Guarding a Subspace in High-Dimensional Space With Two Defenders and One Attacker

Area of Science:

• Game Theory
• Control Theory
• High-Dimensional Geometry

Background:

• Subspace guarding games involve defenders protecting a target area from an attacker.
• Traditional methods like Hamilton-Jacobi-Isaacs are computationally intensive for high dimensions.

Purpose of the Study:

• To develop a novel, analytical solution for subspace guarding games in arbitrary high-dimensional spaces.
• To predict game outcomes and derive optimal strategies for defenders and attackers.

Main Methods:

• Utilized the attack subspace (AS) method to construct a barrier for outcome prediction.
• Formulated optimal strategies as a saddle-point equilibrium.
• Developed a critical payoff function for defender-winning scenarios.

Main Results:

• Provided a closed-form solution for subspace guarding games.
• Successfully predicted game winners before the game commenced.
• Demonstrated analytical results applicable to 2D and 3D guarding games.

Conclusions:

• The proposed analytical approach is computationally efficient and suitable for real-time updates.
• This work represents the first analytical solution for arbitrary high-dimensional target guarding games.
• The AS method offers a significant advancement over traditional computational approaches.