Selective Smooth Fictitious Play: An approach based on game theory for patrolling infrastructures with a multi-robot system Erik Hernández ⇑ , Antonio Barrientos, Jaime del Cerro Centre for Robotics and Automation, UPM-CSIC, C/ José Gutiérrez Abascal, 2, Madrid 28006, Spain article info Keywords: Game theory Multi-robot patrolling Distributed systems abstract The multi-robot patrolling problem is defined as the activity of traversing a given environment. In this activity, a fleet of robots visits some places at irregular intervals of time for security purpose. To date, this problem has been solved with different approaches. However, the approaches that obtain the best results are unfeasible for security applications because they are centralized and deterministic. To overcome the disadvantages of previous work, this paper presents a new distributed and non-deterministic approach based on a model from game theory called Smooth Fictitious Play. To this end, the multi-robot patrolling problem is formulated by using concepts of graph theory to represent an environment. In this formula- tion, several normal-form games are defined at each node of the graph. This approach is validated by comparison with best suited literature approaches by using a patrolling simulator. The results for the pro- posed approach turn out to be better than previous literature approaches in as many as 88% of the cases of study. Moreover, the novel approach presented in this work has many advantages over other approaches of the literature such distribution, robustness, scalability, and dynamism. The achievements obtained in this work validate the potential of game theory to protect infrastructures. Ó 2013 Elsevier Ltd. All rights reserved. 1. Introduction Multi-robot systems have recently become a focus of consider- able interest due to their applicability in several areas. Generally speaking, a multi-robot system is defined as a set of homogeneous or heterogeneous robots, which operate in the same environment. However, robotic systems may range from simple sensors, acquir- ing and processing data, to complex human-like machines, able to interact with the environment in fairly complex ways. Currently, these systems represent a field of research within robotics and arti- ficial intelligence (Farinelli, Iocchi, & Nardi, 2004). This field is fo- cused on designing solutions to coordinate the decision-making among robots. There are several advantages of multi-robot systems over single-robot systems. Firstly, a multi-robot system performs a given task more eficiently. Secondly, multiple robots increase robustness and reliability. Thirdly, multi-robot systems enhance performance in complex and distributed tasks. Finally, several ro- bots with limited capabilities are cheaper and easier to build than a single powerful robot (Parker, 2008). The multi-robot systems can be used to solve several applica- tions in adversarial domains such as robotic security (Veloso & Nardi, 2006). In these domains, a robotic security platform repre- sents a powerful defensive tool for mitigating threats (Everett, 2003). The multi-robot patrolling problem addresses a task within robotic security. In this task, a group of robots visit points of inter- est defined around an area for security purpose (Portugal & Rocha, 2011a). A fair solution for this problem must reduce the time be- tween two visits to the same point (Chevaleyre, 2004) as well as avoid predictability. The multi-robot patrolling problem has received much atten- tion in recent years, specially works that develop algorithms to coordinate decision-making among robots (Portugal & Rocha, 2011a). Those works have implemented different principles such as reinforcement learning (Santana, Ramalho, Corruble, & Ratitch, 2004); negotiations methods (Menezes, Tedesco, & Ramalho, 2006; Hwang, 2009); swarm optimization (Glad & Simonin, 2008; Chu & Glad, 2007; Wagner, 2000; Glad & Buffet, 2009); cy- cles and partitioning (Chevaleyre, 2004; Elmaliach, Agmon, & Kam- inka, 2007; Portugal & Rocha, 2010, 2011b); and adaptive solutions (Sempé & Drogoul, 2003; Chu & Glad, 2007). A description of all of them can be found at a recent survey in Portugal and Rocha (2011a). The results obtained by those works have demonstrated the effectiveness of the methods that implement solutions based on cycles and partitioning (Menezes et al., 2006; Chevaleyre, 2004). The suitable performance of those methods can be explained by their centralized coordinator scheme (Almeida et al., 2004). How- ever, those methods have three disadvantages. Firstly, a centralized solution has several problems such as lack of scalability, low 0957-4174/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.eswa.2013.10.024 ⇑ Corresponding author. Tel.: +34 660705718. E-mail address: ehernandez@industriales.upm.es (E. Hernández). Expert Systems with Applications 41 (2014) 2897–2913 Contents lists available at ScienceDirect Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa