Adversarial Risk Analysis in support of defensive resource allocation for counterterrorism

Zawadzki, Marcelo; Costa, André Negrão; Belderrain, Mischel Carmen Neyra; Montibeller, Gilberto

doi:10.1590/0104-530X1814-16

Marcelo Zawadzki

Subdivisão de Sistemas de Apoio à Decisão, Instituto de Estudos Avançados – IEAv, Trevo Coronel Aviador José Alberto Albano do Amarante, 1, CEP 12228-001, São José dos Campos, SP, Brazil, e-mail: mzawa@ieav.cta.br, negrao@ieav.cta.br

André Negrão Costa

Subdivisão de Sistemas de Apoio à Decisão, Instituto de Estudos Avançados – IEAv, Trevo Coronel Aviador José Alberto Albano do Amarante, 1, CEP 12228-001, São José dos Campos, SP, Brazil, e-mail: mzawa@ieav.cta.br, negrao@ieav.cta.br

Mischel Carmen Neyra Belderrain

Departamento de Produção, Instituto Tecnológico de Aeronáutica – ITA, Praça Marechal Eduardo Gomes, 50, CEP 12228-900, São José dos Campos, SP, Brazil, e-mail: carmen@ita.br

Gilberto Montibeller

Management Science and Operations Management Group, Loughborough University, LE11 3TU, Leicestershire, England, United Kingdom, e-mail: G.Montibeller@lboro.ac.uk

Topic	Variations	Comments/Conclusion	References
Distribution of resources for protection against terrorist actions	Strategic threats	The attacker's preferences change according to the prior allocation of defense resources.	(Powell, 2007aPowell, R. (2007a). Defending against terrorist attacks with limited resources. The American Political Science Review, 101(3), 527-541. http://dx.doi.org/10.1017/S0003055407070244. http://dx.doi.org/10.1017/S0003055407070... )
	Non-strategic threats	The opponent is not attacked where it is the weakest and where the expected gains will be greater. Preferences of attackers are unchanged.	(Banks & Anderson, 2006Banks, D., & Anderson, S. (2006). Combining game theory and risk analysis in counterterrorism: a smallpox example. In A. G. Wilson, G. D. Wilson & D. H. Olwell (Eds.), Statistical methods in counterterrorism (pp. 9-22). New York: Springer.; Farrow, 2007Farrow, S. (2007). The Economics of homeland security expenditures: foundational expected cost-effectiveness approaches. Contemporary Economic Policy, 25(1), 14-26. http://dx.doi.org/10.1111/j.1465-7287.2006.00029.x. http://dx.doi.org/10.1111/j.1465-7287.20... ; Powell, 2007aPowell, R. (2007a). Defending against terrorist attacks with limited resources. The American Political Science Review, 101(3), 527-541. http://dx.doi.org/10.1017/S0003055407070244. http://dx.doi.org/10.1017/S0003055407070... ; Zhuang & Bier, 2007Zhuang, J., & Bier, V. M. (2007). Balancing terrorism and natural disasters: defensive strategy with endogenous attacker effort. Operations Research, 55(5), 976-991. http://dx.doi.org/10.1287/opre.1070.0434. http://dx.doi.org/10.1287/opre.1070.0434... )
	Synergic defensive measures	A single defense measure may result in reducing the vulnerability of more than one target.	(Farrow, 2007Farrow, S. (2007). The Economics of homeland security expenditures: foundational expected cost-effectiveness approaches. Contemporary Economic Policy, 25(1), 14-26. http://dx.doi.org/10.1111/j.1465-7287.2006.00029.x. http://dx.doi.org/10.1111/j.1465-7287.20... ; Powell, 2007aPowell, R. (2007a). Defending against terrorist attacks with limited resources. The American Political Science Review, 101(3), 527-541. http://dx.doi.org/10.1017/S0003055407070244. http://dx.doi.org/10.1017/S0003055407070... )
Uncertainties inherent to attackers and defenders	Defender uncertainties about how the attacker's preferences change over targets.	Centralization of resources protecting high valued targets performs better than those acting in a decentralized manner.	(Bier et al., 2007Bier, V. M., Oliveros, S., & Samuelson, L. (2007). Choosing what to protect: strategic defensive allocation against an unknown attacker. Journal of Public Economic Theory, 9(4), 563-587. http://dx.doi.org/10.1111/j.1467-9779.2007.00320.x. http://dx.doi.org/10.1111/j.1467-9779.20... ; Major, 2002Major, J. A. (2002). Advanced techniques for modeling terrorism risk. The Journal of Risk Finance, 4(1), 15-24. http://dx.doi.org/10.1108/eb022950. http://dx.doi.org/10.1108/eb022950... )
		With scarce resources the uncertainty becomes more significant and it is more difficult to protect targets that are more valuable.	(Wang & Bier, 2011Wang, C., & Bier, V. M. (2011). Target-hardening decisions based on uncertain multiattribute terrorist utility. Decision Analysis, 8(4), 286-302. http://dx.doi.org/10.1287/deca.1110.0218. http://dx.doi.org/10.1287/deca.1110.0218... )
		A robust type optimization can circumvent the uncertainties about the attacker's parameters.	(Nikoofal & Zhuang, 2011Nikoofal, M. E., & Zhuang, J. (2011). Robust allocation of a defensive budget considering an attacker’s private information. Risk Analysis: An Official Publication of the Society for Risk Analysis, 32(5), 930-943. http://dx.doi.org/10.1111/j.1539-6924.2011.01702.x. http://dx.doi.org/10.1111/j.1539-6924.20... )
	Attacker’s uncertainty related to the targets.	Game modeling can be performed as signaling games. Allocating too much resource to protect a target can signal high value target.	(Powell, 2007bPowell, R. (2007b). Allocating defensive resources with private information about vulnerability. The American Political Science Review, 101(04), 799-809. http://dx.doi.org/10.1017/S0003055407070530. http://dx.doi.org/10.1017/S0003055407070... )
	Uncertainties about the targets’ values and about the attacker’s target preference.	Uncertainty about attackers' preferences has little impact on how defense resources are allocated.	(Bier et al., 2008Bier, V. M., Haphuriwat, N., Menoyo, J., Zimmerman, R., & Culpen, A. M. (2008). Optimal resource allocation for defense of targets based on differing measures of attractiveness. Risk Analysis: An Official Publication of the Society for Risk Analysis, 28(3), 763-770. http://dx.doi.org/10.1111/j.1539-6924.2008.01053.x. http://dx.doi.org/10.1111/j.1539-6924.20... )
Advantages and disadvantages of making resource allocation public		It is better for the defense to make its strategy public than to make it a secret.	(Farrow, 2007Farrow, S. (2007). The Economics of homeland security expenditures: foundational expected cost-effectiveness approaches. Contemporary Economic Policy, 25(1), 14-26. http://dx.doi.org/10.1111/j.1465-7287.2006.00029.x. http://dx.doi.org/10.1111/j.1465-7287.20... ; Major, 2002Major, J. A. (2002). Advanced techniques for modeling terrorism risk. The Journal of Risk Finance, 4(1), 15-24. http://dx.doi.org/10.1108/eb022950. http://dx.doi.org/10.1108/eb022950... ; Zhuang & Bier, 2010Zhuang, J., & Bier, V. M. (2010). Reasons for secrecy and deception in homeland-security resource allocation. Risk Analysis: An Official Publication of the Society for Risk Analysis, 30(12), 1737-1743. http://dx.doi.org/10.1111/j.1539-6924.2010.01455.x. http://dx.doi.org/10.1111/j.1539-6924.20... )
		The balance between disclosure of defense strategies and maintaining the secrecy of this information is the best strategy.	(Brown et al., 2005Brown, G. G., Carlyle, M. W., Salmerón, J., & Wood, K. (2005). Analyzing the vulnerability of critical infrastructure to attack and planning defenses. In H. J. Greenberg & J. C. Smith (Eds.), Tutorials in operations research: emerging theory, methods, and applications (pp. 102-123). Catonsville: INFORMS.)
Redundant protection of critical targets		It is more advantageous to protect more critical components than to opt for redundancy.	(Brown et al., 2005Brown, G. G., Carlyle, M. W., Salmerón, J., & Wood, K. (2005). Analyzing the vulnerability of critical infrastructure to attack and planning defenses. In H. J. Greenberg & J. C. Smith (Eds.), Tutorials in operations research: emerging theory, methods, and applications (pp. 102-123). Catonsville: INFORMS.)
Redundant protection of critical targets		The author emphasizes the validity of redundancy as a defensive strategy.	(Bier, 2006Bier, V. M. (2006). Game-theoretic and reliability methods in counter-terrorism and security. In In A. G. Wilson, G. D. Wilson & D. H. Olwell (Eds.), Statistical methods in counterterrorism (pp. 23-40). New York: Springer.)
Tradeoff analysis in the distribution of resources for the protection of several targets		Analysis of how considering equity and efficiency impacts on the distribution of resources for the protection of various targets.	(Shan & Zhuang, 2012Shan, X., & Zhuang, J. (2012). Cost of equity in homeland security resource allocation in the face of a strategic attacker. Risk Analysis: An Official Publication of the Society for Risk Analysis, 33(6), 1083-1099. https://doi.org/10.1111/j.1539-6924.2012.01919.x. https://doi.org/10.1111/j.1539-6924.2012... )

Protection	s₀ (utility)	s₁ (utility)
d₁	55	–45
d₂	40	–60
d₃	65	–35

Attack Types	s₁ (probability)	s₀ (probability)
a₁	Tri(35; 55; 75)	100-[u(a₁;s₁)]
a₂	Tri(20; 40; 60)	100-[u(a₂;s₁)]
a₃	Tri(45; 65; 85)	100-[u(a₃;s₁)]

Success and failure probabilities		s₀	s₁
d₁	a₁	30%	70%
	a₂	20%	80%
	a₃	55%	45%
d₂	a₁	35%	65%
	a₂	25%	75%
	a₃	35%	65%
d₃	a₁	35%	65%
	a₂	40%	60%
	a₃	25%	75%

Probability distribution		s₀	s₁
d₁	a₁	U(0.1, 0.5)	Complementary
	a₂	U(0.2, 0.6)	Complementary
	a₃	U(0.35, 0.75)	Complementary
d₂	a₁	U(0.15, 0.55)	Complementary
	a₂	U(0.05, 0.45)	Complementary
	a₃	U(0.15, 0.55)	Complementary
d₃	a₁	U(0.15, 0.55)	Complementary
	a₂	U(0.20, 0.60)	Complementary
	a₃	U(0.05, 0.45)	Complementary

	µ	A	B	Η	${\prod^{}}_{D} (A = a)$
a₁	0.0885	91.06	937.88	1028.94	$B e (91,06; 937,88)$
a₂	0.0463	95.32	1963.50	2058.83	$B e (95,32; 1963,50)$
a₃	0.8652	12.61	1.9654	14.58	$B e (12,61; 1,9654)$

Probability Distribution		s₀	s₁
d₁	a₁	Tri (0.10; 0.30; 0,50)	Complementary
	a₂	Tri (0; 0.20; 0.40)	Complementary
	a₃	Tri (0.35; 0.55; 0.75)	Complementary
d₂	a₁	Tri (0.15; 0.35; 0.55)	Complementary
	a₂	Tri (0.05; 0.25; 0.45)	Complementary
	a₃	Tri (0.15; 0.35; 0.55)	Complementary
d₃	a₁	Tri (0.15; 0.35; 0.55)	Complementary
	a₂	Tri (0.20; 0.40; 0.60)	Complementary
	a₃	Tri (0.05; 0.25; 0.45)	Complementary

d * = \arg \max_{d \in D} \sum_{a \in A} [\sum_{s \in {0, 1}} u_{D} (d, s) p_{D} (S = s | d, a)] \times π_{D} (A = a)

a * = \arg \max_{a \in A} \sum_{d \in D} [\sum_{s \in {0, 1}} u_{A} (a, s) p_{A} (S = s | d, a)] \times π_{A} (D = d)

A | D ~ \arg \max_{a \in A} \sum_{d \in D} [\sum_{s \in {0, 1}} U_{A} (a, s) P_{A} (S = s | d, a)] \times {\prod^{​}}_{A} (D = d)

D | A^{1} ~ \arg \max_{d \in D} \sum_{a \in A} [\sum_{s \in {0, 1}} U_{D} (d, s) P_{D} (S = s | d, a)] \times {\prod^{​}}_{D} (A^{1} = a)

D | A ~ \arg \max_{d \in D} \sum_{a \in A} [\sum_{s \in {0, 1}} u_{D} (d, s) P_{D} (S = s | d, a)] \times {\prod^{​}}_{D}

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Adversarial Risk Analysis in support of defensive resource allocation for counterterrorism

Abstract