Fair Division of Mixed Divisible and Indivisible Goods

TL;DR

This paper introduces a new fairness concept, envy-freeness for mixed goods (EFM), applicable to resources with both divisible and indivisible items, and provides algorithms for computing such allocations.

Contribution

It proposes the EFM fairness notion, proves its universal existence, and develops efficient algorithms for computing EFM and approximate EFM allocations.

Findings

EFM always exists for any number of agents.

Efficient algorithms are provided for two agents and for n agents with piecewise linear valuations.

An algorithm for approximate EFM runs in polynomial time.

Abstract

We study the problem of fair division when the resources contain both divisible and indivisible goods. Classic fairness notions such as envy-freeness (EF) and envy-freeness up to one good (EF1) cannot be directly applied to the mixed goods setting. In this work, we propose a new fairness notion envy-freeness for mixed goods (EFM), which is a direct generalization of both EF and EF1 to the mixed goods setting. We prove that an EFM allocation always exists for any number of agents. We also propose efficient algorithms to compute an EFM allocation for two agents and for $n$ agents with piecewise linear valuations over the divisible goods. Finally, we relax the envy-free requirement, instead asking for $\epsilon$-envy-freeness for mixed goods ($\epsilon$-EFM), and present an algorithm that finds an $\epsilon$-EFM allocation in time polynomial in the number of agents, the number of indivisible goods, and $1/\epsilon$.