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A Stochastic Surveillance Stackelberg Game: Co-Optimizing Defense Placement and Patrol Strategy

Yohan John, Gilberto Diaz-Garcia, Xiaoming Duan, Jason R. Marden, Francesco Bullo

TL;DR

This work adopts a worst-case omniscient adversary model from the literature and extends the formulation to accommodate heterogeneous defenses at the various nodes of the graph to identify efficient methods for computing these strategies in certain classes of graphs.

Abstract

Stochastic patrol routing is known to be advantageous in adversarial settings; however, the optimal choice of stochastic routing strategy is dependent on a model of the adversary. We adopt a worst-case omniscient adversary model from the literature and extend the formulation to accommodate heterogeneous defenses at the various nodes of the graph. Introducing this heterogeneity leads to interesting new patrol strategies. We identify efficient methods for computing these strategies in certain classes of graphs. We assess the effectiveness of these strategies via comparison to an upper bound on the value of the game. Finally, we leverage the heterogeneous defense formulation to develop novel defense placement algorithms that complement the patrol strategies.

A Stochastic Surveillance Stackelberg Game: Co-Optimizing Defense Placement and Patrol Strategy

TL;DR

This work adopts a worst-case omniscient adversary model from the literature and extends the formulation to accommodate heterogeneous defenses at the various nodes of the graph to identify efficient methods for computing these strategies in certain classes of graphs.

Abstract

Stochastic patrol routing is known to be advantageous in adversarial settings; however, the optimal choice of stochastic routing strategy is dependent on a model of the adversary. We adopt a worst-case omniscient adversary model from the literature and extend the formulation to accommodate heterogeneous defenses at the various nodes of the graph. Introducing this heterogeneity leads to interesting new patrol strategies. We identify efficient methods for computing these strategies in certain classes of graphs. We assess the effectiveness of these strategies via comparison to an upper bound on the value of the game. Finally, we leverage the heterogeneous defense formulation to develop novel defense placement algorithms that complement the patrol strategies.
Paper Structure (14 sections, 8 theorems, 67 equations, 1 figure, 1 algorithm)

This paper contains 14 sections, 8 theorems, 67 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Markov Chains
  4. Stackelberg Game
  5. Heterogeneous Attack Duration
  6. Upper Bound for Optimal Capture Probability
  7. Complete Graphs
  8. Complete Bipartite Graphs
  9. Star Graphs
  10. Optimal Defense Placement
  11. Complete Graphs
  12. Complete Bipartite Graphs
  13. Numerical Example
  14. Conclusions

Key Result

Theorem 1

Given a strongly connected digraph $\mathcal{G} = (\mathcal{V},\mathcal{E})$ with $\lvert \mathcal{V} \rvert = n$ and a vector $\pmb{\tau} \in {\mathbb{N}}^n$ of attack durations corresponding to each node in $\mathcal{V}$, the optimal strategy $P^*$ has a unique stationary distribution $\pmb{\pi}^*

Figures (1)

  • Figure 1: We consider a complete bipartite graph where $n_p = 3, n_q = 2$ and the defense budget $B = 20$. Optimized defense allocations $\tau$, transition matrices, and resulting capture probabilities $\mu$ come from (a) fmincon and (b) the proposed method.

Theorems & Definitions (16)

  • Theorem 1: Upper Bound for Optimal Capture Probability
  • proof
  • Lemma 2: Capture Probability for Complete Graph Strategy
  • proof
  • Lemma 3: Capture Probability for Complete Bipartite Graph Strategy
  • proof
  • Theorem 4: Optimal Strategy for Star Graphs
  • proof
  • Lemma 5: Lower Bound for Objective Function Value
  • proof
  • ...and 6 more