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Heterogeneous Roles against Assignment Based Policies in Two vs Two Target Defense Game

Goutam Das, Violetta Rostobaya, James Berneburg, Zachary I. Bell, Michael Dorothy, Daigo Shishika

TL;DR

The paper studies a two-attacker two-defender target-defense differential game on a plane and shows that the common 1v1 decomposition with Linear Bottleneck Assignment policy can be suboptimal when attackers adopt cooperative, heterogeneous roles. It develops attacker strategies that coordinate roles to counter defenders’ assignment-based strategies, providing sufficient conditions for single- and double-attacker deviations to improve the payoff and potentially win. The authors derive geometric and phase-based analysis arguments, including interception points and region-bounded reasoning, and validate the results with numerical examples illustrating attacker-win scenarios. Overall, the work highlights the need for true team-based equilibrium solutions in TT-TDG and motivates future research on teaming-aware strategies and guarantees.

Abstract

In this paper, we consider a target defense game in which the attacker team seeks to reach a high-value target while the defender team seeks to prevent that by capturing them away from the target. To address the curse of dimensionality, a popular approach to solve such team-vs-team game is to decompose it into a set of one-vs-one games. Such an approximation assumes independence between teammates assigned to different one-vs-one games, ignoring the possibility of a richer set of cooperative behaviors, ultimately leading to suboptimality. In this paper, we provide teammate-aware strategies for the attacker team and show that they can outperform the assignment-based strategy, if the defenders still employ an assignment-based strategy. More specifically, the attacker strategy involves heterogeneous roles where one attacker actively intercepts a defender to help its teammate reach the target. We provide sufficient conditions under which such a strategy benefits the attackers, and we validate the results using numerical simulations.

Heterogeneous Roles against Assignment Based Policies in Two vs Two Target Defense Game

TL;DR

The paper studies a two-attacker two-defender target-defense differential game on a plane and shows that the common 1v1 decomposition with Linear Bottleneck Assignment policy can be suboptimal when attackers adopt cooperative, heterogeneous roles. It develops attacker strategies that coordinate roles to counter defenders’ assignment-based strategies, providing sufficient conditions for single- and double-attacker deviations to improve the payoff and potentially win. The authors derive geometric and phase-based analysis arguments, including interception points and region-bounded reasoning, and validate the results with numerical examples illustrating attacker-win scenarios. Overall, the work highlights the need for true team-based equilibrium solutions in TT-TDG and motivates future research on teaming-aware strategies and guarantees.

Abstract

In this paper, we consider a target defense game in which the attacker team seeks to reach a high-value target while the defender team seeks to prevent that by capturing them away from the target. To address the curse of dimensionality, a popular approach to solve such team-vs-team game is to decompose it into a set of one-vs-one games. Such an approximation assumes independence between teammates assigned to different one-vs-one games, ignoring the possibility of a richer set of cooperative behaviors, ultimately leading to suboptimality. In this paper, we provide teammate-aware strategies for the attacker team and show that they can outperform the assignment-based strategy, if the defenders still employ an assignment-based strategy. More specifically, the attacker strategy involves heterogeneous roles where one attacker actively intercepts a defender to help its teammate reach the target. We provide sufficient conditions under which such a strategy benefits the attackers, and we validate the results using numerical simulations.
Paper Structure (16 sections, 8 theorems, 33 equations, 7 figures)

This paper contains 16 sections, 8 theorems, 33 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Notations
  4. Problem Statement
  5. Existing Solution
  6. Solution of 1v1 Game
  7. Linear Bottleneck Assignment Policy (LBAP)
  8. Nominal Strategies in 2v2 Game
  9. Main Results
  10. One Attacker Deviation
  11. Two Attacker Deviation
  12. Examples
  13. Example 1 Nominal Strategies
  14. Example 2 One Attacker Deviation
  15. Example 3 Two Attacker Deviation
  16. ...and 1 more sections

Key Result

Lemma 1

The time derivative of the optimal capture point, $\mathbf{x}_{B} \triangleq \mathbf{x}_{B_{ij}}$ can be written as follows: where $\mathbf{a}, \mathbf{b} \in \mathbb{R}^2$ are vectors defined as $\mathbf{a} =\mathbf{x}_{A_i}-\mathbf{x}_{D_j}$, and $\mathbf{b} =\mathbf{x}_T-\mathbf{x}_{C_{ij}}$, respectively. If both the attacker and the defender employ their respective state-feedback strategies

Figures (7)

  • Figure 1: Example of 2v2 game with two possible defender-attacker assignments, both of which resulting in defender's win but with different payoff. The assignment 1 is preferred by the defender team.
  • Figure 2: Illustration of $A_2$'s deviation from the nominal strategy to intercept $D_1$ at $\mathbf{x}_I$.
  • Figure 3: Illustration of two-attacker deviation scenario, where $A_1$ moves towards the target with a straight-line path, and $A_2$ seeks to intercept its pursuer $D_1$.
  • Figure 4: Illustration of points $P_1,P_2, P_3$, and bounded regions $\Omega_B$ and $\Omega_D$ for 1v1 scenario between $D_1$ and $A_1$.
  • Figure 5: Nominal trajectory of the players. Defender team wins by capturing $A_1$ and $A_2$, at $\mathbf{x}_{B_{11}}$ and $\mathbf{x}_{B_{22}}$, respectively, before they reach the target.
  • ...and 2 more figures

Theorems & Definitions (17)

  • Lemma 1
  • proof
  • Remark 1
  • Theorem 1: Condition for Non-Critical Attacker Deviation
  • proof
  • Remark 2
  • Corollary 1: Straight-line Interception Strategy for $A_2$
  • Lemma 2
  • proof
  • Lemma 3
  • ...and 7 more