N-player and mean field games among fund managers considering excess logarithmic returns

ArXiv ID: 2503.02722 “View on arXiv”

Authors: Unknown

Abstract

This paper studies the competition among multiple fund managers with relative performance over the excess logarithmic return. Fund managers compete with each other and have expected utility or mean-variance criteria for excess logarithmic return. Each fund manager possesses a unique risky asset, and all fund managers can also invest in a public risk-free asset and a public risk asset. We construct both an $n$-player game and a mean field game (MFG) to address the competition problem under these two criteria. We explicitly define and rigorously solve the equilibrium and mean field equilibrium (MFE) for each criteria. In the four models, the excess logarithmic return as the evaluation criterion of the fund leads to the {" allocation fractions"} being constant. The introduction of the public risky asset yields different outcomes, with competition primarily affecting the investment in public assets, particularly evident in the MFG. We demonstrate that the MFE of the MFG represents the limit of the $n$-player game’s equilibrium as the competitive scale $n$ approaches infinity. Finally, the sensitivity analyses of the equilibrium are given.

Keywords: Mean Field Game, Nash Equilibrium, Relative Performance, Excess Logarithmic Return, Optimal Portfolio Allocation, General Financial Assets

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats
• Why: The paper is highly mathematically complex, featuring rigorous derivation of n-player games and mean field games using advanced stochastic control and PDEs, but it is purely theoretical with no backtests, real data, or implementation details provided.

  flowchart TD
    Start["Research Goal<br>How does competition affect<br>fund manager portfolios?"] --> Method["Methodology<br>N-player Nash Equilibrium &<br>Mean Field Game MFG"]
    
    Method --> Inputs["Inputs & Assumptions<br>- Excess Logarithmic Return<br>- Relative Performance<br>- Public Risky Asset &<br>Public Risk-Free Asset"]
    
    Inputs --> Process["Computational Process<br>Solve HJB/PDE Equations for<br>1. n-Player Equilibrium<br>2. Mean Field Equilibrium"]
    
    Process --> Findings["Key Findings & Outcomes<br>- Optimal Allocation is Constant<br>- Competition shifts investment to<br>Public Risky Asset<br>- MFG is limit of n-player as n→∞"]
    
    Findings --> Sensitivity["Sensitivity Analysis<br>Impact of Volatility &<br>Risk Aversion on Allocation"]
    
    Sensitivity --> Final["Final Outcomes<br>Rigorous characterization of<br>Equilibrium and MFE"]