Weighted Envy-Freeness in House Allocation

TL;DR

This paper studies weighted envy-freeness in house allocation, providing algorithms to determine existence, methods to achieve fairness via subsidies, and characterizations of when such fair allocations are possible.

Contribution

It introduces polynomial-time algorithms for weighted envy-freeness, explores subsidy-based fairness, and characterizes conditions for existence in weighted house allocation.

Findings

Weighted envy-free allocations do not always exist.

Polynomial-time algorithms can determine the existence of such allocations.

Subsidies can sometimes achieve weighted envy-freeness.

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.