Mean Field Game of Optimal Tracking Portfolio

2025-05-03

AI Summary

This paper studies a mean-field game (MFG) arising in large-scale fund competition under a novel relative performance evaluation scheme incorporating benchmark tracking: each fund manager must ensure their wealth outperforms a composite benchmark formed by a weighted average of the population’s mean wealth and an exogenous market index. To address the benchmark-tracking problem with a lower bound (floor) constraint, we formulate it for the first time as a reflected MFG and transform it into an equivalent unconstrained problem via Lagrangian duality. Leveraging reflected diffusion processes, PDE analysis, and mean-field equilibrium theory, we rigorously establish existence of solutions, derive an explicit optimal strategy for a representative agent, and characterize a mean-field equilibrium with clear economic interpretation. Our work extends classical unconstrained MFG frameworks by proving equilibrium consistency in piecewise domains, thereby providing a new theoretical foundation for asset management under relative performance incentives.

Technology Category

• Multiagent Systems: Mechanism Design
• Game Theory and Economic Paradigms: Mechanism Design
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Economics, Online Markets and Human Computation: Uses of LLMs and GenAI for marketplace design, bidding, and strategic interactions
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

This paper studies the mean field game (MFG) problem arising from a large population competition in fund management, featuring a new type of relative performance via the benchmark tracking constraint. In the n-agent model, each agent can strategically inject capital to ensure that the total wealth outperforms the benchmark process, which is modeled as a linear combination of the population's average wealth process and an exogenous market index process. That is, each agent is concerned about the performance of her competitors captured by the floor constraint. With a continuum of agents, we formulate the constrained MFG problem and transform it into an equivalent unconstrained MFG problem with a reflected state process. We establish the existence of the mean field equilibrium (MFE) using the PDE approach. Firstly, by applying the dual transform, the best response control of the representative agent can be characterized in analytical form in terms of a dual reflected diffusion process. As a novel contribution, we verify the consistency condition of the MFE in separated domains with the help of the duality relationship and properties of the dual process.

Problem

• Study MFG problem in fund management competition
• Analyze tracking constraint on benchmark performance
• Establish mean field equilibrium existence via PDE

Innovation

• Dual transform for analytical control characterization
• Reflected state process in unconstrained MFG
• PDE approach for mean field equilibrium existence