Mean Field Games for Optimal Investment Under Relative Performance Criteria

Paper: arXiv 2307.10540

Authors: Ananya Parashar

Abstract

In this paper, we study the portfolio optimization problem formulated by Lacker and Soret. They formulate a finite time horizon model that allows agents to be competitive, measuring their utility not only by their absolute wealth but also relative performance compared to the average of other agents. While the finite population or $n$-player game is tractable in some cases, the authors present the Mean Field Game framework to solve this problem. Here, we seek to use this framework to clearly detail the optimal investment and consumption strategies in the CRRA utility case as was briefly outlined in Lacker and Soret, but also derive a solution in the CARA utility case.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 1.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper relies heavily on rigorous mathematical frameworks, specifically Mean Field Games, Hamilton-Jacobi-Bellman equations, and stochastic control, to extend the Lacker and Soret model to CARA utility functions. However, it is a purely theoretical paper with no empirical backtesting, real market data validation, or implementation code provided.

Research Flowchart

  flowchart TD
    A[Research Goal: Solve optimal investment/consumption under relative performance via Mean Field Games] --> B[Inputs: CRRA & CARA Utility Functions, Wealth, Relative Performance]
    B --> C[Methodology: Formulate N-Player Game & Transition to Mean Field Game Framework]
    C --> D[Computational Process: Derive HJB & FP Equations for MFG Equilibrium]
    D --> E[Outcomes: Closed-Form Strategies for CRRA & New Analytical Solution for CARA Utility]