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Bandit Learning in Housing Markets

Shiyun Lin

TL;DR

Bandit Learning in Housing Markets treats unknown agent preferences in one-sided housing markets as a multiplayer bandit problem with collisions and a core-based allocation objective. It introduces per-player core regret and develops two algorithms: a decentralized Explor-then-Commit (ETC) and a centralized anytime UCB-TTC, both achieving an order-optimal $Reg_i(T)=O\left(\dfrac{N\log T}{\Delta_{\min}^2}\right)$ and matching lower bounds in the decentralized setting. An offline TTC baseline anchors core optimality for known preferences, while the centralized approach leverages a platform to coordinate TTC using learned rankings. These results advance sublinear, core-stable learning in housing markets and point to future work on existing tenants, indifference, and incentive issues.

Abstract

The housing market, also known as one-sided matching market, is a classic exchange economy model where each agent on the demand side initially owns an indivisible good (a house) and has a personal preference over all goods. The goal is to find a core-stable allocation that exhausts all mutually beneficial exchanges among subgroups of agents. While this model has been extensively studied in economics and computer science due to its broad applications, little attention has been paid to settings where preferences are unknown and must be learned through repeated interactions. In this paper, we propose a statistical learning model within the multi-player multi-armed bandit framework, where players (agents) learn their preferences over arms (goods) from stochastic rewards. We introduce the notion of core regret for each player as the market objective. We study both centralized and decentralized approaches, proving $O(N \log T / Δ^2)$ upper bounds on regret, where $N$ is the number of players, $T$ is the time horizon and $Δ$ is the minimum preference gap among players. For the decentralized setting, we also establish a matching lower bound, demonstrating that our algorithm is order-optimal.

Bandit Learning in Housing Markets

TL;DR

Bandit Learning in Housing Markets treats unknown agent preferences in one-sided housing markets as a multiplayer bandit problem with collisions and a core-based allocation objective. It introduces per-player core regret and develops two algorithms: a decentralized Explor-then-Commit (ETC) and a centralized anytime UCB-TTC, both achieving an order-optimal and matching lower bounds in the decentralized setting. An offline TTC baseline anchors core optimality for known preferences, while the centralized approach leverages a platform to coordinate TTC using learned rankings. These results advance sublinear, core-stable learning in housing markets and point to future work on existing tenants, indifference, and incentive issues.

Abstract

The housing market, also known as one-sided matching market, is a classic exchange economy model where each agent on the demand side initially owns an indivisible good (a house) and has a personal preference over all goods. The goal is to find a core-stable allocation that exhausts all mutually beneficial exchanges among subgroups of agents. While this model has been extensively studied in economics and computer science due to its broad applications, little attention has been paid to settings where preferences are unknown and must be learned through repeated interactions. In this paper, we propose a statistical learning model within the multi-player multi-armed bandit framework, where players (agents) learn their preferences over arms (goods) from stochastic rewards. We introduce the notion of core regret for each player as the market objective. We study both centralized and decentralized approaches, proving upper bounds on regret, where is the number of players, is the time horizon and is the minimum preference gap among players. For the decentralized setting, we also establish a matching lower bound, demonstrating that our algorithm is order-optimal.

Paper Structure

This paper contains 29 sections, 15 theorems, 37 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Multi-player Bandit Learning
  4. Competing Bandits in Two-sided Matching Markets
  5. Problem Setting
  6. Offline Top-Trading-Cycle Algorithm
  7. Decentralized Algorithm
  8. Theoretical Analysis
  9. Regret Lower Bound
  10. Single Top Trading Cycle Bandits
  11. Centralized Anytime Algorithm
  12. Conclusion and Discussion
  13. Housing Markets with Existing Tenants
  14. Housing Markets with Indifference
  15. Incentive Compatibility in the Learning Setting
  16. ...and 14 more sections

Key Result

Theorem 1

Following Algorithm alg:decentralize, the core regret of each player $p_i \in {\mathcal{N}}$ satisfies

Theorems & Definitions (32)

  • Definition 1: Core
  • Remark 1
  • Definition 2: Minimum Preference Gap
  • Theorem 1
  • proof : Proof of Theorem \ref{['thm:decentralize_upper_bound']}
  • Lemma 1: Bad Concentration Event
  • Lemma 2
  • Lemma 3
  • Definition 3: Uniformly Consistent Policies
  • Lemma 4: Regret Decomposition
  • ...and 22 more