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The Degree of Fairness in Efficient House Allocation

Hadi Hosseini, Medha Kumar, Sanjukta Roy

TL;DR

The paper studies the degree to which fairness and efficiency can be reconciled in the classic house allocation problem. It analyzes four fairness notions—envy-freeness (EF), min #envy, min total envy, and minimax total envy—under three welfare criteria: max size, max utilitarian welfare ($USW$), and max egalitarian welfare ($ESW$). It shows that envy-free allocations maximizing welfare can be computed in polynomial time when EF exists, but uncovers a stark difference between utilitarian and egalitarian welfare under fairness constraints, with several NP-hardness results for egalitarian settings and polynomial-time results under certain conditions. The work is complemented by empirical experiments illustrating the trade-offs in randomly generated valuation graphs and 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.

The Degree of Fairness in Efficient House Allocation

TL;DR

The paper studies the degree to which fairness and efficiency can be reconciled in the classic house allocation problem. It analyzes four fairness notions—envy-freeness (EF), min #envy, min total envy, and minimax total envy—under three welfare criteria: max size, max utilitarian welfare (), and max egalitarian welfare (). It shows that envy-free allocations maximizing welfare can be computed in polynomial time when EF exists, but uncovers a stark difference between utilitarian and egalitarian welfare under fairness constraints, with several NP-hardness results for egalitarian settings and polynomial-time results under certain conditions. The work is complemented by empirical experiments illustrating the trade-offs in randomly generated valuation graphs and 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.
Paper Structure (1 section, 1 figure, 1 table)

This paper contains 1 section, 1 figure, 1 table.

Table of Contents

  1. Introduction

Figures (1)

  • Figure 1: An envy-free allocation of maximum size is shown in wavy orange; a minimum #envy complete allocation is shown in dashed green; no envy-free allocation has the maximum welfare $2$; and the allocation in blue achieves minimum #envy among the matchings that has the maximum welfare.

Theorems & Definitions (1)

  • Example 1: Fairness in House Allocation