Fair allocation of combinations of indivisible goods and chores

TL;DR

This paper explores fair division of items that can be either goods or chores with agents having positive or negative utilities, introducing new algorithms and analyzing the complexity of fair allocations in this generalized setting.

Contribution

It extends fair division theory to include mixed utilities, providing new algorithms and identifying gaps in existing literature on fairness and efficiency.

Findings

Some positive axiomatic results extend to mixed utilities.

New efficient algorithms for fair allocation in the generalized setting.

Identified gaps and complexity issues in existing fair division literature.

Abstract

We consider the problem of fairly dividing a set of items. Much of the fair division literature assumes that the items are `goods' i.e., they yield positive utility for the agents. There is also some work where the items are `chores' that yield negative utility for the agents. In this paper, we consider a more general scenario where an agent may have negative or positive utility for each item. This framework captures, e.g., fair task assignment, where agents can have both positive and negative utilities for each task. We show that whereas some of the positive axiomatic and computational results extend to this more general setting, others do not. We present several new and efficient algorithms for finding fair allocations in this general setting. We also point out several gaps in the literature regarding the existence of allocations satisfying certain fairness and efficiency properties and further study the complexity of computing such allocations.