Paper: arXiv 2511.05270

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.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 1.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: 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.

Research Flowchart

  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