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Weighted Envy-free Allocation with Subsidy

Haris Aziz, Xin Huang, Kei Kimura, Indrajit Saha, Zhaohong Sun, Mashbat Suzuki, Makoto Yokoo

Abstract

We consider the problem of fair allocation of indivisible items with subsidies when agents have weighted entitlements. After highlighting several important differences from the unweighted case, we present several results concerning weighted envy-freeability including general characterizations, algorithms for achieving and testing weighted envy-freeability, lower and upper bounds of the amount of subsidies for envy-freeable allocations, and algorithms for achieving weighted envy-freeability along with other properties.

Weighted Envy-free Allocation with Subsidy

Abstract

We consider the problem of fair allocation of indivisible items with subsidies when agents have weighted entitlements. After highlighting several important differences from the unweighted case, we present several results concerning weighted envy-freeability including general characterizations, algorithms for achieving and testing weighted envy-freeability, lower and upper bounds of the amount of subsidies for envy-freeable allocations, and algorithms for achieving weighted envy-freeability along with other properties.
Paper Structure (19 sections, 23 theorems, 18 equations, 1 figure, 1 table, 3 algorithms)

This paper contains 19 sections, 23 theorems, 18 equations, 1 figure, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Contributions
  3. Related Work
  4. Envy-free allocation with money
  5. Fair allocation with entitlements
  6. Model
  7. A General Characterization and its Implications
  8. Restricted Domains
  9. Super-modular valuation
  10. Additive valuation
  11. General additive valuation
  12. Identical additive valuation
  13. Binary additive valuation
  14. Identical items
  15. Approximate envy-free allocations that are weighted envy-freeable
  16. ...and 4 more sections

Key Result

Theorem 1

The following conditions are equivalent for a given allocation:

Figures (1)

  • Figure 1: A diagram for various variables in the proof of Lemma \ref{['lem:path-weight-1/w']}.

Theorems & Definitions (63)

  • Definition 1: Weighted envy-freeability
  • Example 1
  • Definition 2: Pareto efficiency
  • Definition 3: Non-wastefulness
  • Definition 4: Weighted welfare maximizing allocation
  • Definition 5: Non-zero social welfare
  • Definition 6: Weighted reassignment-stability
  • Definition 7: Weighted envy-graph
  • Theorem 1
  • proof
  • ...and 53 more