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Limit Order Book Dynamics in Matching Markets: Microstructure, Spread, and Execution Slippage

Yao Wu

TL;DR

This paper reframes matching markets as limit order book–like systems where an internal, reference-based gap $\Delta V = V_{\text{uncond}} - V_{\text{reach}}$ acts as a structural liquidity constraint that monetary transfers cannot close unless a regime switch occurs. It develops a dynamic discrete-choice model where execution happens when the market-to-book ratio $\theta = V_{\text{reach}}/V_{\text{uncond}}$ crosses a time-decaying liquidity threshold $T(t)$, linking settlement to inventory effects and threshold decay rather than continuous price improvement. Key contributions include the Threshold Impossibility Theorem, a formal isomorphism between social matching and financial order books, and a suite of numerical tests showing persistent slippage, regional invariance of rankings, and high-tier zero-spread executions. The model offers a unified microstructure explanation for non-clearing matches, compensation inefficiency, and post-match regret in illiquid settings, with implications for labor, education, and OTC markets where reference-dependent utility and liquidity constraints are pivotal.

Abstract

Conventional models of matching markets assume that monetary transfers can clear markets by compensating for utility differentials. However, empirical patterns show that such transfers often fail to close structural preference gaps. This paper introduces a market microstructure framework that models matching decisions as a limit order book system with rigid bid ask spreads. Individual preferences are represented by a latent preference state matrix, where the spread between an agent's internal ask price (the unconditional maximum) and the market's best bid (the reachable maximum) creates a structural liquidity constraint. We establish a Threshold Impossibility Theorem showing that linear compensation cannot close these spreads unless it induces a categorical identity shift. A dynamic discrete choice execution model further demonstrates that matches occur only when the market to book ratio crosses a time decaying liquidity threshold, analogous to order execution under inventory pressure. Numerical experiments validate persistent slippage, regional invariance of preference orderings, and high tier zero spread executions. The model provides a unified microstructure explanation for matching failures, compensation inefficiency, and post match regret in illiquid order driven environments.

Limit Order Book Dynamics in Matching Markets: Microstructure, Spread, and Execution Slippage

TL;DR

This paper reframes matching markets as limit order book–like systems where an internal, reference-based gap acts as a structural liquidity constraint that monetary transfers cannot close unless a regime switch occurs. It develops a dynamic discrete-choice model where execution happens when the market-to-book ratio crosses a time-decaying liquidity threshold , linking settlement to inventory effects and threshold decay rather than continuous price improvement. Key contributions include the Threshold Impossibility Theorem, a formal isomorphism between social matching and financial order books, and a suite of numerical tests showing persistent slippage, regional invariance of rankings, and high-tier zero-spread executions. The model offers a unified microstructure explanation for non-clearing matches, compensation inefficiency, and post-match regret in illiquid settings, with implications for labor, education, and OTC markets where reference-dependent utility and liquidity constraints are pivotal.

Abstract

Conventional models of matching markets assume that monetary transfers can clear markets by compensating for utility differentials. However, empirical patterns show that such transfers often fail to close structural preference gaps. This paper introduces a market microstructure framework that models matching decisions as a limit order book system with rigid bid ask spreads. Individual preferences are represented by a latent preference state matrix, where the spread between an agent's internal ask price (the unconditional maximum) and the market's best bid (the reachable maximum) creates a structural liquidity constraint. We establish a Threshold Impossibility Theorem showing that linear compensation cannot close these spreads unless it induces a categorical identity shift. A dynamic discrete choice execution model further demonstrates that matches occur only when the market to book ratio crosses a time decaying liquidity threshold, analogous to order execution under inventory pressure. Numerical experiments validate persistent slippage, regional invariance of preference orderings, and high tier zero spread executions. The model provides a unified microstructure explanation for matching failures, compensation inefficiency, and post match regret in illiquid order driven environments.

Paper Structure

This paper contains 70 sections, 1 theorem, 34 equations, 10 figures, 4 tables.

Table of Contents

  1. Introduction
  2. The Internal Differential Model
  3. Preliminaries: Valuation Kernels and Attribute Matrices
  4. The Structural Spread ($\Delta V$)
  5. Compensation as Price Improvement
  6. The Regime Switching Threshold $C^\star$
  7. The Threshold Impossibility Theorem
  8. Implications for Market Clearing
  9. Persistent Slippage Post-Execution
  10. Zero-Spread Execution in High-Tier Matches
  11. Dynamic Execution and Liquidity Thresholds
  12. States, Transitions, and the Search Process
  13. State definitions
  14. The Liquidity Threshold $T(t)$
  15. The Execution Condition
  16. ...and 55 more sections

Key Result

Theorem 1

For any partner $M$ with intrinsic value $V_F(M)$, if the compensation $C < C^\star$, then the structural spread $\Delta V$ remains invariant. The preference ordering is preserved: Only if $C \ge C^\star$ does the agent undergo a categorical regime switch. Thus, linear compensation cannot close a spread derived from ordinal categorization without inducing an identity collapse.

Figures (10)

  • Figure 1: Graphical Abstract of the Unified Framework.(A) The internal preference differential $\Delta V$ creates a structural gap between the Unconditional Max (Ideal) and Reachable Max (Reality) that monetary compensation ($C$) cannot structurally close. (B) Marriage decisions are governed by a state-machine dynamic where commitment occurs only when the reality-to-ideal ratio $\theta$ crosses the agent's willingness threshold $T$. (C) This system is structurally isomorphic to a financial Limit Order Book, where matching failures represent liquidity droughts due to wide bid-ask spreads.
  • Figure 2: The Matching Market Order Book. Visualizing the Internal Preference Differential $\Delta V$ as a rigid bid--ask spread. The market fails to clear because the highest bid ($V_{\text{reach}}$) does not cross the lowest ask ($V_{\text{uncond}}$).
  • Figure 3: Dynamic Execution Process. The diagram illustrates the agent's decision flow. Execution occurs when the market-to-book ratio $\theta_t$ crosses the time-decaying liquidity threshold $T_t$. External shocks (e.g., peer comparison) can trigger post-execution volatility by repricing the internal ask.
  • Figure 4: Simulation Results of the Gini-Cone Model. This figure shows the outcome of the Python-based simulation integrating the Internal Differential Model with the Gini-conical partner distribution. High-tier scarcity emerges naturally under the Beta-distributed value space, producing slippage levels consistent with Appendix F.
  • Figure 5: Marriage-Market Order-Book Structure. This schematic illustrates the structural equivalence between a financial order book and the LPSM. A match executes when the ratio of the highest bid to the lowest ask exceeds $T$.
  • ...and 5 more figures

Theorems & Definitions (2)

  • Remark 1
  • Theorem 1: Threshold Impossibility