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.     

Robust allocation of a defensive budget considering an attacker's private information

Attackers' private information is one of the main issues in defensive resource allocation games in homeland security. The outcome of a defense resource allocation decision critically depends on the accuracy of estimations about the attacker's attributes. However, terrorists' goals may be unknown to the defender, necessitating robust decisions by the defender. This article develops a robust-optimization game-theoretical model for identifying optimal defense resource allocation strategies for a rational defender facing a strategic attacker while the attacker's valuation of targets, being the most critical attribute of the attacker, is unknown but belongs to bounded distribution-free intervals. To our best knowledge, no previous research has applied robust optimization in homeland security resource allocation when uncertainty is defined in bounded distribution-free intervals. The key features of our model include (1) modeling uncertainty in attackers' attributes, where uncertainty is characterized by bounded intervals; (2) finding the robust-optimization equilibrium for the defender using concepts dealing with budget of uncertainty and price of robustness; and (3) applying the proposed model to real data.     

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.     

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.     

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.     

Modeling the Cost Effectiveness of Fire Protection Resource Allocation in the United States: Models and a 1980–2014 Case Study

The estimated cost of fire in the United States is about $329 billion a year, yet there are gaps in the literature to measure the effectiveness of investment and to allocate resources optimally in fire protection. This article fills these gaps by creating data‐driven empirical and theoretical models to study the effectiveness of nationwide fire protection investment in reducing economic and human losses. The regression between investment and loss vulnerability shows high R² values (≈0.93). This article also contributes to the literature by modeling strategic (national‐level or state‐level) resource allocation (RA) for fire protection with equity‐efficiency trade‐off considerations, while existing literature focuses on operational‐level RA. This model and its numerical analyses provide techniques and insights to aid the strategic decision‐making process. The results from this model are used to calculate fire risk scores for various geographic regions, which can be used as an indicator of fire risk. A case study of federal fire grant allocation is used to validate and show the utility of the optimal RA model. The results also identify potential underinvestment and overinvestment in fire protection in certain regions. This article presents scenarios in which the model presented outperforms the existing RA scheme, when compared in terms of the correlation of resources allocated with actual number of fire incidents. This article provides some novel insights to policymakers and analysts in fire protection and safety that would help in mitigating economic costs and saving lives.     

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.     

Implementation of Equity in Resource Allocation for Regional Earthquake Risk Mitigation Using Two‐Stage Stochastic Programming

This article presents a new methodology to implement the concept of equity in regional earthquake risk mitigation programs using an optimization framework. It presents a framework that could be used by decisionmakers (government and authorities) to structure budget allocation strategy toward different seismic risk mitigation measures, i.e., structural retrofitting for different building structural types in different locations and planning horizons. A two‐stage stochastic model is developed here to seek optimal mitigation measures based on minimizing mitigation expenditures, reconstruction expenditures, and especially large losses in highly seismically active countries. To consider fairness in the distribution of financial resources among different groups of people, the equity concept is incorporated using constraints in model formulation. These constraints limit inequity to the user‐defined level to achieve the equity‐efficiency tradeoff in the decision‐making process. To present practical application of the proposed model, it is applied to a pilot area in Tehran, the capital city of Iran. Building stocks, structural vulnerability functions, and regional seismic hazard characteristics are incorporated to compile a probabilistic seismic risk model for the pilot area. Results illustrate the variation of mitigation expenditures by location and structural type for buildings. These expenditures are sensitive to the amount of available budget and equity consideration for the constant risk aversion. Most significantly, equity is more easily achieved if the budget is unlimited. Conversely, increasing equity where the budget is limited decreases the efficiency. The risk‐return tradeoff, equity‐reconstruction expenditures tradeoff, and variation of per‐capita expected earthquake loss in different income classes are also presented.     

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.     

Equity sharing in international joint ventures: an empirical analysis of strategic and environmental determinants

Equity sharing in international joint ventures (IJVs) is a critical issue with implications for risk sharing, resource allocation, knowledge commitment, and organizational control. As a departure from previous research, this study aims to explore strategic antecedents and environmental dimensions underlying sharing arrangements. It argues that the structure of equity sharing controls the ex post degree of economic exposure or risk propensity, and boosts risk-adjusted returns earned from strategic resources or global integration. Analysis of data collected from IJVs in China suggests that strategic orientation, resource dependence, knowledge protection, and global integration as sought by MNEs are important antecedents of equity sharing. Equity sharing is an inverse function of host-country environmental complexity and hostility as perceived by MNE managers.     

A Value Model for Evaluating Homeland Security Decisions

One of the most challenging tasks of homeland security policymakers is to allocate their limited resources to reduce terrorism risks cost effectively. To accomplish this task, it is useful to develop a comprehensive set of homeland security objectives, metrics to measure each objective, a utility function, and value tradeoffs relevant for making homeland security investments. Together, these elements form a homeland security value model. This article develops a homeland security value model based on literature reviews, a survey, and experience with building value models. The purposes of the article are to motivate the use of a value model for homeland security decision making and to illustrate its use to assess terrorism risks, assess the benefits of countermeasures, and develop a severity index for terrorism attacks.     

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.     

不对称信息下反恐阻止网络设计

当前，我国面临的恐怖主义威胁日益严峻。为防止境外恐怖分子潜入，政府可设计反恐阻止网络，通过在交通网络中有效地分配例如安检仪器、传感设备等阻断资源，来提前识别和拦截正在潜入的恐怖分子。特别地，考虑信息不对称情形，把阻断资源分为"公开"和"隐蔽"两种类型，并假设恐怖分子观察不到"隐蔽"阻断。主要研究政府应如何同时优化两类阻断方案，才能发挥信息优势，设置"陷阱"并降低袭击分析。首先，将该问题构造为双层规划模型，上层规划是关于政府的阻止网络设计问题，下层规划则是关于恐怖分子的袭击节点选择和入侵路径优化问题。随后，设计一类用改进遗传算法处理上层规划，并结合下层规划直接求解的混合算法。其中，改进体现于杂交算子和变异算子的设计。最后，结合喀什地区进行算例分析，并分析"隐蔽"阻断的作用机理。     

Risk Analysis for Critical Asset Protection

This article proposes a quantitative risk assessment and management framework that supports strategic asset-level resource allocation decision making for critical infrastructure and key resource protection. The proposed framework consists of five phases: scenario identification, consequence and criticality assessment, security vulnerability assessment, threat likelihood assessment, and benefit-cost analysis. Key innovations in this methodology include its initial focus on fundamental asset characteristics to generate an exhaustive set of plausible threat scenarios based on a target susceptibility matrix (which we refer to as asset-driven analysis) and an approach to threat likelihood assessment that captures adversary tendencies to shift their preferences in response to security investments based on the expected utilities of alternative attack profiles assessed from the adversary perspective. A notional example is provided to demonstrate an application of the proposed framework. Extensions of this model to support strategic portfolio-level analysis and tactical risk analysis are suggested.     

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。     

Measuring and Achieving Equity in Multiperiod Emergency Material Allocation

Emergency material allocation is an important part of postdisaster emergency logistics that is significant for improving rescue effectiveness and reducing disaster losses. However, the traditional single‐period allocation model often causes local surpluses or shortages and high cost, and prevents the system from achieving an equitable or optimal multiperiod allocation. To achieve equitable allocation of emergency materials in the case of serious shortages relative to the demand by victims, this article introduces a multiperiod model for allocation of emergency materials to multiple affected locations (using an exponential utility function to reflect the disutility loss due to material shortfalls), and illustrates the relationship between equity of allocations and the cost of emergency response. Finally, numerical examples are presented to demonstrate both the feasibility and the usefulness of the proposed model for achieving multiperiod equitable allocation of emergency material among multiple disaster locations. The results indicate that the introduction of a nonlinear utility function to reflect the disutility of large shortfalls can make the material allocation fairer, and minimize large losses due to shortfalls. We found that achieving equity has a significant but not unreasonable impact on emergency costs. We also illustrate that using differing utility functions for different types of materials adds an important dimension of flexibility.     

基于信号博弈模型的公立医院利益补偿机制

针对医疗服务市场的特点,讨论了医药分业后政府补贴公立医院的利益补偿机制,以优化资源配置为目标建立医院与政府的信号博弈模型并进行均衡分析,发现市场的均衡效率随作假的伪装成本和期望风险成本减少而降低;实现市场完全成功的分离均衡的关键条件为保持较高伪装成本和期望风险成本;相应的政策应集中于建立医疗服务行业规范和提升医疗监察效率.     

Defender–Attacker Decision Tree Analysis to Combat Terrorism

We propose a methodology, called defender–attacker decision tree analysis, to evaluate defensive actions against terrorist attacks in a dynamic and hostile environment. Like most game‐theoretic formulations of this problem, we assume that the defenders act rationally by maximizing their expected utility or minimizing their expected costs. However, we do not assume that attackers maximize their expected utilities. Instead, we encode the defender's limited knowledge about the attacker's motivations and capabilities as a conditional probability distribution over the attacker's decisions. We apply this methodology to the problem of defending against possible terrorist attacks on commercial airplanes, using one of three weapons: infrared‐guided MANPADS (man‐portable air defense systems), laser‐guided MANPADS, or visually targeted RPGs (rocket propelled grenades). We also evaluate three countermeasures against these weapons: DIRCMs (directional infrared countermeasures), perimeter control around the airport, and hardening airplanes. The model includes deterrence effects, the effectiveness of the countermeasures, and the substitution of weapons and targets once a specific countermeasure is selected. It also includes a second stage of defensive decisions after an attack occurs. Key findings are: (1) due to the high cost of the countermeasures, not implementing countermeasures is the preferred defensive alternative for a large range of parameters; (2) if the probability of an attack and the associated consequences are large, a combination of DIRCMs and ground perimeter control are preferred over any single countermeasure.