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Fair Orientations: Proportionality and Equitability

Ankang Sun, Ruijie Wang, Bo Li

TL;DR

This work extends the fair allocation of indivisible items under relevance constraints to other key fairness criteria -- such as proportionality, equitability, and their relaxations -- in settings where the items may be goods, chores, or a mixture of both.

Abstract

We study the fair allocation of indivisible items under relevance constraints, where each agent has a set of relevant items and can only receive items that are relevant to them. While the relevance constraint has been studied in recent years, existing work has largely focused on envy-freeness. Our work extends this study to other key fairness criteria -- such as proportionality, equitability, and their relaxations -- in settings where the items may be goods, chores, or a mixture of both. We complement the literature by presenting a picture of the existence and computational complexity of the considered criteria.

Fair Orientations: Proportionality and Equitability

TL;DR

This work extends the fair allocation of indivisible items under relevance constraints to other key fairness criteria -- such as proportionality, equitability, and their relaxations -- in settings where the items may be goods, chores, or a mixture of both.

Abstract

We study the fair allocation of indivisible items under relevance constraints, where each agent has a set of relevant items and can only receive items that are relevant to them. While the relevance constraint has been studied in recent years, existing work has largely focused on envy-freeness. Our work extends this study to other key fairness criteria -- such as proportionality, equitability, and their relaxations -- in settings where the items may be goods, chores, or a mixture of both. We complement the literature by presenting a picture of the existence and computational complexity of the considered criteria.
Paper Structure (17 sections, 15 theorems, 16 equations, 7 figures)

This paper contains 17 sections, 15 theorems, 16 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Our Contribution
  3. Proportionality
  4. Equitability
  5. Envy-freeness
  6. Other Related Work
  7. Preliminaries
  8. The Orientation Model
  9. Solution Concepts
  10. Proportionality and Its Relaxations
  11. PROP Fairness
  12. PROP1 and PROPX Fairness
  13. Strongly PROP1 and MMS Fairness
  14. Equitability and Its Relaxations
  15. EF1 Orientations for Chores
  16. ...and 2 more sections

Key Result

Theorem 3.1

Deciding the existence of PROP orientations is NP-complete, even for simple graphs where (1) all items are goods, valuations are normalized, and $v_i(e)\in \{1,2\}$ for all $i$ and $e\in E_i$; or all items are chores, valuations are normalized, and $v_i(e)\in \{-1,-2\}$ for all $i$ and $e\in E_i$.

Figures (7)

  • Figure 1: Combined illustration of the clause gadget and variable gadget.
  • Figure 2: The illustration of the goods-instance where PROPX orientations do not exist. The label on each edge represents the value of the edge to both endpoints
  • Figure 3: The illustration for goods-instance of the clause gadget for $C_j$. For each edge, if there is only one label, it represents the value of the edge to both endpoints. If there are two labels, the label closer to a vertex represents that vertex's value for the edge.
  • Figure 4: The illustration of the goods-instance for the reduction from EQ. The label closer to a vertex represents that vertex’s value for the edge.
  • Figure 5: The illustration of the goods-instance where EQ1 does not exist. The label on each edge represents the value of the edge to both endpoints.
  • ...and 2 more figures

Theorems & Definitions (36)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4: EQX
  • Definition 2.5: EQ1
  • Definition 2.6: EF1
  • Theorem 3.1
  • proof : Proof for the goods-instance
  • Theorem 3.2
  • proof
  • ...and 26 more