Biblio

Filters: Author is Yevgeniy Vorobeychik  [Clear All Filters]
2019-05-31
2019-05-29
Amin Ghafouri, Xenofon Koutsoukos, Yevgeniy Vorobeychik, Waseem Abbas, Aron Laszka.  2019.  A game-theoretic approach for selecting optimal time-dependent thresholds for anomaly detection. International Foundation for Autonomous Agents and Multi-Agent Systems Journal. 33

Adversaries may cause significant damage to smart infrastructure using malicious attacks. To detect and mitigate these attacks before they can cause physical damage, operators can deploy anomaly detection systems (ADS), which can alarm operators to suspicious activities. However, detection thresholds of ADS need to be configured properly, as an oversensitive detector raises a prohibitively large number of false alarms, while an undersensitive detector may miss actual attacks. This is an especially challenging problem in dynamical environments, where the impact of attacks may significantly vary over time. Using a game-theoretic approach, we formulate the problem of computing optimal detection thresholds which minimize both the number of false alarms and the probability of missing actual attacks as a two-player Stackelberg security game. We provide an efficient dynamic programming-based algorithm for solving the game, thereby finding optimal detection thresholds. We analyze the performance of the proposed algorithm and show that its running time scales polynomially as the length of the time horizon of interest increases. In addition, we study the problem of finding optimal thresholds in the presence of both random faults and attacks. Finally, we evaluate our result using a case study of contamination attacks in water networks, and show that our optimal thresholds significantly outperform fixed thresholds that do not consider that the environment is dynamical.

2018-10-09
Amin Ghafouri, Yevgeniy Vorobeychik, Xenofon D. Koutsoukos.  2018.  Adversarial Regression for Detecting Attacks in Cyber-Physical Systems. CoRR. abs/1804.11022

Attacks in cyber-physical systems (CPS) which manipulate sensor readings can cause enormous physical damage if undetected. Detection of attacks on sensors is crucial to mitigate this issue. We study supervised regression as a means to detect anomalous sensor readings, where each sensor's measurement is predicted as a function of other sensors. We show that several common learning approaches in this context are still vulnerable to \emph{stealthy attacks}, which carefully modify readings of compromised sensors to cause desired damage while remaining undetected. Next, we model the interaction between the CPS defender and attacker as a Stackelberg game in which the defender chooses detection thresholds, while the attacker deploys a stealthy attack in response. We present a heuristic algorithm for finding an approximately optimal threshold for the defender in this game, and show that it increases system resilience to attacks without significantly increasing the false alarm rate.

2019-05-29
Amin Ghafouri, Xenofon Koutsoukos, Yevgeniy Vorobeychik.  2018.  Adversarial Regression for Detecting Attacks in Cyber-Physical Systems. Twenty-Seventh International Joint Conference on Artificial Intelligence.

Attacks in cyber-physical systems (CPS) which manipulate sensor readings can cause enormous physical damage if undetected. Detection of attacks on sensors is crucial to mitigate this issue. We study supervised regression as a means to detect anoma- lous sensor readings, where each sensor’s measure- ment is predicted as a function of other sensors. We show that several common learning approaches in this context are still vulnerable to stealthy at- tacks, which carefully modify readings of compro- mised sensors to cause desired damage while re- maining undetected. Next, we model the interac- tion between the CPS defender and attacker as a Stackelberg game in which the defender chooses detection thresholds, while the attacker deploys a stealthy attack in response. We present a heuris- tic algorithm for finding an approximately optimal threshold for the defender in this game, and show that it increases system resilience to attacks without significantly increasing the false alarm rate.

2018-10-09
Aron Laszka, Waseem Abbas, Yevgeniy Vorobeychik, Xenofon Koutsoukos.  2018.  Synergistic Security for the Industrial Internet of Things: Integrating Redundancy, Diversity, and Hardening.

As the Industrial Internet of Things (IIot) becomes more prevalent in critical application domains, ensuring security and resilience in the face of cyber-attacks is becoming an issue of paramount importance. Cyber-attacks against critical infrastructures, for example, against smart water-distribution and transportation systems, pose serious threats to public health and safety. Owing to the severity of these threats, a variety of security techniques are available. However, no single technique can address the whole spectrum of cyber-attacks that may be launched by a determined and resourceful attacker. In light of this, we consider a multi-pronged approach for designing secure and resilient IIoT systems, which integrates redundancy, diversity, and hardening techniques. We introduce a framework for quantifying cyber-security risks and optimizing IIoT design by determining security investments in redundancy, diversity, and hardening. To demonstrate the applicability of our framework, we present two case studies in water distribution and transportation a case study in water-distribution systems. Our numerical evaluation shows that integrating redundancy, diversity, and hardening can lead to reduced security risk at the same cost.

2019-05-30
Xenofon Koutsoukos, Gabor Karsai, Aron Laszka, Himanshu Neema, Bradley Potteiger, Peter Volgyesi, Yevgeniy Vorobeychik, Janos Sztipanovits.  2018.  SURE: A Modeling and Simulation Integration Platform for Evaluation of Secure and Resilient Cyber–Physical Systems. Proceedings of the IEEE. 106:93-112.

The exponential growth of information and communication technologies have caused a profound shift in the way humans engineer systems leading to the emergence of closed-loop systems involving strong integration and coordination of physical and cyber components, often referred to as cyber-physical systems (CPSs). Because of these disruptive changes, physical systems can now be attacked through cyberspace and cyberspace can be attacked through physical means. The paper considers security and resilience as system properties emerging from the intersection of system dynamics and the computing architecture. A modeling and simulation integration platform for experimentation and evaluation of resilient CPSs is presented using smart transportation systems as the application domain. Evaluation of resilience is based on attacker-defender games using simulations of sufficient fidelity. The platform integrates 1) realistic models of cyber and physical components and their interactions; 2) cyber attack models that focus on the impact of attacks to CPS behavior and operation; and 3) operational scenarios that can be used for evaluation of cybersecurity risks. Three case studies are presented to demonstrate the advantages of the platform: 1) vulnerability analysis of transportation networks to traffic signal tampering; 2) resilient sensor selection for forecasting traffic flow; and 3) resilient traffic signal control in the presence of denial-of-service attacks.

Aron Laszka, Waseem Abbas, Yevgeniy Vorobeychik, Xenofon Koutsoukos.  2018.  Synergistic Security for the Industrial Internet of Things: Integrating Redundancy, Diversity, and Hardening. IEEE International Conference on Industrial Internet (ICII). :153-158.

As the Industrial Internet of Things (IIot) becomes more prevalent in critical application domains, ensuring security and resilience in the face of cyber-attacks is becoming an issue of paramount importance. Cyber-attacks against critical infrastructures, for example, against smart water-distribution and transportation systems, pose serious threats to public health and safety. Owing to the severity of these threats, a variety of security techniques are available. However, no single technique can address the whole spectrum of cyber-attacks that may be launched by a determined and resourceful attacker. In light of this, we consider a multi-pronged approach for designing secure and resilient IIoT systems, which integrates redundancy, diversity, and hardening techniques. We introduce a framework for quantifying cyber-security risks and optimizing IIoT design by determining security investments in redundancy, diversity, and hardening. To demonstrate the applicability of our framework, we present a case study in water-distribution systems. Our numerical evaluation shows that integrating redundancy, diversity, and hardening can lead to reduced security risk at the same cost.

2017-10-27
Bo Li, Yining Wang, Aarti Singh, Yevgeniy Vorobeychik.  2017.  Data poisoning attacks on factorization-based collaborative filtering. Neural Information Processing Systems (NIPS 2016).
Recommendation and collaborative filtering systems are important in modern information and e-commerce applications. As these systems are becoming increasingly popular in the industry, their outputs could affect business decision making, introducing incentives for an adversarial party to compromise the availability or integrity of such systems. We introduce a data poisoning attack on collaborative filtering systems. We demonstrate how a powerful attacker with full knowledge of the learner can generate malicious data so as to maximize his/her malicious objectives, while at the same time mimicking normal user behavior to avoid being detected. While the complete knowledge assumption seems extreme, it enables a robust assessment of the vulnerability of collaborative filtering schemes to highly motivated attacks. We present efficient solutions for two popular factorization-based collaborative filtering algorithms: the alternative minimization formulation and the nuclear norm minimization method. Finally, we test the effectiveness of our proposed algorithms on real-world data and discuss potential defensive strategies.
Bo Li, Kevin Roundy, Chris Gates, Yevgeniy Vorobeychik.  2017.  Large-scale identification of malicious singleton files. ACM Conference on Data and Application Security and Privacy.
We study a dataset of billions of program binary files that appeared on 100 million computers over the course of 12 months, discovering that 94% of these files were present on a single machine. Though malware polymorphism is one cause for the large number of singleton files, additional factors also contribute to polymorphism, given that the ratio of benign to malicious singleton files is 80:1. The huge number of benign singletons makes it challenging to reliably identify the minority of malicious singletons. We present a large-scale study of the properties, characteristics, and distribution of benign and malicious singleton files. We leverage the insights from this study to build a classifier based purely on static features to identify 92% of the remaining malicious singletons at a 1.4% percent false positive rate, despite heavy use of obfuscation and packing techniques by most malicious singleton files that we make no attempt to de-obfuscate. Finally, we demonstrate robustness of our classifier to important classes of automated evasion attacks.
Waseem Abbas, Aron Laszka, Yevgeniy Vorobeychik, Xenofon Koutsoukos.  2017.  Scheduling Resource-Bounded Monitoring Devices for Event Detection and Isolation in Networks. IEEE Transactions on Network Science and Engineering.
In networked systems, monitoring devices such as sensors are typically deployed to monitor various target locations. Targets are the points in the physical space at which events of some interest, such as random faults or attacks, can occur. Most often, these devices have limited energy supplies, and they can operate for a limited duration. As a result, energyefficient monitoring of various target locations through a set of monitoring devices with limited energy supplies is a crucial problem in networked systems. In this paper, we study optimal scheduling of monitoring devices to maximize network coverage for detecting and isolating events on targets for a given network lifetime. The monitoring devices considered could remain active only for a fraction of the overall network lifetime. We formulate the problem of scheduling of monitoring devices as a graph labeling problem, which unlike other existing solutions, allows us to directly utilize the underlying network structure to explore the trade-off between coverage and network lifetime. In this direction, first we propose a greedy heuristic to solve the graph labeling problem, and then provide a game-theoretic solution to achieve optimal graph labeling. Moreover, the proposed setup can be used to simultaneously solve the scheduling and placement of monitoring devices, which yields improved performance as compared to separately solving the placement and scheduling problems. Finally, we illustrate our results on various networks, including real-world water distribution networks.
Jiarui Gan, Bo An, Yevgeniy Vorobeychik, Brian Gauch.  2017.  Security games on a plane. AAAI Conference on Artificial Intelligence.
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 zerosum 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.
Aron Laszka, Waseem Abbas, Yevgeniy Vorobeychik, Xenofon Koutsoukos.  2017.  Synergic Security for Smart Water Networks: Redundancy, Diversity, and Hardening. 3rd International Workshop on Cyber-Physical Systems for Smart Water Networks (CySWater 2017).
Smart water networks can provide great benefits to our society in terms of efficiency and sustainability. However, smart capabilities and connectivity also expose these systems to a wide range of cyber attacks, which enable cyber-terrorists and hostile nation states to mount cyber-physical attacks. Cyber-physical attacks against critical infrastructure, such as water treatment and distribution systems, pose a serious threat to public safety and health. Consequently, it is imperative that we improve the resilience of smart water networks. We consider three approaches for improving resilience: redundancy, diversity, and hardening. Even though each one of these “canonical” approaches has been thoroughly studied in prior work, a unified theory on how to combine them in the most efficient way has not yet been established. In this paper, we address this problem by studying the synergy of these approaches in the context of protecting smart water networks from cyber-physical contamination attacks.
Aron Laszka, Yevgeniy Vorobeychik, Daniel Fabbri, Chao Yan, Bradley Malin.  2017.  A Game-Theoretic Approach for Alert Prioritization. AAAI-17 Workshop on Artificial Intelligence for Cyber Security (AICS).
The quantity of information that is collected and stored in computer systems continues to grow rapidly. At the same time, the sensitivity of such information (e.g., detailed medical records) often makes such information valuable to both external attackers, who may obtain information by compromising a system, and malicious insiders, who may misuse information by exercising their authorization. To mitigate compromises and deter misuse, the security administrators of these resources often deploy various types of intrusion and misuse detection systems, which provide alerts of suspicious events that are worthy of follow-up review. However, in practice, these systems may generate a large number of false alerts, wasting the time of investigators. Given that security administrators have limited budget for investigating alerts, they must prioritize certain types of alerts over others. An important challenge in alert prioritization is that adversaries may take advantage of such behavior to evade detection - specifically by mounting attacks that trigger alerts that are less likely to be investigated. In this paper, we model alert prioritization with adaptive adversaries using a Stackelberg game and introduce an approach to compute the optimal prioritization of alert types. We evaluate our approach using both synthetic data and a real-world dataset of alerts generated from the audit logs of an electronic medical record system in use at a large academic medical center.
Nika Haghtalab, Aron Laszka, Ariel Procaccia, Yevgeniy Vorobeychik, Xenofon Koutsoukos.  2017.  Monitoring Stealthy Diffusion. Knowledge and Information Systems.
(No abstract.)
Aron Laszka, Yevgeniy Vorobeychik, Xenofon Koutsoukos.  2017.  A game-theoretic approach for integrity assurance in resource-bounded systems. International Journal of Information Security.

Assuring communication integrity is a central problem in security. However, overhead costs associated with cryptographic primitives used towards this end introduce significant practical implementation challenges for resource-bounded systems, such as cyberphysical systems. For example, many control systems are built on legacy components which are computationally limited but have strict timing constraints. If integrity protection is a binary decision, it may simply be infeasible to introduce into such systems; without it, however, an adversary can forge malicious messages, which can cause significant physical or financial harm. To bridge the gap between such binary decisions, we propose a stochastic message authentication approach that can explicitly trade computational cost off for security. We introduce a formal game-theoretic framework for optimal stochastic message authentication, providing provable guarantees for resource-bounded systems based on an existing message authentication scheme. We use our framework to investigate attacker deterrence, as well as optimal stochastic message authentication when deterrence is impossible, in both short-term and long-term equilibria. Additionally, we propose two schemes for implementing stochastic message authentication in practice, one for saving computation only at the receiver and one for saving computation at both ends, and demonstrate the associated computational savings using an actual implementation.

Waseem Abbas, Aron Laszka, Yevgeniy Vorobeychik, Xenofon Koutsoukos.  2017.  Improving Network Connectivity Using Trusted Nodes and Edges. American Control Conference (ACC 2017).

Network connectivity is a primary attribute and a characteristic phenomenon of any networked system. A high connectivity is often desired within networks; for instance to increase robustness to failures, and resilience against attacks. A typical approach to increasing network connectivity is to strategically add links; however, adding links is not always the most suitable option. In this paper, we propose an alternative approach to improving network connectivity, that is by making a small subset of nodes and edges “trusted,” which means that such nodes and edges remain intact at all times and are insusceptible to failures. We then show that by controlling the number of trusted nodes and edges, any desired level of network connectivity can be obtained. Along with characterizing network connectivity with trusted nodes and edges, we present heuristics to compute a small number of such nodes and edges. Finally, we illustrate our results on various networks.

Yue Yin, Yevgeniy Vorobeychik, Bo An, Noam Hazon.  2016.  Optimally protecting elections. International Joint Conference on Artificial Intelligence.
(No abstract.)
Jian Lou, Yevgeniy Vorobeychik.  2016.  Decentralization and security in dynamic traffic light control. Symposium and Bootcamp on Science of Security.

Complex traffic networks include a number of controlled intersections, and, commonly, multiple districts or municipalities. The result is that the overall traffic control problem is extremely complex computationally. Moreover, given that different municipalities may have distinct, non-aligned, interests, traffic light controller design is inherently decentralized, a consideration that is almost entirely absent from related literature. Both complexity and decentralization have great bearing both on the quality of the traffic network overall, as well as on its security. We consider both of these issues in a dynamic traffic network. First, we propose an effective local search algorithm to efficiently design system-wide control logic for a collection of intersections. Second, we propose a game theoretic (Stackelberg game) model of traffic network security in which an attacker can deploy denial-of-service attacks on sensors, and develop a resilient control algorithm to mitigate such threats. Finally, we propose a game theoretic model of decentralization, and investigate this model both in the context of baseline traffic network design, as well as resilient design accounting for attacks. Our methods are implemented and evaluated using a simple traffic network scenario in SUMO.

Aron Laszka, Waseem Abbas, Shankar Sastry, Yevgeniy Vorobeychik, Xenofon Koutsoukos.  2016.  Optimal Thresholds for Intrusion Detection Systems. 3rd Annual Symposium and Bootcamp on the Science of Security (HotSoS).

In recent years, we have seen a number of successful attacks against high-profile targets, some of which have even caused severe physical damage. These examples have shown us that resourceful and determined attackers can penetrate virtually any system, even those that are secured by the "air-gap." Consequently, in order to minimize the impact of stealthy attacks, defenders have to focus not only on strengthening the first lines of defense but also on deploying effective intrusion-detection systems. Intrusion-detection systems can play a key role in protecting sensitive computer systems since they give defenders a chance to detect and mitigate attacks before they could cause substantial losses. However, an over-sensitive intrusion-detection system, which produces a large number of false alarms, imposes prohibitively high operational costs on a defender since alarms need to be manually investigated. Thus, defenders have to strike the right balance between maximizing security and minimizing costs. Optimizing the sensitivity of intrusion detection systems is especially challenging in the case when multiple interdependent computer systems have to be defended against a strategic attacker, who can target computer systems in order to maximize losses and minimize the probability of detection. We model this scenario as an attacker-defender security game and study the problem of finding optimal intrusion detection thresholds.

Aron Laszka, Bradley Potteiger, Yevgeniy Vorobeychik, Saurabh Amin, Xenofon Koutsoukos.  2016.  Vulnerability of Transportation Networks to Traffic-Signal Tampering. 7th ACM/IEEE International Conference on Cyber-Physical Systems (ICCPS).

Traffic signals were originally standalone hardware devices running on fixed schedules, but by now, they have evolved into complex networked systems. As a consequence, traffic signals have become susceptible to attacks through wireless interfaces or even remote attacks through the Internet. Indeed, recent studies have shown that many traffic lights deployed in practice have easily exploitable vulnerabilities, which allow an attacker to tamper with the configuration of the signal. Due to hardware-based failsafes, these vulnerabilities cannot be used to cause accidents. However, they may be used to cause disastrous traffic congestions. Building on Daganzo's well-known traffic model, we introduce an approach for evaluating vulnerabilities of transportation networks, identifying traffic signals that have the greatest impact on congestion and which, therefore, make natural targets for attacks. While we prove that finding an attack that maximally impacts congestion is NP-hard, we also exhibit a polynomial-time heuristic algorithm for computing approximately optimal attacks. We then use numerical experiments to show that our algorithm is extremely efficient in practice. Finally, we also evaluate our approach using the SUMO traffic simulator with a real-world transportation network, demonstrating vulnerabilities of this network. These simulation results extend the numerical experiments by showing that our algorithm is extremely efficient in a microsimulation model as well.

2016-04-11
Aron Laszka, Bradley Potteiger, Yevgeniy Vorobeychik, Saurabh Amin, Xenofon Koutsoukos.  2016.  Vulnerability of Transportation Networks to Traffic-Signal Tampering. 7th ACM/IEEE International Conference on Cyber-Physical Systems (ICCPS).

Traffic signals were originally standalone hardware devices running on fixed schedules, but by now, they have evolved into complex networked systems. As a consequence, traffic signals have become susceptible to attacks through wireless interfaces or even remote attacks through the Internet. Indeed, recent studies have shown that many traffic lights deployed in practice have easily exploitable vulnerabilities, which allow an attacker to tamper with the configuration of the signal. Due to hardware-based failsafes, these vulnerabilities cannot be used to cause accidents. However, they may be used to cause disastrous traffic congestions. Building on Daganzo's well-known traffic model, we introduce an approach for evaluating vulnerabilities of transportation networks, identifying traffic signals that have the greatest impact on congestion and which, therefore, make natural targets for attacks. While we prove that finding an attack that maximally impacts congestion is NP-hard, we also exhibit a polynomial-time heuristic algorithm for computing approximately optimal attacks. We then use numerical experiments to show that our algorithm is extremely efficient in practice. Finally, we also evaluate our approach using the SUMO traffic simulator with a real-world transportation network, demonstrating vulnerabilities of this network. These simulation results extend the numerical experiments by showing that our algorithm is extremely efficient in a microsimulation model as well.

2017-10-27
Amin Ghafouri, Waseem Abbas, Yevgeniy Vorobeychik, Xenofon Koutsoukos.  2016.  Vulnerability of Fixed-Time Control of Signalized Intersections to Cyber-Tampering. 9th International Symposium on Resilient Control Systems.

— Recent experimental studies have shown that traf- fic management systems are vulnerable to cyber-attacks on sensor data. This paper studies the vulnerability of fixedtime control of signalized intersections when sensors measuring traffic flow information are compromised and perturbed by an adversary. The problems are formulated by considering three malicious objectives: 1) worst-case network accumulation, which aims to destabilize the overall network as much as possible; 2) worst-case lane accumulation, which aims to cause worstcase accumulation on some target lanes; and 3) risk-averse target accumulation, which aims to reach a target accumulation by making the minimum perturbation to sensor data. The problems are solved using bilevel programming optimization methods. Finally, a case study of a real network is used to illustrate the results.

Aron Laszka, Jian Lou, Yevgeniy Vorobeychik.  2016.  Multi-Defender Strategic Filtering Against Spear-Phishing Attacks. 30th AAAI Conference on Artificial Intelligence (AAAI).
Spear-phishing attacks pose a serious threat to sensitive computer systems, since they sidestep technical security mechanisms by exploiting the carelessness of authorized users. A common way to mitigate such attacks is to use e-mail filters which block e-mails with a maliciousness score above a chosen threshold. Optimal choice of such a threshold involves a tradeoff between the risk from delivered malicious emails and the cost of blocking benign traffic. A further complicating factor is the strategic nature of an attacker, who may selectively target users offering the best value in terms of likelihood of success and resulting access privileges. Previous work on strategic threshold-selection considered a single organization choosing thresholds for all users. In reality, many organizations are potential targets of such attacks, and their incentives need not be well aligned. We therefore consider the problem of strategic threshold-selection by a collection of independent self-interested users. We characterize both Stackelberg multi-defender equilibria, corresponding to short-term strategic dynamics, as well as Nash equilibria of the simultaneous game between all users and the attacker, modeling long-term dynamics, and exhibit a polynomial-time algorithm for computing short-term (Stackelberg) equilibria. We find that while Stackelberg multi-defender equilibrium need not exist, Nash equilibrium always exists, and remarkably, both equilibria are unique and socially optimal.
2016-04-07
Aron Laszka, Jian Lou, Yevgeniy Vorobeychik.  2016.  Multi-Defender Strategic Filtering Against Spear-Phishing Attacks. 30th AAAI Conference on Artificial Intelligence (AAAI).

Spear-phishing attacks pose a serious threat to sensitive computer systems, since they sidestep technical security mechanisms by exploiting the carelessness of authorized users. A common way to mitigate such attacks is to use e-mail filters which block e-mails with a maliciousness score above a chosen threshold. Optimal choice of such a threshold involves a tradeoff between the risk from delivered malicious emails and the cost of blocking benign traffic. A further complicating factor is the strategic nature of an attacker, who may selectively target users offering the best value in terms of likelihood of success and resulting access privileges. Previous work on strategic threshold-selection considered a single organization choosing thresholds for all users. In reality, many organizations are potential targets of such attacks, and their incentives need not be well aligned. We therefore consider the problem of strategic threshold-selection by a collection of independent self-interested users. We characterize both Stackelberg multi-defender equilibria, corresponding to short-term strategic dynamics, as well as Nash equilibria of the simultaneous game between all users and the attacker, modeling long-term dynamics, and exhibit a polynomial-time algorithm for computing short-term (Stackelberg) equilibria. We find that while Stackelberg multi-defender equilibrium need not exist, Nash equilibrium always exists, and remarkably, both equilibria are unique and socially optimal.