Reforming an Unfair Allocation by Exchanging Goods

TL;DR

This paper investigates the complexity and bounds of reforming initial allocations of indivisible goods into envy-free up to one good (EF1) allocations through exchanges, considering various agent and utility scenarios.

Contribution

It provides complexity results, bounds, and distinctions for transforming initial allocations into EF1 allocations via exchanges, advancing understanding of the process's computational aspects.

Findings

Complexity classifications for reaching EF1 allocations.

Tight bounds on the number of exchanges needed.

Differences based on number of agents and utility functions.

Abstract

Fairly allocating indivisible goods is a frequently occurring task in everyday life. Given an initial allocation of the goods, we consider the problem of reforming it via a sequence of exchanges to attain fairness in the form of envy-freeness up to one good (EF1). We present a vast array of results on the complexity of determining whether it is possible to reach an EF1 allocation from the initial allocation and, if so, the minimum number of exchanges required. In particular, we uncover several distinctions based on the number of agents involved and their utility functions. Furthermore, we derive essentially tight bounds on the worst-case number of exchanges needed to achieve EF1.