Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Optimal Matching Strategies in Two-sided Markets: A Mean Field Approach

Erhan Bayraktar, Dantong Chu, Bohan Li, Ho Man Tai

TL;DR

We develop a mean field game framework for dynamic two-sided matching where populations with quality attributes meet according to Poisson processes and agents choose mutual acceptance thresholds. The equilibrium is characterized by a fully coupled forward–backward $HJB$–$FP$ system with nonlocal coupling across populations, and the matching process follows FCFS under a mutual-acceptance rule. We prove global well-posedness and a verification theorem for mean field Nash equilibria, with a graphon-structured perspective that clarifies the block-interaction pattern. Numerical experiments in a labor-market setting illustrate how dynamic thresholds and unmatched densities evolve, revealing imperfect sorting and potential upward mobility, thereby linking micro-level strategic decisions to macro-level matching outcomes and policy implications.

Abstract

This paper develops a mean field game framework for dynamic two-sided matching markets, extending existing matching theory by integrating micro-macro dynamics in two-sided environments. Unlike traditional matching models focusing on static equilibrium or unilateral optimization, our framework simultaneously captures dynamic interactions and strategic behaviors of both market sides, as well as the equilibrium. We model two types of agents who meet each other via Poisson processes and make simultaneous matching decisions to maximize their respective objective functionals, and find the corresponding equilibrium. Our approach formulates the equilibrium as a fully coupled Hamilton-Jacobi-Bellman and Fokker-Planck system with nonlocal structure coupling two distinct populations. The mathematical analysis addresses significant challenges from the dual-layered coupling structure and nonlocal structure. We also provide insights into individual behaviors shaping aggregate patterns in labor markets through numerical experiments.

Optimal Matching Strategies in Two-sided Markets: A Mean Field Approach

TL;DR

We develop a mean field game framework for dynamic two-sided matching where populations with quality attributes meet according to Poisson processes and agents choose mutual acceptance thresholds. The equilibrium is characterized by a fully coupled forward–backward – system with nonlocal coupling across populations, and the matching process follows FCFS under a mutual-acceptance rule. We prove global well-posedness and a verification theorem for mean field Nash equilibria, with a graphon-structured perspective that clarifies the block-interaction pattern. Numerical experiments in a labor-market setting illustrate how dynamic thresholds and unmatched densities evolve, revealing imperfect sorting and potential upward mobility, thereby linking micro-level strategic decisions to macro-level matching outcomes and policy implications.

Abstract

This paper develops a mean field game framework for dynamic two-sided matching markets, extending existing matching theory by integrating micro-macro dynamics in two-sided environments. Unlike traditional matching models focusing on static equilibrium or unilateral optimization, our framework simultaneously captures dynamic interactions and strategic behaviors of both market sides, as well as the equilibrium. We model two types of agents who meet each other via Poisson processes and make simultaneous matching decisions to maximize their respective objective functionals, and find the corresponding equilibrium. Our approach formulates the equilibrium as a fully coupled Hamilton-Jacobi-Bellman and Fokker-Planck system with nonlocal structure coupling two distinct populations. The mathematical analysis addresses significant challenges from the dual-layered coupling structure and nonlocal structure. We also provide insights into individual behaviors shaping aggregate patterns in labor markets through numerical experiments.

Paper Structure

This paper contains 32 sections, 24 theorems, 229 equations, 7 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Our Model and Main Contributions
  4. Organization of the paper:
  5. Problem Settings
  6. Probabilistic Setting and Meeting Mechanism
  7. Dynamic Matching Mechanism and Objective Functionals
  8. Optimal Threshold, Stopping Time Problems, and Associated Mean Field Games
  9. Applications
  10. Main Results
  11. Derivation of HJB and FP Equations
  12. Evolution of Defective Probability Density Functions
  13. Derivation of HJB Equations
  14. Existence and Uniqueness of Mean Field Nash Equilibrium of Problem \ref{['def Nash']}
  15. Properties of Matching Equilibrium and Interpretations
  16. ...and 17 more sections

Key Result

Lemma 2.1

Let $\mathbf{I}, \mathbf{J}\in \{\mathbf{A},\mathbf{B}\}$ with $\mathbf{I} \neq \mathbf{J}$ and $t\in[0,T]$, the defective probability distribution $\mu_{\mathbf{I} ,t}(\mathbin{\vcenter{\hbox{$\cdot$}}}):=\pi_{\mathbf{I} ,t}(\mathbin{\vcenter{\hbox{$\cdot$}}},0)$ defined in (6) of sec. Probabilisti where $\tau_{\space\mathbf{I}} (x,0)$ is the matching time of the type-$\mathbf{I}$ agent $(x,0)$ d

Figures (7)

  • Figure 2: Dynamic mean field Nash game
  • Figure 3: Fitted cumulative distribution functions and data quantile values: The left panel corresponds to $f_{\mathbf{A},0}$, and the right panel to $f_{\mathbf{B},0}$
  • Figure 4: Value functions (optimal thresholds) $V_\mathbf{A}$ and $V_\mathbf{B}$ against quality levels $x$ and $y$, for selected values of time $t$
  • Figure 5: Defective probability densities $f_\mathbf{A}$ and $f_\mathbf{B}$ against quality levels $x$ and $y$, for selected times
  • Figure 6: Unmatched rates $F_\mathbf{A}$ and $F_\mathbf{B}$ against time $t$, for selected percentile bands of quality levels
  • ...and 2 more figures

Theorems & Definitions (53)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Remark 2.4
  • proof : Proof of \ref{['lem eq. of P1 P2']}
  • ...and 43 more