# Envy-Freeness in House Allocation Problems

## TL;DR

This paper introduces a polynomial-time algorithm to determine the existence of envy-free house allocations and shows such allocations are likely when houses slightly outnumber agents.

## Contribution

The paper provides the first efficient method for checking envy-freeness in house allocation and probabilistic guarantees for existence under certain conditions.

## Key findings

- Polynomial-time algorithm for envy-free assignment existence
- High probability of envy-free solutions when houses exceed agents by a logarithmic factor
- Envy-freeness can be efficiently verified and computed

## Abstract

We consider the house allocation problem, where $m$ houses are to be assigned to $n$ agents so that each agent gets exactly one house. We present a polynomial-time algorithm that determines whether an envy-free assignment exists, and if so, computes one such assignment. We also show that an envy-free assignment exists with high probability if the number of houses exceeds the number of agents by a logarithmic factor.