On the Production of Homeland Security Under True Uncertainty

Transcription

1 Uiversity of Massahusetts Aherst Deartet of Resoure Eoois Workig Paer No htt:// O the Produtio of Hoelad Seurity Uder True Uertaity Joh K. Stralud 1 ad Barry C. Field Abstrat: Hoelad seurity agaist ossible terrorist attaks ivolves akig deisios uder true uertaity. Not oly are we igorat of the for, lae, ad tie of otetial terrorist attaks, we are also largely igorat of the likelihood of these attaks. I this aer, we oetualize hoelad seurity uder true uertaity as soiety s iuity to uaetable losses. We illustrate ad aalyze the osequees of this otio of seurity with a sile odel of alloatig a fixed budget for hoelad seurity to defedig the athways through whih a terrorist ay lauh a attak ad to itigatig the daage fro a attak that evades this defese. I this roble, iuity is the rage of uertaity about the likelihood of a attak withi whih the atual exeted loss will ot exeed soe ritial value. We aalyze the alloatio of a fixed hoelad seurity budget to defesive ad itigative efforts to axiize iuity to alterative levels of exeted loss. We show that the rodutio of hoelad seurity ivolves a fudaetal trade-off betwee iuity ad aetable loss; that is, for fixed resoures that are otially alloated to defese ad itigatio, ireasig iuity requires aetig higher exeted losses, ad reduig aetable exeted losses requires lower iuity. Greater ivestets i hoelad seurity allow soiety to irease its iuity to a artiular exeted loss, redue the exeted losses to whih we are iue while holdig the degree of iuity ostat, or soe obiatio of ireased iuity to a lower ritial exeted loss. Keywords: Hoelad Seurity, Terroris, True Uertaity. JEL Classifiatio: D0, D81, H56 1 Joh K Stralud, Deartet of Resoure Eoois Uiversity of Massahusetts, Stokbridge Hall, 80 Caus Ceter Way, Aherst, MA E: [email protected] P: F: Barry C. Field, Deartet of Resoure Eoois Uiversity of Massahusetts, Stokbridge Hall, 80 Caus Ceter Way, Aherst, MA E: [email protected] P: F:

2 August 006 O the Produtio of Hoelad Seurity Uder True Uertaity JOHN K. STRANLUND * Deartet of Resoure Eoois Uiversity of Massahusetts-Aherst BARRY C. FIELD Deartet of Resoure Eoois Uiversity of Massahusetts-Aherst Akowledgeets: Fudig for this researh was rovided by the U. S. Deartet of Agriulture uder USDA/ERS/PREISM Cooerative Agreeet No. 43-3AEM , ad the Cooerative State Researh Extesio, Eduatio Servie, U. S. Deartet of Agriulture, Massahusetts Agriultural Exeriet Statio uder Projet No. MAS The authors gratefully akowledge helful oets fro L. Joe Moffitt ad Yakov Be-Hai. * Corresodee to Joh K. Stralud, Deartet of Resoure Eoois, 80 Caus Ceter Way, 14 Stokbridge Hall, Uiversity of Massahusetts-Aherst, Aherst, MA 01003, USA. Phoe: (413) , Fax: (413) , E-ail: [email protected]. 1

3 O the Produtio of Hoelad Seurity Uder True Uertaity Abstrat: Hoelad seurity agaist ossible terrorist attaks ivolves akig deisios uder true uertaity. Not oly are we igorat of the for, lae, ad tie of otetial terrorist attaks, we are also largely igorat of the likelihood of these attaks. I this aer, we oetualize hoelad seurity uder true uertaity as soiety s iuity to uaetable losses. We illustrate ad aalyze the osequees of this otio of seurity with a sile odel of alloatig a fixed budget for hoelad seurity to defedig the athways through whih a terrorist ay lauh a attak ad to itigatig the daage fro a attak that evades this defese. I this roble, iuity is the rage of uertaity about the likelihood of a attak withi whih the atual exeted loss will ot exeed soe ritial value. We aalyze the alloatio of a fixed hoelad seurity budget to defesive ad itigative efforts to axiize iuity to alterative levels of exeted loss. We show that the rodutio of hoelad seurity ivolves a fudaetal trade-off betwee iuity ad aetable loss; that is, for fixed resoures that are otially alloated to defese ad itigatio, ireasig iuity requires aetig higher exeted losses, ad reduig aetable exeted losses requires lower iuity. Greater ivestets i hoelad seurity allow soiety to irease its iuity to a artiular exeted loss, redue the exeted losses to whih we are iue while holdig the degree of iuity ostat, or soe obiatio of ireased iuity to a lower ritial exeted loss. Keywords: Hoelad Seurity, Terroris, True Uertaity. JEL Classifiatios: D0, D81, H56 1. Itrodutio There are essetially three ways to rotet a oulatio fro terrorist attaks: (1) eutralizig terrorists before they a out attaks, () stoig attaks after they have bee started but before they are oleted, ad (3) takig stes to redue the severity of suessful attaks. There learly exists a eooi roble i deidig how to divide fixed resoures aog these differet futios. The last two, defedig athways through whih a attak ay our ad itigatig the effets of a failure of this defese, geerally orresod to the resosibilities of the U.S. Deartet of Hoelad Seurity (DHS). A substatial art of its budget goes to a variety of efforts to iteret terrorist attaks before they a be osuated. Chief aog these are the isetio rogras ut ito lae i the air, lad, ad sea etry orts of the outry. Aother ortio of its budget goes to hardeig otetial targets ad develoig strategies to itigate daage fro attaks that elude detetio.

4 Give kow, or ofidetly estiated robability desity futios over the for, lae, ad tie of otetial terrorist attaks, oe ould ast the roble of ivestig i seurity agaist terrorist attaks i the failiar ters of risk aalysis. 1 For oe exale aog several ossibilities, oe ould odel the hoies of defese ad itigatio to iiize the exeted losses fro terrorist attaks. I riile, robability distributio futios over terrorist attaks a be estiated fro the frequey of ast attaks. But while there has bee sigifiat work to develo ad exaie tie series of terrorist evets over reasoably log eriods (Eders ad Sadler, 00; Mikolus et al., 1989 ad 1993; Eders et al., 199; O Brie, 1996) it is ot straightforward to tur frequeies of ast attaks ito urret attak robabilities. Terrorists a hoose iovative tatis, as haeed o , i ways that are ot readily reditable fro ast atios. Moreover, tehologial hage (e.g., the iteret), the raid ae of globalizatio, ad the ebb ad flow of olitial oveets rodue ovel oortuities that a oly be roughly haraterized by ast attaks. Uder oteorary irustaes, we are extreely essiisti about our ability to estiate robability distributio futios over terrorist attaks with ay degree of ofidee. What this strogly suggests is that it ay ot be useful to thik about hoelad seurity agaist terroris as deisios ivolvig gables with kow robabilities. Istead, with reset to hoelad seurity, we are truly i a world of Kightia uertaity; that is, ot oly are we igorat of the for, lae, ad tie of otetial terrorist attaks, we are also igorat of the likelihood of these attaks. Thus, ay useful haraterizatio of the defiitio ad soial hoie of hoelad seurity ust aout for this uertaity. I this aer we roose that seurity uder true uertaity a be usefully thought of as the degree of iuity agaist uaetable exeted losses fro terrorist attaks. Moreover, it is reasoable to assue that soiety s referee for seurity is ootoially ireasig i the iuity to a artiular loss, ad dereasig i this loss while holdig iuity ostat. That is, we are ore seure if we a ahieve greater iuity to a artiular ritial loss, if we a redue the ritial loss without affetig the degree of iuity to this loss, or if we a ahieve both ireased iuity to lower loss. 1 This aroah has bee take by Kureuther ad Heal (003), Heal ad Kureuther (005), Keohae ad Zekhauser (003), Lakdawalla ad Zajai (005), Bueo de Mesquita (005), ad others. Kight was oered "with situatios whih are far too uique, geerally seakig, for ay sort of statistial tabulatio to have ay value for guidae. The oetio of a objetively easurable robability or hae is sily ialiable." Kight referred to this as true or ueasurable uertaity (Kight, 191, hater 7). 3

5 We illustrate ad aalyze the osequees of this otio of seurity with a sile odel of alloatig a fixed budget to hoelad seurity for efforts to sto a terrorist attak ad efforts to itigate the daage fro a attak that evades our defeses. Withi this roble iuity is the rage of uertaity about the likelihood of a attak withi whih the atual exeted loss will ot exeed soe ritial value. With this defiitio of seurity, we aalyze the alloatio of a fixed hoelad seurity budget to defesive ad itigative efforts to axiize seurity to a truly uertai terrorist attak. We deostrate that defese is a oral iut ito the rodutio of hoelad seurity, while itigatio is a iferior iut. That is, as soiety ireases its ivestet i hoelad seurity, ore of these resoures should be devoted to defese ad less to itigatio. Moreover, we show that the rodutio of hoelad seurity ivolves a fudaetal trade-off betwee iuity ad aetable loss; that is, for fixed resoures that are otially alloated to defese ad itigatio, ireasig iuity to loss requires aetig higher exeted losses, ad reduig aetable exeted losses requires lower iuity. Ireasig seurity i the sese of ireasig iuity to a artiular exeted loss, reduig the ritial exeted loss for the sae degree of iuity, or a obiatio of ireased iuity to a lower ritial exeted loss, is ahievable oly with a greater ivestet i hoelad seurity. Several aroahes have bee develoed to aalyze deisio akig uder true uertaity. These aroahes ilude aliatio of the axii, axiax, Lalae, ad Hurwitz riteria (Reder et al., 003). While oe of these riteria require kowledge of robability distributios for aliatio, the first two rereset olar extrees i ters of otiis ad essiis while the latter two require iforatio siilar to robabilities i order to be alied. Siilarly, quatifiatio of other otios related to uertaity suh as igorae ad surrise have also required the seifiatio of futios ofied to the uit iterval (Katzer, 1998; Hora et al., 00). Additioally, Kelsey (1993) has roosed a deisio theory requirig a rakig of evet robabilities rather tha a seifi robability distributio. Noe of these deisio riteria uder uertaity have ahieved the widesread aliatio i eoois afforded traditioal risk riteria. More iortatly for our uroses, oe of these deisio riteria rovide a atural oetualizatio of the defiitio ad ursuit of seurity uder true uertaity. 4

6 Our aalysis is a aliatio of Be-Hai s (006) iforatio-ga deisio theory. 3 The heart of Be-Hai s aroah is the ursuit of deisios that are robust i the sese that they axiize the rage of uertaity about odel araeters withi whih the deisio aker is ertai to ahieve a erforae riterio. 4 I our roble of hoelad seurity, we are uertai about the likelihood of a terrorist attak but we seek to axiize the rage of this uertaity over whih the exeted loss fro a attak will ot exeed soe ritial value. Thus, Be-Hai s aroah rovides a useful way to defie hoelad seurity as iuity to uaetable losses ad to aalyze its rodutio.. Hoelad Seurity Uder True Uertaity The rovisio of hoelad seurity agaist terroris is exeedigly olex, ivolvig defedig the literally thousads of aveues through whih terrorists ight oeivably attak ad ivestig i at least as ay ethods by whih the effets of a suessful attak a be itigated. Our urose, however, is ot to odel hoelad seurity i all its olexity, but rather to foralize a useful defiitio of seurity agaist uertai terrorist attaks ad to aalyze ertai harateristis of its rodutio. To that ed we exaie a situatio i whih a terrorist attak ay be lauhed with a ukow robability through a large uber of otetial athways. The uber of these athways is large eough that defedig all of the is rohibitively ostly. Sie ot all athways a be defeded, there is soe likelihood that a terrorist attak will be suessful, hee soiety ivests i efforts to itigate the loss fro a suessful attak. Though highly stylized, our aroah is aliable to ay situatio ivolvig defedig oe s borders agaist a attak that ours with ukow robability ad itigatig the effets of a failure i this defese. 3 Be-Hai s deisio theory has bee alied to a wide variety of robles, iludig fiaial risk assesset (Be-Hai 005), searh behavior i aial foragig odels (Carel ad Be-Hai 005), oliy deisios i arie reserve desig (Haler et al. 006), atural resoure oservatio deisios (Moilae et al. 006), isetio deisios by ort authorities to detet terrorist weaos (Moffitt et al. 005a) ad ivasive seies (Moffitt et al. 005b), tehologial fault diagosis (Piere et al. 006) ad egieerig odel-testig (Viot et al. 005). 4 This aroah is related to reet attets to develo robust oetary oliy, where the soure of uertaity is ukow variatio i the uderlyig oetary odels that are used to derive oliy rules. See the aers o robust deisios i Maroeooi Dyais, Vol. 6, No., 1, February 00. Poliy rules are evaluated usig a referee odel of the workigs of the oetary syste. Iortat araeters of the odel are the erturbed, leadig to variatios i outoes uder the oliy rule. The robustess of a rule is the axiu erturbatio that a be allowed while keeig the outoe of the rule withi seified bouds. By oarig alterative rules uder this roedure oe a idetify the oe with axial robustess. 5

7 .1 A odel of hoelad seurity Let N deote the uber of athways through whih a attak o a atio ay our. The robability of a attak that ay our through ay oe of these athways is. The uertaity i this odel is about the robability that a terrorist has lauhed a attak is oletely ukow. The athways are idetial i all regards, so a terrorist is idifferet about whih athway to attak. To defed agaist a otetial attak, N of the athways are defeded. Sie they are hoogeeous, defeded athways are hose at rado. A attak o a defeded athway will be thwarted. Thus, the robability of a suessful attak is the robability that a attak has bee lauhed ties the robability that the weao gets through the atio s defeses, ( N )/ N. A suessful terrorist attak results i a ertai loss L. 5 Mitigatio efforts,, redue this otetial loss, but at a dereasig rate. That is, the loss fro a suessful attak is L(), with L ( ) < 0 ad L ( ) > 0. Give the robability of a attak,, the exeted loss fro a attak is LN ( ) ( ) / N. This value is ukow beause the robability of a attak is oletely ukow. We ight, however, have olete ofidee that the robability of a attak is o ore tha soe value 1; that is, we ay be uertai of the true robability of a attak, but we are ertai that it does ot exeed.. The defiitio of hoelad seurity uder true uertaity We are ow ready to foralize a defiitio of hoelad seurity as the degree of iuity agaist uaetable loss. Let L be a ritial value for the exeted loss fro a terrorist attak, ad osider the roble of axiizig the rage of the robability of a attak withi whih the exeted loss fro a ossible attak does ot exeed L ; that is, ax LN ( ) ( ) / N L, [0, N], 0. [1] [0, ] 5 There are likely to be rado eleets of the daage aused by a suessfully deloyed weao, ad these eleets ay eve be truly uertai. We assue that this loss is kow with ertaity i order to fous o oe uertai eleet, that is, o the robability that a attak has bee lauhed. 6

8 Give defese ad itigatio efforts, LN ( ) ( ) / N, is ireasig i. Therefore, the solutio to [1] is to set LN ( ) ( )/ N= L ad solve for to obtai LN L (, ;, ) =, [0, ], [0, N], 0. [] L ( )( N ) The futio ( L, ;, ) haraterizes seurity agaist a uertai attak i the sese of iuity to otetial loss. Seifially, give defese ad itigatio efforts, ( L, ;, ) is the axiu robability of a attak for whih we are ertai that the exeted loss does ot exeed the ritial value L. I other words, soiety is iue to exeted loss L as log as the atual robability of a attak does ot exeed ( L, ;, )..3 The tehology of seurity Equatio [] desribes the tehology of hoelad seurity uder true uertaity. It haraterizes soiety s oortuities for takig atios (defese ad itigatio) to ahieve alterative degrees of iuity to alterative levels of exeted loss. I the ext setio, we will haraterize the rodutio of hoelad seurity as the alloatio of defese ad itigatio efforts that axiizes [], give the resoures devoted to hoelad seurity. Toward this ed we first eed to desribe how ( L, ;, ) varies with the uber of defeded athways ad efforts to itigate the effets of a suessful attak. The followig roositio, whih is roved i the aedix, rovides this desritio: Proositio 1: For ( L, ;, ) <, (0, N), ad > 0, ( L, ;, ) has the followig harateristis: i) > 0, > 0, ad > 0; ii) > 0, while the sig of is ideteriate; iii) ( L, ;, ) is stritly quasi-oave i (, ) if ad oly if < 0. ( L, ;, ) is stritly quasi-ovex if ad oly if > 0. 7

9 Part i) of the roositio idiates that ( L, ;, ) is otoially ireasig i ad so that ireasig the uber of defeded athways ad ireasig itigatio efforts both irease soiety s iuity to the ritial exeted loss L. Moreover, ilies that defese ad itigatio are oleets i the sese that ireased itigatio ireases the argial rodutivity of defese, ad vie-versa. Part ii) of the roositio reveals that the argial rodutivity of defese ireases as ore athways are defeded ( other had, it is ulear how ireased itigatio affets the argial rodutivity of itigatio. If but at a dereasig rate. O the other had, if > itigatio ireases with greater itigatio effort. Part iii) of the roositio reveals that whether the argial rodutivity of itigatio is ireasig or dereasig deteries whether 0 > 0). O the < 0, the ireasig itigatio ireases soiety s iuity to a ritial exeted loss, > 0 the the argial rodutivity of ( L, ;, ) is quasi-oave or quasi-ovex. This distitio has iliatios for the otial alloatio of hoelad seurity resoures to defese ad itigatio, whih we exlore ext. 3. The Produtio of Hoelad Seurity Obviously, levels of hoelad seurity deed o the resoures devoted to it. Moreover, give the resoures devoted to seurity, axial seurity is attaied by the effiiet alloatio of these resoures to defese ad itigatio efforts. Let R deote the oetary resoures devoted to hoelad seurity, ad let w ad deote the uit osts of defese ad itigatio efforts, resetively. The, the effiiet alloatio of resoures to defese ad itigatio to axiize iuity to a artiular exeted loss is the solutio to: w, ( ) ax ( L,,, ) = LN L ( )( N ) s.t. R w + w ( L, ;, ) [0, N], 0. [3] 8

10 At the outset this otiizatio roble a be silified a bit. First, the resoure ostrait always bids. The ootoiity of ( L, ;, ) i ad ilies that if R > w+ w, the ( L, ;, ) =. But the seurity a be ireased by dereasig L while ireasig ad/or util the budget is exhausted. There is a ossible solutio to [3] tha ivolves defedig all athways agaist ossible attak. Sie ( L,,, ) LN( L ( )( N ) ) =, the ostrait ( L, ;, ) a be writte as L L( )( N ) N. Settig = N so that all athways are defeded allows us to be erfetly iue to ay exeted loss; that is, we a ahieve ( L, ;, ) = for L = 0. It sees to us, though, that defedig all otetial targets of a terrorist attak is likely to be rohibitively exesive. Therefore, fro here o, we oly exaie solutios to [3] that ivolve < N. Moreover, let us assue that ( L, ;, ) is stritly quasi-oave i (, ) so that we a fous o solutios to [3] that ivolve > 0 ad > 0. 6 (We will briefly osider situatios ivolvig = 0 or = 0 later). The the hoies of defese ad itigatio solve ax ( L,,, ) subjet to R w w = 0. Let L deote the Lagrage equatio for this, roble ad let λ deote the ultilier for the budget ostrait. Uder our assutios the followig first-order oditios are both eessary ad suffiiet to solve [3]: LN L = w 0; L ( )( N ) λ = [4] LNL ( ) L = λw = 0; [5] [ L ( )] ( N ) L λ = R w w = 0. [6] 6 Quasi-oavity or quasi-ovexity deteries the urvature of the level urves of L (, ;, ). If L (, ;, ) is stritly quasi-ovex, the its level urves are stritly oave. I these ases, the otial hoies of defese ad itigatio will ertaily be a orer solutio. 9

11 Deote the solutio to [4] [6] as (,, λ ). The followig roositio haraterizes how seurity-axiizig hoies of defese ad itigatio deed o the ritial loss L ad the resoures devoted to seurity R. It is roved i the aedix. Proositio : Provided that ( L, ;, ) is stritly quasi-oave ad > 0 ad > 0 : i) ad are ideedet of L ; ii) is ireasig i R while is dereasig i R. Part i) idiates that otial defese ad itigatio are ideedet of the ritial exeted loss. This is true beause ( L, ;, ) is a liear futio of L. Part ii) of the roositio reveals that the uber of defeded athways is a oral iut i the rodutio of seurity, while itigatio is a iferior iut. Thus, ireased resoures alloated to hoelad seurity should be devoted to defese, while at the sae tie dereasig itigatio efforts. The ituitio behid this result is straightforward. Clearly, sie the argial rodutivity of defese is ireasig i higher levels of defese ( ), soiety should exloit this by alloatig at least a art of a irease i seurity resoures devoted to ireased defese. However, doig so ust be aoaied by a derease i itigatio efforts. To uderstad why this ust be the ase, ote that the first-order oditios [4] ad [5] ily that defese ad itigatio be hose so that the ratio of the argial roduts of these efforts i roduig seurity is equal to the ratio of the ries of these efforts. That is, [4] ad [5] a be obied to yield i.e., > 0 / = w / w. Alloatig at least a art of a irease i seurity resoures to additioal defese ireases, beause > 0. To aitai / = w / w, the, itigatio ust hage to irease by the sae aout as the irease i. Sie the argial rodutivity of itigatio is dereasig i this effort ( < 0 ), ireasig is aolished by reduig itigatio. 10

12 We will ot be aalyzig the effets of hages i the osts of defese ad itigatio o the otial solutio. 7 Therefore, Proositio 3 allows us to sily write the otial values for defese ad itigatio i ters of the resoures devoted to hoelad seurity; that is ad = ( R). The, axial iuity to exeted loss L is = ( R) LN ( LR,, ) = (, ; L, ) =. [7] L ( ( R))( N ( R)) Clearly, with reset to L, LN ( L( )( N )) has a zero iteret ad is liearly ireasig (this last follows beause ad are ideedet of L ). Figure 1 is a grah of LN ( L( )( N )) that we use to further refie the rodutio futio for seurity. As see i the grah, for suffiietly high values of L, ( ( )( )) LN L N exeeds. A exale of suh a outoe is ( +, L+ ). Clearly, sie + >, iuity to L + exeeds the axiu robability of a terrorist attak. The, oe ay be teted to sily say that axial seurity is give by the oit (, L + ) i the grah. Doig so suggests that otial seurity ( LR,, ) ireases u to ad the is ostat at this level for higher levels of ritial loss L. However, a oit like (, L + ) aot rereset the axial seurity attaiable with resoures R, beause suh a outoe ilies that soiety is willig to aet higher exeted losses tha it eeds to. Reduig the exeted ritial loss value fro L + to L k while holdig iuity to reresets greater seurity without additioal osts. Therefore, the otial rodutio of seurity is defied oly over exeted ritial losses betwee zero ad L k, ilusive. Thus, the followig roositio oletely haraterizes the otial rodutio of hoelad seurity. 7 As oe would guess, both defese ad itigatio are dereasig i their resetive uit osts. The ross-ost effets the effet o defese of a irease i the uit ost of itigatio ad the effet o itigatio of a irease i the uit ost of defese are abiguous. 11

13 Proositio 3: Give resoures R devoted to hoelad seurity, its otial rodutio is: where LN ( LR,, ) =, L [0, Lk( R, )], L ( ( R))( N ( R)) NL Lk( R, ) = L = L( ( R))( N ( R)) = L ( ( R))( N ( R)) N. [8] Our fial roositio rovides the fudaetal harateristis of the otial rodutio of hoelad seurity. It is roved i the aedix. Proositio 4: i) LR (,, ) = 0 for L = 0 ; ii) ( LR,, ) is liearly ireasig i L u to ; iii) ( LR,, ) is ireasig ad stritly ovex i R, for L < Lk( R, ) ; iv) LR (,, ) RL > 0, for L Lk( R, ) < ; Part i) of the roositio idiates zero iuity agaist zero exeted loss; that is, soiety has o ofidee that the exeted loss fro a terrorist attak is zero. Part ii) idiates a fudaetal tradeoff betwee iuity ad exeted loss. Give resoures devoted to hoelad seurity that are otially alloated to defese ad itigatio, ireased iuity agaist uaetable exeted losses is attaied oly by toleratig higher exeted losses. However, arts iii) ad iv) idiate that devotig ore resoures to hoelad seurity allows the attaiet of greater iuity to a artiular exeted loss, lower ritial exeted loss for the sae degree of iuity, or soe obiatio of greater iuity to lower ritial exeted loss. Moreover, give a ritial exeted loss L (0, L ( R, )), iuity to this loss ireases at a ireasig rate with greater resoures devoted to hoelad seurity. k 1

14 Figure is a grah of ( LR,, ) for two resoure levels, R0 < R1, whih we a use to illustrate the ai results of this aer. I the grah, L0 = Lk( R0, ) ad L1 = Lk( R1, ) as defied by [8]. Hoelad seurity uder true uertaity about the robability of a terrorist attak is the degree of iuity to uaetable exeted losses. More rigorously, a oit like ( A, L A) i Figure idiates the axiu robability of a terrorist attak, A, u to whih we are ertai that the exeted loss fro suh a attak does ot exeed L A. Suose that soiety devotes R 0 resoures to defedig agaist a terrorist attak ad itigatig the effets of a suessful attak, but ahieves oly ( A, L A). Clearly, the alloatio of R 0 to defese ad itigatio is ieffiiet beause greater seurity a be ahieved with these sae resoures. Cobiatios of iuity ad ritial exeted loss o the ab seget of ( LR, 0, ) rereset oits of greater seurity tha ( A, L A), beause they ivolve greater iuity to exeted loss L A, redued exeted loss while keeig iuity to that loss ostat at lower aetable exeted loss like oit ( B, L B). A, or soe obiatio of greater iuity to Give that soiety effiietly alloates R 0 to defese ad itigatio, hages i seurity ivolve a fudaetal tradeoff betwee iuity ad loss. Ahievig greater iuity requires toleratig a higher otetial exeted loss, ad vie versa. (Ideed, reduig the allowable loss to zero requires reduig iuity to zero). Ahievig greater seurity requires a greater ivestet i hoelad seurity. This is illustrated i Figure where a irease i this ivestet fro R 0 to R 1, allows the ahieveet of oits of greater seurity tha ( B, LB) o the d seget of ( LR, 1, ). Iterestigly, sie defese is oral iut ito the rodutio of seurity ad itigatio is a iferior iut (Proositio, art ii)), the irease i seurity resoures fro R 0 to R 1 alls for alloatig ore of these resoures to defese ad less to itigatio. To olete the aalysis let us very briefly exaie how ossible orer solutios for the hoie of defese or itigatio affet our ai results. These situatios ay our whe ( L, ;, ) is quasi-ovex i (, ). Or, give that we kow that defese is a oral iut ito the rodutio of seurity while itigatio i a iferior iut, very high levels of hoelad 13

15 seurity resoures ay all for devotig all of these resoures to defese ad oe to itigatio. Eve i these orer-solutio ases, ( LR,, ) retais its essetial harateristis rovided i Proositio 4. If = 0, all seurity resoures are alloated to itigatio so that = R w. / The, ( LR,, ) = LLR ( / w). As i Proositio 4, LR (,, ) = 0 for L= 0ad is liearly ireasig i L. Moreover, it is easy to show LR (,, ) R= LNL ( ) w[ L ( )] > 0, idiatig that ( LR,, ) is ireasig i R. O the other had, if = 0, the = R/ w ad ( LR,, ) = LNL(0)( N R/ w). It is straightforward to show that i these ases, ( LR,, ) retais the sae basi harateristis. The ushot the is that orer hoies of defese ad itigatio do ot hage the fudaetal struture of the rodutio of hoelad seurity as haraterized by Proositio Coludig Rearks: The Soial Choie of Hoelad Seurity We have exaied the defiitio ad rodutio of hoelad seurity uder true uertaity about terrorist attaks. We have argued that the degree of iuity to uaetable exeted losses fro terrorist attaks is a useful way to oetualize seurity uder true uertaity. We have illustrated this oet of seurity with a odel of alloatig a fixed budget for hoelad seurity to defedig the athways through whih a terrorist ay lauh a attak ad to efforts to itigate the daage fro a attak that evades this defese. Iuity to uaetable losses i this roble is the rage of uertaity about the likelihood of a attak withi whih the atual exeted loss will ot exeed soe ritial value. Hoelad seurity resoures are otially alloated to defese ad itigatio to axiize iuity to alterative levels of exeted loss. Our ost iortat result is that the rodutio of hoelad seurity ivolves a fudaetal trade-off betwee iuity ad aetable loss; that is, for fixed resoures that are otially alloated to defese ad itigatio, ireasig iuity to loss requires aetig higher exeted losses, ad reduig aetable exeted losses requires lower iuity. Greater ivestets i hoelad seurity allow soiety to irease seurity by ireasig iuity to soe ritial exeted loss, reduig the exeted loss we are willig to tolerate while holdig iuity to this loss ostat, or soe obiatio of ireased iuity to a lower ritial exeted loss. 14

16 Although we have show a useful way to thik about the roble of hoelad seurity ad have aalyzed harateristis of its rodutio, we have said little about soiety s referee ad hoie of seurity. Ideed, the variables that are iortat for defiig ad roduig seurity degree of iuity, ritial exeted loss, ad the resoures devoted to hoelad seurity are all atters of soial hoie. We have assued that soiety refers greater iuity to lower exeted loss, but we have show that ahievig both with fixed resoures is ot ossible. Therefore, a fuller desritio of the relative values that soiety laes o iuity ad ritial exeted loss is eessary to aalyze the soial hoie over these eleets of hoelad seurity. 15

17 Aedix Proof of Proositio 1: For arts i) ad ii) use [] to alulate = LN L ( )( N ) > 0 ; LNL ( ) = > 0 ; [ L ( )] ( N ) = LN 0 3 L ( )( N ) > LNL ( ) = > 0 ; [ L ( )] ( N ) ; { [ ( )] ( ) ( )} LN L L L = 3 3 [ L ( )] ( N ). Note that the sig of is equal to the sig of [ L ( )] L ( ) L( ), ad hee is ideteriate. For art iii), strit quasi-oavity requires ( ) ( ) > 0. Use the alulatios above to show that ( ) = 0. Therefore, ( L, ;, ) is stritly quasi-oave i (, ) if ad oly if >, whih requires < 0. Clearly, if ( ) 0 > 0, the ( L, ;, ) is stritly quasi-ovex, ad ( L, ;, ) is stritly quasi-ovex oly if > 0. QED. Proof of Proositio : For art i), set [4] ad [5] equal to eah other ad silify the result to obtai wl ( ) + wl ( )( N ) = 0. [A.1] This equatio ad the resoure ostrait, R w w = 0, deterie the otial values for ad. Note that L does ot eter either of these equatios, ilyig that the otial values of ad are ideedet of this ritial loss. To rove art ii), ote that the Hessia atrix assoiated with [4], [5], ad [6] is 16

18 w H = w. w w 0 The seod order oditio for a otiu is that the deteriat of this atrix be stritly ositive; that is H > 0. The oarative statis with reset to the budget are the solutios to the syste of equatios R 0 H R 0 =. λ R 1 [A.] Fro [A.] obtai R = ( w w) H > 0, The sig of R follows beause > 0 fro < 0, whih is required if ( L, ;, ) is stritly quasi- art ii) of Proositio 1 ad oave. Fro [A.] obtai = ( w w ) H. Calulate R w w = [ ( )( ) ( )] LN w L N w L [ L ( )] ( N ) 3. Fro [A.1], wl ( )( N ) = wl ( ). Substitutig this ito w w reveals that w w < 0. Cosequetly, R < 0. The roof is olete. QED. Proof of Proositio 4: Part i) follows fro [7] ad the fat that ad are ideedet of L (Proositio ). For art ii), use the eveloe theore to obtai LR (,, ) R= λ. If the first-order oditios [4] ad [5] hold, the λ > 0 ad LR (,, ) R> 0. Note that LR (,, ) R = λ R. Fro the first order oditios [4], [5], ad [6], ad [A.] obtai λ R = [( ) )] H > 0. The sig of R λ follows beause > 0 fro art ii) of 17

19 Proositio 1 ad < 0, whih is required for ( L, ;, ) to be stritly quasi-oave. Fially, to rove art iv) of the roositio ote that LR (,, ) RL = λ. Fro the first- L order oditio [4], λ ( )( ) 0 λ = LN w L( )( N ). Sie ad = N w L N >. This oletes the roof. QED. L are ideedet of L, 18

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21 Mikolus, E. F.; T. Sadler, J.M. Murdok, ad P. Fleig Iteratioal Terroris: Attributes of Terrorist Evets, (ITERATE 4). Du Lorig, VA. Viyard Software. Moffitt, L. Joe; Joh K. Stralud ad Barry C. Field. 005a. Isetios to Avert Terroris: Robustess uder Severe Uertaity. Joural of Hoelad Seurity ad Eergey Maageet (3), Artile 3. htt:// Moffitt L. Joe, Joh K. Stralud, Barry C. Field, ad Craig D. Ostee. 005b. "Robust Isetio for Ivasive Seies with a Liited Budget." Forthoig i The Eoois of Plat Health. Alfos Oude Lasik (ed.). Sriger. O Brie, S. P Foreig Poliy Crisis ad the Resort to Terroris: A Tie Series Aalysis of Coflit Likages. Joural of Coflit Resolutio 40(), Piere, S.G.; K. Worde ad G. Maso A Novel Iforatio-Ga Tehique to Assess Reliability of Neural Network-Based Daage Detetio. Joural of Soud ad Vibratio 93(1-), Reder, B.; R. M. Stair Jr. ad M. E. Haa Quatitative Aalysis for Maageet. Nith editio. Pretie Hall. Eglewood Cliffs, NJ. Viot, P.; S. Coga ad V. Ciolla A Robust Model-Based Test Plaig Proedure. Joural of Soud ad Vibratio 88(3),

22 ( ( )( )) LN L N (, L ) + + (, L + ) (, L ) k L L k Figure 1: Refiig the Produtio Futio for Hoelad Seurity. 1

23 ( LR,, ) LR (, 1, ) LR (, 0, ) a d b (, L ) B B (, L ) A A L1 L0 L Figure : The Produtio of Hoelad Seurity.