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Optimal Investment under Mutual Strategy Influence among Agents

Huisheng Wang, H. Vicky Zhao

TL;DR

The paper addresses how mutual influence among $n$ agents affects investment decisions by formulating a multi-agent differential game with a deviation penalty that captures alignment of strategies. It derives analytical optimal strategies showing each $P_j^*(t)$ as a blend of the rational strategy and the common asymptotic path; it also introduces a fast approximation algorithm to compute solutions efficiently. A key contribution is the asymptotic strategy under strong mutual influence, where all agents converge to a social-average behavior with risk aversion $\tilde{\alpha}$, alongside a precise relationship between rational, optimal, and asymptotic strategies. Numerical experiments demonstrate both high accuracy of the fast algorithm (relative error $<10^{-31}$) and the significant computational gains, while also illustrating how increasing mutual influence alters mean-variance terminal wealth depending on agents’ relative risk aversion.

Abstract

In financial markets, agents often mutually influence each other's investment strategies and adjust their strategies to align with others. However, there is limited quantitative study of agents' investment strategies in such scenarios. In this work, we formulate the optimal investment differential game problem to study the mutual influence among agents. We derive the analytical solutions for agents' optimal strategies and propose a fast algorithm to find approximate solutions with low computational complexity. We theoretically analyze the impact of mutual influence on agents' optimal strategies and terminal wealth. When the mutual influence is strong and approaches infinity, we show that agents' optimal strategies converge to the asymptotic strategy. Furthermore, in general cases, we prove that agents' optimal strategies are linear combinations of the asymptotic strategy and their rational strategies without others' influence. We validate the performance of the fast algorithm and verify the correctness of our analysis using numerical experiments. This work is crucial to comprehend mutual influence among agents and design effective mechanisms to guide their strategies in financial markets.

Optimal Investment under Mutual Strategy Influence among Agents

TL;DR

The paper addresses how mutual influence among agents affects investment decisions by formulating a multi-agent differential game with a deviation penalty that captures alignment of strategies. It derives analytical optimal strategies showing each as a blend of the rational strategy and the common asymptotic path; it also introduces a fast approximation algorithm to compute solutions efficiently. A key contribution is the asymptotic strategy under strong mutual influence, where all agents converge to a social-average behavior with risk aversion , alongside a precise relationship between rational, optimal, and asymptotic strategies. Numerical experiments demonstrate both high accuracy of the fast algorithm (relative error ) and the significant computational gains, while also illustrating how increasing mutual influence alters mean-variance terminal wealth depending on agents’ relative risk aversion.

Abstract

In financial markets, agents often mutually influence each other's investment strategies and adjust their strategies to align with others. However, there is limited quantitative study of agents' investment strategies in such scenarios. In this work, we formulate the optimal investment differential game problem to study the mutual influence among agents. We derive the analytical solutions for agents' optimal strategies and propose a fast algorithm to find approximate solutions with low computational complexity. We theoretically analyze the impact of mutual influence on agents' optimal strategies and terminal wealth. When the mutual influence is strong and approaches infinity, we show that agents' optimal strategies converge to the asymptotic strategy. Furthermore, in general cases, we prove that agents' optimal strategies are linear combinations of the asymptotic strategy and their rational strategies without others' influence. We validate the performance of the fast algorithm and verify the correctness of our analysis using numerical experiments. This work is crucial to comprehend mutual influence among agents and design effective mechanisms to guide their strategies in financial markets.
Paper Structure (18 sections, 5 theorems, 26 equations, 1 figure, 1 table, 1 algorithm)

This paper contains 18 sections, 5 theorems, 26 equations, 1 figure, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Optimal Investment Strategies
  4. The Analytical Solution of Agents' Optimal Strategies
  5. The Proposed Fast Algorithm
  6. Influence of Mutual Influence on Agents' Optimal Strategies and Terminal Wealth
  7. The Optimal Strategies
  8. The Asymptotic Strategy
  9. The Relationship Among Optimal Strategy, Rational Strategy, and Asymptotic Strategy
  10. The Terminal Wealth
  11. Numerical Experiments
  12. The Accuracy and Efficiency of the Fast Algorithm
  13. Experiment setup
  14. Experimental results
  15. The Impact of Mutual Influence on Agents' Optimal Strategies and Terminal Wealth
  16. ...and 3 more sections

Key Result

Theorem 1

The $j$-th agent's optimal strategy $\{\boldsymbol{P}_j^*(t)\}_{t\in\mathcal{T}}$ in the differential game problem eq:game_problem is where $\boldsymbol{Q}^*_j(t)$ is the average followees' optimal strategy with $\boldsymbol{Z}_j$ is called the $j$-th agent's investment opinion with and $\eta_j$ is the integral constant with

Figures (1)

  • Figure 1: The impact of mutual influence on agents' optimal strategies and terminal wealth.

Theorems & Definitions (10)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Theorem 5
  • proof