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The Computational Complexity of the Housing Market

Edwin Lock, Zephyr Qiu, Alexander Teytelboym

TL;DR

The paper establishes that finding approximate competitive equilibria in Gale's housing market with income effects is $PPAD$-complete, even for $n=3$ agents with identical preferences. It achieves this by proving a computational equivalence between HousingMarket and RainbowKKM, and by constructing polynomial-time reductions from CakeCutting and 2D-Sperner to RainbowKKM. The results hold in both white-box (polynomial-time preference descriptions) and black-box (query oracle) models, with exponential query bounds in the black-box setting for $n\ge 4$. This work situates equilibrium computation with income effects among $PPAD$-complete problems, highlighting fundamental barriers to efficient computation in realistic market designs and outlining several avenues for future research, including the role of utility representations and practical approximation methods.

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.

The Computational Complexity of the Housing Market

TL;DR

The paper establishes that finding approximate competitive equilibria in Gale's housing market with income effects is -complete, even for agents with identical preferences. It achieves this by proving a computational equivalence between HousingMarket and RainbowKKM, and by constructing polynomial-time reductions from CakeCutting and 2D-Sperner to RainbowKKM. The results hold in both white-box (polynomial-time preference descriptions) and black-box (query oracle) models, with exponential query bounds in the black-box setting for . This work situates equilibrium computation with income effects among -complete problems, highlighting fundamental barriers to efficient computation in realistic market designs and outlining several avenues for future research, including the role of utility representations and practical approximation methods.

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.
Paper Structure (20 sections, 30 theorems, 34 equations, 3 figures, 1 table)

This paper contains 20 sections, 30 theorems, 34 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Organization.
  3. The PPAD complexity class.
  4. Notation.
  5. The Housing Market and Rainbow-KKM problems
  6. The housing market
  7. The KKM and RainbowKKM problems
  8. Sparse KKM coverings
  9. Reductions between HousingMarket and RainbowKKM
  10. Connecting the domains of HousingMarket and RainbowKKM
  11. The computational equivalence
  12. PPAD-Hardness of HousingMarket and RainbowKKM
  13. Reducing from the CakeCutting problem
  14. The reduction from the 2D-Sperner problem
  15. The 2D-Sperner problem.
  16. ...and 5 more sections

Key Result

Theorem 2.1

Under assumptions (i) to (iii), there exists a market equilibrium.

Figures (3)

  • Figure 1: Illustration of the main reductions developed in this paper. The three problems shown in blue boxes are known to be PPAD-complete; CakeCutting was shown to be PPAD-complete by deng2012algorithmichollender2023envy, 2D-Sperner in chen2009complexity and Sperner is a canonical PPAD-complete problem introduced in papadimitriou1994complexity. Arrows denote reductions from origin to destination.
  • Figure 2: Triangulation of simplex $\Delta_{2}$ with side length $4$ into regular triangles with side length $1$, drawn using solid black lines. The three corresponding covering sets $C_1, C_2$ and $C_3$ consist of the yellow, red and blue regions, respectively.
  • Figure 3: The Kuhn triangulation of the unit cube.

Theorems & Definitions (47)

  • Definition 1.1
  • Theorem 2.1: gale1984equilibrium
  • Theorem 2.2
  • proof
  • Lemma 2.3: knaster1929beweis
  • Lemma 2.4: gale1984equilibrium
  • Proposition 2.5
  • proof
  • Lemma 3.1
  • proof
  • ...and 37 more