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

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

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

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. 1/12/2019

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. 1/12/2019

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. 1/12/2019

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. 1/12/2019

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

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

The model when the attacker attacks all targets Nomenclature The model Solving the model Illustrating the optimal defender’s decisions 1/12/2019

The model when the attacker attacks all targets -Nomenclature 1/12/2019

The model when the attacker attacks all targets -Nomenclature 1/12/2019

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. 1/12/2019

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. 1/12/2019

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’’. 1/12/2019

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 1/12/2019

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. 1/12/2019

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 1/12/2019

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. 1/12/2019

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): 1/12/2019

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*)}. 1/12/2019

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. 1/12/2019

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 1/12/2019

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 1/12/2019

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 1/12/2019

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

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

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. 1/12/2019

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 1/12/2019

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: 1/12/2019

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 1/12/2019

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*)}. 1/12/2019

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

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

Illustrating the solution of the game z=0.7 δ=ρ=2 m=1 σ=c/r 1/12/2019

Illustrating the solution of the game z=0.7 σ=0.1 µ=1 ρ=2 δ=R/D 1/12/2019

Illustrating the solution of the game z=0.7 σ=0.1 δ=ρ=2 1/12/2019

Illustrating the solution of the game z=0.9 σ=0.1 m=µ=1 ρ=r/R δ=R/D 1/12/2019

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

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

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. 1/12/20191/12/2019

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) : 1/12/20191/12/2019

Conservative defense strategy under uncertain contest intensities In order to solve the minmax game, the following procedure should be applied: 1/12/2019

Conservative defense strategy under uncertain contest intensities 0.5≤m,µ≤1.5 0.1≤m,µ≤1.9 1/12/20191/12/2019

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

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

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. 1/12/20191/12/2019

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. 1/12/20191/12/2019

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 1/12/20191/12/2019

Thanks so much for your listening.  1/12/20191/12/2019