Transiting Areas Patrolled by a Mobile Adversary Ondˇ rej Vanˇ ek, Branislav Boˇ sansk´ y, Michal Jakob and Michal Pˇ echouˇ cek Abstract—We study the problem of a mobile agent trying to cross an area patrolled by a mobile adversary. The transiting agent aims to choose its route so as to minimize the probability of hostile encounter; the patroller agent, controlling one or more patrol units, aims at the opposite. We model the problem as a two-player zero-sum game (termed transit game) and search for an optimum route selection strategy as a mixed Nash equilibrium of the game. In contrast to existing game-theoretic models of this kind, we explicitly consider the limited endurance of patrols and the notion of bases to which the patrols need to repeatedly return. Noting the prohibitive size of the transit game, we employ two techniques for reducing the complexity of finding Nash equilibria – a compact network-flow-based representation of transit routes and iterative single- and double- oracle algorithms for incremental game matrix construction. We measure the computational time of all the methods on a range of transit game instances. In order to assess the practical relevance of the approach, we apply the transit game model and its solution to the real-world case of ship transit through areas affected by piracy. The results obtained using an agent-based simulation of maritime traffic show that the randomized game- theoretic transit routing strategy results in a lower number of pirate attacks than the currently employed method based on static transit corridors. I. I NTRODUCTION A common problem faced by agents operating in areas controlled by an adversarial mobile agent is how to move while avoiding detection and interception. An important vari- ant of this problem arises when an agent needs to safely cross an area from one side to another and the mobile adversary wants to thwart the attempt by finding and intercepting the transiting agent. Real-world examples of such situations include logistics in insecure regions, illegal border crossing and/or smuggling interdiction. In such cases, a rational transiting agent aims to choose a route for which the probability of encounter is minimal; analogously, a rational patrolling agent aims to choose a route maximizing the probability. The situation can then be modeled as a game and the best route selection strategy can be sought as a solution of the game. Game-theoretic approach has been applied to similar problems in the past, resulting in a family of various games, each reflecting specific assumptions and modeling decisions made regarding the capability of the agents and the properties of the environment (see the analysis in the Related Work section). In this paper, we model the problem of transiting an area with a mobile adversary as a zero-sum game between two players: the first player – the Evader – chooses a route from one side of the area to the other side; the second player – the Patroller – controls one or more mobile patrols for Agent Technology Center, Dept. of Cybernetics, FEE, Czech Technical University, Technick´ a 2, 16627 Prague 6, Czech Republic, {vanek, bosansky, jakob, pechoucek}@agents.felk.cvut.cz which it chooses a closed-walk starting and ending in a given location in the area, termed base . If, following their chosen routes, some patrol and the Evader meet, the Patroller wins; otherwise the Evader wins. The concept of patrol bases and the closed-walk nature of patrols’ routes are salient features of our model, termed transit game. Despite their importance to many real-world scenarios, these elements have not been, to our best knowl- edge, considered in existing game-theoretic work. Formalizing a strategic conflict as a game is only the first step in addressing the problem. In order to determine a Nash equilibrium of the game, i.e., the route selection strategy that a rational agent should execute, we employ linear programming in a standard construction used for solving two-player zero-sum games. In its basic form, however, the approach becomes quickly intractable due to the prohibitive size of the resulting linear programs when the size of the transit area or the number of patrols increases. To tackle this problem, we employ two complexity reduction techniques: (1) using a compact network-flow-based representation of the Evader’s strategy space (described in [1] and later used e.g. in [2]) and (2) using an single- and double-oracle iterative algorithms (presented in [3]) for finding a Nash equilibrium. Employing the two techniques, individually and in concert, and analyzing their computational requirements is the second important contribution of the work. We do not stop here, however. Taking advantage of our work on transport security [4], [5], we apply the transit game model to a real-world problem of ship transit through areas affected by maritime piracy. Employing an agent-based simulation of maritime traffic, we validate the presented model and evaluate the effectiveness of the game-theoretic approach on a very timely use case. II. RELATED WORK As already mentioned, game theory has been applied to model and study strategic behavior in scenarios involving mobile patrols. The models developed share several char- acteristics – they are usually defined on a discrete space (i.e. graph), consist of one player trying to avoid contact (termed hider, evader, or infiltrator) and the other player (termed searcher, seeker, pursuer, patroller or guard) trying to discover and intercept the first player. All such games can be expressed in the normal form if the current position and strategies of one player are unknown to the other players. In further text we refer to these type of games as the patrol games. Despite certain similarities, there are also characteristics distinguishing individual classes of patrol games. In ambush games [6] the evading player tries to transit an area, whereas 978-1-4244-6297-1/10/$26.00 c 2010 IEEE 9