Patrol Security Game: Defending Against Adversary with Freedom in Attack Timing, Location, and Duration

TL;DR

This paper models a robotic patrolling problem as a Stackelberg game, proposing efficient algorithms to optimize patrol schedules against adversaries with complex attack strategies, validated on synthetic and real datasets.

Contribution

It introduces a novel PSG model with a closed-form minimax formulation, and develops graph-based and reinforcement learning algorithms to optimize patrol strategies.

Findings

Optimal patrol strategies depend on attack penalties and randomness levels.

Increasing patrol randomness reduces attacker payoff in high-penalty scenarios.

Proposed algorithms outperform baselines on synthetic and real-world datasets.

Abstract

We explored the Patrol Security Game (PSG), a robotic patrolling problem modeled as an extensive-form Stackelberg game, where the attacker determines the timing, location, and duration of their attack. Our objective is to devise a patrolling schedule with an infinite time horizon that minimizes the attacker's payoff. We demonstrated that PSG can be transformed into a combinatorial minimax problem with a closed-form objective function. By constraining the defender's strategy to a time-homogeneous first-order Markov chain (i.e., the patroller's next move depends solely on their current location), we proved that the optimal solution in cases of zero penalty involves either minimizing the expected hitting time or return time, depending on the attacker model, and that these solutions can be computed efficiently. Additionally, we observed that increasing the randomness in the patrol schedule reduces the attacker's expected payoff in high-penalty cases. However, the minimax problem becomes non-convex in other scenarios. To address this, we formulated a bi-criteria optimization problem incorporating two objectives: expected maximum reward and entropy. We proposed three graph-based algorithms and one deep reinforcement learning model, designed to efficiently balance the trade-off between these two objectives. Notably, the third algorithm can identify the optimal deterministic patrol schedule, though its runtime grows exponentially with the number of patrol spots. Experimental results validate the effectiveness and scalability of our solutions, demonstrating that our approaches outperform state-of-the-art baselines on both synthetic and real-world crime datasets.