Strong Stackelberg Equilibrium Research Articles

Protecting Infrastructure Networks: Solving the Stackelberg Game with Interval-Valued Intuitionistic Fuzzy Number Payoffs

Critical infrastructure is essential for the stability and development of modern society, and a combination of complex network theory and game theory has become a new research direction in the field of infrastructure protection. However, existing studies do not consider the fuzziness and subjective factors of human judgment, leading to challenges when analyzing strategic interactions between decision makers. This paper employs interval-valued intuitionistic fuzzy numbers (IVIFN) to depict the uncertain payoffs in a Stackelberg game of infrastructure networks and then proposes an algorithm to solve it. First, we construct IVIFN payoffs by considering the different complex network metrics and subjective preferences of decision makers. Next, we propose a lexicographic algorithm to solve this game based on the concept of a strong Stackelberg equilibrium (SSE). Finally, we conduct experiments on target scale-free networks. Our results illustrate that in an SSE, for the defender in a weak position, it is better to defend nodes with high degrees. The experiments also indicate that taking fuzziness into account leads to higher SSE payoffs for the defender. Our work aims to solve a Stackelberg game with IVIFN payoffs and apply it to enhance the protection of infrastructure networks, thereby improving their overall security.

Equilibrium analysis for multi-defender Stackelberg security games under a logit tie-breaking rule

We introduce a general multi-defender Stackelberg security game where multiple independent defenders jointly protect a same set of targets from being attacked by a common attacker. In the game, Strong Stackelberg Equilibrium is fundamentally problematic, because the notion of ‘breaking ties in defender's favour’ is no longer well defined, as we must specify which defender will receive the favour. To address this issue, we define a new equilibrium concept under a newly defined tie-breaking rule. We characterise Logit Stackelberg Multi-Defender Equilibrium, corresponding to a logit tie-breaking rule, as well as an equivalent Nash Equilibrium among defenders, and exhibit algorithms for computing the equilibrium solutions. We find that Logit Stackelberg Multi-Defender Equilibrium and its' equivalent Nash Equilibrium may not exist, which motivates us to find an approximate equilibrium. We design a revised exclusion algorithm to find the approximate ε-Nash Equilibrium in which no defender gains more than ε by deviating.

Stationary Strong Stackelberg Equilibrium in Discounted Stochastic Games

Stationary Strong Stackelberg Equilibrium in Discounted Stochastic Games

Zero-determinant strategy in stochastic Stackelberg asymmetric security game

In a stochastic Stackelberg asymmetric security game, the strong Stackelberg equilibrium (SSE) strategy is a popular option for the defender to get the highest utility against an attacker with the best response (BR) strategy. However, the attacker may be a boundedly rational player, who adopts a combination of the BR strategy and a fixed stubborn one. In such a condition, the SSE strategy may not maintain the defensive performance due to the stubbornness. In this paper, we focus on how the defender can adopt the unilateral-control zero-determinate (ZD) strategy to confront the boundedly rational attacker. At first, we verify the existence of ZD strategies for the defender. We then investigate the performance of the defender’s ZD strategy against a boundedly rational attacker, with a comparison of the SSE strategy. Specifically, when the attacker’s strategy is close to the BR strategy, the ZD strategy admits a bounded loss for the defender compared with the SSE strategy. Conversely, when the attacker’s strategy is close to the stubborn strategy, the ZD strategy can bring higher defensive performance for the defender than the SSE strategy does.

Computing Stackelberg Equilibrium with Memory in Sequential Games

Stackelberg equilibrium is a solution concept that describes optimal strategies to commit: Player 1 (the leader) first commits to a strategy that is publicly announced, then Player 2 (the follower) plays a best response to the leader's commitment. We study the problem of computing Stackelberg equilibria in sequential games with finite and indefinite horizons, when players can play history-dependent strategies. Using the alternate formulation called strategies with memory, we establish that strategy profiles with polynomial memory size can be described efficiently. We prove that there exist a polynomial time algorithm which computes the Strong Stackelberg Equilibrium in sequential games defined on directed acyclic graphs, where the strategies depend only on the memory states from a set which is linear in the size of the graph. We extend this result to games on general directed graphs which may contain cycles. We also analyze the setting for approximate version of Strong Stackelberg Equilibrium in the games with chance nodes.

Double-Oracle Sampling Method for Stackelberg Equilibrium Approximation in General-Sum Extensive-Form Games

The paper presents a new method for approximating Strong Stackelberg Equilibrium in general-sum sequential games with imperfect information and perfect recall. The proposed approach is generic as it does not rely on any specific properties of a particular game model. The method is based on iterative interleaving of the two following phases: (1) guided Monte Carlo Tree Search sampling of the Follower's strategy space and (2) building the Leader's behavior strategy tree for which the sampled Follower's strategy is an optimal response. The above solution scheme is evaluated with respect to expected Leader's utility and time requirements on three sets of interception games with variable characteristics, played on graphs. A comparison with three state-of-the-art MILP/LP-based methods shows that in vast majority of test cases proposed simulation-based approach leads to optimal Leader's strategies, while excelling the competitive methods in terms of better time scalability and lower memory requirements.

Coordinating resources in Stackelberg Security Games

In this work we formulate a Stackelberg Security Game that coordinates resources in a border patrol problem. In this security domain, resources from different precincts have to be paired to conduct patrols in the border due to logistic constraints. Given this structure, models that enumerate the pure defender strategies scale poorly. We describe the set of mixed strategies using a polynomial number of variables but exponentially many constraints that come from the matching polytope. We then include this description in a mixed integer formulation to compute the Strong Stackelberg Equilibrium efficiently with a branch and cut scheme. Since the optimal patrol solution is a probability distribution over the set of exponential size, we also introduce an efficient sampling method that can be used to deploy the security resources every shift. Our computational results evaluate the efficiency of the branch and cut scheme developed and the accuracy of the sampling method.

On the Inducibility of Stackelberg Equilibrium for Security Games

Strong Stackelberg equilibrium (SSE) is the standard solution concept of Stackelberg security games. As opposed to the weak Stackelberg equilibrium (WSE), the SSE assumes that the follower breaks ties in favor of the leader and this is widely acknowledged and justified by the assertion that the defender can often induce the attacker to choose a preferred action by making an infinitesimal adjustment to her strategy. Unfortunately, in security games with resource assignment constraints, the assertion might not be valid; it is possible that the defender cannot induce the desired outcome. As a result, many results claimed in the literature may be overly optimistic. To remedy, we first formally define the utility guarantee of a defender strategy and provide examples to show that the utility of SSE can be higher than its utility guarantee. Second, inspired by the analysis of leader’s payoff by Von Stengel and Zamir (2004), we provide the solution concept called the inducible Stackelberg equilibrium (ISE), which owns the highest utility guarantee and always exists. Third, we show the conditions when ISE coincides with SSE and the fact that in general case, SSE can be extremely worse with respect to utility guarantee. Moreover, introducing the ISE does not invalidate existing algorithmic results as the problem of computing an ISE polynomially reduces to that of computing an SSE. We also provide an algorithmic implementation for computing ISE, with which our experiments unveil the empirical advantage of the ISE over the SSE.

A game-theoretic approach for channel security against active time-varying attacks based on artificial noise

To penetrate sensitive communication systems, attackers can attack the channel using an active time-varying (ATV) way, which will lead to a great information loss. The conventional approach is to encrypt the original signal making it difficult for attackers to get information. However, this technology is constrained by the limited wireless terminal equipment. In this paper, we choose to insert artificial noise into the channel, which aims at disturbing the attackers and reducing the loss of the system once attacks occur. However, this technology would produce some side effects and there is a tradeoff between inserting artificial noise and minimizing information loss. In this paper, we deal with this issue and propose a game-theoretic framework to minimize the total losses. We model the problem as a Stackelberg security game between the attacker and the defender. Furthermore, we propose a novel method to reduce the searching space of computing the Strong Stackelberg Equilibrium which is the optimal defense strategy. This algorithm reduces a M-dimensional problem to M 1-dimensional problems so that the complexity is lowered. The experimental results show that our proposed algorithm significantly outperforms other non-strategic strategies in terms of decreasing the total losses against ATV attacks.

Strategic Surveillance Against Primary User Emulation Attacks in Cognitive Radio Networks

Selfish primary user emulation (PUE) is a serious security problem in cognitive radio networks. By emitting emulated incumbent signals, a PUE attacker can selfishly occupy more channels. Consequently, a PUE attacker can prevent other secondary users from accessing radio resources and interfere with nearby primary users. To mitigate the selfish PUE, a surveillance process on occupied channels could be performed. Determining surveillance strategies, particularly in multi-channel context, is necessary for ensuring network operation fairness. Since a rational attacker can learn to adapt to the surveillance strategy, the question is how to formulate an appropriate modeling of the strategic interaction between a defender and an attacker. In this paper, we study the commitment model in which the network manager takes the leadership role by committing to its surveillance strategy and forces the attacker to follow the committed strategy. The relevant strategy is analyzed through the strong Stackelberg equilibrium (SSE). Analytical and numerical results suggest that, by playing the SSE strategy, the network manager significantly improves its utility with respect to playing a Nash equilibrium (NE) strategy, hence obtains a better protection against selfish PUEs. Moreover, the computational effort to compute the SSE strategy is lower than to find an NE strategy.

A Game-theoretic Approach for Channel Security Against Active Time-Varying attacks Based on Artificial Noise.

To penetrate sensitive communication systems, attackers can attack the channel using an Active Time-Varying(ATV) way, which will lead to a great information loss. The conventional approach is to encrypt the original signal making it difficult for attackers to get information. However, this technology is constrained by the limited wireless terminal equipment. In this paper, we choose to insert artificial noise into the channel, which aims at disturbing the attackers and reducing the loss of the system once attacks occur. However this technology has some side effects and tradeoffs. In this paper, we deal with this issue and propose a game-theoretic framework to minimize the total losses. We model the problem as a defender-attacker Stackelberg security game where the communication system plays a defender role. Furthermore we propose a novel binary search based algorithm to efficiently compute the Strong Stackelberg Equilibrium which is the optimal defense strategy. This algorithm reduces a M-dimensional problem to M 1-dimensional problems so that the complexity is lowered. The experimental results show that our proposed algorithm significantly outperforms other non-strategic strategies in terms of decreasing the total losses against ATV attacks.

MTD-Spamguard: a moving target defense-based spammer detection system in social network

Machine learning classifiers are currently the state of the art for spammer detection tasks in SNSs. Note, however, that these classifiers fail to detect adaptive spammers that dynamically change their spamming strategies or behaviors and attempt to pose as legitimate users. In this paper, we propose an efficient spammer detection system (which we call MTD-Spamguard) wherein the notion of MTD is applied to increase the robustness of well-known machine learning classifiers against the adaptive spammers in SNSs. The system introduces a new method of MTD wherein the concept of differential immunity of different classifiers is employed to detect the spammers. To classify a single user in the test dataset, we pick one of the appropriate trained classifiers from multiple classifiers and then use its classification output. To choose the appropriate classifier, we design an effective classifier switching strategy by formulating the interaction of users (normal users and spammers) and detector (which hosts the machine learning classifier) as a repeated Bayesian Stackelberg game. The classifier switching strategy provides strong Stackelberg equilibrium between users and detector, maximizing the accuracy of classification and reducing the misclassification of spammers. The system achieves 30% gain in classification accuracy over the Facebook dataset (constructed in our recent work).

Optimal Personalized Defense Strategy Against Man-In-The-Middle Attack

The Man-In-The-Middle (MITM) attack is one of the most common attacks employed in the network hacking. MITM attackers can successfully invoke attacks such as denial of service (DoS) and port stealing, and lead to surprisingly harmful consequences for users in terms of both financial loss and security issues. The conventional defense approaches mainly consider how to detect and eliminate those attacks or how to prevent those attacks from being launched in the first place. This paper proposes a game-theoretic defense strategy from a different perspective, which aims at minimizing the loss that the whole system sustains given that the MITM attacks are inevitable. We model the interaction between the attacker and the defender as a Stackelberg security game and adopt the Strong Stackelberg Equilibrium (SSE) as the defender's strategy. Since the defender's strategy space is infinite in our model, we employ a novel method to reduce the searching space of computing the optimal defense strategy. Finally, we empirically evaluate our optimal defense strategy by comparing it with non-strategic defense strategies. The results indicate that our game-theoretic defense strategy significantly outperforms other non-strategic defense strategies in terms of decreasing the total losses against MITM attacks.

Security Games on a Plane

Most existing models of Stackelberg security games ignore the underlying topology of the space in which targets and defence resources are located. As a result, allocation of resources is restricted to a discrete collection of exogenously defined targets. However, in many practical security settings, defense resources can be located on a continuous plane. Better defense solutions could therefore be potentially achieved by placing resources in a space outside of actual targets (e.g., between targets). To address this limitation, we propose a model called Security Game on a Plane (SGP) in which targets are distributed on a 2-dimensional plane, and security resources, to be allocated on the same plane, protect targets within a certain effective distance. We investigate the algorithmic aspects of SGP. We find that computing a strong Stackelberg equilibrium of an SGP is NP-hard even for zero-sum games, and these are inapproximable in general. On the positive side, we find an exact solution technique for general SGPs based on an existing approach, and develop a PTAS (polynomial-time approximation scheme) for zero-sum SGP to more fundamentally overcome the computational obstacle. Our experiments demonstrate the value of considering SGP and effectiveness of our algorithms.

Conforming coalitions in Markov Stackelberg security games: Setting max cooperative defenders vs. non-cooperative attackers

The use of multiple agents to perform intruder search or patrolling activities have been widely studied in security Stackelberg games. Although existing methods achieve an efficient way of performing schedules, it is still an open problem that needs substantial further research in developing an algorithmic method of convergence to a Strong Stackelberg/Nash equilibrium, that would positively impact the effectiveness of the patrollers ability to execute their schedules, when acting cooperatively.To address this shortcoming, in this work we present a novel approach for conforming coalitions in multiple agents Stackelberg security games. The concept of coalition coincides in fact with the concept of Strong Stackelberg Equilibrium. Our approach is restricted to a class of a time-discrete ergodic controllable Markov chains. The security model describes a strategic game in which the defenders cooperate and attackers do not cooperate. In case of a metric state space, the coalition of the defenders achieves it synergy by computing the Strong Lp-Stackelberg/Nash equilibrium. We employ the extraproximal method for solving the problem. In addition, we consider the control problem that involves defenders and attackers performing a discrete-time random walk. Actions are selected using the max operator, which returns the maximum value of a set of strategies. We prove that the sequences generated by the two synchronized random walks of both defenders and attackers, converge to the product of the individual probabilities. A realization of the method in a numerical example shows the efficiency of the solution.

Using Correlated Strategies for Computing Stackelberg Equilibria in Extensive-Form Games

Strong Stackelberg Equilibrium (SSE) is a fundamental solution concept in game theory in which one player commits to a strategy, while the other player observes this commitment and plays a best response. We present a new algorithm for computing SSE for two-player extensive-form general-sum games with imperfect information (EFGs) where computing SSE is an NP-hard problem. Our algorithm is based on a correlated version of SSE, known as Stackelberg Extensive-Form Correlated Equilibrium (SEFCE). Our contribution is therefore twofold: (1) we give the first linear program for computing SEFCE in EFGs without chance, (2) we repeatedly solve and modify this linear program in a systematic search until we arrive to SSE. Our new algorithm outperforms the best previous algorithms by several orders of magnitude.

Exploring Information Asymmetry in Two-Stage Security Games

Stackelberg security games have been widely deployed to protect real-word assets. The main solution concept there is the Strong Stackelberg Equilibrium (SSE), which optimizes the defender's random allocation of limited security resources. However, solely deploying the SSE mixed strategy has limitations. In the extreme case, there are security games where the defender is able to defend all the assets ``almost perfectly" at the SSE, but she still sustains significant loss. In this paper, we propose an approach for improving the defender's utility in such scenarios. Perhaps surprisingly, our approach is to strategically reveal to the attacker information about the sampled pure strategy. Specifically, we propose a two-stage security game model, where in the first stage the defender allocates resources and the attacker selects a target to attack, and in the second stage the defender strategically reveals local information about that target, potentially deterring the attacker's attack plan. We then study how the defender can play optimally in both stages. We show, theoretically and experimentally, that the two-stage security game model allows the defender to gain strictly better utility than SSE.

Security Games with Protection Externalities

Stackelberg security games have been widely deployed in recent years to schedule security resources. An assumption in most existing security game models is that one security resource assigned to a target only protects that target. However, in many important real-world security scenarios, when a resource is assigned to a target, it exhibits protection externalities: that is, it also protects other “neighbouring” targets. We investigate such Security Games with Protection Externalities (SPEs). First, we demonstrate that computing a strong Stackelberg equilibrium for an SPE is NP-hard, in contrast with traditional Stackelberg security games which can be solved in polynomial time. On the positive side, we propose a novel column generation based approach—CLASPE—to solve SPEs. CLASPE features the following novelties: 1) a novel mixed-integer linear programming formulation for the slave problem; 2) an extended greedy approach with a constant-factor approximation ratio to speed up the slave problem; and 3) a linear-scale linear programming that efficiently calculates the upper bounds of target-defined subproblems for pruning. Our experimental evaluation demonstrates that CLASPE enable us to scale to realistic-sized SPE problem instances.

Network spot‐checking games: Theory and application to toll enforcing in transportation networks

We introduce the class of spot‐checking games (SC games). These games model problems where the goal is to distribute fare inspectors over a toll network. In an SC game, the pure strategies of network users correspond to paths in a graph, and the pure strategies of the inspectors are subset of arcs to be controlled. Although SC games are not zero‐sum, we show that a Nash equilibrium can be computed by linear programming. The computation of a strong Stackelberg equilibrium (SSE) is more relevant for this problem and we give a mixed integer programming (MIP) formulation for this problem. We show that the computation of such an equilibrium is NP‐hard. More generally, we prove that it is NP‐hard to compute a SSE in a polymatrix game, even if the game is pairwise zero‐sum. Then, we give some bounds on the price of spite, which measures how the payoff of the inspector degrades when committing to a Nash equilibrium. Finally, we report computational experiments on instances constructed from real data, for an application to the enforcement of a truck toll in Germany. These numerical results show the efficiency of the proposed methods, as well as the quality of the bounds derived in this article. © 2015 Wiley Periodicals, Inc. NETWORKS, Vol. 65(4), 312–328 2015

Stackelberg security games: Computing the shortest-path equilibrium

In this paper we consider a game theory approach for representing a real-world attacker–defender Stackelberg security game. In this novel approach the behavior of an ergodic system (repeated stochastic Markov chain game) is represented by a Lyapunov-like function non-decreasing in time. Then, the representation of the Stackelberg security game is transformed in a potential game in terms of Lyapunov. We present a method for constructing a Lyapunov-like function: the function replaces the recursive mechanism with the elements of the ergodic system seeking to drive the underlying finite-state Stackelberg game to an equilibrium point along a least expected cost path. The proposed method analyzes both pure and mixed stationary strategies to find the strong Stackelberg equilibrium. Mixed strategies are found when the resources available for both the defender and the attacker are limited. Lyapunov games model how players are likely to behave in one-shot games and allow finishing during the game whether the applied best-reply strategy (pure or mixed) provides the convergence to a shortest-path equilibrium point (or not). We prove that Lyapunov games truly fit into the framework for deterministic and stochastic shortest-path security games. The convergence rate of the proposed method to a Stackelberg/Nash equilibrium is analyzed. Validity of the proposed method is successfully demonstrated both theoretically and by a simulated experiment.