Two Algorithms for Additive and Fair Division of Mixed Manna

TL;DR

This paper introduces algorithms for fair division of mixed items with positive, zero, and negative utilities, analyzing new fairness properties and providing solutions under various utility models.

Contribution

It presents new algorithms for achieving EFX and EF1$^3$ fairness with Pareto-optimality in mixed utility settings, including cases with identical and mixed utilities.

Findings

Modified EF1$^3$ algorithm for general utilities

Algorithms for EFX$^3$ and PO with mixed utilities

New impossibility results for fair division

Abstract

We consider a fair division model in which agents have positive, zero and negative utilities for items. For this model, we analyse one existing fairness property - EFX - and three new and related properties - EFX$_0$, EFX$^3$ and EF1$^3$ - in combination with Pareto-optimality. With general utilities, we give a modified version of an existing algorithm for computing an EF1$^3$ allocation. With $-\alpha/0/\alpha$ utilities, this algorithm returns an EFX$^3$ and PO allocation. With absolute identical utilities, we give a new algorithm for an EFX and PO allocation. With $-\alpha/0/\beta$ utilities, this algorithm also returns such an allocation. We report some new impossibility results as well.