Extending Algorithms for Mobile Robot Patrolling in the Presence of Adversaries to More Realistic Settings Nicola Basilico, Nicola Gatti, Thomas Rossi, Sofia Ceppi, and Francesco Amigoni Dipartimento di Elettronica e Informazione, Politecnico di Milano, Milano, Italy Email: {basilico, ngatti, ceppi, amigoni}@elet.polimi.it, thomas.rossi@mail.polimi.it Abstract—Patrolling environments by means of autonomous mobile robots has received an increasing attention in the last few years. The interest of the agent community is mainly in the development of effective patrolling strategies. Approaches based on game theory have been demonstrated to be very effective. They model the patrolling situation as a two-player leader-follower game, where the patroller is the leader and the intruder is the follower. These models present several limitations that prevent their use in realistic settings. In this paper, we extend the most general model from the state of the art along two directions, we propose algorithms to solve efficiently our extensions, and we experimentally evaluate them. Index Terms—Algorithmic game theory, games for security, robotic patrolling with adversaries I. I NTRODUCTION The problem of patrolling environments by means of au- tonomous mobile robots have become a topic of increasing interest in the last few years [1], [2], [3], [4]. The scientific challenge for the agent community is the development of effective patrolling strategies. The basic setting considers a patroller aiming at preserving an environment from intrusions. It has some ability to detect the intruder that, in turn, can hide and observe the robot patrolling the environment before attempting to intrude. The problem is particularly interesting when the patroller cannot employ a deterministic strategy for its movements, e.g., a fixed route, see [5]. In these cases unpredictable strategies should be employed. The literature shows a large number of studies that apply game theory [6] to patrolling scenarios to produce effective unpredictable strategies [2], [3], [4]. These strategies usu- ally grant the patrolling robot a larger expected utility than adopting a purely random strategy that does not consider the presence of the intruder [2]. A patrolling situation is commonly modeled as a two-player (i.e., the patroller and the intruder) strategic-form (i.e., simultaneous) game. The fact that the intruder can observe the patroller before acting leads to the adoption of a leader-follower solution concept [7], where the patroller is the leader and the intruder is the follower. The environment is commonly modeled as a set of connected cells that can be traversed by the robot and that may have different values for the patroller and the intruder. The problem of searching for a leader-follower patrolling strategy is cast to an optimization problem and commercial solvers [8], [9] are employed for its resolution. The models proposed in the literature provide a coarse grain description of the patrolling situations, not assuring effective strategies in real-world settings. In this paper, we consider the most general model presented in the literature [3] and we provide two different extensions with the aim to make the patrolling model closer to real-world situations. The first extension captures the movement of the intruder (Section III). In previous models [1], [3], [4], [10], the intruder is assumed to strike a target by directly appearing at it. In our extension, we force the intruder to reach the target moving along paths. We formulate a mathematical programming problem to capture our extension and we reduce the search space by removing all the dominated intruder’s actions (i.e., actions that the intruder would never play independently of the patroller’s strategy). This reduction is, to the best of our knowledge, the first attempt to extend the well-known game theoretic iterated dominance [6] to the case of patrolling. The second extension captures the visibility limitations of the intruder (Section IV). In the previous models, the intruder is assumed to be in the position to observe perfectly the patroller’s movements. In our extension, we relax this assumption, allowing the intruder, be- fore attempting an intrusion, to observe perfectly the patroller only in a portion of the environment. This introduces imperfect information in the game and, to the best of our knowledge, it is the first attempt to introduce imperfect information in leader-follower games. Also in this case, we formulate a mathematical programming problem to capture our extension and we provide some algorithms to reduce the search space. From the experimental point of view, our extensions assure the patroller a larger expected utility than the original model and our reduction algorithms are very effective, saving more than 90% computational time, and, although our models extend the state-of-the-art model, the time needed for solving them is not larger than the time needed for solving the original model. II. STATE OF THE ART A. Mobile Robot Strategic Patrolling A patrolling situation is characterized by one or more patrollers, a possible intruder, and some targets. The targets