Allocating Mixed Goods with Customized Fairness and Indivisibility Ratio

TL;DR

This paper introduces a novel fairness relaxation framework for allocating mixed divisible and indivisible goods, using a fraction-based approach tailored to the indivisibility ratio, bridging existing fairness criteria gaps.

Contribution

It proposes 'up to a fraction' fairness relaxations that adapt to the indivisibility ratio, providing tight bounds and asymmetric conditions for mixed goods allocation.

Findings

Established tight bounds for EF and PROP fairness relaxations.

Designed fractional fairness criteria based on indivisibility ratios.

Provided asymmetric fairness conditions for individual preferences.

Abstract

We consider the problem of fairly allocating a combination of divisible and indivisible goods. While fairness criteria like envy-freeness (EF) and proportionality (PROP) can always be achieved for divisible goods, only their relaxed versions, such as the ''up to one'' relaxations EF1 and PROP1, can be satisfied when the goods are indivisible. The ''up to one'' relaxations require the fairness conditions to be satisfied provided that one good can be completely eliminated or added in the comparison. In this work, we bridge the gap between the two extremes and propose ''up to a fraction'' relaxations for the allocation of mixed divisible and indivisible goods. The fraction is determined based on the proportion of indivisible goods, which we call the indivisibility ratio. The new concepts also introduce asymmetric conditions that are customized for individuals with varying indivisibility ratios. We provide both upper and lower bounds on the fractions of the modified item in order to satisfy the fairness criterion. Our results are tight up to a constant for EF and asymptotically tight for PROP.