An Attacker–defender Resource Allocation Game with Substitution and Complementary Effects

The United States is funding homeland security programs with a large budget (e.g., 74.4 billion for FY 2019). A number of game-theoretic defender–attacker models have been developed to study the optimal defense resource allocation strategies for the government (defender) against the strategic adversary (attacker). However, to the best of our knowledge, the substitution or complementary effects between different types of defensive resources (e.g., human resource, land resource, and capital resource) have not been taken into consideration even though they exist in practice. The article fills this gap by studying a sequential game-theoretical resource allocation model and then exploring how the joint effectiveness of multiple security investments influences the defensive budget allocation among multiple potential targets. Three false belief models have been developed in which only the defender, only the attacker, and both the defender and attacker hold false beliefs about the joint effectiveness of resources. Regression analysis shows that there are significant substitution effects between human and capital resources. The results show that the defender will suffer a higher loss if he fails to consider the substitution or complementary effects. Interestingly, if the attacker holds a false belief while the defender does not, the defender will suffer an even higher loss, especially when the resources are substitutes. However, if both the attacker and defender hold false beliefs, there will be lower loss when resources are complementary. The results also show that the defender should allocate the highly effective resource when the resources substitute each other. This article provides some new insights to the homeland security resource allocation.     

Choosing What to Protect

We study a strategic model in which a defender must allocate defensive resources to a collection of locations, and an attacker must choose a location to attack. The defender does not know the attacker's preferences, while the attacker observes the defender's resource allocation. The defender's problem gives rise to negative externalities, in the sense that increasing the resources allocated to one location increases the likelihood of an attack at other locations. In equilibrium, the defender exploits these externalities to manipulate the attacker's behavior, sometimes optimally leaving a location undefended, and sometimes preferring a higher vulnerability at a particular location even if a lower risk could be achieved at zero cost. Key results of our model are as follows: (1) the defender prefers to allocate resources in a centralized (rather than decentralized) manner; (2) as the number of locations to be defended grows, the defender can cost effectively reduce the probability of a successful attack only if the number of valuable targets is bounded; (3) the optimal allocation of resources can be nonmonotonic in the relative value of the attacker's outside option; and (4) the defender prefers his or her defensive allocation to be public rather than secret.     

Optimal Resource Allocation for Defense of Targets Based on Differing Measures of Attractiveness

This article describes the results of applying a rigorous computational model to the problem of the optimal defensive resource allocation among potential terrorist targets. In particular, our study explores how the optimal budget allocation depends on the cost effectiveness of security investments, the defender's valuations of the various targets, and the extent of the defender's uncertainty about the attacker's target valuations. We use expected property damage, expected fatalities, and two metrics of critical infrastructure (airports and bridges) as our measures of target attractiveness. Our results show that the cost effectiveness of security investment has a large impact on the optimal budget allocation. Also, different measures of target attractiveness yield different optimal budget allocations, emphasizing the importance of developing more realistic terrorist objective functions for use in budget allocation decisions for homeland security.     

Modeling Arbitrary Layers of Continuous‐Level Defenses in Facing with Strategic Attackers

We propose a novel class of game‐theoretic models for the optimal assignment of defensive resources in a game between a defender and an attacker. Compared to the other game‐theoretic models in the literature of defense allocation problems, the novelty of our model is that we allow the defender to assign her continuous‐level defensive resources to any subset (or arbitrary layers) of targets due to functional similarity or geographical proximity. We develop methods to solve for equilibrium, and illustrate our model using numerical examples. Compared to traditional models that only allow for individual target hardening, our results show that our model could significantly increase the defender's payoff, especially when the unit cost of defense is high.     

Defender–Attacker Games with Asymmetric Player Utilities

The presence of strategic attackers has become an important factor in the security and protection of systems, especially since the 9/11/2001 attacks, and considerable efforts have been dedicated to its study. When defending against the strategic attacker, many existing studies assume that the attacker would seek to minimize the defender's utility, which implies that the defender and attacker have symmetric utilities. However, the attacker's objective is determined by its own valuation of the system and target of the attack, which is not necessarily consistent with the defender's utility. If the attacker unexpectedly targets a different utility, then the defense strategy might no longer be optimal. In particular, the defense strategy could be inferior if the attacker's utility is not known to the defender. This study considers a situation where the defender's utility is the system survivability and the attacker's utility is the expected number of destroyed elements in the system. We investigate possible attack strategies under these two different utilities and compare (a) the conservative defense strategy when the attack utility is unknown to the defender with (b) the optimal defense strategy when the attack utility is known to the defender. We show that the conservative protection strategy is still optimal under asymmetric utilities when the contest intensity is smaller than one.     

Intelligent Adversary Risk Analysis: A Bioterrorism Risk Management Model

The tragic events of 9/11 and the concerns about the potential for a terrorist or hostile state attack with weapons of mass destruction have led to an increased emphasis on risk analysis for homeland security. Uncertain hazards (natural and engineering) have been successfully analyzed using probabilistic risk analysis (PRA). Unlike uncertain hazards, terrorists and hostile states are intelligent adversaries who can observe our vulnerabilities and dynamically adapt their plans and actions to achieve their objectives. This article compares uncertain hazard risk analysis with intelligent adversary risk analysis, describes the intelligent adversary risk analysis challenges, and presents a probabilistic defender–attacker–defender model to evaluate the baseline risk and the potential risk reduction provided by defender investments. The model includes defender decisions prior to an attack; attacker decisions during the attack; defender actions after an attack; and the uncertainties of attack implementation, detection, and consequences. The risk management model is demonstrated with an illustrative bioterrorism problem with notional data.     

Deterrence and Risk Preferences in Sequential Attacker–Defender Games with Continuous Efforts

Most attacker–defender games consider players as risk neutral, whereas in reality attackers and defenders may be risk seeking or risk averse. This article studies the impact of players' risk preferences on their equilibrium behavior and its effect on the notion of deterrence. In particular, we study the effects of risk preferences in a single‐period, sequential game where a defender has a continuous range of investment levels that could be strategically chosen to potentially deter an attack. This article presents analytic results related to the effect of attacker and defender risk preferences on the optimal defense effort level and their impact on the deterrence level. Numerical illustrations and some discussion of the effect of risk preferences on deterrence and the utility of using such a model are provided, as well as sensitivity analysis of continuous attack investment levels and uncertainty in the defender's beliefs about the attacker's risk preference. A key contribution of this article is the identification of specific scenarios in which the defender using a model that takes into account risk preferences would be better off than a defender using a traditional risk‐neutral model. This study provides insights that could be used by policy analysts and decisionmakers involved in investment decisions in security and safety.     

Game Theory and Risk Analysis

Risk analysts often analyze adversarial risks from terrorists or other intelligent attackers without mentioning game theory. Why? One reason is that many adversarial situations—those that can be represented as attacker‐defender games, in which the defender first chooses an allocation of defensive resources to protect potential targets, and the attacker, knowing what the defender has done, then decides which targets to attack—can be modeled and analyzed successfully without using most of the concepts and terminology of game theory. However, risk analysis and game theory are also deeply complementary. Game‐theoretic analyses of conflicts require modeling the probable consequences of each choice of strategies by the players and assessing the expected utilities of these probable consequences. Decision and risk analysis methods are well suited to accomplish these tasks. Conversely, game‐theoretic formulations of attack‐defense conflicts (and other adversarial risks) can greatly improve upon some current risk analyses that attempt to model attacker decisions as random variables or uncertain attributes of targets (“threats”) and that seek to elicit their values from the defender's own experts. Game theory models that clarify the nature of the interacting decisions made by attackers and defenders and that distinguish clearly between strategic choices (decision nodes in a game tree) and random variables (chance nodes, not controlled by either attacker or defender) can produce more sensible and effective risk management recommendations for allocating defensive resources than current risk scoring models. Thus, risk analysis and game theory are (or should be) mutually reinforcing.     

Cost of Equity in Homeland Security Resource Allocation in the Face of a Strategic Attacker

Hundreds of billions of dollars have been spent in homeland security since September 11, 2001. Many mathematical models have been developed to study strategic interactions between governments (defenders) and terrorists (attackers). However, few studies have considered the tradeoff between equity and efficiency in homeland security resource allocation. In this article, we fill this gap by developing a novel model in which a government allocates defensive resources among multiple potential targets, while reserving a portion of defensive resources (represented by the equity coefficient) for equal distribution (according to geographical areas, population, density, etc.). Such a way to model equity is one of many alternatives, but was directly inspired by homeland security resource allocation practice. The government is faced with a strategic terrorist (adaptive adversary) whose attack probabilities are endogenously determined in the model. We study the effect of the equity coefficient on the optimal defensive resource allocations and the corresponding expected loss. We find that the cost of equity (in terms of increased expected loss) increases convexly in the equity coefficient. Furthermore, such cost is lower when: (a) government uses per‐valuation equity; (b) the cost‐effectiveness coefficient of defense increases; and (c) the total defense budget increases. Our model, results, and insights could be used to assist policy making.     

Counterterrorism resource allocation during a pandemic: The effects of dynamic target valuations when facing a strategic terrorist

The outbreak of pandemics such as COVID-19 can result in cascading effects for global systemic risk. To combat an ongoing pandemic, governmental resources are largely allocated toward supporting the health of the public and economy. This shift in attention can lead to security vulnerabilities which are exploited by terrorists. In view of this, counterterrorism during a pandemic is of critical interest to the safety and well-being of the global society. Most notably, the population flows among potential targets are likely to change in conjunction with the trend of the health crisis, which leads to fluctuations in target valuations. In this situation, a new challenge for the defender is to optimally allocate his/her resources among targets that have changing valuations, where his/her intention is to minimize the expected losses from potential terrorist attacks. In order to deal with this challenge, in this paper, we first develop a defender–attacker game in sequential form, where the target valuations can change as a result of the pandemic. Then we analyze the effects of a pandemic on counterterrorism resource allocation from the perspective of dynamic target valuations. Finally, we provide some examples to display the theoretical results, and present a case study to illustrate the usability of our proposed model during a pandemic.     

Resource Distribution in Multiple Attacks Against a Single Target

A target is protected by the defender and attacked by an attacker launching sequential attacks. For each attack, a contest intensity measures whether the agents’ efforts have low or high impact on the target vulnerability (low vs. high contest intensity). Both the defender and the attacker have limited resources. It is assumed that the attacker can observe the outcome of each attack and stop the sequence of attacks when the target is destroyed. Two attacker objectives are considered, that is, to maximize the target vulnerability or to minimize the expected attacker resource expenditure. The article addresses the following three questions: whether the attacker should allocate its entire resource into one large attack or distribute it among several attacks; whether geometrically increasing or decreasing resource distribution into a fixed number of sequential attacks is more beneficial than equal resource distribution; and how the optimal attack strategy depends on the contest intensity.     

Reasons for Secrecy and Deception in Homeland‐Security Resource Allocation

In this article, we explore reasons that a defender might prefer secrecy or deception about her defensive resource allocations, rather than disclosure, in a homeland‐security context. Our observations not only summarize and synthesize the results of existing game‐theoretic work, but also provide intuitions about promising future research directions.     

Strategy Alternatives for Homeland Air and Cruise Missile Defense

Air and cruise missile defense of the U.S. homeland is characterized by a requirement to protect a large number of critical assets nonuniformly dispersed over a vast area with relatively few defensive systems. In this article, we explore strategy alternatives to make the best use of existing defense resources and suggest this approach as a means of reducing risk while mitigating the cost of developing and acquiring new systems. We frame the issue as an attacker‐defender problem with simultaneous moves. First, we outline and examine the relatively simple problem of defending comparatively few locations with two surveillance systems. Second, we present our analysis and findings for a more realistic scenario that includes a representative list of U.S. critical assets. Third, we investigate sensitivity to defensive strategic choices in the more realistic scenario. As part of this investigation, we describe two complementary computational methods that, under certain circumstances, allow one to reduce large computational problems to a more manageable size. Finally, we demonstrate that strategic choices can be an important supplement to material solutions and can, in some cases, be a more cost‐effective alternative.     

Efficiency of Even Separation of Parallel Elements with Variable Contest Intensity

The article considers strategic defense and attack of a system that can be separated into parallel elements. The defender distributes its resource between separation and protecting the elements from outside attacks. The vulnerability of each element is determined by an attacker‐defender contest success function, which depends on a contest intensity that may increase or decrease through the separation process. The article determines criteria of separation efficiency for systems without performance redundancy and 1‐out‐of‐N and Q‐out‐of‐N systems with performance redundancy. For the systems with performance redundancy the cases of expected damage proportional to the probability that the demand is not met, and expected damage proportional to the unsupplied demand, are considered.     

Managing Risk at the Tucson Sector of the U.S. Border Patrol

This article describes a risk analysis used to inform resource allocation at the Tucson Sector of the U.S. Border Patrol, the busiest sector for alien and drug trafficking along the Southwest land border with Mexico. The model and methodology that underlie this analysis are generally applicable to many resource allocation decisions regarding the management of frequently occurring hazards, decisions regularly made by officials at all levels of the homeland security enterprise. The analysis was executed by agents without previous risk expertise working under a short time frame, and the findings from the analysis were used to inform several resource allocation decisions.     

Behavioral Modeling of Adversaries with Multiple Objectives in Counterterrorism

Attacker/defender models have primarily assumed that each decisionmaker optimizes the cost of the damage inflicted and its economic repercussions from their own perspective. Two streams of recent research have sought to extend such models. One stream suggests that it is more realistic to consider attackers with multiple objectives, but this research has not included the adaption of the terrorist with multiple objectives to defender actions. The other stream builds off experimental studies that show that decisionmakers deviate from optimal rational behavior. In this article, we extend attacker/defender models to incorporate multiple objectives that a terrorist might consider in planning an attack. This includes the tradeoffs that a terrorist might consider and their adaption to defender actions. However, we must also consider experimental evidence of deviations from the rationality assumed in the commonly used expected utility model in determining such adaption. Thus, we model the attacker's behavior using multiattribute prospect theory to account for the attacker's multiple objectives and deviations from rationality. We evaluate our approach by considering an attacker with multiple objectives who wishes to smuggle radioactive material into the United States and a defender who has the option to implement a screening process to hinder the attacker. We discuss the problems with implementing such an approach, but argue that research in this area must continue to avoid misrepresenting terrorist behavior in determining optimal defensive actions.     

Resource Distribution in Multiple Attacks with Imperfect Detection of the Attack Outcome

This article extends the previous research of consecutive attacks strategy by assuming that an attacker observes the outcome of each attack imperfectly. With given probabilities it may wrongly identify a destroyed target as undestroyed, and wrongly identify an undestroyed target as destroyed. The outcome of each attack is determined by a contest success function that depends on the amount of resources allocated by the defender and the attacker to each attack. The article suggests a probabilistic model of the multiple attacks and analyzes how the target destruction probability and the attacker's relative resource expenditure are impacted by the two probabilities of incorrect observation, the attacker's and defender's resource ratio, the contest intensity, the number of attacks, and the resource distribution across attacks. We analyze how the attacker chooses the number of attacks, the attack stopping rule, and the optimal resource distribution across attacks to maximize its utility.     

不确定需求下考虑供应商参与机制的应急资源配置鲁棒优化研究

突发事件应急资源优化配置是突发事件发生后救援工作有效开展的前提和基本保障。本文在调研国内外相关研究的基础上，以应对地震灾难为背景研究考虑供应商参与机制的应急资源保障策略，在灾前选择供应商建立政府与供应商的合作机制完成政府储备仓库的选址与资源配置，突发事件发生后依托政府储备仓库和供应商生产能力共同满足应急救援对应急资源的分时段需求，以期协调供应商与政府储备、灾前实物采购与灾后生产能力采购的比例，在保障救援效率的同时降低应急资源保障体系的成本。同时由于地震灾难发生具有需求不确定性的特点，本文引入L1范数描述需求的不确定性，建立了备灾与灾害救援两阶段决策的鲁棒优化模型，并给出了鲁棒模型对应问题的转化方法。最后通过算例仿真验证了模型和对应问题转换方法的有效性，为地震的应对提供理论指导和决策支持。     

Adversarial risk analysis for counterterrorism modeling

Recent large-scale terrorist attacks have raised interest in models for resource allocation against terrorist threats. The unifying theme in this area is the need to develop methods for the analysis of allocation decisions when risks stem from the intentional actions of intelligent adversaries. Most approaches to these problems have a game-theoretic flavor although there are also several interesting decision-analytic-based proposals. One of them is the recently introduced framework for adversarial risk analysis, which deals with decision-making problems that involve intelligent opponents and uncertain outcomes. We explore how adversarial risk analysis addresses some standard counterterrorism models: simultaneous defend-attack models, sequential defend-attack-defend models, and sequential defend-attack models with private information. For each model, we first assess critically what would be a typical game-theoretic approach and then provide the corresponding solution proposed by the adversarial risk analysis framework, emphasizing how to coherently assess a predictive probability model of the adversary's actions, in a context in which we aim at supporting decisions of a defender versus an attacker. This illustrates the application of adversarial risk analysis to basic counterterrorism models that may be used as basic building blocks for more complex risk analysis of counterterrorism problems.