Use of game-theoretic models to address patrolling applications has gained increasing interest in the very last years. The patrolling agent is considered playing a game against an hostile possible intruder. Game-theoretic approaches allow the development of patrolling strategies that take into account a model of the intruder. In this paper, we adopt a game-theoretic approach for finding an efficient strategy for a patrolling mobile robot. We consider, differently from most recent approaches presented in literature, patrolling as an extended-form game that can be solved, in an approximated way, by reduction to a sequence of repeated strategic-form games, each one modeling the decision of where the robot should go in the next step. The solution of these games, found with mathematical programming tools, constitutes the patrolling strategy for the robot. Experimental results show the effectiveness of our approach.