1. Resource Distribution in Multiple Attacks Against a Single Target Author: Gregory Levitin,Kjell Hausken Risk Analysis, Vol. 30, No. 8, 2010

2. Agenda Introduction & Background Problem Description (Goal) The model Assumption Target vulnerability(V) Expenditure(E) Resource Distribution Even Resource Distribution(V,E) Geometric Resource Distribution(V,E) Numerical simulations Conclusion

3. Introduction & Background It has been common to consider a nonstrategic attacker, either by assuming a fixed attack or a fixed attack probability. Bier et al. (1) assume that a defender allocates defense to a collection of locations while an attacker chooses a location to attack.

4. Introduction & Background In this article, we consider a target (object) that a defender seeks to protect and an attacker seeks to destroy through multiple sequential attacks. The defender tries to keep the object undestroyed in each attack launched by the attacker.The phenomenon is modeled as a contest between a defender and an attacker.

5. Introduction & Background Problem Description (Goal) The model Assumption Target vulnerability(V) Expenditure(E) Resource Distribution Even Resource Distribution(V,E) Geometric Resource Distribution(V,E) Numerical simulations Conclusion

6. Problem Description Basic definitions: – Vulnerability: Probability of target destruction by the attacker. – Effort: Amount of intentional force aimed at destruction or protection of a system element (in this article, it is measured as the amount of attacker’s resource allocated to each attack and amount of defender’s resource allocated to defense)

7. Problem Description 1. Whether the attacker should allocate its entire resource into one large attack or distribute it among several attacks.

8. Problem Description 1. Whether the attacker should allocate its entire resource into one large attack or distribute it among several attacks. Attack strategy One large attack Several attacks

9. Problem Description

10. Attack strategy One large attack Several attacks Even Resource Distribution Geometric Resource Distribution

11. Problem Description 2.Whether geometrically increasing or decreasing resource distribution into a fixed number of sequential attacks is more beneficial than equal resource distribution Attack strategy One large attack Several attacks Even Resource Distribution Geometric Resource Distribution Geometrically increasing Geometrically decreasing

12. Problem Description 3.How the optimal attack strategy depends on the contest intensity(m). Two objectives: – 1.To maximize the target vulnerability(V). – 2.To minimize the expected attacker resource expenditure(E).

13. Optimal attack straregy Problem Description 3.How the optimal attack strategy depends on the contest intensity(m). Attack strategy One large attack Several attacks Even Resource Distribution Geometric Resource Distribution Geometrically increasing Geometrically decreasing

14. Introduction Problem Description (Goal) The model Assumption Target vulnerability(V) Expenditure(E) Resource Distribution Even Resource Distribution(V,E) Geometric Resource Distribution(V,E) Numerical simulations Conclusion

15. The model- Assumption Assumption: – (1) We consider a target (single target) that a defender seeks to protect and an attacker seeks to destroy through multiple sequential attacks. – (2)Both the defender and the attacker have limited resources. – (3)The attacker can observe the outcome of each attack and stop the sequence of attacks when the target is destroyed. – (4)The attacker distributes its resource over time.

16. The model- Assumption Assumption: – (5)We model the common case that the protection is static and cannot be changed over time. Target is destroyed->The protection is destroyed. Target is not destroyed->the protection remains in place also for the subsequent attack. – (6) We assume that the defender uses the same protection during the series of K attacks and allocates its entire resource into this protection.

17. Introduction & Background Problem Description (Goal) The model Assumption Target vulnerability(V) Expenditure(E) Resource Distribution Even Resource Distribution(V,E) Geometric Resource Distribution(V,E) Numerical simulations Conclusion

18. Target vulnerability(V) For any single attack, the vulnerability of a target is determined by a contest between the defender exerting effort t and the attacker exerting effort T in this attack. ->Contest success function

19. Target vulnerability(V) Contest success function :The attack success probability. T :Attacker’s effort to attack a target. t :Defender’s effort to protect a target. m: Attacker-defender contest intensity.

20. Target vulnerability(V) Contest success function Two factors influence the : – 1.The relation between the resources(t/T) in each attack. – 2.Contest intensity m.

21. Target vulnerability(V) 1.The relation between the resources(t/T) in each attack. – If the attacker exerts high effort(T>t), it is likely to win the contest that gives high vulnerability. – If the defender exerts high effort(T<t), it is likely to win the contest that gives low vulnerability.

22. Target vulnerability(V) 2.Contest intensity m : Measures whether the agents’ efforts have low or high impact on the target vulnerability

23. Target vulnerability(V) According to assumption (6), We assume that the defender uses the same protection during the series of K attacks and allocates its entire resource into this protection: t=r NOMENCLATURE t :Defender’s effort to protect a target r :Defender’s resource

24. Target vulnerability(V) On the contrary, the attacker distributes its entire resource R among K attacks such that the resource allocated to attack NOMENCLATURE R :Attacker’s resource :Attacker’s effort (resource used) in the th attack K :Number of consecutive attacks

25. Target vulnerability(V) The success probability of the th attack according to Contest success function is: The probability that the target survives in th attacks is:

26. Target vulnerability(V) The probability that the target survives all K attacks is: Thus, the target vulnerability in K attacks is:

27. Expenditure(E) According to assumption (3), The attacker can observe the outcome of each attack and stop the sequence of attacks when the target is destroyed. If the target is destroyed in the attack, the attacker spends the resource:

28. Expenditure(E) NOMENCLATURE T : attacker’s effort (resource used) in the attack(for even resource distribution T ≡ T) If the probability that the target is destroyed in the th attack is,the expected attacker’s resource expenditure can be obtained as:

29. Expenditure(E) If the target is destroyed in the th attack, the resource attacker spends.

30. Expenditure(E) The expected attacker’s resource expenditure when target is destroyed.

31. Expenditure(E) The probability that the target survives all K attacks.

32. Expenditure(E) The expected attacker’s resource expenditure when after K attacks the attacker fails to destroy the target.

33. Expenditure(E) The expected attacker’s resource expenditure can be obtained as: We will present the expected resource expenditure as a fraction of the total of attacker’s resource(R):

34. Introduction & Background Problem Description (Goal) The model Assumption Target vulnerability(V) Expenditure(E) Resource Distribution Even Resource Distribution(V,E) Geometric Resource Distribution(V,E) Numerical simulations Conclusion

35. Even Resource Distribution-V The attacker can choose the number of attacks K and distribute its resource evenly among the attacks such that T = R/K and the probability of target destruction in any attack is: =K/R *r = 1/T *r=r/T

36. Even Resource Distribution-V The target vulnerability is: Even resource distribution ->1- are equal in all K attacks,so

37. Even Resource Distribution-V Parameter values exist where the derivative in Equation (6) is negative, but it is often positive.

38. Even Resource Distribution-V Example: – Negative: (m = 2, R = r)

39. Even Resource Distribution-V Example: – Positive: (m = 0 ; m=1,R=r )

40. Even Resource Distribution-V m = 0,V increases concavely from 0 to 1 as a function of K and the attacker benefits from unlimitedly increasing the number of attacks. In realistic situations, the number of attacks is limited by time constraints by limited minimal cost of a single attack, etc. Therefore, the upper limit of K always exists.

41. Even Resource Distribution-V Fig. 1 presents the target vulnerability as a function of the contest intensity m for different K and r/R.

42. Even Resource Distribution-V It can be seen that the smaller the contest intensity(m), the more beneficial it is to increase the number of attacks.

43. Even Resource Distribution-E If the target is destroyed in the th attack, the probability of this event is: The probability that the target survives in all -1 attacks. The probability that the target is destroyed.

44. Even Resource Distribution-E The attacker spends the resource T =R/K*. * T

45. Even Resource Distribution-E R/K* R* R/T* *

46. Even Resource Distribution-E Fig. 2 presents the expected attacker’s resource expenditure as a function of the contest intensity m for different r/R.

47. Introduction & Background Problem Description (Goal) The model Assumption Target vulnerability(V) Expenditure(E) Resource Distribution Even Resource Distribution(V,E) Geometric Resource Distribution(V,E) Numerical simulations Conclusion

48. Geometric Resource Distribution-V Now we assume that the attacker can change the amount of resources allocated to each of the K attacks. To model the resource distribution, we use the geometric progression since it is simple and flexible.

49. Geometric Resource Distribution-V Assume that the attacker allocates effort to the first attack and changes the effort according to the geometric progression.

50. Geometric Resource Distribution-V The parameter q(Attack effort variation factor) determines the strategy of effort variation through the K sequential attacks:

51. Geometric Resource Distribution-V Attack strategy One large attack q=0 Several attacks Even Resource Distribution q=1 Geometric Resource Distribution Geometricall y increasing q>1 Geometricall y decreasing q<1

52. Geometric Resource Distribution-V For the given resource R and effort variation parameter q, we obtain:

53. Geometric Resource Distribution-V We obtain the probability of success in the th attack as:

54. Geometric Resource Distribution-V The total system vulnerability in K sequential attacks with effort variation parameter q is:

55. Geometric Resource Distribution-E The probability that the target is destroyed in the th attack is: Thus, the expected resource expenditure is:

56. Geometric Resource Distribution-E Fig. 3 illustrates how maximum target vulnerability and minimum resource expenditure can be competing attacker objectives.

57. Introduction & Background Problem Description (Goal) The model Assumption Target vulnerability(V) Expenditure(E) Resource Distribution Even Resource Distribution(V,E) Geometric Resource Distribution(V,E) Numerical simulations Conclusion

58. Numerical simulations For low contest intensity m = 0.5 – Maximal V: q=1 (Even distribution) V=0.842 E=0.545 – Minimal E: q=1.79 E=0.47 V=0.8175

59. Numerical simulations For high contest intensity m = 2 – Maximal V: q=0 (Single attack) V=0.5 E=1 – Minimal E: q=0.47 E=0.8855 V=0.2875

60. Numerical simulations Using numerical simulations to demonstrate the methodology of model analysis. Fig. 4 presents the values of q and K that maximize V and the values of q and K that minimize E and the corresponding values of V and E as functions of m for different values of the ratio r/R and K is allowed to vary from 1 to 10. There is a reasonable upper limit of K, which we here set to 10.

61. Numerical simulations-Maximal V r/R=0.5 r/R=1 r/R=2

62. Numerical simulations-Maximal V r/R=0.5 m 1 = 1.08 The optimal number of attacks is maximal for m < m 1

63. Numerical simulations-Maximal V For m> m 2, the single attack with K = 1 and q = 0 becomes preferable for the attacker. m 2 = 1.6

64. Numerical simulations-Maximal V r/R=0.5 r/R=1 r/R=2

65. Numerical simulations-Maximal V The optimal number of attacks is maximal for m < m 1 (where for r/R = 0.5 m 1 = 1.08, for r/R = 1 m 1 = 1.04, and for r/R = 2 m 1 = 1.02). Then it drops gradually and for m < m 2 (where for r/R = 0.5 m 2 = 1.6, for r/R = 1 m 2 = 1.28, and for r/R = 2 m 2 = 1.16) the single attack with K = 1 and q = 0 becomes preferable for the attacker. The values of q between 1 and 0 are never optimal when the system vulnerability is maximized.

66. Numerical simulations-Minimal E r/R=0.5 r/R=1r/R=2

67. Numerical simulations-Minimal E For low contest intensities the attack effort should increase through the K attacks (q > 1), whereas for more intensive contests it should decrease (q < 1).

68. Numerical simulations-Optimal r/R=0.5r/R=1 r/R=2

69. Numerical simulations-Optimal The attacker needs to strike a V versus E balance (tradeoff) for the attack strategy. Fig. 5 presents the differences in V and E obtained for strategies maximizing V and minimizing E.

70. Numerical simulations-Optimal

71. For m = 0.3(Low intensity), E can be reduced by 51.4% by the price of a 1.6% decrease of V. Move from 0.3->1

72. Numerical simulations-Optimal On the contrary, when m = 3(High intensity), E can be reduced by only by 7.74%, which causes 49% decrease of V. Move from 3->1

73. Numerical simulations-Optimal When the contest intensity is low, a great reduction of E without significant sacrifice of V is possible. Highly intensive contests even a small reduction of E causes a drastic reduction of V. The benefit of choosing the minE strategy for small m increases with the growth of the attacker’s resource superiority (decrease of r/R).

74. Numerical simulations-Optimal

75. Introduction & Background Problem Description (Goal) The model Assumption Target vulnerability(V) Expenditure(E) Resource Distribution Even Resource Distribution(V,E) Geometric Resource Distribution(V,E) Numerical simulations Conclusion

76. When the contest intensity is low, the attacker benefits from distributing its resource among several attacks and attacking the target with an effort that increases for each subsequent attack. For highly intensive contests, concentrating the entire attacker’s effort in one single attack is preferable.