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Two-sided Competing Matching Recommendation Markets With Quota and Complementary Preferences Constraints

Yuantong Li, Guang Cheng, Xiaowu Dai

TL;DR

This paper tackles two-sided matching with quota and complementary preferences where firm-type preferences are unknown and must be learned. It casts the problem in a bandit framework and introduces Multi-agent Multi-type Thompson Sampling (MMTS) coupled with a double-matching procedure to ensure stability at every time step. The authors prove stability and incentive-compatibility, and establish a Bayesian regret bound of the form \\\mathfrak{R}(T) \leq 8Q\log(QT) \\sqrt{K_{\max}T} + NK/Q,\\ highlighting near-linear dependence on total quota and square-root dependence on the maximum type size; simulations validate effectiveness in diverse market settings. The work advances stable, data-driven matching in complex markets with heterogeneous quotas and complementarities, with practical relevance for large-scale recommendation and hiring platforms.

Abstract

In this paper, we propose a new recommendation algorithm for addressing the problem of two-sided online matching markets with complementary preferences and quota constraints, where agents' preferences are unknown a priori and must be learned from data. The presence of mixed quota and complementary preferences constraints can lead to instability in the matching process, making this problem challenging to solve. To overcome this challenge, we formulate the problem as a bandit learning framework and propose the Multi-agent Multi-type Thompson Sampling (MMTS) algorithm. The algorithm combines the strengths of Thompson Sampling for exploration with a new double matching technique to provide a stable matching outcome. Our theoretical analysis demonstrates the effectiveness of MMTS as it can achieve stability and has a total $\widetilde{\mathcal{O}}(Q{\sqrt{K_{\max}T}})$-Bayesian regret with high probability, which exhibits linearity with respect to the total firm's quota $Q$, the square root of the maximum size of available type workers $\sqrt{K_{\max}}$ and time horizon $T$. In addition, simulation studies also demonstrate MMTS's effectiveness in various settings. We provide code used in our experiments \url{https://github.com/Likelyt/Double-Matching}.

Two-sided Competing Matching Recommendation Markets With Quota and Complementary Preferences Constraints

TL;DR

This paper tackles two-sided matching with quota and complementary preferences where firm-type preferences are unknown and must be learned. It casts the problem in a bandit framework and introduces Multi-agent Multi-type Thompson Sampling (MMTS) coupled with a double-matching procedure to ensure stability at every time step. The authors prove stability and incentive-compatibility, and establish a Bayesian regret bound of the form \\\mathfrak{R}(T) \leq 8Q\log(QT) \\sqrt{K_{\max}T} + NK/Q,\\ highlighting near-linear dependence on total quota and square-root dependence on the maximum type size; simulations validate effectiveness in diverse market settings. The work advances stable, data-driven matching in complex markets with heterogeneous quotas and complementarities, with practical relevance for large-scale recommendation and hiring platforms.

Abstract

In this paper, we propose a new recommendation algorithm for addressing the problem of two-sided online matching markets with complementary preferences and quota constraints, where agents' preferences are unknown a priori and must be learned from data. The presence of mixed quota and complementary preferences constraints can lead to instability in the matching process, making this problem challenging to solve. To overcome this challenge, we formulate the problem as a bandit learning framework and propose the Multi-agent Multi-type Thompson Sampling (MMTS) algorithm. The algorithm combines the strengths of Thompson Sampling for exploration with a new double matching technique to provide a stable matching outcome. Our theoretical analysis demonstrates the effectiveness of MMTS as it can achieve stability and has a total -Bayesian regret with high probability, which exhibits linearity with respect to the total firm's quota , the square root of the maximum size of available type workers and time horizon . In addition, simulation studies also demonstrate MMTS's effectiveness in various settings. We provide code used in our experiments \url{https://github.com/Likelyt/Double-Matching}.
Paper Structure (43 sections, 13 theorems, 37 equations, 5 figures, 2 tables, 5 algorithms)

This paper contains 43 sections, 13 theorems, 37 equations, 5 figures, 2 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Problem
  3. Environment
  4. Policy
  5. Challenges and Solutions
  6. Algorithms
  7. Algorithm Description
  8. a). First Match:
  9. b). Sanitize Quota:
  10. c). Second Match:
  11. Incapable Exploration
  12. Properties of MMTS: Stability and Regret
  13. Stability
  14. Bayesian Regret Upper Bound
  15. Incentive-Compatibility
  16. ...and 28 more sections

Key Result

Theorem 5.1

Given two sides' preferences from firms and $M$ types of workers. The double-matching procedure can provide a firm-optimal stable matching solution $\forall t \in [T]$.

Figures (5)

  • Figure 1: $\text{MMTS}$ Algorithm for $\text{CMCPR}$ with its application in the job market, including five stages: preference learning, ranking construction, matching, recommendation, feedback collection.
  • Figure 2: A comparison of centralized UCB and TS that demonstrates the incapable exploration of UCB.
  • Figure 3: Firms and their sub-types regret for Example 1 and, firms and their sub-types regret for Example 2.
  • Figure 4: Posterior distribution of learning parameters for two firms in Example 1.
  • Figure 5: Left: 10 out of 100 randomly selected firms' total regret in Examples 3. Right: all firms' total regret in Example 4.

Theorems & Definitions (29)

  • Definition 1: Blocking pair
  • Definition 2: Stable Matching
  • Definition 3: Valid Match
  • Definition 4: Agent Optimal Match
  • Theorem 5.1
  • proof
  • Theorem 5.2
  • proof
  • Definition 5: Blocking triplet
  • Lemma 5.1
  • ...and 19 more