fair division problems involving multiple groups and item types, extending to chores and continuous domains with a new, simplified technique.

Contribution

It provides the first explicit bounds for arbitrary groups and item types, resolving a key open question and introducing a versatile, constructive method for fair division.

Findings

Explicit bounds for envy-free allocations with multiple groups and item types.

A new simplified technique applicable to goods, chores, and continuous domains.

Extension of results to cake cutting and related fair division problems.

Abstract

We study the fundamental problem of fairly allocating a multiset $\mathcal{M}$ of $t$ types of indivisible items among $d$ groups of agents, where all agents within a group have identical additive valuations. Gorantla et al. [GMV23] showed that for every such instance, there exists a finite number $\mu$ such that, if each item type appears at least $\mu$ times, an envy-free allocation exists. Their proof is non-constructive and only provides explicit upper bounds on $\mu$ for the cases of two groups ($d=2$) or two item types ($t=2$). In this work, we resolve one of the main open questions posed by Gorantla et al. [GMV23] by deriving explicit upper bounds on $\mu$ that hold for arbitrary numbers of groups and item types. We introduce a significantly simpler, yet powerful technique that not only yields constructive guarantees for indivisible goods but also extends naturally to chores and continuous domains, leading to new results in related fair division settings such as cake cutting.