Weighted Envy-Freeness in House Allocation

TL;DR

This work introduces weighted envy-freeness in the house allocation problem, where agents have positive weights representing entitlements. It provides a polynomial-time algorithm to decide the existence of weighted envy-free allocations (WEF) and to compute one when possible, and develops a framework to study WEF allocations with subsidies, including a weighted envy graph that characterizes when subsidies can eliminate envy. The paper proves that WEFable allocations do not always exist, even with subsidies, and offers precise characterizations and efficient algorithms for special cases such as identical utilities, two agent types, and bi-valued utilities. It also discusses normalization effects and compares with unweighted settings, growing the understanding of fairness under weighted entitlements in indivisible good allocations. The results have implications for fair housing and subsidy-design problems, highlighting when fairness can be achieved in polynomial time and when it cannot, under various structural assumptions and utility models.

Abstract

The classic house allocation problem involves assigning $m$ houses to $n$ agents based on their utility functions, ensuring each agent receives exactly one house. A key criterion in these problems is satisfying fairness constraints such as envy-freeness. We extend this problem by considering agents with arbitrary weights, focusing on the concept of weighted envy-freeness, which has been extensively studied in fair division. We present a polynomial-time algorithm to determine whether weighted envy-free allocations exist and, if so, to compute one. Since weighted envy-free allocations do not always exist, we also investigate the potential of achieving such allocations through the use of subsidies. We provide several characterizations for weighted envy-freeable allocations (allocations that can be turned weighted envy-free by introducing subsidies) and show that they do not always exist, which is different from the unweighted setting. Furthermore, we explore the existence of weighted envy-freeable allocations in specific scenarios and outline the conditions under which they exist.

Figures (4)

• Figure 1: The corresponding Weighted Envy Graph.
• Figure 2: An example of path compression, where orange nodes represent the large agents and the green nodes represent the small agents. The edge $(i_{k-1}, i_k)$ and $(i_{0}, i_k)$ are the cross-type edges.
• Figure 3: Example of $\overleftarrow{\mathcal{C}}$ and $\mathcal{C}'$, where the black edges are from $\mathcal{C}'$ and blue edges are from $\overleftarrow{\mathcal{C}}$.
• Figure 4: Example of augmenting path with $k=2$, where black edges represent assignments in $\mathbf{B}$, blue edges represent assignments in $\mathbf{A}$. Orange nodes present agents and green nodes represent houses.

Theorems & Definitions (42)

• Definition 2.1: WEF Allocation
• Definition 2.2: WEF Outcome
• Definition 2.3: WEFable
• Definition 2.5: PO
• Definition 3.1: Hall Violator
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4