Security Games have been widely adopted to model scenarios in which one player, the Defender, has to decide how to deploy her resources to minimize the loss that can be caused by an attack performed by another player, the Attacker, aiming at maximizing such loss. In the present paper, we focus on scenarios in which the Defender has lexicographic-like preferences on the targets, being primarily interested in defending the integrity of a subset of the targets and, only secondarily, to reduce the amount of the other damaged targets. Our central motivation for studying this problem comes from the need to reduce the impact of malicious flows in networks, that can be either physical, like cities, or virtual, e.g., social networks. In this work, we introduce a new class of security games to model these scenarios, characterizing it and proving the NP-hardness of computing a leader-follower equilibrium, which is the most appropriate solution concept for this setting. To compute such an equilibrium, we then provide an exact exponential-time algorithm, capable of exploiting the topological properties of the network. Finally, we show that, with opportune optimizations, this algorithm can work efficiently even on network of 10000 nodes.

Leadership games provide a powerful paradigm to model many real-world settings. Most literature focuses on games with a single follower who acts optimistically, breaking ties in favour of the leader. Unfortunately, for real-world applications, this is unlikely. In this paper, we look for efficiently solvable games with multiple followers who play either optimistically or pessimistically, i.e., breaking ties in favour or against the leader. We study the computational complexity of finding or approximating an optimistic or pessimistic leader-follower equilibrium in specific classes of succinct games—polymatrix like—which are equivalent to 2-player Bayesian games with uncertainty over the follower, with interdependent or independent types. Furthermore, we provide an exact algorithm to find a pessimistic equilibrium for those game classes. Finally, we show that in general polymatrix games the computation is harder even when players are forced to play pure strategies.

The Team-maxmin equilibrium prescribes the optimal strategies for a team of rational players sharing the same goal and without the capability of correlating their strategies in strategic games against an adversary. This solution concept can capture situations in which an agent controls multiple resources - corresponding to the team members - that cannot communicate. It is known that such equilibrium always exists and it is unique (except degenerate cases) and these properties make it a credible solution concept to be used in real-world applications, especially in security scenarios. Nevertheless, to the best of our knowledge, the Team-maxmin equilibrium is almost completely unexplored in the literature. In this paper, we investigate bounds of (in)efficiency of the Team-maxmin equilibrium w.r.t. the Nash equilibria and w.r.t. the Maxmin equilibrium when the team members can play correlated strategies. Furthermore, we study a number of algorithms to find and/or approximate an equilibrium, discussing their theoretical guarantees and evaluating their performance by using a standard testbed of game instances.

Motivated by a number of security applications, among which border patrolling, we study, to the best of our knowledge, the first Security Game model in which patrolling strategies need to be combined with responses to signals raised by an alarm system, which is spatially uncertain (i.e., it is uncertain over the exact location the attack is ongoing) and is affected by false negatives (i.e., the missed detection rate of an attack may be positive). Ours is an infinite-horizon patrolling scenario on a graph, where a single patroller moves. We study the properties of the game model in terms of computational issues and form of the optimal strategies and we provide an approach to solve it. Finally, we provide an experimental analysis of our techniques.