Competitive optimal portfolio selection under mean-variance criterion

ArXiv ID: 2511.05270 “View on arXiv”

Authors: Guojiang Shao, Zuo Quan Xu, Qi Zhang

Abstract

We investigate a portfolio selection problem involving multi competitive agents, each exhibiting mean-variance preferences. Unlike classical models, each agent’s utility is determined by their relative wealth compared to the average wealth of all agents, introducing a competitive dynamic into the optimization framework. To address this game-theoretic problem, we first reformulate the mean-variance criterion as a constrained, non-homogeneous stochastic linear-quadratic control problem and derive the corresponding optimal feedback strategies. The existence of Nash equilibria is shown to depend on the well-posedness of a complex, coupled system of equations. Employing decoupling techniques, we reduce the well-posedness analysis to the solvability of a novel class of multi-dimensional linear backward stochastic differential equations (BSDEs). We solve a new type of nonlinear BSDEs (including the above linear one as a special case) using fixed-point theory. Depending on the interplay between market and competition parameters, three distinct scenarios arise: (i) the existence of a unique Nash equilibrium, (ii) the absence of any Nash equilibrium, and (iii) the existence of infinitely many Nash equilibria. These scenarios are rigorously characterized and discussed in detail.

Keywords: Nash Equilibrium, Backward Stochastic Differential Equations (BSDEs), Mean-Variance Optimization, Game Theory, Stochastic Control, Portfolio Management

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 1.0/10
• Quadrant: Lab Rats
• Why: The paper is dominated by advanced mathematical tools such as stochastic linear-quadratic control, backward stochastic differential equations (BSDEs), and fixed-point theory, with heavy LaTeX and derivations. It lacks any mention of backtesting, empirical data, or implementation details, focusing purely on theoretical analysis of Nash equilibria.

  flowchart TD
    A["Research Goal:<br>Portfolio Selection with Multi-Agent<br>Mean-Variance Preferences & Relative Wealth"] --> B["Reformulation & Optimization:<br>Non-homogeneous Stochastic<br>Linear-Quadratic Control"]
    B --> C["Decoupling & Analysis:<br>Multi-dimensional Linear<br>Backward SDEs (BSDEs)"]
    C --> D["Solving Nonlinear BSDEs:<br>Fixed-Point Theory &<br>Existence Theorems"]
    D --> E["Key Findings:<br>Three Nash Equilibrium Scenarios"]
    
    E --> E1((Unique Nash Equilibrium))
    E --> E2((No Nash Equilibrium))
    E --> E3((Infinitely Many Nash Equilibria))

    style A fill:#e1f5fe
    style E fill:#fff3e0
    style E1 fill:#e8f5e8
    style E2 fill:#ffebee
    style E3 fill:#e8f5e8