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Weighted Fairness Notions for Indivisible Items Revisited

Mithun Chakraborty, Erel Segal-Halevi, Warut Suksompong

TL;DR

This work proposes a parameterized family of relaxations for weighted envy-freeness and the same for weighted proportionality and introduces an intuitive weighted generalization of maximin share fairness that can be guaranteed.

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 smaller-weight 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

TL;DR

This work proposes a parameterized family of relaxations for weighted envy-freeness and the same for weighted proportionality and introduces an intuitive weighted generalization of maximin share fairness that can be guaranteed.

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 smaller-weight 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.
Paper Structure (21 sections, 43 theorems, 32 equations, 4 figures, 3 tables)

This paper contains 21 sections, 43 theorems, 32 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Our contributions
  3. Related work
  4. Preliminaries
  5. Fairness Up to One Item
  6. Weighted envy-freeness notions
  7. Weighted proportionality notions
  8. Weighted Nash welfare
  9. Share-Based Notions
  10. Definitions
  11. Ordinal versus cardinal notions
  12. Approximation of NMMS
  13. Maximum weighted Nash welfare and share-based notions
  14. Ordered-EF1 allocations
  15. Other relationships between fairness notions
  16. ...and 6 more sections

Key Result

Theorem 3.2

WWEF1 is equivalent to the following condition: for every pair of agents $i,j$, there exists some $x_{i,j} \in [0,1]$ such that the condition for WEF$(x_{i,j},1-x_{i,j})$ is satisfied for that pair.

Figures (4)

  • Figure 1: Percentage of allocations produced by the divisor picking sequence with function $f(t) = t+y$ that satisfy weighted envy-freeness for three agents. The $x$-axis indicates the value of $y$ in the function.
  • Figure 2: Percentage of allocations produced by the divisor picking sequence with function $f(t) = t+y$ that satisfy weighted proportionality for three agents. The $x$-axis indicates the value of $y$ in the function.
  • Figure 3: Percentage of allocations produced by the divisor picking sequence with function $f(t) = t+y$ that satisfy WMMS-fairness for three agents. The $x$-axis indicates the value of $y$ in the function.
  • Figure 4: Percentage of allocations produced by the divisor picking sequence with function $f(t) = t+y$ that satisfy NMMS-fairness for three agents. The $x$-axis indicates the value of $y$ in the function.

Theorems & Definitions (56)

  • Definition 3.1: WEF$(x,y)$
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 3.4
  • Theorem 3.5
  • Definition 3.6: WPROP$(x,y)$
  • Definition 3.7: WPROP$^*(x,y)$
  • Lemma 3.8
  • Lemma 3.9
  • Corollary 3.10
  • ...and 46 more