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Weighted Envy-Freeness Revisited: Indivisible Resource and House Allocations

Yuxi Liu, Mingyu Xiao

Abstract

Envy-Freeness is one of the most fundamental and important concepts in fair allocation. Some recent studies have focused on the concept of weighted envy-freeness. Under this concept, each agent is assigned a weight, and their valuations are divided by their weights when assessing fairness. This concept can promote more fairness in some scenarios. But on the other hand, experimental research has shown that this weighted envy-freeness significantly reduces the likelihood of fair allocations. When we must allocate the resources, we may propose fairness concepts with lower requirements that are potentially more feasible to implement. In this paper, we revisit weighted envy-freeness and propose a new concept called SumAvg-envy-freeness, which substantially increases the existence of fair allocations. This new concept can be seen as a complement of the normal weighted envy-fairness. Furthermore, we systematically study the computational complexity of finding fair allocations under the old and new weighted fairness concepts in two types of classic problems: Indivisible Resource Allocation and House Allocation. Our study provides a comprehensive characterization of various properties of weighted envy-freeness.

Weighted Envy-Freeness Revisited: Indivisible Resource and House Allocations

Abstract

Envy-Freeness is one of the most fundamental and important concepts in fair allocation. Some recent studies have focused on the concept of weighted envy-freeness. Under this concept, each agent is assigned a weight, and their valuations are divided by their weights when assessing fairness. This concept can promote more fairness in some scenarios. But on the other hand, experimental research has shown that this weighted envy-freeness significantly reduces the likelihood of fair allocations. When we must allocate the resources, we may propose fairness concepts with lower requirements that are potentially more feasible to implement. In this paper, we revisit weighted envy-freeness and propose a new concept called SumAvg-envy-freeness, which substantially increases the existence of fair allocations. This new concept can be seen as a complement of the normal weighted envy-fairness. Furthermore, we systematically study the computational complexity of finding fair allocations under the old and new weighted fairness concepts in two types of classic problems: Indivisible Resource Allocation and House Allocation. Our study provides a comprehensive characterization of various properties of weighted envy-freeness.
Paper Structure (13 sections, 7 theorems, 17 equations, 3 figures, 3 tables)

This paper contains 13 sections, 7 theorems, 17 equations, 3 figures, 3 tables.

Table of Contents

  1. More Related Work
  2. Weighted Models.
  3. House Allocation.
  4. Our Model and Contributions
  5. New Concept of Envy-Freeness
  6. Computational Complexity
  7. Preliminaries
  8. Indivisible Resource Allocations
  9. Polynomial solvable cases
  10. Few agents or few resources
  11. House Allocations
  12. Experiments
  13. Conclusion

Key Result

Theorem 1

SAEF-Allocation under identical and 0/1 preferences can be solved in $O(nm^3)$ time.

Figures (3)

  • Figure 1: An illustration for $(A, R, U)$, where $r_1 = x_1\vee x_2\vee \bar{x}_n$. For each agent $a$ and resource $r$, (i) there is a solid edge between $a$ and $r$ if and only if $u_a(r) = M$; (ii) there is a dashed edge between $a$ and $r$ if and only if $u_a(r) = 1$; (ii) there is no edge between $a$ and $r$ if and only if $u_a(r) = 0$.
  • Figure 2: The number of instances that admit SumAvg-envy-free allocations(resp. Sum-envy-free allocations or Avg-envy-free allocations) in four different settings. The term $1\leq w\leq 100$ means that the weights of agents are drawn uniformly and independently from 1 to 100, while $101\leq w \leq 200$ means that the weights of agents are drawn uniformly and independently from 101 to 200. The term "IC" means the linear preferences of the agents are generated from impartial culture and "SPUP" means the linear preferences of the agents are generated from the single-peaked uniform peak culture.
  • Figure 3: The number of instances that admit SumAvg-envy-free allocations(resp. Sum-envy-free allocations or Avg-envy-free allocations) in four different settings. The term $1\leq w\leq 100$ means that the weights of agents are drawn uniformly and independently from 1 to 100, while $101\leq w \leq 200$ means that the weights of agents are drawn uniformly and independently from 101 to 200. The term "IC" means the linear preferences of the agents are generated from impartial culture and "SPUP" means the linear preferences of the agents are generated from the single-peaked uniform peak culture.

Theorems & Definitions (20)

  • Example 1
  • Example 2
  • Definition 1
  • Definition 2
  • proof
  • proof
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • ...and 10 more