Almost envy-freeness in group resource allocation

Abstract

We study the problem of fairly allocating indivisible goods between groups of agents using the recently introduced relaxations of envy-freeness. We consider the existence of fair allocations under different assumptions on the valuations of the agents. In particular, our results cover cases of arbitrary monotonic, responsive, and additive valuations, while for the case of binary valuations we fully characterize the cardinalities of two groups of agents for which a fair allocation can be guaranteed with respect to both envy-freeness up to one good (EF1) and envy-freeness up to any good (EFX). Moreover, we introduce a new model where the agents are not partitioned into groups in advance, but instead the partition can be chosen in conjunction with the allocation of the goods. In this model, we show that for agents with arbitrary monotonic valuations, there is always a partition of the agents into two groups of any given sizes along with an EF1 allocation of the goods. We also provide an extension of this result to any number of groups.

(Almost Full) EFX Exists for Four Agents (and Beyond)

“…Other extensions such as group EFX allocations and EF1 for matroid settings were considered respectively by [KSV20] and [GMT14].…”

Maximum Nash welfare and other stories about EFX

“…ere are two variants of EFX used in the related literature, depending on whether only the positively valued goods are considered or not; for the la er case we adopt the name EFX 0 suggested by Kyropoulou et al [2019]. We start by establishing a strong algorithmic connection between the two variants (Proposition 2.3).…”

Almost Envy-Freeness for Groups: Improved Bounds via Discrepancy Theory

“…While fair division has a long and storied history, several fairness notions for the indivisible goods setting, including envy-freeness relaxations, have only been proposed and studied in the past few years [Bouveret et al, 2016;Markakis, 2017]. In the group setting, Kyropoulou et al [2020] showed that EF1 can be guaranteed for all agents only when the groups are small-for instance, with two groups, an EF1 allocation does not always exist when both groups have size at least three. Segal-Halevi and Suksompong [2019] investigated democratic fairness, where the goal is to satisfy a certain fraction of the agents in each group.…”

“…The group aspect of fair division has been addressed in a number of papers in the past few years [Manurangsi and Suksompong, 2017;Ghodsi et al, 2018;Suksompong, 2018;Segal-Halevi and Nitzan, 2019;Segal-Halevi and Suksompong, 2019;Kyropoulou et al, 2020;Segal-Halevi and Suksompong, 2021]. Most of these papers studied the important fairness notion of envy-freeness: an agent is said to be envy-free if she values the goods allocated to her group at least as much as those allocated to any other group.…”