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

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

2. Agenda • Introduction • The model • Sensitivity analysis • Sensitivity results and discussion • Protecting critical assets in Wisconsin • Conclusions • Future research directions

3. Agenda • Introduction • The model • Sensitivity analysis • Sensitivity results and discussion • Protecting critical assets in Wisconsin • Conclusions • Future research directions

4. Introduction • In 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.

5. Introduction • Azaiez 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.

6. Introduction • Zhuang 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 defender’s target valuations are common knowledge, but the defender does not know the attacker’s target valuations. • Wang and Bier (2009) consider a dynamic game inwhich the defender is uncertain about the attacker’s target valuations. • Zhuanget al. (2010) use a signaling game to model resource allocationsover multiple time periods, allowing the attacker to update hisknowledge based on the defender’s signals. defender signals—truthful disclosure, secrecy, and deception.

7. Introduction • One 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 smugglein multiple nuclear bombs.

8. Introduction(game theory) • 1. 賽局的要素:參與者(Player)、採取的行動(Strategies)、 報酬(payoff)以及資訊(information) • 2. 賽局的分類: • 依各參與者的行動是同時或依序可分為 同時賽局(simultaneous game，又稱靜態賽局) 順序賽局(sequential game，又稱動態賽局) • 依各參與者的利益是相衝突或互利可分為 定和賽局(constant-sum game)-payoff加起來為一個定數 零和賽局 (zero-sum game)-策略組合payoff的和為0 非零和賽局(nonzero-sum game)-策略組合payoff的和不為0

9. Introduction(game theory) • 2. 賽局的分類: • 依賽局是否重複可分為 一次性賽局(one-shot game) 重複賽局(repeat game) • 依參與者是否瞭解賽局要素所包括的所有知識可分為 完全訊息賽局-賽局規則成為共有知識的賽局 不完全訊息賽局-賽局規則並未成為共有知識的賽局 • 不完全訊息的賽局又可分為 a. 傳訊(signals)-參與者未共享資訊時，策略性的將自己所知的資訊傳達給其他參與者 b. 篩選(screening)-刻意採取一些作為，好揭露對方所隱藏的真正意圖

10. Introduction(game theory) • 2. 賽局的分類: • 按照參與者之間是否合作可分為 合作賽局 非合作賽局 • 3. 賽局的解:當賽局中的參與者都覺得不需要改變策略時， 此時就是均衡，且一場賽局中可能有不只一個均衡。 • 4. 賽局的表現方式: 標準型  展開型

11. Introduction(game theory) • 賽局依參與者的互動關係可分為靜態與動態賽局，根據訊息的掌握又可分為完全訊息賽局與不完全訊息賽局，而形成下列4種不同的賽局，其分別對應不同的均衡

13. Introduction • For simplicity, this paper applies game theory to the problem of discreteattacker target choice,and neglects the defender’s uncertainty about the attacker’sobjectives. • This paper considers only a single defender and a single attacker, assumes that the defender’s 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 targetthat would result in the highest expected damage, afterobserving any defensive investments. • The defender chooses howmuch to spend both on target hardening and on overarching protectionin order to minimize expected damage against both an intentionalattack and a natural disaster, subject to a budget constraint.

14. Introduction • This paper hypothesizes that target hardening will tend to be moredesirable when the number of targets to be protected is relativelysmall, 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 arehypothesized to be more desirable when there are large numbersof comparably-valued targets to be protected.

15. Agenda • Introduction • The model • Sensitivity analysis • Sensitivity results and discussion • Protecting critical assets in Wisconsin • Conclusions • Future research directions

16. The model • This paper considers investments in all-hazardsprotection as wellas investments in protection from intentional threats, patterningour 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. • Thispaper assumes that all intentional threats originate externally tothe system, and must penetrate any overarching defenses in orderto be effective.

17. The model

18. The model • This paper assumes that the attacker and the defender have the samevaluations Vj for all targets, that the attacker can observe thedefensive allocations cj, and that the attacker will choose toattack 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.

19. The model • the defender’s optimization problem

20. The model • This paper represents the success probability of an attack and the failureprobability of all-hazards protection by power-law functions; i.e., where are positive-valued parameters that determine the cost effectiveness of defensive investment.

21. The model • Unfortunately, the Hessian of the Lagrangian of this optimizationproblem is not positive semidefinite, implying that theproblem is not convex. • Hence, this paper solves this problem by numericalapproximation using the branch-and-reduce optimization navigator(Sahinidis and Tawarmalani, 2002) in the General AlgebraicModeling System (GAMS).

22. Agenda • Introduction • The model • Sensitivity analysis • Sensitivity results and discussion • Protecting critical assets in Wisconsin • Conclusions • Future research directions

23. Sensitivity analysis • This 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.

24. Sensitivity analysis

25. Sensitivity analysis • Therefore,this paper considers range factors of 1.2, 3, 30, and 60 in their sensitivityanalysis. • This paper chose to generate data sets from distributions thatgenerate exclusively positive target valuations; in particular, thePearson, 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. ω = 0, dN+1 = 0, and QN+1 = 1, ρ = 1

26. Sensitivity analysis • With regard to the success probability of attacks, this paper holds the parameter κj in the success-probability function constant for all targets j, and also for overarching protection. • αj in this success-probability function is the same for all targets.i.e., αj = αfor j = 1,. . . ,N • They selected values of the parametersκj, α, and αN+1 based on the range of cost effectiveness used in Bieret al. (2008); in particular, they let κj = 7, and let the parameters αand αN+1 take on values of 50, 200, and 600.

27. Sensitivity analysis • The primary output of interest in the sensitivity analysis is theoptimal 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.

28. Agenda • Introduction • The model • Sensitivity analysis • Sensitivity results and discussion • Protecting critical assets in Wisconsin • Conclusions • Future research directions

29. Sensitivity results and discussion

30. Sensitivity results and discussion

31. Sensitivity results and discussion

32. Agenda • Introduction • The model • Sensitivity analysis • Sensitivity results and discussion • Protecting critical assets in Wisconsin • Conclusions • Future research directions

33. Protecting critical assets in Wisconsin • The Office of Justice Assistance isresponsible for distributing federal funds to protect critical infrastructurewithin the state of Wisconsin against both natural disastersand intentional threats . • In FY 2007, the Office of Justice Assistance requested funding forthirteen different types of defensive investments from the Departmentof Homeland Security.

34. Protecting critical assets in Wisconsin • They assume that investment in Catastrophic Planning andPreparedness is a form of overarching protection that can protectagainst 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.

35. Protecting critical assets in Wisconsin 2005年 2007年

36. Protecting critical assets in Wisconsin • They used sensitivity analysis to explorethe effects of different possible parameter values. • For the probabilitiesof intentional attack and natural disaster, this paper considers the entirerange between zero and one, with 0.1 increments. • In order to getsimilar behavior as in Bier et al. (2008), they allowed the values ofαj, αN+1, and ηN+1 to range over 0.18, 0.6, and 2.4, to represent high,moderate, and low levels of cost effectiveness, respectively.

37. Protecting critical assets in Wisconsin • Thay use the same cost effectiveness levelfor all targets (i.e., αj = αfor j = 1,. . . ,N), and set κj and γN+1 equalto 7. For convenience, they fix the total budget at 1.0. • They compare the defender’s expected loss from the actualhistorical budget-allocation decision with that from the optimaldecision 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.

38. Protecting critical assets in Wisconsin • Letting T be the total resources allocated to targethardening, they also consider cases when the defender optimallyallocates resources among the three types of investments (T, cN+1,and dN+1), but either sets cj = δjT for j = 1,. . . ,N, where δj representsthe actual fraction of the total resources for target hardening receivedby target j in 2007, or sets cj = T/Nfor j = 1,. . . ,N (i.e., equal allocation of resources for targethardening).

39. Protecting critical assets in Wisconsin • optimizing Infrastructure Protection only

40. Protecting critical assets in Wisconsin • Letting all targets receive eithertheir actual historical percentage allocations, or equal allocations. • The expected losses of both suboptimal strategies were quite similarto those obtained from the fully optimal strategy. • As in thefully optimal solutions, however, Infrastructure Protection againreceives less than 10% of the available funding, with CatastrophicPlanning and Preparedness receiving most of the budget.

41. Protecting critical assets in Wisconsin • This paper conducted an analysis where target attractivenessis represented by the exponent of those risk scores yielding a range factor of 5.4 (compared to only 1.16 for theuntransformed risk scores).

42. Agenda • Introduction • The model • Sensitivity analysis • Sensitivity results and discussion • Protecting critical assets in Wisconsin • Conclusions • Future research directions

43. Conclusions • Thispaperhas applied game theory to model resource-allocationproblems with two levels of protection, in which the inner levelrepresents hardening of individual targets, and the outer level isoverarching protection. • This paper studied how the tradeoff between target hardening and overarching protection depends on various parameters. • The results showedthat as the number of targets increases, target hardeningbecomes less desirable. Moreover,investment in target hardening increases as target hardening itselfbecomes more cost effective.

44. Conclusions • The results for how investment in target hardening depends onthe distribution of target valuations turned out to be more complicatedthan we had expected. • Since the valuations of the criticalassets in Wisconsin were quite similar, the optimal budget allocationdevoted most of the budget to overarching protection.

45. Agenda • Introduction • The model • Sensitivity analysis • Sensitivity results and discussion • Protecting critical assets in Wisconsin • Conclusions • Future research directions

46. Future research directions • It maybe worthwhile to extend the model to consider forms of overarchingprotection that provide less than 100% protection. • Moreover, this model could be made more realistic by addingmore detail within the broad categories of intentional attacksand natural disasters. • However, perhaps the major need for this paper’s method to be applicablein practice is better techniques for quantifying key parametersof the model, such as target valuations and the cost effectiveness ofdefensive investments.

47. Thanks for your listening.