The Computational Complexity of the Housing Market

TL;DR

This paper establishes that finding a competitive equilibrium in a housing market with indivisible goods, money, and income effects is PPAD-complete, highlighting the computational difficulty of such economic problems.

Contribution

It proves the PPAD-completeness of the classic housing market equilibrium problem with income effects, contrasting with known polynomial-time algorithms for related simpler problems.

Findings

Proves PPAD-completeness of the housing market equilibrium problem.

Shows the Rainbow-KKM problem is PPAD-complete.

Provides bounds on query complexity for finding equilibria.

Abstract

We prove that the classic problem of finding a competitive equilibrium in an exchange economy with indivisible goods, money, and unit-demand agents is PPAD-complete. In this "housing market", agents have preferences over the house and amount of money they end up with, but can experience income effects. Our results contrast with the existence of polynomial-time algorithms for related problems: Top Trading Cycles for the "housing exchange" problem in which there are no transfers and the Hungarian algorithm for the "housing assignment" problem in which agents' utilities are linear in money. Along the way, we prove that the Rainbow-KKM problem, a total search problem based on a generalization by Gale of the Knaster-Kuratowski-Mazurkiewicz lemma, is PPAD-complete. Our reductions also imply bounds on the query complexity of finding competitive equilibrium.