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EF1 for Mixed Manna with Unequal Entitlements

Jugal Garg, Eklavya Sharma

Abstract

We study fair division of indivisible mixed manna when agents have unequal entitlements, with weighted envy-freeness up to one item (WEF1) as our primary notion of fairness. We identify several shortcomings of existing techniques to achieve WEF1. Hence, we relax WEF1 to weighted envy-freeness up to 1 transfer (WEF1T), and give a polynomial-time algorithm for achieving it. We also generalize Fisher markets to the mixed manna setting, and use them to get a polynomial-time algorithm for two agents that outputs a WEF1 allocation.

EF1 for Mixed Manna with Unequal Entitlements

Abstract

We study fair division of indivisible mixed manna when agents have unequal entitlements, with weighted envy-freeness up to one item (WEF1) as our primary notion of fairness. We identify several shortcomings of existing techniques to achieve WEF1. Hence, we relax WEF1 to weighted envy-freeness up to 1 transfer (WEF1T), and give a polynomial-time algorithm for achieving it. We also generalize Fisher markets to the mixed manna setting, and use them to get a polynomial-time algorithm for two agents that outputs a WEF1 allocation.

Paper Structure

This paper contains 15 sections, 13 theorems, 10 equations, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our Results
  3. Related Work
  4. Preliminaries
  5. Fair Division Instances
  6. Fairness
  7. Efficiency
  8. WEF1 and WEF1T
  9. Algorithm for WEF1T
  10. Impossibility for Ordinal WEF1
  11. Hard Examples for Two-Phase Algorithms
  12. WEF1+PO Alloction Using Fisher Market
  13. Fisher Market
  14. Algorithm for Two Agents
  15. Conclusion and Open Problems

Key Result

Theorem 1

Let $([n], G \cup C, (v_i)_{i=1}^n, w)$ be a fair division instance, where $G$ is the set of goods and neutral items, and $C$ is the set of chores. Let $A^{(G)}$ be a WEF1 allocation of $G$ and $A^{(C)}$ be a WEF1 allocation of $C$. Then allocation $A$ is WEF1T, where $A_i := A^{(G)}_i \cup A^{(C)}_

Theorems & Definitions (34)

  • Definition 1
  • Definition 2: PO and fPO
  • Theorem 1
  • proof
  • Definition 3
  • Theorem 2
  • proof
  • Example 4
  • Example 5
  • Definition 6: market equilibrium
  • ...and 24 more