Trade-offs between target hardening and overarching protection N. Haphuriwat, V.M. Bier

Trade-offs between target hardening and overarching protectionN. Haphuriwat, V.M. Bier Advisor: Yeong-Sung LinPresented by I-Ju Shih

2011/5/3011AgendaIntroductionThe modelSensitivity analysis Sensitivity results and discussion Protecting critical assets in WisconsinConclusionsFuture research directions

2011/5/3022AgendaIntroductionThe modelSensitivity analysis Sensitivity results and discussion Protecting critical assets in WisconsinConclusionsFuture research directions

2011/5/3033Introduction2011/5/304In principle, defenders concerned about protecting multiple targets could choose to protect them individually (through target hardening), or collectively (through overarching protections).Game theory has been widely used in the study of resource allocation.Major (2002) allows the attacker to choose the level of attack effort to spend on each target, while the defender chooses how much to spend on protecting each target.Woo (2002) provides a methodology for estimating the likelihood that each target will be attacked.4Introduction2011/5/305Azaiez and Bier (2006) apply game theory to model optimal investments in both series and parallel reliability systems.Heal and Kunreuther (2007) study a model in which there are multiple targets, with each defender simultaneously allocating resources to protect her own target. They also explore tipping effects, and cascading effects.Zhuang et al. (2007) consider the case when players have different discount rates.5Introduction2011/5/306Zhuang and Bier (2007) consider the case when the defender must allocate her resources to protect against both natural disasters and terrorism.Bier et al. (2007) model a scenario where the defenders target valuations are common knowledge, but the defender does not know the attackers target valuations.Wang and Bier (2009) consider a dynamic game in which the defender is uncertain about the attackers target valuations.Zhuang et al. (2010) use a signaling game to model resource allocations over multiple time periods, allowing the attacker to update his knowledge based on the defenders signals.defender signalstruthful disclosure, secrecy, and deception. 6Introduction2011/5/307One form of overarching protection is border security, due to concerns about illegal immigration and smuggling.Bier and Haphuriwat, (2009) apply a game-theoretic model to analytically determine conditions under which partial inspection is sufficient to deter smuggling attempts.Haphuriwat et al. (2011) revise the model in Bier and Haphuriwat(2009), to address the case of a single attacker attempting to smuggle in multiple nuclear bombs.

7Introduction(game theory)2011/5/3081. : (Player)(Strategies) (payoff)(information)2. :(simultaneous game)(sequential game)(constant-sum game)-payoff (zero-sum game)-payoff0(nonzero-sum game)-payoff0

8Introduction(game theory)2011/5/3092. :(one-shot game)(repeat game)--a. (signals)-b. (screening)-9Introduction(game theory)2011/5/30102. :3. : 4. :

10Introduction(game theory)2011/5/30114

112011/5/3012ModelAzaiez and Bier (2006)seriesparallelZhuang et al. (2007)Zhuang and Bier (2007)Bier et al. (2007)Bier et al. (2008)two-parameter Rayleigh distributionWang and Bier (2009)targetZhuang et al. (2010)3Hausken and Bier (2011)12Introduction2011/5/3013For simplicity, this paper applies game theory to the problem of discrete attacker target choice, and neglects the defenders uncertainty about the attackers objectives.This paper considers only a single defender and a single attacker, assumes that the defenders defensive resource allocation is fully disclosed, and consider a single-period game rather than a dynamic game.In the model, the attacker is assumed to attack the target that would result in the highest expected damage, after observing any defensive investments.The defender chooses how much to spend both on target hardening and on overarching protection in order to minimize expected damage against both an intentional attack and a natural disaster, subject to a budget constraint.13Introduction2011/5/3014This paper hypothesizes that target hardening will tend to be more desirable when the number of targets to be protected is relatively small, when the cost effectiveness of defensive investment is high, and when there are relatively few high-value targets. By contrast, border security and other forms of overarching protection are hypothesized to be more desirable when there are large numbers of comparably-valued targets to be protected.14AgendaIntroductionThe modelSensitivity analysis Sensitivity results and discussion Protecting critical assets in WisconsinConclusionsFuture research directions

2011/5/301515The model2011/5/3016This paper considers investments in all-hazards protection as well as investments in protection from intentional threats, patterning our model roughly on that in Zhuang and Bier (2007).This paper allows the targets to be heterogeneous in value.The attacker is assumed to attack the most attractive target, taking into account any defensive investments that have been made.This paper assumes that all intentional threats originate externally to the system, and must penetrate any overarching defenses in order to be effective.16The model2011/5/3017

17The model2011/5/3018This paper assumes that the attacker and the defender have the same valuations Vj for all targets, that the attacker can observe the defensive allocations cj, and that the attacker will choose to attack the target j* with the highest expected damage; i.e., .This paper also assumes that the natural disaster will affect only a single target (and omit the possibility of multiple attacks, or natural disasters affecting multiple targets).This model is designed to apply to situations in which both natural disaster and intentional attack are relatively unlikely.

18The model2011/5/3019the defenders optimization problem

19The model2011/5/3020This paper represents the success probability of an attack and the failure probability of all-hazards protection by power-law functions; i.e.,

where are positive-valued parameters that determine the cost effectiveness of defensive investment.

20The model2011/5/3021Unfortunately, the Hessian of the Lagrangian of this optimization problem is not positive semidefinite, implying that the problem is not convex.Hence, this paper solves this problem by numerical approximation using the branch-and-reduce optimization navigator (Sahinidis and Tawarmalani, 2002) in the General Algebraic Modeling System (GAMS).21AgendaIntroductionThe modelSensitivity analysis Sensitivity results and discussion Protecting critical assets in WisconsinConclusionsFuture research directions

2011/5/302222Sensitivity analysis 2011/5/3023This paper used simulation to generate randomly sampled data sets from distributions of target values Vj with specified parameters.The distribution characteristic that had the strongest relationship to the desirability of target hardening was the ratio of the 95th percentile to the 50th percentile, sometimes called the range factor.Bier et al. (2008), considering the top ten urban areas in the US, yielded range factors of 1.92 for air departures and 2.03 for average daily bridge traffic, respectively.Willis et al. (2005), considering the 46 urban areas in the US that received 2004 funding from the Urban Areas Security Initiative, yielded range factors of 24.14 for property damage, 55 for fatalities, and 59 for injuries, respectively.23Sensitivity analysis 2011/5/3024

24Sensitivity analysis 2011/5/3025Therefore, this paper considers range factors of 1.2, 3, 30, and 60 in their sensitivity analysis.This paper chose to generate data sets from distributions that generate exclusively positive target valuations; in particular, the Pearson, beta, gamma, and lognormal distributions.In the sensitivity analysis, this paper considers only intentional attacks, in order to focus on the trade-offs between target hardening and overarching protection. = 0, dN+1 = 0, and QN+1 = 1, = 125Sensitivity analysis 2011/5/3026With regard to the success probability of attacks, this paper holds the parameter j in the success-probability function constant for all targets j, and also for overarching protection.j in this success-probability function is the same for all targets. i.e., j = for j = 1,. . . ,NThey selected values of the parameters j, , and N+1 based on the range of cost effectiveness used in Bier et al. (2008); in particular, they let j = 7, and let the parameters and N+1 take on values of 50, 200, and 600.

26Sensitivity analysis 2011/5/3027The primary output of interest in the sensitivity analysis is the optimal percentage investment in target hardening.To keep the number of sensitivity runs manageable, they began by simulating 200 sets of target valuations for each sensitivity run from a given distribution.27AgendaIntroductionThe modelSensitivity analysis Sensitivity results and discussion Protecting critical assets in WisconsinConclusionsFuture research directions

2011/5/302828Sensitivity results and discussion 2011/5/3029

29Sensitivity results and discussion 2011/5/3030

30Sensitivity results and discussion 2011/5/3031

31AgendaIntroductionThe modelSensitivity analysis Sensitivity results and discussion Protecting critical assets in WisconsinConclusionsFuture research directions

2011/5/303232Protecting critical assets in Wisconsin2011/5/3033The Office of Justice Assistance is responsible for distributing federal funds to protect critical infrastructure within the state of Wisconsin against both natural disasters and intentional threats .In FY 2007, the Office of Justice Assistance requested funding for thirteen different types of defensive investments from the Department of Homeland Security.33Protecting critical assets in Wisconsin2011/5/3034They assume that investment in Catastrophic Planning and Preparedness is a form of overarching protection that can protect against all hazards.They treat investment in Infrastructure Protection as a form of target hardening that can protect only against terrorism.They assume that investment in the Wisconsin Statewide Intelligence Center provides overarching protection against terrorism, but not against natural disasters.

34Protecting critical assets in Wisconsin2011/5/3035

2005200735Protecting critical assets in Wisconsin2011/5/3036They used sensitivity analysis to explore the effects of different possible parameter values.For the probabilities of intentional attack and natural disaster, this paper considers the entire range between zero and one, with 0.1 increments.In order to get similar behavior as in Bier et al. (2008), they allowed the values of j, N+1, and N+1 to range over 0.18, 0.6, and 2.4, to represent high, moderate, and low levels of cost effectiveness, respectively.36Protecting critical assets in Wisconsin2011/5/3037Thay use the same cost effectiveness level for all targets (i.e., j = for j = 1,. . . ,N), and set j and N+1 equal to 7. For convenience, they fix the total budget at 1.0.They compare the defenders expected loss from the actual historical budget-allocation decision with that from the optimal decision obtained by solving the model.They consider the case when the allocations to different investment types are as specified (i.e., cN+1 = 0.25 and dN+1 = 0.35), but the allocation of the remaining budget to individual targets is chosen optimally while keeping cj =0.4.37Protecting critical assets in Wisconsin2011/5/3038Letting T be the total resources allocated to target hardening, they also consider cases when the defender optimally allocates resources among the three types of investments (T, cN+1, and dN+1), but either sets cj = jT for j = 1,. . . ,N, where j represents the actual fraction of the total resources for target hardening received by target j in 2007, or sets cj = T/N for j = 1,. . . ,N (i.e., equal allocation of resources for target hardening).38Protecting critical assets in Wisconsin2011/5/3039optimizing Infrastructure Protection only

39Protecting critical assets in Wisconsin2011/5/3040Letting all targets receive either their actual historical percentage allocations, or equal allocations.The expected losses of both suboptimal strategies were quite similar to those obtained from the fully optimal strategy.As in the fully optimal solutions, however, Infrastructure Protection again receives less than 10% of the available funding, with Catastrophic Planning and Preparedness receiving most of the budget.40Protecting critical assets in Wisconsin2011/5/3041This paper conducted an analysis where target attractiveness is represented by the exponent of those risk scores yielding a range factor of 5.4 (compared to only 1.16 for the untransformed risk scores).41AgendaIntroductionThe modelSensitivity analysis Sensitivity results and discussion Protecting critical assets in WisconsinConclusionsFuture research directions

2011/5/304242Conclusions2011/5/3043This paper has applied game theory to model resource-allocation problems with two levels of protection, in which the inner level represents hardening of individual targets, and the outer level is overarching protection.This paper studied how the tradeoff between target hardening and overarching protection depends on various parameters.The results showed that as the number of targets increases, target hardening becomes less desirable. Moreover, investment in target hardening increases as target hardening itself becomes more cost effective.43Conclusions2011/5/3044The results for how investment in target hardening depends on the distribution of target valuations turned out to be more complicated than we had expected.Since the valuations of the critical assets in Wisconsin were quite similar, the optimal budget allocation devoted most of the budget to overarching protection.44AgendaIntroductionThe modelSensitivity analysis Sensitivity results and discussion Protecting critical assets in WisconsinConclusionsFuture research directions

2011/5/304545Future research directions2011/5/3046It may be worthwhile to extend the model to consider forms of overarching protection that provide less than 100% protection.Moreover, this model could be made more realistic by adding more detail within the broad categories of intentional attacks and natural disasters.However, perhaps the major need for this papers method to be applicable in practice is better techniques for quantifying key parameters of the model, such as target valuations and the cost effectiveness of defensive investments.462011/5/3047 Thanks for your listening.