We consider a security game in a setting consisting of two players (an attacker and a defender), each with a given budget to allocate towards attack and defense, respectively, of a set of nodes. Each node has a certain value to the attacker and the defender, along with a probability of being successfully compromised, which is a function of the investments in that node by both players. For such games, we characterize the optimal investment strategies by the players at the (unique) Nash Equilibrium. We then investigate the impacts of behavioral probability weighting on the investment strategies; such probability weighting, where humans overweight low probabilities and underweight high probabilities, has been identified by behavioral economists to be a common feature of human decision-making. We show via numerical experiments that behavioral decision-making by the defender causes the Nash Equilibrium investments in each node to change (where the defender overinvests in the high-value nodes and underinvests in the low-value nodes).
This work studies a dynamic, adversarial resource allocation problem in environments modeled as graphs. A blue team of defender robots are deployed in the environment to protect the nodes from a red team of attacker robots. We formulate the engagement as a discrete-time dynamic game, where the robots can move at most one hop in each time step. The game terminates with the attacker's win if any location has more attacker robots than defender robots at any time. The goal is to identify dynamic resource allocation strategies, as well as the conditions that determines the winner: graph structure, …
Multiplayer Reach-Avoid Differential Games with Defender-Side Information Delay
State Key Laboratory of Submarine Geoscience, School of Automation and Intelligent Sensing, Shanghai Jiao Tong University（上海交通大学生物海洋科学与工程学院）；School of Artificial Intelligence, Beihang University（北京航空航天大学人工智能学院）
This study investigates general model-based incident handler's asymptotic behaviors in time against cyber attacks to control systems. The attacker's and the defender's dynamic decision making is modeled as an equilibrium of a dynamic signaling game. It is shown that the defender's belief on existence of an attacker converges over time for any attacker's strategy provided that the stochastic dynamics of the control system is known to the defender. This fact implies that the rational behavior of the attacker converges to a harmless action as long as the defender possesses an effective counteract…
In this paper we consider a Target-guarding differential game where the Defender must protect a linearly moving line segment by intercepting the Attacker who tries to reach it. In contrast to common Target-guarding problems, we assume that the Defender is attached to the Target and moves along with it. This assumption affects the Defender's maximum speed depending on its heading direction. A zero-sum differential game of degree for the Attacker-winning scenario is studied, where the payoff is defined to be the distance between the two agents at the time of reaching the Target. We derive the eq…
In this paper we consider a target-guarding differential game where the defender must protect a linearly translating line-segment by intercepting an attacker who tries to reach it. In contrast to common target-guarding problems, we assume that the defender is attached to the target and moves along with it. This assumption affects the defenders' maximum speed in inertial frame, which depends on the target's direction of motion. Zero-sum differential game of degree for both the attacker-win and defender-win scenarios are studied, where the payoff is defined to be the distance between the two age…
This study investigates the relationship between resilience of control systems to attacks and the information available to malicious attackers. Specifically, it is shown that control systems are guaranteed to be secure in an asymptotic manner by rendering reactions against potentially harmful actions covert. The behaviors of the attacker and the defender are analyzed through a repeated signaling game with an undisclosed belief under covert reactions. In the typical setting of signaling games, reactions conducted by the defender are supposed to be public information and the measurability enable…
Motivated by applications in cyber security, we develop a simple game model for describing how a learning agent's private information influences an observing agent's inference process. The model describes a situation in which one of the agents (attacker) is deciding which of two targets to attack, one with a known reward and another with uncertain reward. The attacker receives a single private sample from the uncertain target's distribution and updates its belief of the target quality. The other agent (defender) knows the true rewards, but does not see the sample that the attacker has received…
We present a moving target defense strategy to reduce the impact of stealthy sensor attacks on feedback systems. The defender periodically and randomly switches between thresholds from a discrete set to increase the uncertainty for the attacker and make stealthy attacks detectable. However, the defender does not know the exact goal of the attacker but only the prior of the possible attacker goals. Here, we model one period with a constant threshold as a Bayesian game and use the Bayesian Nash equilibrium concept to find the distribution for the choice of the threshold in that period, which tak…
In this paper, we study a Target-Attacker-Defender (TAD) differential game involving one attacker, one target and multiple defenders. We consider two variations where (a) the attacker and the target have unlimited observation range and the defenders are visibility constrained (b) only the attacker has unlimited observation range and the remaining players are visibility constrained. We model the players' interactions as a dynamic game with asymmetric information. Here, the visibility constraints of the players induce a visibility network which encapsulates the visibility information during the …
We formulate a Stackelberg game between an attacker and a defender of a power system. The attacker attempts to alter the load setpoints of the power system covertly and intelligently, so that the voltage stability margin of the grid is reduced, driving the entire system towards a voltage collapse. The defender, or the system operator, aims to compensate for this reduction by retuning the reactive power injection to the grid by switching on control devices, such as a bank of shunt capacitors. A modified Backward Induction method is proposed to find a cost-based Stackelberg equilibrium (CBSE) of…
This paper studies a defense approach against one or more swarms of adversarial agents. In our earlier work, we employ a closed formation (`StringNet') of defending agents (defenders) around a swarm of adversarial agents (attackers) to confine their motion within given bounds, and guide them to a safe area. The control design relies on the assumption that the adversarial agents remain close enough to each other, i.e., within a prescribed connectivity region. To handle situations when the attackers no longer stay within such a connectivity region, but rather split into smaller swarms (clusters)…
Consensus and cluster forming of multiagent systems in the face of jamming attacks along with reactive recovery actions by a defender are discussed. The attacker is capable to disable some of the edges of the network with the objective to divide the agents into a smaller size of clusters while, in response, the defender recovers some of the edges by increasing the transmission power. We consider repeated games where the resulting optimal strategies for the two players are derived in a rolling horizon fashion. The attacker and the defender possess different computational abilities to calculate …
In this paper, we consider a linear-quadratic-Gaussian defending assets differential game (DADG) where the attacker and the defender do not know each other's state information while they know the trajectory of a moving asset. Both players can choose to observe the other player's state information by paying a cost. The defender and the attacker have to craft both control strategies and observation strategies. We obtain a closed-form feedback solution that characterizes the Nash control strategies. We show that the trajectory of the asset does not affect both players' observation choices. Moreov…
We consider the probabilistic planning problem for a defender (P1) who can jointly query the sensors and take control actions to reach a set of goal states while being aware of possible sensor attacks by an adversary (P2) who has perfect observations. To synthesize a provably-correct, attack-aware joint control and active sensing strategy for P1, we construct a stochastic game on graph with augmented states that include the actual game state (known only to the attacker), the belief of the defender about the game state (constructed by the attacker based on his knowledge of defender's observatio…
This work establishes a game-theoretic framework to study cross-layer coordinated attacks on cyber-physical systems (CPSs). The attacker can interfere with the physical process and launch jamming attacks on the communication channels simultaneously. At the same time, the defender can dodge the jamming by dispensing with observations. The generic framework captures a wide variety of classic attack models on CPSs. Leveraging dynamic programming techniques, we fully characterize the Subgame Perfect Equilibrium (SPE) control strategies. We also derive the SPE observation and jamming strategies and…
