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A Mean-Field Game Model For Large-Scale Attrition in Attacker-Defender Systems

Avetik Arakelyan, Tigran Bakaryan, Davit Alaverdyan, Naira Hovakimyan, Isaac Kaminer

Abstract

This paper proposes a novel Mean-Field Game (MFG) framework for large-scale attacker-defender systems aimed at protecting one or multiple High-Value Units (HVUs). Motivated by classical agent-wise attrition models, we introduce a population-wise attrition mechanism formulated by statistical distance between populations, enabling a macroscopic description of weapon-based interactions between large populations. Leveraging this and Lions derivative on the space of probability measures, we derive the associated MFG system, which characterizes optimal strategies and the evolution of population distributions in attacker-defender interactions. We analyze the model by establishing upper and lower bounds on the defender density, ensuring physical consistency by preventing concentration and depletion. For numerical investigation, we develop a numerical scheme combining physics-informed neural networks with Sinkhorn method to solve attacker-defender MFG system. Simulations confirm the effectiveness of the framework and reveal key insights, including sensitivity to weapon strengths and population dispersion.

A Mean-Field Game Model For Large-Scale Attrition in Attacker-Defender Systems

Abstract

This paper proposes a novel Mean-Field Game (MFG) framework for large-scale attacker-defender systems aimed at protecting one or multiple High-Value Units (HVUs). Motivated by classical agent-wise attrition models, we introduce a population-wise attrition mechanism formulated by statistical distance between populations, enabling a macroscopic description of weapon-based interactions between large populations. Leveraging this and Lions derivative on the space of probability measures, we derive the associated MFG system, which characterizes optimal strategies and the evolution of population distributions in attacker-defender interactions. We analyze the model by establishing upper and lower bounds on the defender density, ensuring physical consistency by preventing concentration and depletion. For numerical investigation, we develop a numerical scheme combining physics-informed neural networks with Sinkhorn method to solve attacker-defender MFG system. Simulations confirm the effectiveness of the framework and reveal key insights, including sensitivity to weapon strengths and population dispersion.

Paper Structure

This paper contains 12 sections, 1 theorem, 45 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Population-wise Setting
  4. Population-wise Attrition Rate Function
  5. Optimization Problem
  6. Mean-Field Game Model
  7. Model Derivation
  8. Flow bounds
  9. Simulations
  10. Numerical Method and Simulation Setup
  11. Simulation Scenarios
  12. Conclusion and Future Work

Key Result

Theorem 1

Let $T > 0$ and let $(w,m)$ be a classical solution to Problem prob-MFG, with initial condition $m(0, \cdot) = m_0$ such that $0 < \min_{\mathbb{T}^d} m_0 \le \max_{\mathbb{T}^d} m_0 < \infty$. Then, for all $(t,x) \in [0,T] \times \mathbb{T}^d$, the solution $m$ satisfies the following bounds where $K=\underset{t\in[0,T]}{\sup}\|\Delta u\|_{L^\infty(\mathbb{T}^d)}.$$\blacktriangleleft$$\blacktri

Figures (6)

  • Figure C1: Comparison of statistical distances. Left: moving distributions. Right: varying variance.
  • Figure E1: The blue curve represents the survival probability of the HVU, while the orange curve corresponds to the attackers.The initial position of the defenders is $(-3,-3)$ with variance $\sigma^2 = 0.85$. The attackers start from $(-4,4)$ with variance $\sigma^2 = 0.85$ and move toward $(1,1)$, where the HVU is located with variance $\sigma^2 = 0.1$. For the first panel, the attrition rate parameters are $\sigma_A = \sigma_H = 5,\ \lambda_A = 14,\ \lambda_H = 1$. For the second panel, $\sigma_A = \sigma_H = 5,\ \lambda_A =\lambda_H = 7$. For the third panel, $\sigma_A = \sigma_H = 5,\ \lambda_A = 2,\ \lambda_H = 7$.
  • Figure E2: Comparison of survival probabilities for three values of the variance of the initial defender distribution $m_0$: $\sigma^2=0.35,\,1.4,\,$ and $1.8$. The defenders are initially centered at $(0,0)$, while the attackers start from $(-4,0)$ with variance $\sigma^2=1.5$ and move toward $(-4,4)$, where the HVU is located with variance $\sigma^2=0.2$. The attrition rate parameters are $\sigma_A=\sigma_H=2$ and $\lambda_A=3$, $\lambda_H=10$.
  • Figure E3: Top-down view of the evolution of the attacker and defender over time of first scenario ($\sigma^2 = 0.35$) of Fig. \ref{['surv_variances']}.
  • Figure E4: Top-down view of the evolution of the attacker and defender over time for the second scenario ($\sigma^2 = 1.4$) in Fig. \ref{['surv_variances']}.
  • ...and 1 more figures

Theorems & Definitions (2)

  • Theorem 1
  • proof