Weighted Fairness Notions for Indivisible Items Revisited

Abstract

We revisit the setting of fairly allocating indivisible items when agents have different weights representing their entitlements. First, we propose a parameterized family of relaxations for weighted envy-freeness and the same for weighted proportionality; the parameters indicate whether smallerweight or larger-weight agents should be given a higher priority. We show that each notion in these families can always be satisfied, but any two cannot necessarily be fulfilled simultaneously. We then introduce an intuitive weighted generalization of maximin share fairness and establish the optimal approximation of it that can be guaranteed. Furthermore, we characterize the implication relations between the various weighted fairness notions introduced in this and prior work, and relate them to the lower and upper quota axioms from apportionment.

Weighted Fairness Notions for Indivisible Items Revisited

“…This also contrasts with the unweighted setting, where maximum Nash welfare implies Θ(1/ √ n)-MMS (Caragiannis et al 2019). The proofs for all of these results can be found in the full version of our paper (Chakraborty, Segal-Halevi, and Suksompong 2021). Farhadi et al (2019, Thm.…”

“…For brevity, we say that a picking sequence satisfies a fairness notion if its output always satisfies that notion. The proof of this theorem, along with all other missing proofs, can be found in the full version of our paper (Chakraborty, Segal-Halevi, and Suksompong 2021). Theorem 3.2.…”

“…In particular, WEF(0, 0) is the same as weighted envy-freeness, WEF(1, 0) is equivalent to the notion WEF1 proposed by Chakraborty et al (2020), and WEF(1, 1) corresponds to what these authors called "transfer weighted envy-freeness up to one item". The WWEF1 condition of Chakraborty et al (2020) is strictly weaker than WEF(x, 1−x) for every x ∈ [0, 1]; see the full version of our paper (Chakraborty, Segal-Halevi, and Suksompong 2021).…”

“…We determine whether each fairness notion implies lower quota (resp., upper quota) in the identical-item setting, i.e., whether it ensures that each agent i ∈ N receives at least q i items (resp., at most q i items). Our results are summarized in Table 1; all proofs are in the full version of our paper (Chakraborty, Segal-Halevi, and Suksompong 2021).…”