1. Adviser: Frank,Yeong-Sung Lin Present by 瀅如
Preventive strike vs. false targets and protection in defense strategy Gregory Levitin and Kjell Hausken
Adviser: Frank,Yeong-Sung Lin
Present by 瀅如

2. Agenda Introduction The model when the attacker attacks all targets
Attacker chooses a subset of targets to attack
Illustrating the solution of the game
Conservative defense strategy under uncertain contest intensities
Conclusion
1/12/2019

3. Agenda Introduction The model when the attacker attacks all targets
Attacker chooses a subset of targets to attack
Illustrating the solution of the game
Conservative defense strategy under uncertain contest intensities
Conclusion
1/12/2019

4. Introduction
A defender allocates its resource between defending an object passively and striking preventively against an attacker seeking to destroy the object.
With no preventive strike the defender distributes its entire resource between deploying false targets and protecting the object.
If the defender strikes preventively, the attacker’s vulnerability depends on its protection and on the defender’s resource allocated to the strike.
If the attacker survives, the object’s vulnerability depends on the attacker’s revenge attack resource allocated to the attacked object.
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5. Introduction
The defender has a fixed resource, which can be used passively or actively.
The attacker has two resources, one resource that is used for attack, and one that is used to protect against a preventive strike by the defender.
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6. Introduction
Consider an airborne bomber that has a mission to destroy some camouflaged object. The bomber can detect the targets with a given probability.
The defender deploys false targets to dissipate the bomb strike and protects the object to reduce the probability of its destruction in the case of the strike.
If the targets are detected, the bomber distributes its load among a subset of targets it chooses.
The defender can strike preventively using short range anti- aircraft missiles.
As the missile launchers are located near the defended targets, the preventive strike reveals the locations of targets and, therefore, if the bomber, protected by an anti-missile system survives the strike, it attacks for certain.
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7. Introduction
The optimal defense resource distribution is analyzed between
striking preventively
deploying the false targets
protecting the object
Two cases of the attacker strategy are considered:
when the attacker attacks all of the targets
when the attacker chooses a number of targets to attack
An optimization model is presented for making a decision about the efficiency of the preventive strike based on the estimated attack probability, dependent on a variety of model parameters.
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8. Agenda Introduction The model when the attacker attacks all targets
Attacker chooses a subset of targets to attack
Illustrating the solution of the game
Conservative defense strategy under uncertain contest intensities
Conclusion
1/12/2019

9. Agenda Introduction The model when the attacker attacks all targets
Attacker chooses a subset of targets to attack
Illustrating the solution of the game
Conservative defense strategy under uncertain contest intensities
Conclusion
1/12/2019

10. The model when the attacker attacks all targets
Nomenclature
The model
Solving the model
Illustrating the optimal defender’s decisions
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11. The model when the attacker attacks all targets -Nomenclature
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12. The model when the attacker attacks all targets -Nomenclature
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13. The model when the attacker attacks all targets -The Model
The estimated probability of the attack is z.
This paper considers z as an exogenously given fixed parameter.
The defender has two free choice variables H and t.
The attacker has no free choice variables.
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14. The model when the attacker attacks all targets -The Model
The vulnerability of the attacked object is determined by the common ratio form of the attacker–defender contest success function.
Where T is the attacker’s effort, t is the defender’s effort,
∂v/∂T≥0, ∂v/∂t≤0 and µ≥0 is a parameter for the contest intensity.
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15. The model when the attacker attacks all targets -The Model
Contest success function:
µ=0, t and T have no impact on v regardless of their size, which gives vulnerability v=0.5 for any T>0 and t>0.
0<µ<1, exerting more effort than one’s opponent gives less advantage in terms of vulnerability. For example, T=2, t=1, µ=0.5 gives v=0.59<2/3.
µ=1, the efforts have proportional impact.
µ>1, exerting more effort than one’s opponent gives more advantage in terms of vulnerability. For example, T=2, t=1, µ=2 gives v=0.8>2/3.
µ=∞, giving a step function where ‘‘winner-takes-all’’.
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16. The model when the attacker attacks all targets -The Model
In the case of no preventive strike：
The defender allocates H≤h= and protects the object using the entire remaining resource: t=r-cH.
The attacker distributes its resource evenly among H+1 targets achieving the per-target effort T=R/(H+1).
The probability of the destruction of the defended object p(H) is
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17. The model when the attacker attacks all targets -The Model
In the case of preventive strike：
The defender exerts the fraction x (0≤x<1) of the effort remaining after deploying H FTs into object protection such that t=x(r–Hc) and the remaining resource (1-x)(r-Hc) into the strike.
The attacker exerts the effort D to defend its facility.
The vulnerability of the attacker’s facility is
Where m has the same interpretation as µ.
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18. The model when the attacker attacks all targets -The Model
In the revenge attack the attacker exerts the per-target effort R/(H+1).
The vulnerability of the defended object in the revenge attack given the attacker survives the preventive strike is
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19. The model when the attacker attacks all targets -The Model
The probability of destruction of the defended object is
The defender has to choose H and x while ρ, σ, and δ are exogenously given.
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20. The model when the attacker attacks all targets -Solving the model
In the case of no preventive strike the defender must choose H=H˝ that minimizes (2):
In the case of preventive strike the defender optimizes its resource distribution in order to minimize (5):
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21. The model when the attacker attacks all targets -Solving the model
The preventive strike is justified if:
P(H*,x*)<p(H˝), i.e.
Estimated probability of the attack against the defended object z exceeds the threshold value zmin, where
The probability of the object destruction given the optimal defense strategy is W=min{p(H˝), P(H*,x*)}.
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22. The model when the attacker attacks all targets -Solving the model
Now proceed to analyze the impact of the variation in the six parameters σ, δ, ρ, z, m and µ on the decision variables H and x, and on the dependent variables P, W and zmin.
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23. The model when the attacker attacks all targets -Illustrating the optimal defender’s decisions
z=0.7
δ=ρ=2
m=µ=1
σ=c/r
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24. The model when the attacker attacks all targets -Illustrating the optimal defender’s decisions
z=0.9
σ=0.1
δ=2
m=µ=1
ρ=r/R
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25. The model when the attacker attacks all targets -Illustrating the optimal defender’s decisions
z=0.9
σ=0.1=c/r
ρ=2=r/R
m=µ=1
δ=R/D
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26. Agenda Introduction The model when the attacker attacks all targets
Attacker chooses a subset of targets to attack
Illustrating the solution of the game
Conservative defense strategy under uncertain contest intensities
Conclusion
1/12/2019

27. Agenda Introduction The model when the attacker attacks all targets
Attacker chooses a subset of targets to attack
Illustrating the solution of the game
Conservative defense strategy under uncertain contest intensities
Conclusion
1/12/2019

28. Attacker chooses a subset of targets to attack
If the attacker survives the preventive strike, it observes H+1 possible targets and cannot distinguish the object and the FTs.
The attacker can decide to attack a randomly chosen subset of targets concentrating its resource in the attack against fewer FTs and hoping that the defended object is among the attacked targets.
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29. Attacker chooses a subset of targets to attack
If the attacker attacks Q targets, Q≤H+1:
Q is a free choice variable for the attacker
The per-target attack effort is T=R/Q
The vulnerability of the object in the case when it is attacked is v(Q)=1/(1+(Qτ)µ), where τ(H,x)=t/R=(1-Hσ)xρ.
The overall probability of the object destruction in the case of the revenge attack is
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30. Attacker chooses a subset of targets to attack
Considering the worst possible scenario for the defender:
Assume that for any H and x, the attacker can always choose or guess the value of Q that maximizes the probability of the object destruction:
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31. Attacker chooses a subset of targets to attack
In the case of no preventive strike (x=1) the probability of the destruction of the defended object p is
In the case of preventive strike the probability of destruction of the defended object is
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32. Attacker chooses a subset of targets to attack
The preventive strike is justified if
P(H*,x*)<p(H˝), i.e.
Estimated probability of the attack against the defended object z exceeds the threshold value zmin, where
The probability of the object destruction given the optimal defense strategy is W=min{p(H˝), P(H*,x*)}.
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33. Agenda Introduction The model when the attacker attacks all targets
Attacker chooses a subset of targets to attack
Illustrating the solution of the game
Conservative defense strategy under uncertain contest intensities
Conclusion
1/12/2019

34. Agenda Introduction The model when the attacker attacks all targets
Attacker chooses a subset of targets to attack
Illustrating the solution of the game
Conservative defense strategy under uncertain contest intensities
Conclusion
1/12/2019

35. Illustrating the solution of the game
z=0.7
δ=ρ=2
m=1
σ=c/r
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36. Illustrating the solution of the game
z=0.7
σ=0.1
µ=1
ρ=2
δ=R/D
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37. Illustrating the solution of the game
z=0.7
σ=0.1
δ=ρ=2
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38. Illustrating the solution of the game
z=0.9
σ=0.1
m=µ=1
ρ=r/R
δ=R/D
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39. Agenda Introduction The model when the attacker attacks all targets
Attacker chooses a subset of targets to attack
Illustrating the solution of the game
Conservative defense strategy under uncertain contest intensities
Conclusion
1/12/20191/12/2019

40. Agenda Introduction The model when the attacker attacks all targets
Attacker chooses a subset of targets to attack
Illustrating the solution of the game
Conservative defense strategy under uncertain contest intensities
Conclusion
1/12/20191/12/2019

41. Conservative defense strategy under uncertain contest intensities
In many practical situations the values of the contest intensities cannot be exactly determined.
The most conservative defense strategy is to assume that the actual values of m and µ (belonging to exogenously defined intervals) are the most favorable for the attacker.
Equivalent to assuming that the attacker can choose m and µ within the given interval as free strategic variables.
The minmax defense strategy is to minimize the maximal probability of the object destruction W associated with a combination of the most unfavorable circumstances (contest intensities m and µ) and the most harmful attacker’s choice of Q.
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42. Conservative defense strategy under uncertain contest intensities
Let Q*(H,x,m,µ) be the number of the attacked targets that maximizes the object destruction probability for the given H, x, m, and µ.
The defender’s strategy is to choose the number of FTs H* and resource distribution parameter x* that minimize W in the range mmin≤m≤ mmax, µmin≤µ≤ µmax of contest intensities assuming that the attacker always chooses it’s best response Q*(H,x,m,µ) :
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43. Conservative defense strategy under uncertain contest intensities
In order to solve the minmax game, the following procedure should be applied:
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44. Conservative defense strategy under uncertain contest intensities
0.5≤m,µ≤1.5
0.1≤m,µ≤1.9
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45. Agenda Introduction The model when the attacker attacks all targets
Attacker chooses a subset of targets to attack
Illustrating the solution of the game
Conservative defense strategy under uncertain contest intensities
Conclusion
1/12/20191/12/2019

46. Agenda Introduction The model when the attacker attacks all targets
Attacker chooses a subset of targets to attack
Illustrating the solution of the game
Conservative defense strategy under uncertain contest intensities
Conclusion
1/12/20191/12/2019

47. Conclusion
The article analyzes how a defender determines a balance between defending an object passively and striking preventively.
If the defender strikes preventively, the attacker’s vulnerability depends on its protection and on the defender’s resource allocated to the strike.
If the attacker survives, the object’s vulnerability depends on the attacker’s revenge attack resource allocated to the attacked object and on the object protection effort.
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48. Conclusion
It is shown that the preventive strike is beneficial for the defender when
the deployment of FTs becomes expensive
the probability of unprovoked attacker’s strike is high
the attacker’s protection effort is low compared with the total defender’s resource, or the attacker’s protection is relatively weak
It is shown that the contest intensity parameters m and µ, for the attacker’s vulnerability and the object’s vulnerability, respectively, strongly influence the optimal defense strategy.
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49. Conclusion
The minimal estimated attacker’s strike probability when the preventive strike is justified can depend on the contest intensity parameters non-monotonically, which complicates the analysis and makes intuition based decision making impossible.
We demonstrate the most conservative approach of handling the uncertainty of the contest intensities in which m and µ can be considered as additional strategic variables that the attacker can choose within the specified ranges
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50. Thanks so much for your listening. 
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