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Multi-Robot Coordination Induced in an Adversarial Graph-Traversal Game

James Berneburg, Xuan Wang, Xuesu Xiao, Daigo Shishika

TL;DR

Numerical simulations demonstrate the results and the effectiveness of this game theoretic formulation of a graph traversal problem, with applications to robots moving through hazardous environments in the presence of an adversary, as in military and security scenarios.

Abstract

This paper presents a game theoretic formulation of a graph traversal problem, with applications to robots moving through hazardous environments in the presence of an adversary, as in military and security scenarios. The blue team of robots moves in an environment modeled by a time-varying graph, attempting to reach some goal with minimum cost, while the red team controls how the graph changes to maximize the cost. The problem is formulated as a stochastic game, so that Nash equilibrium strategies can be computed numerically. Bounds are provided for the game value, with a guarantee that it solves the original problem. Numerical simulations demonstrate the results and the effectiveness of this method, particularly showing the benefit of mixing actions for both players, as well as beneficial coordinated behavior, where blue robots split up and/or synchronize to traverse risky edges.

Multi-Robot Coordination Induced in an Adversarial Graph-Traversal Game

TL;DR

Numerical simulations demonstrate the results and the effectiveness of this game theoretic formulation of a graph traversal problem, with applications to robots moving through hazardous environments in the presence of an adversary, as in military and security scenarios.

Abstract

This paper presents a game theoretic formulation of a graph traversal problem, with applications to robots moving through hazardous environments in the presence of an adversary, as in military and security scenarios. The blue team of robots moves in an environment modeled by a time-varying graph, attempting to reach some goal with minimum cost, while the red team controls how the graph changes to maximize the cost. The problem is formulated as a stochastic game, so that Nash equilibrium strategies can be computed numerically. Bounds are provided for the game value, with a guarantee that it solves the original problem. Numerical simulations demonstrate the results and the effectiveness of this method, particularly showing the benefit of mixing actions for both players, as well as beneficial coordinated behavior, where blue robots split up and/or synchronize to traverse risky edges.
Paper Structure (20 sections, 3 theorems, 21 equations, 8 figures, 1 table)

This paper contains 20 sections, 3 theorems, 21 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Theoretical Analysis
  4. Handling Multiple Blue Robots
  5. Game Solution and Value Preliminaries
  6. Bounds for the Outcome
  7. Blue Player's Security Strategy
  8. Red Player's Security Strategy
  9. Game Solution
  10. Numerical Methodology
  11. Simplifying Computation
  12. Dominated Blue Actions
  13. Sub-Game Formulation
  14. Numerical Results
  15. Illustrative Examples
  16. ...and 5 more sections

Key Result

Theorem 1

For $M=1$, if the blue player follows the policy $\hat{\pi}_\text{blue}$ defined by eq:blueSecurityStratAction, then for all $s \in \mathcal{S}$ and $\pi_\text{red}$, with $\overline{J}\left( s \right)$ defined in eq:blueSecurityStratAction.

Figures (8)

  • Figure 1: An example scenario motivating our problem. The team of robots intend to reach their destination node 7 with minimal risk, while an adversary controls the condition of the terrain, with the capability to destroy the bridge or down trees on the road.
  • Figure 2: This shows an example of red's action graph, where each square node corresponds to a position graph for the blue player, with different edge weights, shown inside. The outlined node indicates ${G^{}}(t)$ and the arrow indicates the action of the red player to select ${G^{}}(t+1)={G^{3}}$.
  • Figure 3: (a) shows a simple example graph, and (b) shows its joint state graph with two robots. The position of each node of the JSG indicates what nodes in the original graph it corresponds to. The markers show equivalent positions and the arrows equivalent actions in both graphs.
  • Figure 4: This demonstrates mixed equilibrium policies of both players. The arrows indicate the possible actions of the blue player under that policy from node $1$. On graph $1$, the red player also mixes over its three options, but on graph $2$ or $3$, it only mixes between selecting graph $2$ and graph $3$.
  • Figure 5: Here, it is better for the blue player to have multiple robots which split up to reach the goal node $7$. The blue arrows on graph $1$ show the policy of a single robot, with the speech bubbles indicating the time of each action. The blue arrows on graph $2$ indicate the paths followed by two robots. For $M=1$, and $\alpha(0) = 1$, the red player will always change the graph so that the blue player will get a cost of $16$ for taking the $(6,7)$ edge, but for two robots exactly one robot will take the cheaper path to the goal, no matter the red player's action.
  • ...and 3 more figures

Theorems & Definitions (6)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof