Impact of Intelligence on Target-Hardening Decisions

Transcription

1 CREATE Research Archve Publshed Artcles & Papers Impact of Intellgence on Target-Hardenng Decsons Vck M. Ber Unversty of Wsconsn Madson, Chen Wang Unversty of Wsconsn - Madson, cwang37@wsc.edu Follow ths and addtonal works at: Recommended Ctaton Ber, Vck M. and Wang, Chen, "Impact of Intellgence on Target-Hardenng Decsons" (29). Publshed Artcles & Papers. Paper Ths Artcle s brought to you for free and open access by CREATE Research Archve. It has been accepted for ncluson n Publshed Artcles & Papers by an authorzed admnstrator of CREATE Research Archve. For more nformaton, please contact grbben@usc.edu.

2 Impact Of Intellgence On Target-Hardenng Decsons Chen Wang, Vck M. Ber Department of Industral and Systems Engneerng, Unversty of Wsconsn-Madson, Madson WI, USA Abstract We present a game-theoretc model to explore how ntellgence about terrorst preferences can affect optmal resource allocatons for nfrastructure protecton. We consder a dynamc game wth ncomplete nformaton n whch the defender chooses how to allocate her defensve resources and then an attacker chooses whch target to attack accordng to a mult-attrbute utlty functon. Our model allows for both defender uncertanty about the attacker weghts on the varous attrbutes n the attacker utlty functon, and also defender gnorance about unobserved attrbutes that may be mportant to the attacker, but have not been dentfed by the defender. In the second perod, the defender also has an opportunty to update her knowledge n a Bayesan manner from actual (or attempted) attacks observed n the frst perod. Usng ths model, we can explore how much the defender would be wllng to pay for perfect ntellgence that resolves all of her uncertanty, as well as the value of mperfect ntellgence about attacker preferences ganed by observng prevous attacks.. ITRODUCTIO Due to the strategc nature of the terrorst threat, numerous researchers have used game theory to study resourceallocaton decsons for protectng potental targets aganst terrorsm; see revews by Sandler and Squera [] and Gukema [2]. In partcular, rgorous models have been establshed to address terrorst adaptablty; e.g., [3-5]. Much of ths work s based on games of complete nformaton, n whch the payoff functons of the partcpants are known to each other. However, some work allows the players to have uncertantes about ther rvals. For example, Lapan and Sandler [6] model a sgnalng game n whch the defender may not be well-nformed about the terrorst group s capabltes. Conversely, Zhuang et al. [7] use a sgnalng game to model attacker uncertanty, allowng the defender to mplement sgnals to mslead the attacker. As noted by Ber [8], defender uncertanty about attacker goals and values plays a central role n contemporary dscussons of terrorsm, and s therefore mportant to capture n a model of optmal defensve allocatons. In fact, -- the lack of knowledge about attacker values, goals, and motvatons s one of the prmary reasons for gatherng ntellgence about potental attackers. Moreover, Intellgence and warnngs are dentfed as one of sx crtcal msson areas n The atonal Strategy for Homeland Securty (Whte House, 22). Thus, there s a need to explctly consder the defender s uncertanty about terrorst goals and motvatons n games of ncomplete nformaton, n order to explore how the defender s optmal resource allocatons vary n the face of greater or lesser uncertanty. Recently, there has also been work on the nterface between ntellgence analyss and rsk analyss n homeland securty. For example, Arce and Sandler [9] consder the avodance of msunderstandng about the attacker s type n a sgnalng game as a measure of the value of ntellgence. From a practcal perspectve, Wlls [] descrbes how rsk analyss can be ntegrated nto the ntellgence cycle for producng terrorsm threat assessments and warnngs. In partcular, Wlls ndcates that rsk analyss can help n dentfyng the attacker scenaros that should be of greatest concern. In ths paper, we defne the value of ntellgence analogously to the value of nformaton n decson analyss n [];.e., the hghest prce that the decson maker would be wllng to pay for addtonal nformaton. In partcular, the defender can obtan mperfect ntellgence by observng the attacker s choce of targets n prevous attacks (or attempted attacks), and then update her knowledge about attacker preferences n subsequent perods of a dynamc (repeated) game. Counter-terrorsm polcy can then be characterzed n terms of a tradeoff between resources spent on ntellgence gatherng and target hardenng. For example, gatherng addtonal nformaton can help dentfy whch targets are at greatest rsk, thereby allowng resources for target hardenng to be allocated more effectvely, and potentally even reducng the number of targets that need to be protected. Hypotheszed terrorst objectve functons are one way to model terrorst goals and motvatons. As shown n Ber et al. [2], dfferent measures of target attractveness can yeld sgnfcantly dfferent optmal budget allocatons. Ths suggests that a sngle unvarate measure of attractveness may be nadequate, and more realstc mult-attrbute terrorst objectve functons, such as those developed by Betel et al. [3] and John and Rosoff [4], would be preferable /9/$ IEEE 373

3 In ths paper, we propose a two-stage dynamc (repeated) game wth two players, a defender and an attacker, n whch the defender has ncomplete nformaton about the attacker type. In our model, the attacker s assumed to choose targets accordng to a mult-attrbute utlty functon. We allow for both defender uncertanty about the attacker weghts on known attrbutes n the attacker s utlty functon, and also defender gnorance about unobserved attrbutes that may be mportant to the attacker. Based on the probablty dstrbutons defnng the defender uncertanty about the attacker attrbutes, we can assess the defender s subjectve probabltes of attacks on varous targets endogenously. We also show how the defender can updatee her knowledge about attacker preferences based on ntellgence ganed from observng prevous attacks (or attempted attacks), and dentfy how much the defender would be wllng to pay for such ntellgence. 2. MODEL AD ASSUMPTIOS 2. Basc Model Our basc model s a two-perod dynamc (repeated) game wth ncomplete nformaton, where each perod s a sequental game n whch defender plays frst (Fgure ). The frst-perod game looks lke the one-perod game from [4] wth some mnor revsons. ature () frst draws a type for the attacker from a dstrbuton f. The attacker (A) observes ths type, but the defender (D) observes only the dstrbuton f. The defender then decdes on an allocaton of her defensve resources c (f) among a heterogeneous collecton of potental targets, and c (f) s chosen n the face of the dstrbuton f. The attacker observes the defensve allocaton, chooses whether to attack, and f makng an attack, chooses an attack target; a (c ) represents the attacker decson and target choce (f any) ). An attack may turn out to be ether a success or a falure. In the second perod, the defender observes the target of any (successful or faled) attack, updates the dstrbutonn f over attacker types to obtan a posteror dstrbuton f', and makes a new resource-allocaton decson based on the new dstrbuton. The defender s new allocaton s c 2 [f'(a )], to whch the attacker responds wth a new attack decson and target choce a 2 (c 2 ). In a sequental game, the defender knows or assumes that the attacker would play hs best response for any gven defensve allocaton; therefore, she can choose her allocaton n order to deter an attack, or deflect an attack to less mportant targets. However, wth uncertanty about the attacker s utlty functon, the defender cannot predct the attacker s best response for sure; therefore, the defender s assumed n ths model to mnmze her expected total loss. In prncple, a tradtonal game-theoryy model would consder the long-term total expected loss for both perods;.e., the defender and attacker would choose ther frst- perod payoffs takng nto account later-perod payoffs. However, n ths paper, we assume a myopc attacker and defender for smplcty. Frst-Perod Game Second-Perod Game Fgure Game Tree In the frst perod, the (myopc) defender s d objectve s to mnmze the total expected loss from an attack, takng nto account threat (quantfed by the probabltes of attacks on the varous targets), vulnerablty (quantfed by the success probabltes of those attacks, as a functon f of the allocaton of defensve resources), and consequences (quantfed by the defender s valuatons of the targets). The defender s optmzaton problem s thus gven by b mn L( c) = h ( c) p( c ) v subject s to c = where: = number of targets c = defender s resource allocaton on target B = defender s total budget L(c) = total expected loss to thee defender v = defender s valuaton of target h (c) = probablty of an attack on target p(c ) = success probablty of an a attack on target, as a functon of the budget allocated to target t As n [2], we assume that the success probablty of an attack on target s an exponental functon of the defender s nvestment n that target, p(c ) = e λc, where λ s the cost effectveness of defensve nvestment. For example, at λ =.2, f the c are measured n mllons of dollars, then every mllon dollars of defensvee nvestment wll reduce the success probablty of an attack by about 2%. For convenence, we ntroduce a target to represent no attack, n whch case the defenderr suffers no loss and the attacker gans only a reserve utlty. Snce a sngle attrbute s not enough to realstcally model attacker preferences (e.g., see [2]), we assume that targets are valued by attacker accordng to a mult-attrbute utlty functon wth m attrbutes. The attacker s assumed to observe the defender s resource allocaton, and then choose a target to attack n order to maxmze hs utlty functon: = c B 374

4 where: u = m j= A j x j max = I ( c) p( c ) u + ε x m = attacker s valuaton of target A j = value of the jth attrbute for target A = ( + ) m matrx of the attrbute values A j ε = random varable representng the effects of unobserved attrbutes on the attacker valuaton of target x = (x,, x m ) = attacker weghts on the m attrbutes, m where x = = I (c) = an ndcator varable showng whether target s attacked, where I ( c) = and I (c) = means that no attack s launched = ote that the A j represent attrbute values that are known to the defender, whle the ε represent unobserved attrbutes that may nonetheless be mportant to the attacker. Snce the defender cannot observe these attrbutes, the ε are modeled as dentcally and ndependently-dstrbuted (..d.) random varables, to reflect the fact that the defender has no knowledge of whch targets are preferred n the unobserved attrbutes. In the above model, A j = ε = U s the attacker s reserve utlty. For example, U could represent the cost of an attack (whch would be saved by the attacker f no attack s launched). For smplcty, we set U = n ths paper. 2.2 Uncertanty Wth perfect nformaton about attacker preferences, the defender wll choose a resource allocaton to equalze the attacker utltes among all defended targets, snce otherwse, defensve resources spent protectng a less attractve target wll be wasted [4]. Therefore, n ths case optmal allocatons depend only on the attacker valuatons. However, Jenelus et al. [5] have shown that when the defender s uncertan about the attacker utltes, optmal allocatons depend not only on the attacker s utltes, but also on the defender s own target valuatons. To specfy the defender s uncertanty about the attacker target valuatons, we defne the attacker s type to be the weghts x placed by the attacker on varous attrbutes and error terms ε capturng the effects of unobserved attrbutes (nspred by the dscrete-choce model n [5]). Wth properly chosen dstrbutons, we can explore the effect of the defender s uncertanty n a straghtforward and parsmonous way by varyng the dstrbuton s parameters. In partcular, we assume that the attacker weghts x over the known attrbutes of target attractveness follow a Drchlet dstrbuton g(x,, x m ) ~ Drchlet(α,, α m ) We also assume error terms ε correspondng to the effects of unobserved attrbutes follow an exponental dstrbuton q(ε ) ~ exponental(μ) If we assume that the x j and ε are ndependent of each other, then the jont dstrbuton of the x j and ε for a gven attacker type can be gven by: f(x, ε) = g(x,, x m ) q (ε ) Changng the value of the parameter α = m α, whle holdng the expected values of the attrbute weghts (gven by the ratos α /α) constant, enables us to vary the extent of the defender s uncertanty by changng a sngle parameter, wth larger α correspondng to smaller uncertanty. In addton, we can vary the extent of defender uncertanty about the unobserved attrbutes by changng the parameter μ. In the frst perod, for a gven set of resource allocatons c = ( c,, c ), the probablty that target k wll be attacked s gven by = h k (c) = P{attack on k c } = P{ p ( c k ) u = k max p ( c ) u } =,..., Smlarly, the probablty that no target wll be attacked s gven by h (c) = P{no attack c } = P{ U = max =,..., p ( c ) u } In the second perod, assumng that an attack (or attempted attack) on asset has occurred n the frst perod, the defender can update her knowledge about the attacker s weghts on varous attrbutes through Bayesan updatng. For resource allocatons c n the frst perod and c 2 n the second perod, the probablty that target k wll be attacked n the second perod s gven by h' k (c, c 2 ) = P{attack on k gven c 2 attack on gven c } l{k c 2, x, ε} l{ c, x, ε} f (x, ε) dxdε where l{k c, x, ε} s the lkelhood that an attack on target k wll occur gven the values of x, ε, and c. ote that for specfc values of x, ε, and c, ths lkelhood wll be ether or, whch makes the posteror dstrbuton a truncated verson of the pror dstrbuton. It s only when x and ε are allowed to be uncertan (as of course they wll be to the defender) that the attacker s choce of target becomes probablstc. 375

5 Defensve allocatons c (f) Attacker types f(x,ε) Total losses L(c, f) Attacker types f(x,ε) Defensve allocatons c (x,ε) Total losses L(c, f) Defender s orgnal decson tree n st -perod game Defender s reversed decson tree n st -perod game Fgure 2 Orgnal and reversed decson tree n the frst-perod game To summarze, the defender s assessed probablty of an attack on target k s based on the defender s uncertanty about the attacker type (expressed by the pror dstrbuton f n the frst perod and the posteror dstrbuton f' n the second perod). Ths allows us to explore explctly how the defender s level of uncertanty (as measured by the breadth of the dstrbuton) affects both the defender s optmal resource allocatons and the optmal total expected losses born by the defender. 2.3 Value of Intellgence Uncertanty about attacker values, goals, and motvatons s one of the prmary reasons for gatherng ntellgence about potental attackers. Usng the above model of defender uncertanty, we can explore how much the defender should be wllng to pay for addtonal ntellgence at a gven level of pror uncertanty (.e., for a gven pror dstrbuton of attacker types). In our model, we defne the value of ntellgence analogously to the value of nformaton n decson analyss [];.e., the hghest prce that the decson maker would be wllng to pay for addtonal ntellgence. We begn wth the value of perfect ntellgence. In the frstperod game, how much would the defender be wllng to pay for perfect ntellgence that would resolve all of her uncertantes about the attacker s attrbute weghts and unknown attrbutes? To compute the value of perfect ntellgence, we can calculate the dfference of expected payoffs between the orgnal decson tree n the frst-perod game and the reversed decson tree (where the attacker type s revealed mmedately, as shown n Fgure 2). The expected payoff n the orgnal decson tree s just the optmal expected loss to the defender n the frst-perod game: EL = mn L( c ) = h ( c ) p( c ) v c subject to = c B = Moreover, for a gven set of attrbute weghts x and error terms ε, the reversed decson tree (.e., n the absence of uncertanty about x and ε) s equvalent to a sequental game where the defender moves frst wth complete nformaton. At equlbrum, the attacker should be ndfferent between all defended targets that are recevng non-zero defensve allocatons. The resultng expected payoff n the reversed decson tree s gven by EL r = + { mn( p[ c ( x, ε )] v} f ( x, ε ) dxdε where c + (x, ε) s the optmal allocaton n a sequental game + λc wth complete nformaton, whch satsfes e u ( x, ε ) = e λ c j + u ( x, ε) for all j for gven x and ε. Then the value j of perfect ntellgence (VOPI) n the frst perod s gven by VOPI = EL EL r In practce, of course, the defender wll not have access to perfect ntellgence. However, the attacker s acton n the frst perod may reveal some nformaton about attacker s preferences. In our model, the defender can narrow her dstrbuton over attacker types by applyng Bayes rule after observng the attacker s choce n the frst perod. Then she can make a better resource-allocaton decson n the second perod. How valuable s the ntellgence learned from a prevous (actual or attempted) attack? Let EL 2 () denote the optmal total expected losses to the defender n the second perod, gven that target has been attacked under the defender s optmal budget allocatons c * n the frst perod. Then the value of mperfect ntellgence (VOII) from a prevous (actual or attempted) attack s gven by VOII = EL h ( c *) EL 2 () = where h (c *) s the probablty of an attack on target under the defender s optmal budget allocaton n the frst perod. Quantfyng the value of ntellgence n ths way wll allow us to explore the defender s tradeoff between ntellgence gatherng and target hardenng. 376

6 3. SAMPLE APPLICATIO We now apply our model to sample data from three urban areas: ew York Cty (YC); the Boston area (ncludng parts of ew Hampshre); and Houston. Those ctes are all among the urban areas wth the hghest expected damage from terrorsm, accordng to Wlls et al. [6], who provde estmates of the expected annual terrorsm losses resultng from attacks on dfferent urban areas n the U.S. In the analyss n ths secton, we assume that the defender valuatons of these ctes are gven by the expected property losses n [6]. We begn by consderng two attrbutes of attractveness to the attacker, as gven by [2]: arport departures; and average daly brdge traffc. In order to make the ctes roughly comparable n these attrbutes, we scale the numbers of ar departures by, and the values of average daly brdge traffc by 3,. We assume that the defender knows the values of these attrbutes (as shown n Table ), but s uncertan about how much weght the attacker puts on each one. Table Expected terrorsm losses, attrbute values, and UASI budget allocatons for selected urban areas Urban Area Expected Property Losses V ($ mllon) Ar Departures A (,) Average Daly Brdge Traffc A 2 (3,) 26 UASI Allocatons C ($ mllon) YC Boston Houston In partcular, we analyze two cty pars: Boston-Houston; and YC-Houston. In the frst par of ctes, Boston scores hgher on average daly brdge traffc, whle Houston scores hgher on arport departures. In the second par, YC s overwhelmngly more attractve to the attacker, because t scores better on both attrbutes. Table also shows the budget allocated to the three ctes by the Urban Area Securty Intatve (UASI) for fscal year 26. We consder two levels of total budget: a hgh of $4 mllon (whch equals the sum of the UASI allocatons to YC and Houston n 26); and a low of $2 mllon (whch s less than the UASI allocaton to any of the three ctes n 26). We start by settng α /α 2 = 4, whch means the defender beleves that arports are roughly four tmes as mportant as brdges to the attacker. We consder two cases: wthout unobserved attrbutes; and wth unobserved attrbutes (where the defender beleves that the attacker may care about attrbutes other than ar departures and brdge traffc). In the second case, we fx the level of certanty about the attrbute weghts at a moderate value (α = ), whle keepng the pror means of the known attrbutes the same (α /α 2 = 4). We assume the mpact of unobserved attrbutes as hgh as the brdge traffc (the less mportant known attrbute) by settng the pror mean weght assgned to the unobserved attrbutes at α 3 = α Attack probablty Wth perfect nformaton about attacker preferences, we would never want to contnue spendng on a cty that was already guaranteed not to be attacked, snce that would be wasteful. However, wth only partal nformaton about attacker preferences, we may want to hedge, especally when one cty s much more valuable to the defender than the other. For example, YC s dsproportonally more valuable to the defender than to the attacker. If two ctes were equally attractve to the attacker, then the attacker would presumably choose randomly whch one to attack. However, snce the defender cares much more about YC than about Houston, ths would be bad for the defender; the defender would rather sacrfce Houston than YC, so mght prefer to nvest suffcently n YC to make t somewhat less attractve than Houston. How much less attractve the defender wants YC to be wll presumably depend on two factors: () how uncertan the defender s about the attacker s asset valuatons (hgh uncertanty wll lead the defender to spend more on YC, to make sure that t becomes less attractve than Houston); and (2) how much more valuable YC s to the defender (wth large dfferences n valuatons agan leadng the defender to spend more on YC). ote, however, ths stuaton wll arse only when the defender has ether a large enough budget, or a hgh enough cost effectveness of defensve nvestment, that she s capable of makng YC less attractve than Houston. Otherwse, whle the defender may stll nvest prmarly n protecton of YC rather than Houston, YC wll reman more attractve to the attacker, due to budget lmtatons. Frst consder the case wthout unobserved attrbutes. When the total budget s not large enough for the defender to make Houston more attractve than YC, then at low levels of certanty, YC s stll reasonably lkely to be attacked (e.g., -2% chance, as shown n Fgure 3a) even f t receves the entre avalable budget. otce that n the absence of unobserved attrbutes, the probabltes of an attack on Houston vs. Boston and YC both become extremely hgh at hgh levels of certanty (Fgure 3a). Ths means that a model wthout unobserved attrbutes leaves lttle or no room for defender learnng n the second-perod game. Therefore, we now consder the case wth unobserved attrbutes that may be mportant to the attacker, but have not been dentfed by the defender. 377

7 Houston vs. Boston Houston vs. YC Certanty about attrbute weghts (α) 3a. Wthout unobserved attrbutes Houston vs. Boston Houston vs. YC Certanty about unobserved attrbutes (μ) 3b. Wth unobserved attrbutes Fgure 3 Optmal attack probabltes wth low budget n the frst-perod game From Fgure 3b, we see that n ths case, Houston becomes more and more lkely to be attacked than Boston as the level of certanty about the unobserved attrbutes grows. Fgure 4 shows that the optmal allocaton to Houston n the frst perod also ncreases as the level of certanty grows. Ths s because as the defender becomes more confdent that Houston s gong to be attacked, she wll want to spend more money protectng Houston. Snce Houston s more lkely to be attacked than Boson (e.g., 55-95% chance, as shown n Fgure 3b), observng an attack on Houston wll not provde much new knowledge about attacker preferences, and therefore, hardly change the optmal resource allocaton. Conversely, an attack on Boston would represent a surprse (snce that cty was beleved to be suffcently well protected to be of lttle nterest to the attacker). Such an attack would lead to a change between the defender s pror and posteror allocatons, wth less resources beng allocated to Houston n the second-perod game (see the sold curve n Fgure 4). 3.3 Optmal total expected loss In ths secton, we show the optmal total expected losses wth low budget for both cty pars (Fgure 5); a large budget yelds smlar patterns, although wth smaller expected losses from terrorsm. Fgure 3b shows the attack probabltes on Houston vs. Boston and YC for the case wth unobserved attrbutes. ote that the defender s not able to make Houston more attractve to the attacker wth hgh confdence at low levels of certanty about the unobserved attrbutes; Boston and YC are stll left to be attacked wth 4%-5% chance n that case. 3.2 Optmal resource allocaton Fgure 4 shows the optmal resource allocatons to Houston vs. Boston n the case wth unobserved attrbutes wth low budget. Expected Total Loss ($ mllon) Houston vs. YC Houston vs. Boston Certanty about attrbute weghts (α) 5a. Wthout unobserved attrbutes Allocaton to Houston vs. Boston % 9% 8% 7% 6% 5% 4% 3% 2% % % frst perod second perod (f Houston was attacked) second perod (f Boston was attacked) Certanty about unobserved attrbutes (μ) Fgure 4 Optmal allocatons to Houston wth unobserved attrbutes wth low budget Expected Total Loss ($ mllon) Houston vs. YC Houston vs. Boston Certanty about unobserved attrbutes (μ) 5b. Wth unobserved attrbutes Fgure 5 Optmal total expected losses wth low budgets 378

8 Snce YC s much more valuable to the defender than the other two ctes, the defender experences a larger expected loss from Houston vs. YC than from Houston vs. Boston. Moreover, the more confdent the defender s about her pror estmates (that s, the hgher level of certanty she has), the less she can expect to lose at optmalty. At low levels of defender certanty, expected losses are hgher, because the uncertanty about attacker preferences creates the lkelhood that some resources wll be wasted on targets that are not hghly attractve to the attacker. The presence of unobserved attrbutes adds to the defender s uncertanty about the attacker s preferences. Therefore, the defender suffers larger expected losses n the case wth unobserved attrbutes (Fgure 5b) than n the case wthout unobserved attrbutes (Fgure 5a). 3.4 Value of ntellgence Perhaps surprsngly, for the sample data we chose, the two cty pars share more or less the same VOPI, especally at large budget levels, so we show only the results for Houston vs. Boston. Fgure 6 shows the VOPI and VOII for Houston vs. Boston for the case wth unobserved attrbutes resultng from the prevous analyss. Value of Intellgence ($ mllon) Value of Intellgence ($ mllon) VOPI VOII Certanty about unobserved attrbutes (μ) 6a. Wth hgh budget VOPI VOII Certanty about unobserved attrbutes (μ) 6b. Wth low budget Fgure 6 Values of ntellgence for Houston vs. Boston wth unobserved attrbutes Values of ntellgence are hgher wth low budget (Fgure 6b) than wth hgh budget (Fgure 6a). Ths s reasonable, snce wth scarce defensve resources, the defender s wllng to pay more to resolve the uncertanty about attacker preferences than wth hgh budgets, n order to avod wastng scarce resources on targets that are already relatvely unlkely to be attacked. The defender can also gan mperfect ntellgence from observng prevous (actual or attempted) attacks. In these sample cases, the VOII can count for up to roughly 3% of the VOPI (at low levels of certanty). The VOII s relatvely large at low levels of certanty, and decreases as the defender becomes more confdent. 4. COCLUSIOS By usng mult-attrbute terrorst utlty functons (rather than the types of sngle-attrbute objectve functons used n most past work, such as maxmze fataltes, or maxmze economc losses ), we are able to acheve a more realstc representaton of defender uncertanty about attacker preferences. Wthn our dynamc model, we allow the defender to update her knowledge from observng a prevous (actual or attempted) attack n a Bayesan manner. Based on ths model, we can also explore how valuable ntellgence about attacker preferences s to the defender. The more confdent the defender s about the attacker s preferences, the less she s expected to lose at optmalty, and the less she s wllng to pay for ether perfect or mperfect ntellgence. More specfcally, ntellgence becomes much more valuable n the face of unknown attacker attrbutes a stuaton that s lkely to pertan n practce, and to our knowledge has rarely been modeled. The magntude of the defender s budget also has a sgnfcant effect on the value of ntellgence. When the budget s suffcently large, the defender can effectvely deter attacks aganst the targets that are most valuable to her. However, wth scarce resources, ths becomes mpossble; n ths case, ntellgence about attacker preferences s especally valuable to the defender, to prevent wastng defensve resources on targets that are unlkely to be attacked. Ths work provdes an mportant buldng block for extendng past game-theoretc models of optmal defensve resource allocatons from smple toy problems or story problems to more realstc mult-attrbute attacker objectve functons (e.g., see [3]). Ths could pave the way for the use of game-theoretc methods (whch have hstorcally been vewed as not yet ready for prme tme ) n makng real-world resource-allocaton decsons. 379

9 ACKOWLEDGMETS Ths research was supported by the Unted States Department of Homeland Securty through the atonal Center for Rsk and Economc Analyss of Terrorsm Events (CREATE) under grant number 27-ST-6-. However, any opnons, fndngs, and conclusons or recommendatons n ths document are those of the authors and do not necessarly reflect vews of the Unted States Department of Homeland Securty. REFERECES [] T. Sandler, K. Squera, Games and Terrorsm: Recent Developments, Smulaton and Gamng, 4(2), 64-92, Apr 29. [2] S.D. Gukema, Modelng Intellgent Actors n Relablty Analyss: An Overvew of the State of the Art, V.M. Ber,. Azaez, eds. Combnng Relablty and Game Theory, Sprnger Seres on Relablty Engneerng, 26. [3] R. Powell, Defendng aganst terrorst attacks wth lmted resources, Amercan Poltcal Scence Revew (3), , Aug 27. [4] V.M. Ber, S. Olveros, L. Samuelson, Choosng What to Protect: Strategc Defensve Allocaton Aganst an Unknown Attacker, Journal of Publc Economc Theory 9(4), , Aug 27. [5] J. Zhuang, V.M. Ber, Balancng Terrorsm and atural Dsasters Defensve Strategy wth Endogenous Attacker Effort, Operatons Research 55(5),976-99, Sep-Oct 27. [6] H.E. Lapan, T. Sandler, Terrorsm and Sgnalng, European Journal of Poltcal Economy 9(3), , Aug 993. [7] J. Zhuang, V.M. Ber, O. Alagoz, "Modelng Secrecy and Decepton n a Multple-perod Attacker-Defender Sgnalng Game," submtted, Aug 28. [8] V.M. Ber, Choosng what to protect, Rsk Analyss 27(3), 67-62, Jul 25. [9] D.G. Arce and T. Sandler, Terrorst Sgnallng and the Value of Intellgence, Brtsh Journal of Poltcal Scence 37(4), , Oct 27. [] H.H. Wlls, Usng Rsk Analyss to Inform Intellgence Analyss, RAD Corporaton WR-464-ISE, Feb 27. [] H. Raffa and R. Schlafer, Appled statstcal decson theory, Cambrdge, Mass.: MIT Press, 96. [2] V.M. Ber,. Haphurwat, J. Menoyo, R. Zmmerman, and A.M. Culpen, Optmal Resource Allocaton for Defense of Targets Based on Dfferng Measures of Attractveness, Rsk Analyss 28(3), , Jun 28. [3] G.A. Betel, D. I. Gertman, and M. M. Plum, Balanced Scorecard Method for Predctng the Probablty of a Terrorst Attack, Rsk Analyss IV, , 24. [4] R. John and H. Rosoff, Estmatng lkelhood of terrorst attacks by modelng terrorst objectves and values, IFORMS Annual Meetng, Pttsburgh, ov 5-8, 26. [5] E. Jenelus, J. Westn, and Å.J. Holmgren, Systems Defense Aganst Antagonstc Threats Usng Dscrete Choce Modelng, submtted, 28. [6] H.H. Wlls, A.R. Morral, T.K. Kelly, and J.J. Medby, Estmatng Terrorsm Rsk, RAD Corporaton MG388,