Preventive Strike vs. False Targets in Defense Strategy

International Journal of Perforability Engineering Vol. 8, No., July, pp. -. RAMS Consultants Printed in India Preventive Strike vs. False Targets in Defense Strategy GREGORY LEVITIN and KJELL HAUSKEN. The Israel Electric Corporation Ltd., Israel. Faculty of Social Sciences, University of Stavanger, Norway (Received on March, and revised on Septeber, and May 9, ) Abstract: A defender allocates its resource between defending an object passively by deploying false targets, and striking preventively against an attacker seeking to destroy the object. With no preventive strike the defender allocates its entire resource to deploying false targets, which the attacker cannot distinguish fro the genuine 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. An optiization odel is presented for aking a decision about the efficiency of the preventive strike based on the estiated attack probability, dependent on a variety of odel paraeters. The optial nuber of false targets to deploy and the optial subset of targets to attack are deterined. Keywords: Vulnerability, active defense, passive defense, attack, protection, contest intensity. Introduction Defending against external ipacts and especially against intentional external ipacts has attracted considerable research effort in recent years. One can distinguish between active and passive defense. Soe easures aied at itigating the effect of external attacks, such as protective shields, are by their nature defensive. Other easures can generate active defense which eans exerting effort when certain conditions are et. See [] for a review of earlier research in this area, classifying according to syste structure, defense easures, and attack tactics and circustances. Two subcategories within defense easures are false targets and preventive strike. The preventive strike can be an effective easure of active defense aied at destroying the potential attacker and, thus, preventing the defended object fro destruction. However, the preventive strike can inflict a revenge strike which causes expenditure of the defender's resources that could be used for passive defense. Thus, the optial balance between the passive and active defense can considerably iprove the survivability of the defended object. Earlier research has considered how a defender balances between protecting an object passively and striking preventively against an attacker, equipped with one or ultiple attack facilities, seeking to destroy the object. In this paper the defender deterines a balance between striking preventively and deploying false targets to distract the attacker. Unlike the previous works, we assue that the object already has soe given protection and the defender distributes the reaining resource between the preventive strike and the false targets. The attacker cannot distinguish the genuine object fro the false targets. First we assue that the attacker attacks all targets. Thereafter we assue that the attacker ay attack a subset of the targets. In any attack against a group of targets the attacker distributes its effort evenly aong the targets. This corresponds to the case when the attacker cannot direct the attack exactly against certain targets, but against a group of targets (for exaple, area coverage weapon attack against a group of separated targets). * Corresponding author s eail: kjell.hausken@uis.no

Gregory Levitin and Kjell Hausken This paper analyzes the defender s objective of iniizing the probability of destruction of an object it controls. The object ay be an asset, a collection of assets, an infrastructure, a country, etc. The defender s two strategies are to defend passively or strike preventively, and, if the latter, which resource fraction to allocate to the preventive strike. It is soeties suggested that attack is the best defense, but not always. This paper seeks to deterine when it is optial to stay on the defensive and await the blow, and when it is optial to go on the offensive and strike preventively. Our focus is on the quite challenging defense optiization where 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. As an exaple, consider an airborne bober that has a ission to destroy soe caouflaged object. To dissipate the bob strike the defender deploys false targets. The bober can detect the targets with a given probability. If the targets are detected the bober distributes its load aong a subset of targets it chooses. The defender can strike preventively using short range anti-aircraft issiles. As the issile launchers are located near the defended targets, the preventive strike reveals the locations of targets and, therefore, if the bober, protected by an anti-issile syste survives the strike, it attacks for certain. The defender has to ake a choice between the passive defense (hoping that either it is not detected or that the object protection can survive the strike weakened by dissipation aong several targets) and active defense (hoping that the attacker is destroyed). When the defender builds its defense it should decide how its liited budget is distributed between deploying the false targets and deploying anti-aircraft systes. In this study we assue that the level of the object protection is a fixed exogenously given paraeter, which eans that the protection resource is independent fro the resource distributed between the false targets and the preventive strike. We analyze how the object protection affects the decision regarding the PS in the presence of the FT deployent option. Reference [] states that a preventive war is initiated in the belief that ilitary conflict, while not iinent, is inevitable, and that to delay would involve greater risk. In contrast, preeption is defined as an attack initiated on the basis of incontrovertible evidence that an eney attack is iinent. According to these two definitions, the focus in this paper is on preventive strike, and not on preeption. In related research [] draws on sociological and psychological research to consider risk governance, [] introduces huan factors into safety analysis, and [] balances hard and soft issues for the perforability of work teas. Section presents the odel when the attacker attacks all targets. Section assues that the attacker attacks a subset of the targets. Section illustrates the solution. Section considers the conservative defense strategy under uncertain contest intensities. Section concludes.. The Model (When the Attacker attacks all Targets) Noenclature PS FT r R t preventive strike false target total defender's resource total attacker's offensive resource effort with which the defended object is protected

Preventive Strike vs. False Targets in Defense Strategy H nuber of FTs deployed by the defender H* optial value of H Q nuber of targets attacked at rando by the attacker * Q optial value of Q given H FTs are deployed H D attacker's protection resource c the cost of single false target T attacker's effort (resource) per attacked target σ ratio c/r between cost of FT and the total defender s resource τ ratio t/r between defender s protection effort and attacker s attack resource δ ratio D/r between attacker s protection effort and total defender s resource h axial possible nuber of deployed FTs (h= /σ ) v(h) conditional probability of the defended object destruction given it is attacked as well as H FTs V(H) conditional probability of the attacker's destruction given it is preventively attacked contest intensity in strike against the attacker contest intensity in attack against the defender z estiated probability of attack against the defender if there is no preventive strike threshold value of attack probability (iniu estiated attack probability when the preventive strike is justified) P ~ probability of destruction of the defended object in the case of no preventive strike P(H) probability of destruction of the defended object in the case of preventive strike against all targets G probability of destruction of the defended object in the revenge attack against Q randoly chosen targets W probability of destruction of the defended object under the optial defense strategy.. The Model Suppose the defender anticipates an attack fro the attacker. The attack can be directed against an object owned or controlled by the defender, or against the defender itself. The estiated probability of the attack is z. This probability can be elicited fro intelligence data, expert judgents, and previous statistics (this paper considers z as an exogenously given fixed paraeter). The defender can defend its object in two ways: ipleenting the preventive strike against the potential attacker (active defense) and deploying false targets to dissipate the attacker's resources (passive defense). In the case of the preventive strike, the defender distributes its resource r between strike effort and false targets. The cost of a single FT is c. Therefore, the axial nuber of FTs that can be deployed given the defender's resource r is r/c = /σ, defined as the largest integer not greater than /σ. If H /σ false targets are deployed, the reaining resource r-hc can be allocated to the preventive strike. If the attacker survives the preventive strike, it attacks the defender with probability (revenge attack), evenly distributing its attack resource aong all the targets. If x= (no preventive strike) the defender allocates its entire resource r into deploying h= /σ FTs. The defender has one free choice variable H, which iplies the aount r-hc allocated to the preventive strike. Hence the choice about the preventive strike deterines

Gregory Levitin and Kjell Hausken the nuber of FTs deployed and vice versa. To focus explicitly on these strategic choices, we abstract away fro the defender s protection which is exogenously given. First, the defender akes only one decision at one point in tie, which either gives no preventive strike or a preventive strike with a resource allocation between the strike and FTs. This decision takes into account the agnitude of the attacker s attack which gets distributed across the genuine object and the FTs. Second, an alternative odel where the defender can decide different protections before and after its preventive strike gets ore coplicated, and is unrealistic when the attacker s revenge attack coes so quickly that the defender does not have the tie to change its protection. Especially protections such as bunkers, protective casings, and shields of various kinds in reote areas are not easily changed quickly. Consequently, first, we consider the cases when the deployent of ready FTs can be accoplished uch ore easily and quickly than protection enhanceent. Second, we consider the cases when the protection is liited by technology (for exaple, oving targets such as tanks or airplanes cannot have unliited protection and the decision about their protection is ade by the designer, not by the field coander) and the only way to iprove the defense of already protected object is to deploy FTs which can be done in battle conditions. The attacker has no free choice variables. We thus analyze the defense optiization. The vulnerability (conditional probability of destruction in the case of attack) of the attacked object is deterined by the coon ratio for of the attacker-defender contest success function [,7] T v =, () T + t where T is the attacker s effort, t is the defender s effort, v / T, v / t, and is a paraeter for the contest intensity. When =, t and T have no ipact on v regardless of their size which gives vulnerability v=. for any T> and t>. When <<, exerting ore effort than one s opponent gives less advantage in ters of vulnerability than the proportionality of the agents efforts specify. For exaple, T=, t=, =. gives v=.9 < /. When =, the efforts have proportional ipact on the v. When >, exerting ore effort than one s opponent gives ore advantage in ters of vulnerability than the proportionality of the agents efforts specify. For exaple, T=, t=, = gives v=.8 > /. Finally, = gives a step function where winner-takes-all. The paraeter can be illustrated by the history of warfare. Low intensity occurs in situations where neither the defender nor the attacker can get a significant upper hand. Exaples are the tie prior to cannons and odern fortifications in the fifteenth century, and entrenchent used with the achine gun in World War I [8]. High occurs when one or the other opponent ore easily can get the upper hand. Airplanes, tanks, and echanized infantry in World War II allowed both the offense and defense to concentrate firepower ore rapidly, which intensified the effect of force superiority. In the case of no preventive strike the defender allocates h= /σ FTs. The attacker distributes its resource evenly aong h+ targets achieving the per-target effort R T =. The probability of the destruction of the defended object P ~ is h + P ~ =zv(h) = T z z z = / =. () T + t + h + t R + τ h + (( ) ) ( ( ))

Preventive Strike vs. False Targets in Defense Strategy In the case of preventive strike the defender exerts effort r-hc reaining after deploying H FTs into strike and the attacker exerts the effort D to defend its facility. The vulnerability of the attacker's facility is ( r Hc) V ( H ) = =, H h. () ( r Hc) + D + ( δ /( Hσ )) where has the sae interpretation as. In the revenge attack the attacker exerts the pertarget effort R/(H+). The vulnerability of the defended object in the revenge attack given the attacker survives the preventive strike is v( H ) =, H h. () + ( H + ) t / R) In the case of preventive strike the probability of destruction of the defended object is P( H) = ( V( H)) v( H) = =. () + (( r Hc)/ D) + ( H+ ) t/ R) + (( Hσ )/ δ) + (( H+ ) τ) In our first forulation the defender has to choose H, while D, t, z and R are exogenously given.. Solving the Model The defender optiizes its resource distribution in order to iniize the probability of destruction of the defended object: H*= arg (P(H) in). Coparing () and (), the H r / c preventive strike is justified if P(H*)< P ~, i.e., + (( H * σ ) / δ ) + (( H * + ) τ ) < z. () + (( h + ) τ ) It follows fro () that the defender should strike preventively if according to its estiates the probability of the attack against the defended object z exceeds the threshold value, where (( h + ) τ ) (( H * ) τ ) + zin =. (7) + (( H * σ ) / δ ) + + The probability of the object destruction given the optial defense strategy is W=in{ P ~, P(H*)}. We assue that the defender's evaluation of the probability z does not change between its choice of H and the attack. Thus, the decision regarding the preventive strike can be ade siultaneously with choosing the value of H based on (). We now proceed to analyze the ipact of the variation in the six paraeters σ, δ, τ, z,, on the decision variables H and Q, and on the dependent variables P, W, and.. Illustrating the Optial Defender's Decisions Fig. presents the values of P ~ (no PS) and P for H= and H= as functions of σ, when δ=., τ =, z=.8, ==. As the FT cost increases relative to the defender s resource, the probability of destruction increases. With no PS, the probability increases as a stepwise function. Indeed, in intervals when h= /σ reains unchanged, σ does not affect P ~. Depending on σ as well as on the nuber of FTs H, the probability P(H) can be lower or greater than P ~. The cobination of stepwise increasing P ~ (σ) and

Gregory Levitin and Kjell Hausken onotonically increasing P(σ) causes situations when the difference P(σ)- P ~ (σ) changes its sign several ties with increase of σ. For exaple, for H=, P(σ)> P ~ (σ) when σ., P(σ)< P ~ (σ) when.9<σ., and again P(σ)> P ~ (σ) when σ>... P........ σ. No PS PS (H=) PS (H=) Figure : P ~ and P for H= and H= as functions of σ, when δ=., τ =, z=.8, ==. Figure presents P ~ for z=.8 and z=., P(H*) and corresponding values of h and H* as functions of σ, when δ=., τ =, ==. Depending on the relation between P ~ and P(H*) the defender decides whether to strike preventively or not. In this exaple the defender avoids the PS when z=. (except when.<σ<.9, where P ~ becoes slightly greater than P(H*)) and always strikes preventively when z=.8, which corresponds to high probability of the unprovoked attacker's strike. The defender always chooses H*<h in order to allocate soe resources to the PS. Both H* and h decrease in σ.. P... 8..... σ. No PS (z=.8) No PS (z=.) PS.... σ. h H* Figure : P ~ for z=.8 and z=., P(H*) and corresponding values of h and H* as functions of σ, when δ=., τ =, ==. Figure presents P ~ for z=.7, P(H*), W and as functions of σ, when δ=., τ =, ==. It can be seen that (thick line) varies non-onotonically and the defender s choice of no PS vs. PS can change any ties as σ increases. Indeed, when >.7 (where.7 is shown with a thin horizontal line) the defender avoids PS, and when <.7 the defender strikes preventively.

Preventive Strike vs. False Targets in Defense Strategy 7. P..9.8.7........ σ. No PS W PS..... σ. Figure : P ~ for z=.7, P(H*), W and as functions of σ, when δ=., τ =, ==. Figure presents W for z=., and H* as functions of σ, for τ =, == and different δ. The destruction probability W increases in σ, especially when the attacker enjoys a large resource advantage δ copared with the defender. has an overall decreasing trend except when δ is large. The non-onotonic behavior of akes the intuitive defender's decision about the optial defense strategy difficult. H* decreases in σ starting out at the largest level H*=9 when the attacker enjoys a large δ=.. The defender copensates for its resource inferiority by deploying any FTs.. W...8.. H* 8..... σ.....8..... σ....8..... σ....8. Figure :, H* and W for z=. as functions of σ, for τ =, == and different δ. Figure presents W for z=., and H* as functions of σ, for δ =., == and different τ. The second and third panels are qualitatively siilar to the second and third panels in Fig.. For the first panel W still increases in σ, as in Fig., but the defender prefers large τ in Fig in contrast to preferring low δ in Fig., and conversely for the attacker.. W..... τ :.... σ..7.....9.7 τ :.... σ. 7 H*...9.7 τ :.... σ....9.7 Figure :, H* and W for z=., as functions of σ, for δ =., == and different τ.

8 Gregory Levitin and Kjell Hausken. Attacker chooses a subset of Targets to attack If the attacker survives the preventive strike, it observes H+ possible targets and cannot distinguish the object and the FTs. However the attacker can decide to attack a randoly chosen subset of targets concentrating its resource in the attack against fewer FTs and hoping that the defended object is aong the attacked targets. If the attacker attacks Q targets, Q H+, where Q is a free choice variable for the attacker, the per-target attack effort is T=R/Q and the probability that the defended object is attacked is Q/(H+). The vulnerability of the object in the case when it is attacked is v(h)=. The overall + ( Qτ ) probability of the object destruction in the case of the revenge attack is G(Q)= Q. (8) ( H + ) + ( Qτ ) Considering the worst possible scenario for the defender, we assue that for any H the attacker can always choose or guess the value of Q that axiizes the probability of the object destruction: Q Q* = arg ax G = [ ]. (9) H / σ ( H + ) + ( Qτ ) Differentiating with respect to Q we get G + ( ) Q τ = Q ( H + ) [ + ( Qτ ) ] () G If, > and axial destruction probability is achieved when Q=H+. In this Q case the attacker attacks all the targets and we have the situation considered in the previous section. G If >, = gives Q * = (for exaple, for = Q*=/τ). For the Q τ convenience of later discussion, we denote q =. Since Q is integer and cannot τ be greater than H+, we have: H + if q H + * Q = q if q < H +, G( q) > G( q + ) () H q + if q < H, G( q) < G( q + ) In the case of no preventive strike the probability of the destruction of the defended object P ~ is P ~ * * =zg( Q h,h) = zq h. () + ( ) + * ( h ) Qhτ In the case of preventive strike the probability of destruction of the defended object is * QH P( H ) = ( V ( H )) G( Q*, H ) =. () ( ) / * + Hσ δ H + + Q τ ( ) ( ) H

Preventive Strike vs. False Targets in Defense Strategy 9 The defender optiizes its resource distribution in order to iniize the probability of destruction of the defended object: H*= arg (P(H) in). Coparing () and H h (), the preventive strike is justified if P(H*)< P ~ *,i.e., Q * * H < zq h () ( + ( H * ) / δ ) H * + + ( Q * τ ) H * ( ) + ( ) * h Qhτ The defender should strike preventively if according to its estiates the probability of the attack against the defended object z exceeds the threshold value, where * * h + QH * + ( Qhτ ) zin =. (( * ) / ) * * * + H σ δ H + Qh + ( QH * τ ) () The probability of object destruction given the optial defense strategy is W=in{ P ~, P(H*)}.. Illustrating the Solution of the Gae Figure presents W for z=.7,, H* and Q* as functions of σ, for δ =., =τ= and different. The destruction probability W increases in σ and decreases in the contest intensity in the attack against the defender. has an overall decreasing trend. Q* decreases in σ since Q* H*+. The attacker chooses Q*=H*+ when, and otherwise chooses Q* H*+ since a large contest intensity is costly inducing the attacker to concentrate its resource into attacking few eleents. When =, the attacker attacks only one eleent regardless of σ, despite any FTs being deployed. Figure 7 presents W for z=.7,, H* and Q* as functions of, for δ =., σ=.8, τ= and different. Again W decreases in, but can be inverse U shaped (e.g., when =) in. Hence when = the attacker benefits fro an interediate, while the defender benefits fro to be either low or high. With low the defender enjoys the deployent of any FTs, whereas with high fewer FTs ake the defender benefit fro concentrating its resource of protection.. W..7... : 7 H* :.... σ.... :.... σ. 8 Q* 7.. :.... σ....... σ... Figure :, H*, Q* and W for z=.7, as functions of σ, for δ =., =τ= and different.

Gregory Levitin and Kjell Hausken However, when both and are low which akes the contest over each target ore egalitarian in both contests, the attacker benefits despite the defender deploying any FTs. This latter result depends strongly on σ=.8 which restricts the defender to deploy axiu five FTs. is inverse U shaped.. W..8.... :..8..... :..8.... H*..8.. Q*..8.. :.. :.. Figure 7:, H*, Q* and W for z=.7, as functions of, for δ =., σ=.8, τ= and different. Figure 8 presents as function of, for =, σ=.8, τ= and different δ (when τ=) and τ (when δ=.). increases in δ and τ, and is increasing or inverse U shaped in. When the attacker s protection effort is low copared with the total defender s resource, or the defender s protection effort is low copared with the attacker s attack resource, the defender justifies the preventive strike for a lower estiated attack probability. When the contest intensity in the strike against the attacker is low, the defender justifies the preventive strike for a low estiated unprovoked attack probability, enjoying the ore egalitarian contest. When δ is large (above. in Fig. 8) and τ is sall (τ= in Fig. 8), increases towards when increases, aking the preventive strike unjustified even when the attacker is relatively certain to attack. The reason is that the defender is inferior in a double sense. Its protection effort is low and the attacker s protection effort is large. In this case the defender relies on deploying FTs instead, as we ll see in Fig. 9. For the reaining cobinations of δ and τ, is inverse U shaped in. For large the defender justifies the preventive strike even when the attack probability is low. The reason is that the defender is no longer inferior in a double sense, and if the attacker is eliinated, its object is secure..9.8.7....8...8.7.... τ :.7....8.... Figure 8: as function of, for =, σ=.8, τ= and different δ (when τ=) and τ (when δ=.).

Preventive Strike vs. False Targets in Defense Strategy Figure 9 presents, H*, Q* and W for z=.8, as functions of τ, for σ =.8, == and different δ. It can be seen that W decreases in τ, causing decreasing destruction probability as the defender s protection effort increases relative to the attacker s attack resource. W is insensitive to δ when δ. The reason is that for δ, z ax becoes greater than z and the defender prefers the passive defense and avoids the PS. In this case the attacker's protection resource has no influence on the outcoe of its ission.. W....9.8..7 H*..8.. τ... Q*..8.. τ....8.. τ....8.. τ.. Figure 9:, H*, Q* and W for z=.8, as functions of τ, for σ =.8, == and different δ. increases in δ and becoes independent of τ for large δ. Indeed with the growth of τ and δ the defender benefits fro increasing the nuber of FTs. At soe point it allocates as any as possible FTs H*=h and () takes the for, z [ ( ) ] in = + ( hσ ) / δ. In this case does not depend onτ. The attacker attacks all targets because Q*=H*+ is the solution of (9) for =.. Conservative Defense Strategy under Uncertain Contest Intensities In any practical situations the values of the contest intensities cannot be exactly deterined. Therefore, it would be useful to suggest a practical way to deterine the optial defense strategy for certain intervals of the contest intensities and. The ost conservative defense strategy is to assue that the actual values of and (belonging to exogenously defined intervals) are the ost favorable for the attacker. This approach is equivalent to assuing that the attacker can choose and within the given interval as free strategic variables. The inax defense strategy, thus, iniizes the axial probability of the object destruction W associated with a cobination of the ost unfavorable circustances (contest intensities and) and the ost harful attacker's choice of Q. Let Q H *(,) be the value of the attacker's effort distribution paraeter that axiizes the object destruction probability for the given H, and. The defender's strategy is to choose the nuber of FTs H* that iniizes W in the range in ax, in ax of contest intensities assuing that the attacker always chooses its best response Q H *(,):

Gregory Levitin and Kjell Hausken ax W H * *, (, ),, ax W H *, (, ),, in ax in ax H H in ax in ax for any H H*. In order to solve the inax gae for any given ranges of the contest intensities the following procedure should be applied:. Find Q h *<h+, and in * ax that axiize P ~ ;. Assign P in =;. For each H=,, h.. Find Q H *<H+, in * ax and in * ax that axiize P(Q,H);.. If P(H)<P ax assign P ax =P(Q H *,H), H*=H;. If P ax < P ~, allocate H* FTs and strike preventively achieving W=P ax, otherwise allocate h FTs and abstain fro striking preventively achieving W= P ~.. Obtain = P ~ /(zp ax ). Figure presents, *, *, H*, Q H * and W for z=.7, as functions of τ and δ for σ =.8 and uncertain contest intensities that can take values in the ranges,. Fig. presents the sae functions for the case of two ties shorter ranges of the uncertain contest intensities (with the sae ean values):..,... One can see that the ost harful for the defender values of and can abruptly change with variation of τ and δ. Indeed, when the gae paraeters change, the defender and the attacker adjust their H and Q H accordingly, which causes variation of the effort balances in the attack and the preventive strike. When (-H*σ)<δ in (), the defender suffers fro the axial possible value of, otherwise the defender suffers fro the inial possible value of. When Q * H τ< in (), the inial possible value of is ost favorable for the defender, otherwise the axial possible value of is ost favorable for the defender. () W..8. H*.......7.9 τ....7.9 τ.. δ.. :.7.7....7.9 τ...7 Q H* * *....7.9 τ....7.9 τ....7.9 τ...7 δ.. :.7 δ.. :.7 Figure :, *, *, H*, Q H * and W for z=.7, as functions of τ and δ for σ =.8, for,.

Preventive Strike vs. False Targets in Defense Strategy W......7.9 τ...7 Q H *.8.......7.9 τ...7 * H*....7.9 τ...7 *....7.9 τ δ.. :.7....7.9 τ δ.. :.7....7.9 τ...7 Figure :, *, *, H*, Q H * and W for z=.7, as functions of τ and δ for σ =.8, for..,... Coparing Fig. with Fig. one can see that the defender benefits fro the reduction of the uncertainty of the contest intensities. Indeed, when..,.., W is always lower than when,.. Conclusion The article analyzes how a defender deterines a balance between defending an object passively by deploying false targets (decoys) and striking preventively against an attacker seeking to destroy the object. With no preventive strike the defender allocates its entire resource to passive protection. 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. The attacker cannot distinguish the false targets fro the genuine object and can choose to attack a subset of the targets. The paper presents an optiization odel that can be used for aking a decision about efficiency of the preventive strike based on the estiated attack probability. The ethodology of analysis of influence of different odel paraeters on the optial defense strategy is deonstrated. It is shown that the preventive strike is beneficial for the defender when the deployent of FTs becoes expensive, when the probability of unprovoked attacker's strike is high and when the attacker s protection effort is low copared with the total defender s resource, or the defender s protection effort is low copared with the attacker s attack resource. It is also shown that the contest intensity paraeters and, for the attacker s vulnerability and the object s vulnerability, respectively, strongly influence the optial defense strategy. The inial estiated attacker's strike probability when the preventive strike is justified can depend on the contest intensity paraeters non-onotonically, which coplicates the analysis and akes intuition based decision aking ipossible. The contest intensity paraeters usually cannot be exactly evaluated in practice. We deonstrate the ost conservative approach of handling the uncertainty of the contest intensities in which the range of possible variation of and is deterined and the

Gregory Levitin and Kjell Hausken "worst case" defense strategy is obtained under the assuption that and take the values that are ost favorable for the attacker (in this case and can be considered as additional strategic variables that the attacker can choose within the specified ranges). References []. Hausken, K. and Levitin, G. Review of Systes Defense and Attack Models. International Journal of Perforability Engineering, ; 8(): -. []. Kroening, V. Prevention or Preeption? Towards a Clarification of Terinology, Project on Defense Alternatives Guest Coentary. Cabridge, MA: Coonwealth Institute,. http://www.cow.org/pda/kroening.htl. []. Renn, O. and Sellke, P. Risk, Society and Policy Making: Risk Governance in a Coplex World. International Journal of Perforability Engineering, ; 7(): 9-. []. Kosowski, K.T. Functional Safety Analysis including Huan Factors. International Journal of Perforability Engineering, ; 7(): -7. []. Kuipers, B.S. Perforability of Work Teas: Balancing Hard and Soft Issues. International Journal of Perforability Engineering, 9; (): -. []. Tullock, G. Efficient Rent-Seeking, in Buchanan, J.M., Tollison, R.D., and Tullock, G., Toward a Theory of the Rent-Seeking Society, Texas A. & M. University Press, College Station, 98: 97-. [7]. Skaperdas, S., Contest success functions. Econoic Theory 7, 99 :8-9. [8]. Hirshleifer, J. Anarchy and Its Breakdown. Journal of Political Econoy, 99; (): -. Gregory Levitin (eail: levitin@iec.co.il) is presently a senior expert at the Reliability Departent of the Israel Electric Corporation and distinguished visiting professor at University of Electronic Science and Technology of China. His current interests are in operations research and artificial intelligence applications in defense and reliability. In this field Prof. Levitin has published ore than 7 papers and four books. He is senior eber of IEEE and chair of the ESRA Technical Coittee on Syste Reliability. He is associate editor of IEEE Transactions on Reliability, Area Coordinator of International Journal of Perforability Engineering and eber of editorial boards of Reliability Engineering & Syste Safety, Journal of Risk and Reliability and Reliability and Quality Perforance. Kjell Hausken (eail: kjell.hausken@uis.no) is Professor of Econoics and Societal Safety at the University of Stavanger, Norway since 999. His research fields are strategic interaction, risk analysis, public choice, conflict, gae theory, terroris, inforation security, econoic risk anageent. He holds a Ph.D. fro the University of Chicago (99 99), was a Postdoc scientist at the Max Planck Institute (Cologne) 99 998, and a Visiting Scholar at Yale University 989 99. He has published articles and is on the Editorial Board for Theory and Decision, and Defence and Peace Econoics. He has refereed for journals, and has advised five Ph.D. students.