Robustness of Optimal Defensive Resource Allocations in the Face of Less Fully Rational Attacker

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1 Proceedigs of the 2009 Idustrial Egieerig Research Coferece Robustess of Optimal Defesive Resource Allocatios i the Face of Less Fully Ratioal Attacker Megra Hao, Shila Ji ad Ju Zhuag Departmet of Idustrial ad Systems Egieerig State Uiversity of New York at Buffalo, Buffalo, NY 14260, USA Abstract Hudreds of billios of U.S. dollars have bee spet o homelad security sice September 11, 2001, while the optimality ad effectiveess of those expeditures remai obscure. I this paper, we develop a umerical model a cetralized defeder (the govermet) optimal resource allocatio amog multiple targets, agaist a attacker (terrorist) who could be either strategic (i.e. ratioal) or o-strategic. We also study the sesitivities of the optimal defeder budget allocatio to: (a) the probability that the attacker is strategic ad (b) the choices of o-strategic attackers. Keywords homelad security, o-strategic behavior, resource allocatio, multiple targets 1. Itroductio Sice September 11, 2001, hudreds of billios of U.S. dollars have bee spet o homelad security. Accordig to the Office of Maagemet ad Budget[1], the total outlay of the U.S. Departmet of Homelad Security s actual total outlay for 2007 is $38 billios ad the expected total outlays for 2008 ad 2009 are $42 ad $44 billio, respectively. However, allocatig those budgets amog multiple cities, urba areas, ad critical ifrastructures (e.g. airports ad bridges), remais a challegig task. The optimality ad effectiveess of these expeditures remai obscure ad have ofte bee criticized. For example, i 2008, Prate ad Bohara[2] metioed that "The distributio of State Homelad Security Grats has bee ofte criticized as pork barrel spedig, where political cosideratios ad ot terrorism risk are determiig the allocatio each state receives." Similarly, Paddock[3] poited out "I 2002, the Homelad Security Grats Program fudig was distributed etirely o a formula basis. The result was that, for the ext three years, the grat dollars were tied up at the state level ad scarcely more tha 30 percet of the fudig was passed through to local first respoders. I some states, that fudig has ever bee spet." Academic iterest i terrorism ad couter-terrorism strategies has also bee sigificatly icreased sice September 11, 2001[4]. Several full-edogeous game-theoretic models (i.e. assumig that both the attacker ad the defeder are fully strategic, ratioal ad have commo kowledge about the rules of the game) i either parallel or series systems have bee developed to study the system reliability (Hauske[5], [6], Bier et al.[7]). Applyig their model to the realworld data from Willis et al.[8], Bier et al.[9] idetifies the attacker ad defeder equilibrium strategies i a sequetial game where the defeder moves first, ad coclude that the cost effectiveess of defesive ivestmet has a sigificat impact o the optimal allocatio of defesive resources. I the real world, of course, attackers may ot be fully strategic; that is, they may be o-strategic or irratioal, for example, pickig target to attack radomly. The strategic attacker will adapt his strategies i respose to the defesive ivestmet, ad therefore may become less iterested to carry out attacks if the target is more defeded. By cotrast, the o-strategic attacker may oly strike certai targets (e.g. most valuable assets), regardless of the observed defese levels. Such o-strategic behavior may sigificatly decrease the robustess of the defeder s optimal resource allocatio, if the allocatio is optimized uder the assumptio that all attackers are fully strategic. Fly[10] cosiders defedig chemical facilities agaist both chemical accidets ad terrorist attacks. Similarly, Chyba[11] poits out that may public health measures iteded to detect ad cotai cotagious diseases i defedig agaist atural outbreaks as well as deliberate bioterrorism attacks. The optimal balace betwee defeses agaist the strategic ad o-strategic threats has bee studied by Zhuag ad Bier[12] who poit out that all else equal, it is less cost effective to protect large umbers of targets from strategic terrorists tha from atural disasters. Similarly, Powell[13] also studies the allocatio of defesive ivestmets betwee full-strategic ad partially strategic attackers. Both Zhuag ad Bier[12] ad Powell[13] use pre-determied probabilities for o-strategic threats, but allow attack

2 Proceedigs of the 2009 Idustrial Egieerig Research Coferece use probabilities for strategic threats to defed o the level of defesive ivestmets. I this paper, we expad the model from Bier et al.[9] by allowig some probability that the attacker is o-strategic. I particular, we itroduce two ew types of parameters: (a) the probability of a attacker beig strategic; ad (b) the probabilities that o-strategic attacker will attack various targets. To our kowledge, ours is the first umerical study to explore how sesitive the defeder s optimal budget allocatio is to these factors, usig realistic data. The ext sectio of this paper itroduces our otatio, assumptios, ad model. Applyig the data itroduced i Sectio 3, Sectio 4 tests the sesitivity of the defeder s optimal budget allocatio to these two ew parameters (the probability of a attacker is o-strategic, ad the probabilities of pre-determied attack strategies). Sectio 5 summarizes the previous sectios, discusses the policy implicatios of our work, ad provides some future research directios. 2. Notatio, Assumptios, ad Problem Formulatio 2.1 Notatio We defie the parameters of our model as follows: q ad 1 q: Probabilities that a attacker is strategic ad o-strategic, respectively. : Number of targets i the system. c i : Defeder s budget allocatio to target i, for i = 1,,. C: Total budget of the defeder. h i (c 1,,c ): Probability that a strategic attacker will attack target i, as a fuctio of the defesive resource allocatios to all targets, for i = 1,,. h i : Probability that a o-strategic attacker will attack target i, for i = 1,,. L(c 1,,c ): Total expected loss due to terrorism. λ: Cost effectiveess parameter of defesive ivestmet. p i (c i ): Success probability of a attack o target i, as a fuctio of the budget allocated to that target, c i. Followig Bier et al.[14], we assume that is expoetially distributed with parameter λ; i.e. p i (c i ) = e λc i, for i = 1,,. x i : Defeder s valuatio of target i, for i = 1,,. y i : Attacker s valuatio of target i, for i = 1,,. 2.2 Assumptios Followig Powell[13] ad Bier et al.[9], we assume that a fully-strategic defeder wishes to allocate a total budget of C amog targets, (c 1,,c ) such that c i = C. A strategic attacker observes the allocatio, ad the chooses a set of attack probabilities (h 1,,h ), where h i is the probability of a strategic attacker of lauchig a attack o target i, such that h i = 1. Followig Powell[13] ad Bier et al.[9], we assume that the attacker will choose to attack at most oe locatio. We also assume that the defeder is fully strategic, which the attacker may be either strategic or o-strategic, with probabilities q ad 1 q, respectively. A o-strategic attacker is assumed to attack the target i with the pre-determied probabilities h i, such that h i = 1. We assume that the strategic attacker wats to maximize the expected damage L(c 1,,c ), while the defeder wats to miimize it. As i Bier et al.[9] we assume that the attacker s valuatio of target i, y i follows a two-parameter Rayleigh distributio, with its mea value equalig to the defeder s valuatio x i. The two-parameter Rayleigh distributio has bee used effectively i modelig stregth ad lifetime data. Importatly for our purposes, the cumulative Rayleigh distributio is ot oly closed form, but also itegrable. 2.3 Problem Formulatio We model the attacker ad defeder iteractios i a sequetial game where the defeder plays first. The defeder s

3 Hao, Ji ad Zhuag objective is to miimize the total expected loss from a attack by assigig a portio of defesive budget C to each target i, c 1,,c, for i = 1,,. That is, where f i (y i ) = 2( y i ε i σ 2 i ε i = x i (1 cv π )e ( y i ε i σ i mi L(c 1,,c ) (1) c 1,,c s.t. = q = h i (c 1,,c )p i (c i )x i + (1 q) h i p i (c i )x i (2) p i (c i )x i [qh i (c 1,,c ) + (1 q)h i] (3) c i = C (4) ) 2 ad F i (y i ) = 1 e ( y i ε i σ ) 2 i, ad ε i is the lowest possible value of y i, which satisfyig 4 π ). Followig Bier et al.[9], we assume all the y i s have the same coefficiet of variace (cv), so σ i = 2x icv 4 π. Assumig all y i are idepedet, the probability that the attacker will attack target i is give by Z h i (c 1,,c ) = ε i f i (y i ) j i F i [ p i(c i )y i p j (c j ) ]dy i (5) 4 For sufficietly large coefficiets of variatio, cv > π 1, the miimum value ε i will be egative. It is assumed that o attack will be made whe targets have egative valuatio to the attacker. 3. Data Sources Accordig to Willis et al.[8], the te urba areas with the highest expected damage from terrorism are: New York; ; Sa Fracisco; the Washigto DC, area (icludig parts of Marylad, Virgiia, ad West Virgiia); Los Ageles ad Log ; the Philadelphia area (icludig parts of New Jersey); the Bosto area (icludig parts of New Hampshire); Housto; Newark; ad the Seattle area (icludig Bellevue ad Everett). Followig Bier et al.[9], we restrict our aalysis to these te urba areas for the purpose of computatioal tractability. Table 1 shows the expected damages from Willis et al.[8] ad the budget allocated to those te areas from the Office of Grats ad Traiig, U.S. Departmet of Homelad Security [15]. Sice the data o expected damages from Willis et al.[8] are from 2004, we use the FY2004 UASI Grat Allocatio as the budget to be allocated. However, data from 2008 is available at [1] ad ready to be used. We assume that the defeder valuatios of these te cities, x i, are give first by the expected property losses (colum 2 i Table 1), ad the by the expected fatalities (colum 3 i Table 1). 4. Sesitivity of Percetage of Strategic Attacker o Optimal Budget Allocatio We apply the model developed i Sectio 2 to the data source discussed i Sectio 3 ad cosider differet levels of the probabilities of a attacker beig o-strategic ad their correspodig pre-determied attack choices. I particular, we let the cv = 0.1, followig Bier et al.[9]. Ad we let the cost effectiveess of defesive ivestmet λ = (for the sesitivities of λ, see Bier et al.[9]). We use the data provided i Tables 1 ad assume this iformatio is commo kowledge to o-strategic attacker. Furthermore, we cosider two scearios describig the behavior of the o-strategic attacker: Sceario (a) probability of attackig the city with the highest attacker valuatio; Sceario (b) probability each of attackig two cities with the two highest expected property losses or fatalities. Each case is combied with the percetage of strategic attacker q at levels of 0,,,,..., ad, respectively.

4 Budget allocatio % Budget allocatio % Hao, Ji ad Zhuag Table 1: Expected property losses, fatalities, ad UASI budget allocatios for the te urba areas with the highest losses Urba Expected Property Expected FY 2004 UASI Losses [8] Fatalities [8] Grat Allocatios [15] ($ millio) ($ millio) New York Sa Fracisco Los Ageles-Log Philadelphia, PA-NJ Bosto, MA-NH Housto Newark Seattle-Bellevue-Everett Total Usig Expected Property Losses as the Measure of Target Attractiveess Figure 1(a) shows the result of case (a), whe the o-strategic attacker is assumed to attack the city with the highest expected property loss, which is New York, with the probability of. As showig i Figure 1(a), whe 1 q = 0, the attacker is fully strategic ad the optimal defese allocatio is well spread over the te cities. As the value of 1 q is icreasig, we ca see there is more moey beig trasferred to New York from other cities at optimality. Evetually whe 1 q = 1, the defeder kows that attacker is surely o-strategic ad will oly attack New York, all the moey goes to New York. Figure 1(b) shows the result of sceario (b), whe the o-strategic attacker is assumed to attack the first two cities with the highest expected property losses, which are New York ad, with the probability of of each. Similar to case (a), as value of 1 q is icreases, the optimal defese allocatio evetually goes to New York ad. Bosto Bosto Los Ageles-Log Los Ageles-Log Probabilities of a attacker beig o-strategic (a) Probabilities of a attacker beig o-strategic (b) Figure 1: Optimal budget allocatio as a fuctio of the probability for a attacker to be o-strategic (usig property losses as the measure of target attractiveess) 4.2 Usig Expected Fatalities as the Measure of Target Attractiveess Figure 2(a) shows the result of sceario (a), whe the o-strategic attacker is assumed to attack the city with the highest fatalities, which is, with the probability of. Similar to Figure 1(a), as value of 1 q is

5 Budget allocatio % Budget allocatio % Hao, Ji ad Zhuag icreasig, all the moey evetually goes to. Bosto Bosto Los Ageles-Log Los Ageles-Log Probabilities of a attacker beig o-strategic (a) Probabilities of a attacker beig o-strategic (b) Figure 2: Optimal budget allocatio as a fuctio of the probability for a attacker to be o-strategic (usig expected fatalities as the measure of target attractiveess) Figure 2(b) shows the result of sceario (b), whe the o-strategic attacker is assumed to attack the first two cities with the highest expected fatalities, which are ad, with the probability of of each. Aalogous to Figure 2(a), as value of 1 q is icreasig, all the moey evetually goes to ad. I summary, from Figures 1-2, we see that the defeder s optimal budget allocatio is sesitive to the probability for attacker to be o-strategic. As this probability icreases, the defeder s optimal budget allocatio evetually goes to targets beig cosidered to be pre-determied chose to attack. 5. Summary ad Future Research Directios Hudreds of billios of U.S. dollars have bee spet o homelad security sice September 11, 2001, while the optimality ad effectiveess of those expeditures remai obscure. I this paper, we develop a umerical model to determie the cetralized defeder (govermet) optimal resource allocatio amog multiple targets, agaist a attacker (terrorist) who could be either strategic (edogeous or ratioal) or o-strategic. We also study the sesitivities of (a) the probability of a attacker beig strategic ad (b) the o-strategic attacker s correspodig pre-determied choices, o the optimal defeder budget allocatio. We fid that the defeder s optimal budget allocatio is sesitive to the percetage of o-strategic attacker. As the probability for a attacker beig o-strategic icreases, the defeder s optimal budget allocatio evetually goes to cities beig cosidered to be pre-determied chose to attack. I the ear future, this model could be exteded to study the sesitivity of the percetage of o-strategic attacker upo the real defeder payoffs, if the defeder believes that the attacker is fully strategic, or fully o-strategic. These extesios would help evaluate the robustess of the may game-theoretical resource allocatio model ad the practically-used o-game-therectical resource allocatio model. Ackowledgemet This research was supported by the Uited States Departmet of Homelad Security through the Natioal Ceter for Risk ad Ecoomic Aalysis of Terrorism Evets (CREATE) uder grat umber 2007-ST However, ay opiios, fidigs, ad coclusios or recommedatios i this documet are those of the authors ad do ot ecessarily reflect views of the Uited States Departmet of Homelad Security. We also thak to Prof. Vicki Bier ad two aoymous referees for helpful commets.

6 Hao, Ji ad Zhuag Refereces [1] U.S.Departmet of Homelad Security, Office of Maagemet ad Budget (access Sep. 2008), [2] T. Prate ad A. Bohara, What determies homelad security spedig? a ecoometric aalysis of the homelad security grat program, The Policy Studies Joural, vol. 36, o. 2, [3] M. Paddock, 2007 homelad security fudig review: Old habits die hard, Homelad Security Today - ews ad aalysis, [4] T. Sadler ad K. Siqueira, Games ad terrorism: Recet developmets, Simulatio ad Gamig, vol. 40, [5] K. Hauske, Strategic defese ad attack for series ad parallel reliability systems, Europea Joural of Operatioal Research, vol. 186, pp , [6] K. Hauske, Strategic defese ad attack for reliability systems, Reliability Egieerig ad System Safety, vol. 93, pp , [7] V. M. Bier ad A. Nagaraja, Protectio of simple series ad parallel systems with compoets of differet values, Reliability Egieerig ad System Safety, vol. 87, pp , [8] H. H. Willis ad A. R. Morral, Estimatig terrorism risk, RAND Corporatio, Sata Moica, MG388.pdf, [9] V. M. Bier ad N. Haphuriwat, Optimal resource allocatio for defese of targets based o differig measures of attractiveess, Risk Aalysis, vol. 28, o. 3, [10] S. Fly, America the vulerable, New York: Harper Collis. [11] C. Chyba, Toward biological security, Foreig Affairs, vol. 122, o. 36, [12] J. Zhuag ad V. M. Bier, Balacig terrorism ad atural disasters defesive strategy with edogeous attacker effort, Operatios Research, vol. 55, pp , [13] R. Powell, Defedig agaist terrorist attacks with limited resources, America Political Sciece Review, vol. 101, o. 3, [14] N. Balakrisha ad V. B. Nevzorov, A primer o statistical distributios (hard copy), Joh Wiley & Sos, Ic. [15] O. o. G. U.S. Departmet of Homelad Security ad Traiig, Overview: Fy 2004 homelad security grat program,

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