Mean-Field Price Formation on Trees with a Network of Relative Performance Concerns

ArXiv ID: 2512.21621 “View on arXiv”

Authors: Masaaki Fujii

Abstract

Financial firms and institutional investors are routinely evaluated based on their performance relative to their peers. These relative performance concerns significantly influence risk-taking behavior and market dynamics. While the literature studying Nash equilibrium under such relative performance competitions is extensive, its effect on asset price formation remains largely unexplored. This paper investigates mean-field equilibrium price formation of a single risky stock in a discrete-time market where agents exhibit exponential utility and relative performance concerns. Unlike existing literature that typically treats asset prices as exogenous, we impose a market-clearing condition to determine the price dynamics endogenously within a relative performance equilibrium. Using a binomial tree framework, we establish the existence and uniqueness of the market-clearing mean-field equilibrium in both single- and multi-population settings. Finally, we provide illustrative numerical examples demonstrating the equilibrium price distributions and agents’ optimal position sizes.

Keywords: market equilibrium, asset pricing, relative performance, mean-field games, binomial tree

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 2.0/10
• Quadrant: Lab Rats
• Why: The paper employs advanced mean-field game theory and recursive backward induction on a binomial tree, involving heavy mathematical derivations and proofs of equilibrium existence/uniqueness. However, it presents only illustrative numerical examples without any backtesting, real data, or statistical metrics, focusing on theoretical framework construction.

  flowchart TD
    Start["Research Goal<br>Formulate and solve mean-field equilibrium<br>price formation with relative performance concerns"]

    Methodology["Methodology<br>Discrete-time binomial tree framework<br>Exponential utility with relative performance"]

    Inputs["Data & Inputs<br>Market-clearing condition<br>Agent heterogeneity (single/multi-population)<br>Risk preferences"]

    Process["Computational Process<br>Derive optimal trading strategies<br>Solve fixed-point problem for<br>endogenous price dynamics"]

    Results["Key Findings & Outcomes<br>Existence & uniqueness of equilibrium<br>Equilibrium price distributions<br>Optimal agent position sizes<br>Illustrative numerical examples"]
    
    Start --> Methodology
    Methodology --> Inputs
    Inputs --> Process
    Process --> Results