On allocations that give intersecting groups their fair share

TL;DR

This paper studies fair item allocations for protected groups with additive valuations, identifying conditions under which fair or approximately fair allocations exist, especially highlighting differences between divisible and indivisible items and between chores and goods.

Contribution

It introduces formal fairness definitions for protected groups, provides examples of non-existence of fair allocations, and offers efficient algorithms for approximate fairness under specific conditions.

Findings

Fair allocations may not exist or be approximable in general.

For identical valuations and laminar groups with chores, a 2-approximate fair allocation exists and is efficiently computable.

No constant approximation guarantee exists for goods under similar conditions.

Abstract

We consider item allocation to individual agents who have additive valuations, in settings in which there are protected groups, and the allocation needs to give each protected group its "fair" share of the total welfare. Informally, within each protected group we consider the total welfare that the allocation gives the members of the group, and compare it to the maximum possible welfare that an allocation can give to the group members. An allocation is fair towards the group if the ratio between these two values is no worse then the relative size of the group. For divisible items, our formal definition of fairness is based on the proportional share, whereas for indivisible items, it is based on the anyprice share. We present examples in which there are no fair allocations, and even not allocations that approximate the fairness requirement within a constant multiplicative factor. We then attempt to identify sufficient conditions for fair or approximately fair allocations to exist. For example, for indivisible items, when agents have identical valuations and the family of protected groups is laminar, we show that if the items are chores, then an allocation that satisfies every fairness requirement within a multiplicative factor no worse than two exists and can be found efficiently, whereas if the items are goods, no constant approximation can be guaranteed.