The Degree of Fairness in Efficient House Allocation

TL;DR

This paper explores the trade-offs between efficiency and fairness in house allocation, showing computational complexities and providing empirical insights into envy-based fairness measures.

Contribution

It characterizes the computational complexity of finding welfare-maximizing fair allocations and compares envy-based fairness measures in house allocation problems.

Findings

Envy-free allocations with maximum welfare are computationally tractable if they exist.

Maximizing utilitarian welfare under fairness constraints is polynomial-time solvable.

Egalitarian welfare maximization remains mostly intractable, even with binary valuations.

Abstract

The classic house allocation problem is primarily concerned with finding a matching between a set of agents and a set of houses that guarantees some notion of economic efficiency (e.g. utilitarian welfare). While recent works have shifted focus on achieving fairness (e.g. minimizing the number of envious agents), they often come with notable costs on efficiency notions such as utilitarian or egalitarian welfare. We investigate the trade-offs between these welfare measures and several natural fairness measures that rely on the number of envious agents, the total (aggregate) envy of all agents, and maximum total envy of an agent. In particular, by focusing on envy-free allocations, we first show that, should one exist, finding an envy-free allocation with maximum utilitarian or egalitarian welfare is computationally tractable. We highlight a rather stark contrast between utilitarian and egalitarian welfare by showing that finding utilitarian welfare maximizing allocations that minimize the aforementioned fairness measures can be done in polynomial time while their egalitarian counterparts remain intractable (for the most part) even under binary valuations. We complement our theoretical findings by giving insights into the relationship between the different fairness measures and conducting empirical analysis.