Weighted Envy-Freeness in House Allocation

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.