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Staffing models for covert counterterrorism agencies
Socio-Economic Planning Sciences 47 (2013) 2e8
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Invited paper
Staffing models for covert counterterrorism agencies Edward H. Kaplan*,1 Yale University, School of Management, 135 Prospect Street, New Haven, Connecticut 06511, United States
a r t i c l e i n f o
a b s t r a c t
Article history: Available online 5 October 2012
This article presents staffing models for covert counterterrorism agencies such as the New York City Police Department, the US Federal Bureau of Investigation, Britain’s Security Service or the Israeli Shin Bet. The models ask how many good guys are needed to catch the bad guys, and how should agents be deployed? Building upon the terror queue model of the detection and interdiction of terror plots by undercover agents, the staffing models developed respond to objectives such as: prevent a specified fraction of terror attacks, maximize the benefits-minus-costs of preventing attacks, staff in expectation that smart terrorists will attack with a rate that optimizes their outcomes, and allocate a fixed number of agents across groups to equalize detection rates, or prevent as many attacks as possible, or prevent as many attack casualties as possible. Numerical examples based on published data describing counterterrorism operations in the United States and Israel are provided throughout. Ó 2012 Elsevier Ltd. All rights reserved.
Keywords: Counterterrorism Staffing models Terror queues Resource allocation
1. Introduction and motivation A recent Congressional Research Service report stated that “.intelligence to counter terrorism depends more on human intelligence (humint) such as spies and informers.” than other forms of intelligence [2]. In addition to such intelligence, the detection and interdiction of terrorists depends crucially upon the ability of covert counterterrorism agents (CCAs) to identify terror plots, infiltrate the relevant terror cell or organization, and intervene before such plots are executed [13]. It follows that the ability of counterterrorism agencies such as the New York Police Department (NYPD), Federal Bureau of Investigation (FBI), Britain’s Security Service (MI5), and Israel’s General Security Service (Shabak) to prevent terror attacks depends rather crucially on how many CCAs these agencies place in the field, and how such agents are allocated to focus upon threats emanating from different regions, population subgroups, or terror organizations. CCA staffing is thus another important area of application for intelligence operations research [14]. Operations researchers have long studied staffing problems in service systems such as police [19], fire [25], hospitals [11] and call centers [8] in addition to other service and manufacturing systems.
* Tel.: þ1 203 432 6031; fax: þ1 203 432 9995. E-mail address: [email protected]. 1 William N. and Marie A. Beach Professor of Management Sciences, Yale School of Management; Professor of Public Health, Yale School of Public Health; and Professor of Engineering, Yale School of Engineering and Applied Science. 0038-0121/$ e see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.seps.2012.09.006
To date no such application to covert counterterrorism agencies has appeared in the open literature. This article provides a first cut at this problem by proposing a collection of relatively simple (and hopefully easy to understand) CCA staffing models. The analysis builds upon the terror queue framework for modeling the detection and interdiction of terror plots by undercover agents introduced in [13]. In terror queue models, the customers are the terror plots, the servers are undercover agents or informants, service commences when a plot is detected, and service concludes when a plot is interdicted. However, unlike traditional queueing systems, the customers in terror queues do not wish to be located or served; indeed servers must find their customers, while customers actually hope to renege before receiving service as such queue abandonment represents the execution of a terror attack. In the next section, the necessary terror queue background is briefly reviewed along with those key results required for the new staffing models that follow. Section 3 presents attack level staffing whereby the problem of determining the number of CCAs required to prevent a specified fraction of attacks is formulated and solved within the terror queue framework. A virtue of this model is that, while of interest in its own right, it serves as a building block for more realistic models. Indeed, Section 4 determines the number of CCAs to deploy for an agency that seeks to maximize the net benefits of preventing terror attacks, that is, the model recognizes the cost of deploying agents in addition to the benefit of averting attacks. To this point in the paper, terrorist behavior is presumed exogenous and fixed, but unlike natural risks, a large literature recognizes terrorists as adversarial opponents who actively choose
E.H. Kaplan / Socio-Economic Planning Sciences 47 (2013) 2e8
x ¼
(3)
2m
and
y ¼
3
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ða rf mr=dÞ þ ða þ rf þ mr=dÞ 4arf
ða þ rf þ mr=dÞ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ða þ rf þ mr=dÞ 4arf 2r
:
(4)
Fig. 1. The flow of terror plots [13].
how, when and where to mount attacks [3,5,6,10]. Section 5 thus considers smart terrorists who choose how often to attack so as to maximize the difference between their benefits and costs of attacking, and illustrates the resulting equilibrium assuming the government responds using its own cost-benefit calculation as outlined earlier in Section 4. Finally, Section 6 considers simple models for allocating a fixed number of CCAs across different regions, subpopulations or terror organizations to achieve objectives ranging from equalizing detection rates to maximizing the number of casualties averted. While the data necessary for applying these models in actual counterterrorism agencies remain highly classified, several numerical examples based on published data describing counterterrorism operations in the United States and Israel are provided throughout. Section 7 summarizes the paper. 2. Terror queue review This section briefly reviews the key terror queue results required for developing CCA staffing models (see [13] for further details). As illustrated in Fig. 1 , new terror plots arise with rate a per unit time. Available CCAs detect ongoing but as yet undiscovered (and unexecuted) terror plots at a rate that is proportional to both the number of undetected plots and the number of available agents. Thus, if there are X undetected terror plots (X is the terror queue of plots waiting to execute or be detected, whichever comes first), and Y out of the total force of f CCAs are occupied and unavailable for detecting plots, then the aggregate rate with which plots are detected (and available CCAs become unavailable for detection) equals dX(fY) per unit time. Terror plots on average require 1/m time units from inception to execution, thus the rate of successful terror attacks equals mX per unit time when there are X undetected plots. Finally, detected plots are interdicted at rate r per detected plot (equivalently per busy CCA), for an aggregate interdiction rate of rY interdicted terror plots per unit time. Section 3 of [13] shows that the Markov population model associated with Fig. 1 above can be very well-approximated by a continuous Ornstein-Uhlenbeck process. In particular, the expected number of undetected (detected) terror plots, E(X) (respectively E(Y)), can be closely approximated by the equilibrium quantities x and y that solve the following deterministic flow equations:
a ¼ mx þ dxðf yÞ
(1)
and
dxðf yÞ ¼ ry:
(2)
Equation (1) states that newly arriving terror plots either result in terror attacks or are detected, while equation (2) says that detected terror plots are interdicted.2 These equations solve to yield
2 Modifying the model to allow some detected plots to proceed to execution without interdiction is straightforward; see [16].
The results above presume that CCAs never commit false positive errors, that is, that while not all terror plots are detected, any detection claim made by a CCA is correct. However, often intelligence agents do apprehend suspects who ultimately have no connections to terrorism [13,16]. The relatively large cost of failing to detect a real terror plot compared to monitoring (and sometimes detaining) suspects innocent of terrorism is one reason for CCAs to “overdetect” [16], while in addition it is advantageous to terrorists to deliberately broadcast false information pointing to fake plots as a means of guarding the secret planning of true plots [21]. This concern is addressed in Section 4 of [13] by keeping track of CCAs who are unavailable for detection on account of being occupied with “fake plots.” Specifically, it is assumed that the rate with which available CCAs commit to following fake plots equals l per available CCA per unit time, while the rate with which CCAs so committed return to availability by disposing of fake plots equals j per unit time per CCA occupied with fake plots. Under these assumptions, it is shown in [13] that after defining
d0 ¼ d
j ; lþj
(5) 0
equations (3) and (4) remain correct upon substituting d for d. Equation (5) shows that if the rate at which CCAs commit to fake plots (l) is very high relative to the rate with which fake plots can be identified as such and disposed (j), the consequence is equivalent to fielding a force of CCAs who never commit false positive errors but have a greatly reduced ability to detect actual plots (as 0 d d), while if fake plots can be identified very quickly (meaning j [ l), there is little loss in CCA effectiveness. The interpretations of equations (3) and (4) remain as the expected number of undetected true terror plots in progress, and the expected number of detected terror plots awaiting interdiction. We will employ equation (4) throughout this paper, but note that all subsequent results are valid presuming the existence of fake plots as per the discussion 0 surrounding equation (5) (and the reader should substitute d for d where appropriate). 3. Attack level staffing The first staffing model considered is also the simplest: how many CCAs are needed in order to prevent a target percentage of terror attacks? Assuming that the terror plot arrival rate and all operational parameters governing terror plot duration and CCA detection and interdiction rates remain constant, the resulting model provides clean qualitative insight into the level of protection offered by different staffing levels while serving as a useful building block for the more complicated models intended to offer greater realism discussed later. We begin by noting that since the rate with which terror plots are interdicted is given by ry (Fig. 1 and equation (2)), the fraction of terror attacks prevented by a force of CCAs, denoted by q, is given by
q ¼
ry : a
(6)
4
E.H. Kaplan / Socio-Economic Planning Sciences 47 (2013) 2e8
Fig. 2. CCA attack level staffing for a ¼ 100; m ¼ 1; d ¼ 0.1 and r ¼ 4.
Substituting equation (4) into equation (6) and solving for f yields
f ¼
a m q qþ : r d1q
(7)
A plot of equation (7) as a function of q for the same parameters employed in producing Fig. 3 in [13] (a ¼ 100; m ¼ 1; d ¼ 0.1; and r ¼ 4) appears in Fig. 2. As clear from both equation (7) and Fig. 2, for values of q 1 (most terror plots succeed), the required number of CCAs grows linearly in q with rate a/r þ m/d, while as q approaches 1 (most terror plots are interdicted), the required number of CCAs grows rapidly, and is unbounded for q ¼ 1. Further insight into equation (7) follows from considering two extreme circumstances first introduced in Section 3.1 of [13]: “heavy traffic” when nearly all of the CCAs are simultaneously occupied interdicting terror plots (y z f), and “light traffic” when nearly all of the CCAs are available for detection (y z 0). In the heavy traffic case where y z f, Fig. 1 and equation (2) imply that terror plots are interdicted at approximately rate rf, which further implies that q, the fraction of plots interdicted, roughly equals rf/a. Thus, if y z f, then the number of CCAs must approximately equal aq/r, which is the first term in equation (7). Such a circumstance can arise if the terror plot initiation rate a becomes very large, in which case it is not at all surprising that all CCAs are almost always occupied (that is, y z f). This extreme circumstance resembles the “efficiency driven regime” familiar to call center staffing models where all servers are busy, and a fixed fraction of calls are lost (see [8] for details). At the other extreme, when y z 0, the system behaves as if virtually all CCAs are available for detection even though many agents could be occupied with fake plots (see again equation (5) and surrounding discussion). In this case, Fig. 1 and equation (1) tell us that the probability a newly arriving terror plot will be detected before it executes, which is the same as the probability a plot will eventually be interdicted, approximately equals df/ (df þ m). Equating this expression to q and solving for f reveals that the number of CCAs required to catch a fraction q of all terror plots is given by mq/(d(1q)) in light traffic; this is the second term in equation (7). This circumstance could arise if a is very small, or if the number of CCAs f is extremely large. In effective counterterrorism agencies, it should not be the case that all CCAs are occupied and unavailable for detecting ongoing terror plots. This suggests that the heavy traffic extreme y z f must be avoided. Indeed, to achieve very high detection levels on the
order of, say, 80% or greater, the second term of equation (7) suggests that many agents are needed, which should push the scenario much closer to the light traffic extreme. To illustrate, consider Israel’s battle against suicide bombers launched by terrorist groups such as Hamas, Islamic Jihad, or the Al Aqsa Martyr’s Brigade. As summarized in Section 4.2 of [13], between 2000 and 2007, Israel faced an average of a ¼ 85 attempted suicide bombings per year. The presumed mean time required to plan such attacks was m1 ¼ 3 months, while the rate with which detected suicide plots were interdicted was taken as r ¼ 16 interdictions per discovered plot per year (or just over 3 weeks on average; this same rate was assumed for the disposal rate j of fake plots). It was further postulated that only 10% of all terror plots interdicted resolved as actual planned suicide bombings, which results in a “self-initiation” fake plot detection rate l approximately equal to 70 per available CCA per year, and a true detection rate d of roughly 1.6 per plot per available CCA per year. Before applying equation (7), it is necessary to adjust the terror plot detection rate to account for the large percentage of CCAs occupied with fake plots. This is easily accomplished via equation (5) which, for this example, yields an adjusted detection rate of 1.6 16/(16 þ 70) z 0.30. While clearly any successful suicide bombing attack is one attack too many, equation (7) makes it clear that preventing all such attacks would in theory require an infinite number of CCAs. Suppose instead that the Israeli General Security Service (also known as the Shin Bet or Shabak) wanted to field a sufficient number of agents (or perhaps teams of agents) to prevent at least q ¼ 95% of all suicide bombing attacks. Following equation (7), the required number of CCAs would equal
f ¼
85 4 0:95 0:95 þ z260: 16 0:3 1 0:95
(8)
It is interesting to note that the actual number of successful suicide bombings in Israel averaged 19 per year between 2000 and 2007, so the actual fraction of such attacks prevented equals 66/ 85 ¼ 0.78. The example in [13] from which the parameter estimates above were taken presumed that there were 50 undercover teams deployed to detect and interdict attacks, so we see that for the parameters in question, while 50 CCAs are sufficient to prevent nearly 80% of attacks, the number of CCAs needed to prevent 95% of attacks is greater by more than a factor of five. As a second example, consider attempted Jihadi terror attacks in the United States since September 11, 2001. As stated on the Federal Bureau of Investigation (FBI) quick facts web page (http://www.fbi. gov/about-us/quick-facts), the FBI’s number one priority is: “Protect the United States from terrorist attack.” While it is difficult to determine the precise number of FBI CCAs operating in the field, according to a 2004 FBI report [4], since 9/11 the FBI “.increased the number of Special Agents working terrorism matters from 1351 to 2398.” Now, not all terror plots in the United States have Jihadi origins (indeed of 1054 terror-related prosecutions between 9/11/ 2001 through 6/30/2011 tracked by the Terrorist Trial Report Card project at New York University [23,24], 55% were motivated by Jihadi ideas), while not all FBI Special Agents operate covertly, though other law enforcement agencies such as the New York Police Department also deploy undercover officers to disrupt terror plots. For this example, we will set the number of CCAs focused on Jihadi threats at 1600 or roughly two-thirds of the reported number of FBI counterterrorism agents. As described in [15], there were an estimated 35 attempted Jihadi terror attacks on American soil between 9/11/2001 and 6/30/ 2011 (a similar estimate was reported in [22]), which provides an estimate of a ¼ 35/9.8 ¼ 3.57 new Jihadi plots per year. Of these 35 attempted attacks, seven succeeded in evading detection
E.H. Kaplan / Socio-Economic Planning Sciences 47 (2013) 2e8
(examples include Faisal Shazahd’s attempted bombing of Times Square and Nidal Hasan’s fatal shooting attack at Fort Hood) while the remaining 28 plots were interdicted, thus a fraction q ¼ 0.8 of Jihadi plots were prevented. Given the relatively low terror plot arrival rate coupled with the presumably high interdiction rate of terror plots post detection, it is clear that the second term in equation (7) will dominate the first for this example (meaning light traffic). This being the case, terror plots are detected at rate df per unit time (since y z 0 in light traffic), the mean time from the inception of a Jihadi terror plot until detection or execution, whichever comes first, equals (df þ m)1, and the probability that a new plot is detected before it is executed is given by q ¼ df/(df þ m). It has already been noted that 80% of Jihadi plots in the US were detected. In addition, a detailed review of news reports, CCA affidavits, indictments and other court documents provided the necessary data for estimating the mean time from the inception of a Jihadi terror plot in the United States until detection or execution, whichever came first; this mean duration was estimated as nine months [15]. This estimate suggests that (in years) (df þ m)1 ¼ 0.75, thus the aggregate detection rate dfzq(df þ m) ¼ 0.8/0.75 ¼ 16/15, while we also obtain m¼(1q)(df þ m) ¼ 4/15. Combining with our earlier assumption that there are 1600 CCAs tasked with detecting and interdicting Jihadi plots, the detection rate per CCA per unit time is given by d ¼ 16/15 ¼ 1/1500. That this figure is so low can be understood as a consequence of the very high rate of false positive detection errors in the United States (for evidence that this is indeed the case see [16]). While the 1600 CCAs in this example suffice to prevent 80% of Jihadi terror plots from executing, how many CCAs would be needed to prevent 95% of attempted attacks? The second term in equation (7) suggests an answer:
fz
m q 4=15 :95 ¼ ¼ 7; 600: d 1 q 1=1500 :05
(9)
Note that including the first term of equation (7) leaves matters essentially unchanged; even if it took as long as one year to interdict a detected Jihadi plot (r ¼ 1), only an additional aq ¼ 3.57 0.95 z 3 agents would be required (while increasing r to reflect more rapid interdiction would further lessen the importance of this term). Finally, while the staffing level above was estimated under light traffic conditions, one should not presume that most CCAs are not busy. Remember, the condition y z 0 means that CCAs are not occupied with real terror threats; the very low detection rate of 1/1500 obtained should be understood (via equation (5)) to imply that there is a very high rate of false positive detections, implying that many (and perhaps most) CCAs are in fact actively following suspicious plots, the overwhelming majority of which turn out to be fake. 4. Cost-benefit staffing While the attack level staffing model is easy to understand, it requires as input the target fraction of terror attacks prevented (q). Equation (7) and Fig. 2 make it clear that providing high levels of protection is very expensive in terms of the number of CCAs required, but the model does not offer guidance for choosing an appropriate value for q. This section presents a simple cost-benefit staffing model that explicitly translates the costs and benefits of preventing terror attacks to the safety level q which can then be used in equation (7) to determine the requisite number of CCAs. Let cG (G for government) represent the annual marginal cost of deploying an additional CCA, and bG denote the benefit of preventing a terror attack. Clearly bG > cG but the question is by how much. According to the FBI, newly hired Special Agents can, depending
5
upon where they work, expect to earn between $60,000 and $70,000 (https://www.fbijobs.gov/113.asp and http://www.opm.gov/oca/ 12tables/indexLEO.asp, both accessed on July 13, 2012). To account for the mix of experience among Special Agents while accounting for fringe benefits and non-salary expenses, the annual cost of deploying an additional CCA is perhaps as large as $300,000. The benefits of preventing terror attacks include saving lives, avoiding injuries and preventing infrastructure damage. Focusing solely on lives saved, different federal agencies offer different estimates for the value of a statistical life. For examples, the Department of Transportation, the Food and Drug Administration, and the Environmental Protection Agency value lives at $6 million, $8 million and $9 million respectively [1]. Furthermore, the Department of Homeland Security has argued that lives lost due to terrorism should be valued 100 times greater than lives lost for other reasons. These figures suggest that the benefits of preventing a terror attack are orders of magnitude larger than the cost of deploying an additional CCA. Given values for cG and bG (or just the benefit to cost ratio rG ¼ bG/cG), if the government seeks to select the fraction of attacks to prevent in order to maximize the net benefits of preventing terror plots from reaching execution, the government must solve
max bG aq cG f ðqÞ
0
(10)
where f(q) is the attack level staffing model of equation (7). This problem is equivalent to selecting q as the solution to
max rG aq f ðqÞ:
0
(11)
Substituting equation (7) into equation (11), differentiating with respect to q and setting the result equal to zero yields
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mr 1 q ¼ 1 : ad rG r 1 *
(12)
Clearly, the larger the benefit-to-cost ratio rG, the larger the fraction of attacks the government seeks to prevent. The optimal number of CCAs to deploy is given by equation (7) using the optimal protection level q*.3 In the case of Jihadi terror plots in the United States, the key parameters deduced in the prior section were m ¼ 4/15; d ¼ 1/1500; and a ¼ 3.57. For purposes of illustration, the interdiction rate r is set equal to 16/CCA/yr as in the Israeli suicide bombing example. With these parameters, and noting the estimate that 80% of Jihadi plots were interdicted in the United States, equation (12) suggests that the implied cost-to-benefit ratio rG equals approximately 2,100. Note that if the cost per CCA equals $300,000, then the implied benefit of preventing an attack, bG, equals $630 million dollars. Using the values of statistical lives saved cited for three different government agencies above, this implies benefits equivalent to saving between 70 and 105 lives per terror attack on average. If instead of preventing 80% of attacks, 95% were stopped, the implied benefit-to-cost ratio would equal 33,600, or in dollar terms benefits in excess of $10 billion per attack prevented, a number moving towards the Department of Homeland Security’s suggestion for how to value lives lost to terror attacks.
3 To ensure a solution to the optimization requires that rGr > 1, a condition that is practically guaranteed since both rG and r will greatly exceed unity in virtually any circumstance one can imagine.
6
E.H. Kaplan / Socio-Economic Planning Sciences 47 (2013) 2e8
5. Smart terrorists: a terror queue staffing game The original terror queue model presumes that new plots arise in accord with a homogeneous Poisson process with rate a [13], and while this assumption was made principally to simplify subsequent analysis, at least in the United States it has some empirical support [9,15,22]. However, terrorists choose their actions in order to achieve their own objectives, which could range from maximizing civilian casualties to winning government concessions to gaining control of territory or even an entire state [3,5,10]. In the realm of intelligence, even if terrorists cannot directly observe CCAs (undercover agents are covert after all), terrorists do see how often their attacks succeed, and on this basis could form inferences regarding the strength and/or effectiveness of the target state’s counterterror operations [6]. Suppose that terrorists strategically select their attack rate a to maximize their net benefits.4 Specifically, assume that terrorists derive a benefit bT for each successful attack, but expend a marginal cost of cT per plot whether successful or not. Further, suppose that the terrorists first select their attack rate after which the government chooses the number of CCAs to deploy, for absent the threat of terror attacks there would be no need for the government to deploy any CCAs.5 Then the terrorists must solve the following optimization problem:
* max bT a 1 q ðaÞ cT a a0
(13)
where q*(a) is given by equation (12) and represents the fraction of terror attacks government CCAs will interdict when terrorists attack with rate a. Letting the terrorists’ benefit-to-cost ratio rT ¼ bT/cT, differentiating equation (13) with respect to a and setting the result equal to zero yields the terrorists’ optimal attack rate a* :
a* ¼
rT2 mr 1 : 4 d rG r 1
(14)
Note that a* is increasing in rT (so the more beneficial terrorists perceive attacks relative to their costs, the greater the attempted number of attacks) and m (so the less time required to plan attacks, the greater the attempted number of attacks), but decreasing in d (better detection leads to fewer attempted attacks), r (better interdiction leads to fewer terror plots, presuming rGr>1 as previously discussed), and rG (so the greater the government’s benefit-tocost ratio of preventing attacks, the fewer the number of attacks attempted). Substituting equation (14) into equation (12) yields the government’s optimal target level (fraction of attacks to prevent) as
q* ¼ 1
2 rT
(15)
providing rT > 2, a mild requirement considering the relatively low costs of past terror attacks relative to their damage [7]. The government’s optimal target level depends solely upon the terrorists’ benefit-to-cost level rT; as this ratio grows, the government prevents a greater fraction of attacks by deploying (in accord with equation (7)) an increasing number of CCAs. That this is a sensible result follows from noting that if rT increases, so does the terrorists’ optimal
4 One could also envision a model where the terrorists control the rate with which available CCAs are exposed to fake plots. 5 Another reason for having the terrorists lead and the government follow is that were the order of play reversed, then for any number of CCAs f the government deploys, the terrorists can always achieve unbounded benefits by setting the attempted attack rate a very large providing that bT > cT; such a result is unrealistic.
Fig. 3. Equilibrium evasion probability as a function of attempted suicide bombing rates.
attack rate a* (equation (14)), while equation (12) makes clear that a net-benefit-maximizing government seeks to prevent a higher fraction of attacks as the rate of attempted attacks increases. To illustrate, data describing the number of attempted suicide bombings and the fraction of such attempts that were successful (that is, the evasion probability 1q) in Israel during the second intifada from 2001 to 2003 were presented in Fig. 1 of [12]. Imagine the suicide terrorists and the government playing the game outlined above during each month of 2001e2003. From equation (12), one should expect that the evasion probability relates to the attempted attack rate a as
k 1 q ¼ pffiffiffi
a
(16)
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi if k ¼ mr=ðdðrG r 1ÞÞ remains constant over time, which it would if government detection and interdiction capabilities remain unchanged (not including changes brought about by deploying additional CCAs) along with the government’s benefit-to-cost ratio of preventing attacks. Fig. 3 plots the data from [12] along with a maximum-likelihood fit of equation (16) which yields b k ¼ 0:74 (standard error ¼ 0.07). While not perfect by any means, the broad pattern in the data is consistent with equation (16).6 Fig. 3 reveals considerable variation in the evasion probability, yet equation (15) suggests that presuming no changes in detection/ interdiction capabilities or in the government’s benefit-to-cost ratio of preventing suicide bombing attacks, the only reason the evasion probability should change is if the terrorists’ benefit-tocost ratio rT also changed, with the evasion probability falling as rT increases. Conversely, equation (14) makes it clear that the equilibrium attempted attack rate a* should increase as rT grows. Fig. 4 reports both attempted suicide bombing rates and evasion probabilities for the 36 months of 2001e2003. While the data are noisy, it is clear that over time, the rate of attempted suicide bombing attacks grew while the evasion probability fell. Both of these could be explained by rT increasing over time, that is, the per attack benefits to the terrorists of successfully orchestrating a suicide bombing targeting Israelis (relative to the cost of mounting such attacks whether successful or not) greatly increased in the face of Israel’s crackdown on suicide terrorism, a crackdown that largely rested on greater deployment of and reliance on CCAs as suggested by equation (7) (for with the evasion probability falling from essentially 100% to very low levels, q and hence f
6 In [12], a negative exponential model for these data was proposed purely as a statistical summary; it is interesting to note that the inverse square root model derived here provides a significantly improved fit to the data (log likelihood ¼ 204) compared to the exponential (log likelihood ¼ 211).
E.H. Kaplan / Socio-Economic Planning Sciences 47 (2013) 2e8
7
attack minimizing allocation could be appropriate. Such an allocation could be achieved by solving
max
n X
ai qi
(20)
fi ðqi Þ f
(21)
i¼1
st
n X i¼1
0 qi 1 for i ¼ 1; 2; .; n:
Fig. 4. Attempted suicide bombing attacks and evasion probabilities over time.
increased greatly), and documented in [18]. Indeed, demonstrating the ability to execute a terror attack could well be more valuable to terrorists in circumstances where it is very difficult to evade detection and mount a successful terror operation.
6. Allocating covert counterterror agents The analysis thus far has been homogeneous in several respects: all terror plots have implicitly been assumed to target some geographic area over which all CCAs have been deployed, while all terror plots have been tacitly assumed to be equally serious (or cause equal damage). In this section we consider the problem of allocating CCAs across disjoint regions, terror organizations, plot types, or other relevant categories generically referred to as groups. To maintain consistency with our earlier notation, group i parameters will be represented by the same symbols used previously with subscript i attached. Suppose that there are f CCAs available, and the problem is to determine the number of these to allocate to each of n different groups. Starting again with a simple model, first suppose that the authorities seek to interdict the same fraction q of terror plots in each group. Defining fi(q) as in equation (7) but with appropriate subscripts for all parameters, this problem reduces to finding the root of the equation n X
fi ðqÞ ¼ f :
(17)
i¼1
Pn Pn Defining A ¼ i¼1 ai =ri and B ¼ i¼1 mi =di , the problem can be restated as finding the root of
Aq þ B
q 1q
¼ f:
(18)
This results in a quadratic equation with solution
q ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 A þ B þ f ðA þ B þ f Þ2 4Af : 2A
(19)
The number of CCAs allocated to group i then equals fi(q). In a sense, equation (19) represents a rough estimate of the detection/ interdiction capability of a force of f CCAs. Of course, other than simplicity and perhaps a desire for interdiction equity, there is no reason to insist upon equal detection fractions in the different groups considered. For example, if one is battling terrorists operating in different regions who tend to favor similar attacks that cause similar damage if successful, an expected
(22)
Or, if allocating agents to focus on different terror groups with some more dangerous than others in terms of, for example, the number of casualties they impose in successful attacks, then letting bi denote the benefit of preventing an attack by the ith group, the objective function in the allocation problem can be reformulated as P max ni¼1 bi ai qi with the same constraints. For example, turning again to the second intifada, data compiled in [20] report that of the 85 suicide bombings carried out within the 1967 borders of the State of Israel between 2001 and 2003, the 32 attacks carried out by Hamas killed 286 civilians (or 8.9 per attack), while the remaining 53 attacks carried out by members of the Palestinian Islamic Jihad, the Al Aqsa Martyrs Brigade and the Popular Front for the Liberation of Palestine killed 185 civilians (or 3.5 per attack; within these three groups, deaths per attack ranged from 3.2 to 3.8). Suppose that Hamas (other terror groups) attempted a1 ¼ 32 (a2 ¼ 53) suicide bombings per year, with deadliness per successful attack as computed above, while all other parameters are set consistently with the suicide bombing example from Section 3 (i.e. d1 ¼ d2 ¼ 0.3; r1 ¼ r2 ¼ 16; and m1 ¼ m2 ¼ 4). Suppose also that f ¼ 250 CCAs are to be divided between Hamas and the other terror groups. Table 1 reports the division of these 250 CCAs between Hamas and the other terror groups, the fraction of attacks detected for each group, and the numbers of attacks and casualties averted for allocations intended to equalize detection rates across groups (via equations (19) and (7)), maximize the number of attacks detected (equations (20)e(22)), and maximize the number of casualties P averted (max 2i¼1 bi ai qi subject to (21e22) where bi is the number of deaths per successful attack from group i). In terms of attacks and casualties averted, all three allocation criteria return comparable results. Regarding the actual deployment of CCAs, at first it might seem odd that both groups receive comparable agent allocations for the objective of equalizing the fraction of attacks detected given that the attempted attack rate is significantly lower for Hamas (32) than the other groups (53). However, this result is easily understood by noting that for the given data, the deployments fi(q) are dominated by the light traffic term (2nd term) in equation (7), and this term does not depend on a. The remaining results accord well with intuition; when the goal is to prevent as many attacks as possible, fewer than half of the CCAs are allocated to counter Hamas as fewer than half of all attacks are committed by that group, while when the goal is to avert as many casualties as possible, more agents are allocated to Hamas given that successful Hamas attacks
Table 1 Results for CCA allocation models. Criterion
f1
f2
q1
q2
Attacks averted
Casualties averted
Equalize q Max Attacks Detected Max Casualties Averted
124 107 139
126 143 111
0.90 0.89 0.91
0.90 0.91 0.89
76.7 76.8 76.3
424 422 425
8
E.H. Kaplan / Socio-Economic Planning Sciences 47 (2013) 2e8
cause more than twice the casualties as successful attacks from other terror groups. To see how strategic considerations might effect the deployment of CCAs across groups, one could also consider gametheoretic versions of the CCA allocation models discussed above. Such extensions would be similar in spirit to the force allocation games proposed in Section 4 of [17] for confronting entrenched insurgents. 7. Summary Operations researchers and management scientists have studied staffing models for a variety of service systems for many years. The present paper has introduced staffing models for covert counterterrorism agencies. Building upon the terror queue model of the detection and interdiction of terror plots, this paper has developed several models that address different staffing objectives, including attack level staffing (deploy sufficient agents to prevent a fraction q of terror attacks), cost-benefit staffing (deploy that number of agents that maximizes the net benefits of preventing terror attacks), a terror queue staffing game (deploy the optimal number of agents presuming that terrorists are smart and will infer your staffing level by observing the fraction of attacks interdicted), and allocating a fixed number of CCAs across different regions or groups to equalize detection probabilities, maximize the number of attacks prevented, or maximize the total benefits of preventing attacks. While several numerical examples informed by available unclassified data were presented to illustrate these models, no claim is being advanced that these models have been implemented in the field. Nonetheless, by highlighting the link between the detection and prevention of attacks and the deployment of undercover agents, our hope is that in view of these models, counterterrorism professionals and researchers will be motivated to think more carefully about counterterrorism staffing decisions.
[5] [6] [7]
[8]
[9] [10]
[11]
[12] [13] [14] [15] [16] [17] [18]
[19] [20]
[21] [22]
[23] [24]
Acknowledgments The author acknowledges the Stanford Graduate School of Business and the Yale School of Management for research support. References [1] Appelbaum B. As U.S. agencies put more value on a life, businesses fret. New York Times. p. A1, http://www.nytimes.com/2011/02/17/business/economy/ 17regulation.html?pagewanted¼all; February 17, 2011 [accessed on 13.7.12]. [2] Best RA. Intelligence to counter terrorism: issues for congress. Congressional research service report RL31292. Washington, DC: Congressional Research Service; 2003. [3] Enders W, Sandler T. The political economy of terrorism. Cambridge: Cambridge University Press; 2011. [4] Federal Bureau of Investigation. The FBI’s counterterrorism program since september 2001. Washington, DC: Federal Bureau of Investigation, http://www.
[25]
fbi.gov/stats-services/publications/fbi_ct_911com_0404.pdf; 2004 [accessed on 13.7.12]. Feinstein JS, Kaplan EH. Analysis of a strategic terror organization. Journal of Conflict Resolution 2010;54(2):281e302. Feinstein JS, Kaplan EH. Counterterror intelligence operations and terror attacks. Public Choice 2011;149(3e4):281e95. Financial Action Task Force. Terrorist financing, 2008. Paris: Organisation for Economic Cooperation and Development, http://www.fatf-gafi.org/dataoecd/ 28/43/40285899.pdf; 2008 [accessed on 15.7.12]. Garnett O, Mandelbaum A, Reiman M. Designing a call center with impatient customers. Manufacturing and Services Operations Management 2002;4(3): 208e27. Gleason JM. A poisson model of incidents of international terrorism in the United States. Terrorism 1980;4(1e4):259e65. Golany B, Kaplan EH, Marmur A, Rothblum UG. Nature plays with dice e terrorists do not: allocating resources to counter strategic versus probabilistic risks. European Journal of Operational Research 2009;192(1):198e208. Green LV. Capacity planning and management in hospitals. In: Brandeau ML, Sainfort F, Pierskalla WP, editors. Operations research and health care. Boston: Kluwer Academic Publishers; 2004. p. 15e41. Jacobson D, Kaplan E. Suicide bombings and targeted killings in (counter) terror games. Journal of Conflict Resolution 2007;51(5):772e92. Kaplan EH. Terror queues. Operations Research 2010;58(4):773e84. Kaplan EH. Intelligence operations research. Operations Research, 2012, in press. Kaplan EH. Estimating the duration of Jihadi terror plots in the United States. Studies in Conflict and Terrorism, 2012; 35(12), in press. Kaplan EH, Feinstein JS. Intel queues. Military Operations Research 2012; 17(1):17e30. Kaplan EH, Kress M, Szechtman R. Confronting entrenched insurgents. Operations Research 2010;58(2):329e41. Kaplan EH, Mintz A, Mishal S, Samban C. What happened to suicide bombings in Israel? insights from a terror stock model. Studies in Conflict and Terrorism 2005;28(3):225e35. Larson RC. Urban police patrol analysis. Cambridge, MA: MIT Press; 1972. Mintz A, Mishal S, Samban C. Suicide bombings in Israel. Data set. United Nations Studies, Yale University, and the Program in Foreign Policy Decision Making, Texas A&M University; 2004. Steele JM. Models for managing secrets. Management Science 1989;35(2): 240e8. Strom K, Hollywood J, Pope M, Weintraub G, Daye C, Gemeinhardt D. Building on clues: examining successes and failures in detecting us terrorist plots, 1999e2099. Research Triangle Park, NC: Institute for Homeland Security Solutions; 2010. Center on Law and Security. Terrorist trial report card: U.S. Edition. New York, NY: New York University School of Law, Center on Law and Security; 2005. Center on Law and Security. Terrorist trial report card: september 11, 2001September 11, 2011. New York, NY: New York University School of Law, Center on Law and Security; 2011. Walker WE, Chaiken JM, Ignall EJ, editors. Fire department deployment analysis. New York: North Holland; 1979.
Edward H. Kaplan is the William N. and Marie A. Beach Professor of Management Sciences, Professor of Public Health, and Professor of Engineering at Yale University Professor Kaplan’s research has been reported on the front pages of the New York Times and the Jerusalem Post, editorialized in the Wall Street Journal, recognized by the New York Times Magazine’s Year in Ideas, and discussed in many other major media outlets. The author of more than 125 research articles, Professor Kaplan received both the Lanchester Prize and the Edelman Award, two top honors in the operations research field, among many other awards. An elected member of both the National Academy of Engineering and the Institute of Medicine of the US National Academies, he has also twice received the prestigious Lady Davis Visiting Professorship at the Hebrew University of Jerusalem, where he has investigated AIDS policy issues facing theState of Israel. Kaplan’s current research focuses on the application of operations research to problems in counterterrorism and homeland security.
Contents lists available at SciVerse ScienceDirect
Socio-Economic Planning Sciences journal homepage: www.elsevier.com/locate/seps
Invited paper
Staffing models for covert counterterrorism agencies Edward H. Kaplan*,1 Yale University, School of Management, 135 Prospect Street, New Haven, Connecticut 06511, United States
a r t i c l e i n f o
a b s t r a c t
Article history: Available online 5 October 2012
This article presents staffing models for covert counterterrorism agencies such as the New York City Police Department, the US Federal Bureau of Investigation, Britain’s Security Service or the Israeli Shin Bet. The models ask how many good guys are needed to catch the bad guys, and how should agents be deployed? Building upon the terror queue model of the detection and interdiction of terror plots by undercover agents, the staffing models developed respond to objectives such as: prevent a specified fraction of terror attacks, maximize the benefits-minus-costs of preventing attacks, staff in expectation that smart terrorists will attack with a rate that optimizes their outcomes, and allocate a fixed number of agents across groups to equalize detection rates, or prevent as many attacks as possible, or prevent as many attack casualties as possible. Numerical examples based on published data describing counterterrorism operations in the United States and Israel are provided throughout. Ó 2012 Elsevier Ltd. All rights reserved.
Keywords: Counterterrorism Staffing models Terror queues Resource allocation
1. Introduction and motivation A recent Congressional Research Service report stated that “.intelligence to counter terrorism depends more on human intelligence (humint) such as spies and informers.” than other forms of intelligence [2]. In addition to such intelligence, the detection and interdiction of terrorists depends crucially upon the ability of covert counterterrorism agents (CCAs) to identify terror plots, infiltrate the relevant terror cell or organization, and intervene before such plots are executed [13]. It follows that the ability of counterterrorism agencies such as the New York Police Department (NYPD), Federal Bureau of Investigation (FBI), Britain’s Security Service (MI5), and Israel’s General Security Service (Shabak) to prevent terror attacks depends rather crucially on how many CCAs these agencies place in the field, and how such agents are allocated to focus upon threats emanating from different regions, population subgroups, or terror organizations. CCA staffing is thus another important area of application for intelligence operations research [14]. Operations researchers have long studied staffing problems in service systems such as police [19], fire [25], hospitals [11] and call centers [8] in addition to other service and manufacturing systems.
* Tel.: þ1 203 432 6031; fax: þ1 203 432 9995. E-mail address: [email protected]. 1 William N. and Marie A. Beach Professor of Management Sciences, Yale School of Management; Professor of Public Health, Yale School of Public Health; and Professor of Engineering, Yale School of Engineering and Applied Science. 0038-0121/$ e see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.seps.2012.09.006
To date no such application to covert counterterrorism agencies has appeared in the open literature. This article provides a first cut at this problem by proposing a collection of relatively simple (and hopefully easy to understand) CCA staffing models. The analysis builds upon the terror queue framework for modeling the detection and interdiction of terror plots by undercover agents introduced in [13]. In terror queue models, the customers are the terror plots, the servers are undercover agents or informants, service commences when a plot is detected, and service concludes when a plot is interdicted. However, unlike traditional queueing systems, the customers in terror queues do not wish to be located or served; indeed servers must find their customers, while customers actually hope to renege before receiving service as such queue abandonment represents the execution of a terror attack. In the next section, the necessary terror queue background is briefly reviewed along with those key results required for the new staffing models that follow. Section 3 presents attack level staffing whereby the problem of determining the number of CCAs required to prevent a specified fraction of attacks is formulated and solved within the terror queue framework. A virtue of this model is that, while of interest in its own right, it serves as a building block for more realistic models. Indeed, Section 4 determines the number of CCAs to deploy for an agency that seeks to maximize the net benefits of preventing terror attacks, that is, the model recognizes the cost of deploying agents in addition to the benefit of averting attacks. To this point in the paper, terrorist behavior is presumed exogenous and fixed, but unlike natural risks, a large literature recognizes terrorists as adversarial opponents who actively choose
E.H. Kaplan / Socio-Economic Planning Sciences 47 (2013) 2e8
x ¼
(3)
2m
and
y ¼
3
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ða rf mr=dÞ þ ða þ rf þ mr=dÞ 4arf
ða þ rf þ mr=dÞ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ða þ rf þ mr=dÞ 4arf 2r
:
(4)
Fig. 1. The flow of terror plots [13].
how, when and where to mount attacks [3,5,6,10]. Section 5 thus considers smart terrorists who choose how often to attack so as to maximize the difference between their benefits and costs of attacking, and illustrates the resulting equilibrium assuming the government responds using its own cost-benefit calculation as outlined earlier in Section 4. Finally, Section 6 considers simple models for allocating a fixed number of CCAs across different regions, subpopulations or terror organizations to achieve objectives ranging from equalizing detection rates to maximizing the number of casualties averted. While the data necessary for applying these models in actual counterterrorism agencies remain highly classified, several numerical examples based on published data describing counterterrorism operations in the United States and Israel are provided throughout. Section 7 summarizes the paper. 2. Terror queue review This section briefly reviews the key terror queue results required for developing CCA staffing models (see [13] for further details). As illustrated in Fig. 1 , new terror plots arise with rate a per unit time. Available CCAs detect ongoing but as yet undiscovered (and unexecuted) terror plots at a rate that is proportional to both the number of undetected plots and the number of available agents. Thus, if there are X undetected terror plots (X is the terror queue of plots waiting to execute or be detected, whichever comes first), and Y out of the total force of f CCAs are occupied and unavailable for detecting plots, then the aggregate rate with which plots are detected (and available CCAs become unavailable for detection) equals dX(fY) per unit time. Terror plots on average require 1/m time units from inception to execution, thus the rate of successful terror attacks equals mX per unit time when there are X undetected plots. Finally, detected plots are interdicted at rate r per detected plot (equivalently per busy CCA), for an aggregate interdiction rate of rY interdicted terror plots per unit time. Section 3 of [13] shows that the Markov population model associated with Fig. 1 above can be very well-approximated by a continuous Ornstein-Uhlenbeck process. In particular, the expected number of undetected (detected) terror plots, E(X) (respectively E(Y)), can be closely approximated by the equilibrium quantities x and y that solve the following deterministic flow equations:
a ¼ mx þ dxðf yÞ
(1)
and
dxðf yÞ ¼ ry:
(2)
Equation (1) states that newly arriving terror plots either result in terror attacks or are detected, while equation (2) says that detected terror plots are interdicted.2 These equations solve to yield
2 Modifying the model to allow some detected plots to proceed to execution without interdiction is straightforward; see [16].
The results above presume that CCAs never commit false positive errors, that is, that while not all terror plots are detected, any detection claim made by a CCA is correct. However, often intelligence agents do apprehend suspects who ultimately have no connections to terrorism [13,16]. The relatively large cost of failing to detect a real terror plot compared to monitoring (and sometimes detaining) suspects innocent of terrorism is one reason for CCAs to “overdetect” [16], while in addition it is advantageous to terrorists to deliberately broadcast false information pointing to fake plots as a means of guarding the secret planning of true plots [21]. This concern is addressed in Section 4 of [13] by keeping track of CCAs who are unavailable for detection on account of being occupied with “fake plots.” Specifically, it is assumed that the rate with which available CCAs commit to following fake plots equals l per available CCA per unit time, while the rate with which CCAs so committed return to availability by disposing of fake plots equals j per unit time per CCA occupied with fake plots. Under these assumptions, it is shown in [13] that after defining
d0 ¼ d
j ; lþj
(5) 0
equations (3) and (4) remain correct upon substituting d for d. Equation (5) shows that if the rate at which CCAs commit to fake plots (l) is very high relative to the rate with which fake plots can be identified as such and disposed (j), the consequence is equivalent to fielding a force of CCAs who never commit false positive errors but have a greatly reduced ability to detect actual plots (as 0 d d), while if fake plots can be identified very quickly (meaning j [ l), there is little loss in CCA effectiveness. The interpretations of equations (3) and (4) remain as the expected number of undetected true terror plots in progress, and the expected number of detected terror plots awaiting interdiction. We will employ equation (4) throughout this paper, but note that all subsequent results are valid presuming the existence of fake plots as per the discussion 0 surrounding equation (5) (and the reader should substitute d for d where appropriate). 3. Attack level staffing The first staffing model considered is also the simplest: how many CCAs are needed in order to prevent a target percentage of terror attacks? Assuming that the terror plot arrival rate and all operational parameters governing terror plot duration and CCA detection and interdiction rates remain constant, the resulting model provides clean qualitative insight into the level of protection offered by different staffing levels while serving as a useful building block for the more complicated models intended to offer greater realism discussed later. We begin by noting that since the rate with which terror plots are interdicted is given by ry (Fig. 1 and equation (2)), the fraction of terror attacks prevented by a force of CCAs, denoted by q, is given by
q ¼
ry : a
(6)
4
E.H. Kaplan / Socio-Economic Planning Sciences 47 (2013) 2e8
Fig. 2. CCA attack level staffing for a ¼ 100; m ¼ 1; d ¼ 0.1 and r ¼ 4.
Substituting equation (4) into equation (6) and solving for f yields
f ¼
a m q qþ : r d1q
(7)
A plot of equation (7) as a function of q for the same parameters employed in producing Fig. 3 in [13] (a ¼ 100; m ¼ 1; d ¼ 0.1; and r ¼ 4) appears in Fig. 2. As clear from both equation (7) and Fig. 2, for values of q 1 (most terror plots succeed), the required number of CCAs grows linearly in q with rate a/r þ m/d, while as q approaches 1 (most terror plots are interdicted), the required number of CCAs grows rapidly, and is unbounded for q ¼ 1. Further insight into equation (7) follows from considering two extreme circumstances first introduced in Section 3.1 of [13]: “heavy traffic” when nearly all of the CCAs are simultaneously occupied interdicting terror plots (y z f), and “light traffic” when nearly all of the CCAs are available for detection (y z 0). In the heavy traffic case where y z f, Fig. 1 and equation (2) imply that terror plots are interdicted at approximately rate rf, which further implies that q, the fraction of plots interdicted, roughly equals rf/a. Thus, if y z f, then the number of CCAs must approximately equal aq/r, which is the first term in equation (7). Such a circumstance can arise if the terror plot initiation rate a becomes very large, in which case it is not at all surprising that all CCAs are almost always occupied (that is, y z f). This extreme circumstance resembles the “efficiency driven regime” familiar to call center staffing models where all servers are busy, and a fixed fraction of calls are lost (see [8] for details). At the other extreme, when y z 0, the system behaves as if virtually all CCAs are available for detection even though many agents could be occupied with fake plots (see again equation (5) and surrounding discussion). In this case, Fig. 1 and equation (1) tell us that the probability a newly arriving terror plot will be detected before it executes, which is the same as the probability a plot will eventually be interdicted, approximately equals df/ (df þ m). Equating this expression to q and solving for f reveals that the number of CCAs required to catch a fraction q of all terror plots is given by mq/(d(1q)) in light traffic; this is the second term in equation (7). This circumstance could arise if a is very small, or if the number of CCAs f is extremely large. In effective counterterrorism agencies, it should not be the case that all CCAs are occupied and unavailable for detecting ongoing terror plots. This suggests that the heavy traffic extreme y z f must be avoided. Indeed, to achieve very high detection levels on the
order of, say, 80% or greater, the second term of equation (7) suggests that many agents are needed, which should push the scenario much closer to the light traffic extreme. To illustrate, consider Israel’s battle against suicide bombers launched by terrorist groups such as Hamas, Islamic Jihad, or the Al Aqsa Martyr’s Brigade. As summarized in Section 4.2 of [13], between 2000 and 2007, Israel faced an average of a ¼ 85 attempted suicide bombings per year. The presumed mean time required to plan such attacks was m1 ¼ 3 months, while the rate with which detected suicide plots were interdicted was taken as r ¼ 16 interdictions per discovered plot per year (or just over 3 weeks on average; this same rate was assumed for the disposal rate j of fake plots). It was further postulated that only 10% of all terror plots interdicted resolved as actual planned suicide bombings, which results in a “self-initiation” fake plot detection rate l approximately equal to 70 per available CCA per year, and a true detection rate d of roughly 1.6 per plot per available CCA per year. Before applying equation (7), it is necessary to adjust the terror plot detection rate to account for the large percentage of CCAs occupied with fake plots. This is easily accomplished via equation (5) which, for this example, yields an adjusted detection rate of 1.6 16/(16 þ 70) z 0.30. While clearly any successful suicide bombing attack is one attack too many, equation (7) makes it clear that preventing all such attacks would in theory require an infinite number of CCAs. Suppose instead that the Israeli General Security Service (also known as the Shin Bet or Shabak) wanted to field a sufficient number of agents (or perhaps teams of agents) to prevent at least q ¼ 95% of all suicide bombing attacks. Following equation (7), the required number of CCAs would equal
f ¼
85 4 0:95 0:95 þ z260: 16 0:3 1 0:95
(8)
It is interesting to note that the actual number of successful suicide bombings in Israel averaged 19 per year between 2000 and 2007, so the actual fraction of such attacks prevented equals 66/ 85 ¼ 0.78. The example in [13] from which the parameter estimates above were taken presumed that there were 50 undercover teams deployed to detect and interdict attacks, so we see that for the parameters in question, while 50 CCAs are sufficient to prevent nearly 80% of attacks, the number of CCAs needed to prevent 95% of attacks is greater by more than a factor of five. As a second example, consider attempted Jihadi terror attacks in the United States since September 11, 2001. As stated on the Federal Bureau of Investigation (FBI) quick facts web page (http://www.fbi. gov/about-us/quick-facts), the FBI’s number one priority is: “Protect the United States from terrorist attack.” While it is difficult to determine the precise number of FBI CCAs operating in the field, according to a 2004 FBI report [4], since 9/11 the FBI “.increased the number of Special Agents working terrorism matters from 1351 to 2398.” Now, not all terror plots in the United States have Jihadi origins (indeed of 1054 terror-related prosecutions between 9/11/ 2001 through 6/30/2011 tracked by the Terrorist Trial Report Card project at New York University [23,24], 55% were motivated by Jihadi ideas), while not all FBI Special Agents operate covertly, though other law enforcement agencies such as the New York Police Department also deploy undercover officers to disrupt terror plots. For this example, we will set the number of CCAs focused on Jihadi threats at 1600 or roughly two-thirds of the reported number of FBI counterterrorism agents. As described in [15], there were an estimated 35 attempted Jihadi terror attacks on American soil between 9/11/2001 and 6/30/ 2011 (a similar estimate was reported in [22]), which provides an estimate of a ¼ 35/9.8 ¼ 3.57 new Jihadi plots per year. Of these 35 attempted attacks, seven succeeded in evading detection
E.H. Kaplan / Socio-Economic Planning Sciences 47 (2013) 2e8
(examples include Faisal Shazahd’s attempted bombing of Times Square and Nidal Hasan’s fatal shooting attack at Fort Hood) while the remaining 28 plots were interdicted, thus a fraction q ¼ 0.8 of Jihadi plots were prevented. Given the relatively low terror plot arrival rate coupled with the presumably high interdiction rate of terror plots post detection, it is clear that the second term in equation (7) will dominate the first for this example (meaning light traffic). This being the case, terror plots are detected at rate df per unit time (since y z 0 in light traffic), the mean time from the inception of a Jihadi terror plot until detection or execution, whichever comes first, equals (df þ m)1, and the probability that a new plot is detected before it is executed is given by q ¼ df/(df þ m). It has already been noted that 80% of Jihadi plots in the US were detected. In addition, a detailed review of news reports, CCA affidavits, indictments and other court documents provided the necessary data for estimating the mean time from the inception of a Jihadi terror plot in the United States until detection or execution, whichever came first; this mean duration was estimated as nine months [15]. This estimate suggests that (in years) (df þ m)1 ¼ 0.75, thus the aggregate detection rate dfzq(df þ m) ¼ 0.8/0.75 ¼ 16/15, while we also obtain m¼(1q)(df þ m) ¼ 4/15. Combining with our earlier assumption that there are 1600 CCAs tasked with detecting and interdicting Jihadi plots, the detection rate per CCA per unit time is given by d ¼ 16/15 ¼ 1/1500. That this figure is so low can be understood as a consequence of the very high rate of false positive detection errors in the United States (for evidence that this is indeed the case see [16]). While the 1600 CCAs in this example suffice to prevent 80% of Jihadi terror plots from executing, how many CCAs would be needed to prevent 95% of attempted attacks? The second term in equation (7) suggests an answer:
fz
m q 4=15 :95 ¼ ¼ 7; 600: d 1 q 1=1500 :05
(9)
Note that including the first term of equation (7) leaves matters essentially unchanged; even if it took as long as one year to interdict a detected Jihadi plot (r ¼ 1), only an additional aq ¼ 3.57 0.95 z 3 agents would be required (while increasing r to reflect more rapid interdiction would further lessen the importance of this term). Finally, while the staffing level above was estimated under light traffic conditions, one should not presume that most CCAs are not busy. Remember, the condition y z 0 means that CCAs are not occupied with real terror threats; the very low detection rate of 1/1500 obtained should be understood (via equation (5)) to imply that there is a very high rate of false positive detections, implying that many (and perhaps most) CCAs are in fact actively following suspicious plots, the overwhelming majority of which turn out to be fake. 4. Cost-benefit staffing While the attack level staffing model is easy to understand, it requires as input the target fraction of terror attacks prevented (q). Equation (7) and Fig. 2 make it clear that providing high levels of protection is very expensive in terms of the number of CCAs required, but the model does not offer guidance for choosing an appropriate value for q. This section presents a simple cost-benefit staffing model that explicitly translates the costs and benefits of preventing terror attacks to the safety level q which can then be used in equation (7) to determine the requisite number of CCAs. Let cG (G for government) represent the annual marginal cost of deploying an additional CCA, and bG denote the benefit of preventing a terror attack. Clearly bG > cG but the question is by how much. According to the FBI, newly hired Special Agents can, depending
5
upon where they work, expect to earn between $60,000 and $70,000 (https://www.fbijobs.gov/113.asp and http://www.opm.gov/oca/ 12tables/indexLEO.asp, both accessed on July 13, 2012). To account for the mix of experience among Special Agents while accounting for fringe benefits and non-salary expenses, the annual cost of deploying an additional CCA is perhaps as large as $300,000. The benefits of preventing terror attacks include saving lives, avoiding injuries and preventing infrastructure damage. Focusing solely on lives saved, different federal agencies offer different estimates for the value of a statistical life. For examples, the Department of Transportation, the Food and Drug Administration, and the Environmental Protection Agency value lives at $6 million, $8 million and $9 million respectively [1]. Furthermore, the Department of Homeland Security has argued that lives lost due to terrorism should be valued 100 times greater than lives lost for other reasons. These figures suggest that the benefits of preventing a terror attack are orders of magnitude larger than the cost of deploying an additional CCA. Given values for cG and bG (or just the benefit to cost ratio rG ¼ bG/cG), if the government seeks to select the fraction of attacks to prevent in order to maximize the net benefits of preventing terror plots from reaching execution, the government must solve
max bG aq cG f ðqÞ
0
(10)
where f(q) is the attack level staffing model of equation (7). This problem is equivalent to selecting q as the solution to
max rG aq f ðqÞ:
0
(11)
Substituting equation (7) into equation (11), differentiating with respect to q and setting the result equal to zero yields
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mr 1 q ¼ 1 : ad rG r 1 *
(12)
Clearly, the larger the benefit-to-cost ratio rG, the larger the fraction of attacks the government seeks to prevent. The optimal number of CCAs to deploy is given by equation (7) using the optimal protection level q*.3 In the case of Jihadi terror plots in the United States, the key parameters deduced in the prior section were m ¼ 4/15; d ¼ 1/1500; and a ¼ 3.57. For purposes of illustration, the interdiction rate r is set equal to 16/CCA/yr as in the Israeli suicide bombing example. With these parameters, and noting the estimate that 80% of Jihadi plots were interdicted in the United States, equation (12) suggests that the implied cost-to-benefit ratio rG equals approximately 2,100. Note that if the cost per CCA equals $300,000, then the implied benefit of preventing an attack, bG, equals $630 million dollars. Using the values of statistical lives saved cited for three different government agencies above, this implies benefits equivalent to saving between 70 and 105 lives per terror attack on average. If instead of preventing 80% of attacks, 95% were stopped, the implied benefit-to-cost ratio would equal 33,600, or in dollar terms benefits in excess of $10 billion per attack prevented, a number moving towards the Department of Homeland Security’s suggestion for how to value lives lost to terror attacks.
3 To ensure a solution to the optimization requires that rGr > 1, a condition that is practically guaranteed since both rG and r will greatly exceed unity in virtually any circumstance one can imagine.
6
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5. Smart terrorists: a terror queue staffing game The original terror queue model presumes that new plots arise in accord with a homogeneous Poisson process with rate a [13], and while this assumption was made principally to simplify subsequent analysis, at least in the United States it has some empirical support [9,15,22]. However, terrorists choose their actions in order to achieve their own objectives, which could range from maximizing civilian casualties to winning government concessions to gaining control of territory or even an entire state [3,5,10]. In the realm of intelligence, even if terrorists cannot directly observe CCAs (undercover agents are covert after all), terrorists do see how often their attacks succeed, and on this basis could form inferences regarding the strength and/or effectiveness of the target state’s counterterror operations [6]. Suppose that terrorists strategically select their attack rate a to maximize their net benefits.4 Specifically, assume that terrorists derive a benefit bT for each successful attack, but expend a marginal cost of cT per plot whether successful or not. Further, suppose that the terrorists first select their attack rate after which the government chooses the number of CCAs to deploy, for absent the threat of terror attacks there would be no need for the government to deploy any CCAs.5 Then the terrorists must solve the following optimization problem:
* max bT a 1 q ðaÞ cT a a0
(13)
where q*(a) is given by equation (12) and represents the fraction of terror attacks government CCAs will interdict when terrorists attack with rate a. Letting the terrorists’ benefit-to-cost ratio rT ¼ bT/cT, differentiating equation (13) with respect to a and setting the result equal to zero yields the terrorists’ optimal attack rate a* :
a* ¼
rT2 mr 1 : 4 d rG r 1
(14)
Note that a* is increasing in rT (so the more beneficial terrorists perceive attacks relative to their costs, the greater the attempted number of attacks) and m (so the less time required to plan attacks, the greater the attempted number of attacks), but decreasing in d (better detection leads to fewer attempted attacks), r (better interdiction leads to fewer terror plots, presuming rGr>1 as previously discussed), and rG (so the greater the government’s benefit-tocost ratio of preventing attacks, the fewer the number of attacks attempted). Substituting equation (14) into equation (12) yields the government’s optimal target level (fraction of attacks to prevent) as
q* ¼ 1
2 rT
(15)
providing rT > 2, a mild requirement considering the relatively low costs of past terror attacks relative to their damage [7]. The government’s optimal target level depends solely upon the terrorists’ benefit-to-cost level rT; as this ratio grows, the government prevents a greater fraction of attacks by deploying (in accord with equation (7)) an increasing number of CCAs. That this is a sensible result follows from noting that if rT increases, so does the terrorists’ optimal
4 One could also envision a model where the terrorists control the rate with which available CCAs are exposed to fake plots. 5 Another reason for having the terrorists lead and the government follow is that were the order of play reversed, then for any number of CCAs f the government deploys, the terrorists can always achieve unbounded benefits by setting the attempted attack rate a very large providing that bT > cT; such a result is unrealistic.
Fig. 3. Equilibrium evasion probability as a function of attempted suicide bombing rates.
attack rate a* (equation (14)), while equation (12) makes clear that a net-benefit-maximizing government seeks to prevent a higher fraction of attacks as the rate of attempted attacks increases. To illustrate, data describing the number of attempted suicide bombings and the fraction of such attempts that were successful (that is, the evasion probability 1q) in Israel during the second intifada from 2001 to 2003 were presented in Fig. 1 of [12]. Imagine the suicide terrorists and the government playing the game outlined above during each month of 2001e2003. From equation (12), one should expect that the evasion probability relates to the attempted attack rate a as
k 1 q ¼ pffiffiffi
a
(16)
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi if k ¼ mr=ðdðrG r 1ÞÞ remains constant over time, which it would if government detection and interdiction capabilities remain unchanged (not including changes brought about by deploying additional CCAs) along with the government’s benefit-to-cost ratio of preventing attacks. Fig. 3 plots the data from [12] along with a maximum-likelihood fit of equation (16) which yields b k ¼ 0:74 (standard error ¼ 0.07). While not perfect by any means, the broad pattern in the data is consistent with equation (16).6 Fig. 3 reveals considerable variation in the evasion probability, yet equation (15) suggests that presuming no changes in detection/ interdiction capabilities or in the government’s benefit-to-cost ratio of preventing suicide bombing attacks, the only reason the evasion probability should change is if the terrorists’ benefit-tocost ratio rT also changed, with the evasion probability falling as rT increases. Conversely, equation (14) makes it clear that the equilibrium attempted attack rate a* should increase as rT grows. Fig. 4 reports both attempted suicide bombing rates and evasion probabilities for the 36 months of 2001e2003. While the data are noisy, it is clear that over time, the rate of attempted suicide bombing attacks grew while the evasion probability fell. Both of these could be explained by rT increasing over time, that is, the per attack benefits to the terrorists of successfully orchestrating a suicide bombing targeting Israelis (relative to the cost of mounting such attacks whether successful or not) greatly increased in the face of Israel’s crackdown on suicide terrorism, a crackdown that largely rested on greater deployment of and reliance on CCAs as suggested by equation (7) (for with the evasion probability falling from essentially 100% to very low levels, q and hence f
6 In [12], a negative exponential model for these data was proposed purely as a statistical summary; it is interesting to note that the inverse square root model derived here provides a significantly improved fit to the data (log likelihood ¼ 204) compared to the exponential (log likelihood ¼ 211).
E.H. Kaplan / Socio-Economic Planning Sciences 47 (2013) 2e8
7
attack minimizing allocation could be appropriate. Such an allocation could be achieved by solving
max
n X
ai qi
(20)
fi ðqi Þ f
(21)
i¼1
st
n X i¼1
0 qi 1 for i ¼ 1; 2; .; n:
Fig. 4. Attempted suicide bombing attacks and evasion probabilities over time.
increased greatly), and documented in [18]. Indeed, demonstrating the ability to execute a terror attack could well be more valuable to terrorists in circumstances where it is very difficult to evade detection and mount a successful terror operation.
6. Allocating covert counterterror agents The analysis thus far has been homogeneous in several respects: all terror plots have implicitly been assumed to target some geographic area over which all CCAs have been deployed, while all terror plots have been tacitly assumed to be equally serious (or cause equal damage). In this section we consider the problem of allocating CCAs across disjoint regions, terror organizations, plot types, or other relevant categories generically referred to as groups. To maintain consistency with our earlier notation, group i parameters will be represented by the same symbols used previously with subscript i attached. Suppose that there are f CCAs available, and the problem is to determine the number of these to allocate to each of n different groups. Starting again with a simple model, first suppose that the authorities seek to interdict the same fraction q of terror plots in each group. Defining fi(q) as in equation (7) but with appropriate subscripts for all parameters, this problem reduces to finding the root of the equation n X
fi ðqÞ ¼ f :
(17)
i¼1
Pn Pn Defining A ¼ i¼1 ai =ri and B ¼ i¼1 mi =di , the problem can be restated as finding the root of
Aq þ B
q 1q
¼ f:
(18)
This results in a quadratic equation with solution
q ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 A þ B þ f ðA þ B þ f Þ2 4Af : 2A
(19)
The number of CCAs allocated to group i then equals fi(q). In a sense, equation (19) represents a rough estimate of the detection/ interdiction capability of a force of f CCAs. Of course, other than simplicity and perhaps a desire for interdiction equity, there is no reason to insist upon equal detection fractions in the different groups considered. For example, if one is battling terrorists operating in different regions who tend to favor similar attacks that cause similar damage if successful, an expected
(22)
Or, if allocating agents to focus on different terror groups with some more dangerous than others in terms of, for example, the number of casualties they impose in successful attacks, then letting bi denote the benefit of preventing an attack by the ith group, the objective function in the allocation problem can be reformulated as P max ni¼1 bi ai qi with the same constraints. For example, turning again to the second intifada, data compiled in [20] report that of the 85 suicide bombings carried out within the 1967 borders of the State of Israel between 2001 and 2003, the 32 attacks carried out by Hamas killed 286 civilians (or 8.9 per attack), while the remaining 53 attacks carried out by members of the Palestinian Islamic Jihad, the Al Aqsa Martyrs Brigade and the Popular Front for the Liberation of Palestine killed 185 civilians (or 3.5 per attack; within these three groups, deaths per attack ranged from 3.2 to 3.8). Suppose that Hamas (other terror groups) attempted a1 ¼ 32 (a2 ¼ 53) suicide bombings per year, with deadliness per successful attack as computed above, while all other parameters are set consistently with the suicide bombing example from Section 3 (i.e. d1 ¼ d2 ¼ 0.3; r1 ¼ r2 ¼ 16; and m1 ¼ m2 ¼ 4). Suppose also that f ¼ 250 CCAs are to be divided between Hamas and the other terror groups. Table 1 reports the division of these 250 CCAs between Hamas and the other terror groups, the fraction of attacks detected for each group, and the numbers of attacks and casualties averted for allocations intended to equalize detection rates across groups (via equations (19) and (7)), maximize the number of attacks detected (equations (20)e(22)), and maximize the number of casualties P averted (max 2i¼1 bi ai qi subject to (21e22) where bi is the number of deaths per successful attack from group i). In terms of attacks and casualties averted, all three allocation criteria return comparable results. Regarding the actual deployment of CCAs, at first it might seem odd that both groups receive comparable agent allocations for the objective of equalizing the fraction of attacks detected given that the attempted attack rate is significantly lower for Hamas (32) than the other groups (53). However, this result is easily understood by noting that for the given data, the deployments fi(q) are dominated by the light traffic term (2nd term) in equation (7), and this term does not depend on a. The remaining results accord well with intuition; when the goal is to prevent as many attacks as possible, fewer than half of the CCAs are allocated to counter Hamas as fewer than half of all attacks are committed by that group, while when the goal is to avert as many casualties as possible, more agents are allocated to Hamas given that successful Hamas attacks
Table 1 Results for CCA allocation models. Criterion
f1
f2
q1
q2
Attacks averted
Casualties averted
Equalize q Max Attacks Detected Max Casualties Averted
124 107 139
126 143 111
0.90 0.89 0.91
0.90 0.91 0.89
76.7 76.8 76.3
424 422 425
8
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cause more than twice the casualties as successful attacks from other terror groups. To see how strategic considerations might effect the deployment of CCAs across groups, one could also consider gametheoretic versions of the CCA allocation models discussed above. Such extensions would be similar in spirit to the force allocation games proposed in Section 4 of [17] for confronting entrenched insurgents. 7. Summary Operations researchers and management scientists have studied staffing models for a variety of service systems for many years. The present paper has introduced staffing models for covert counterterrorism agencies. Building upon the terror queue model of the detection and interdiction of terror plots, this paper has developed several models that address different staffing objectives, including attack level staffing (deploy sufficient agents to prevent a fraction q of terror attacks), cost-benefit staffing (deploy that number of agents that maximizes the net benefits of preventing terror attacks), a terror queue staffing game (deploy the optimal number of agents presuming that terrorists are smart and will infer your staffing level by observing the fraction of attacks interdicted), and allocating a fixed number of CCAs across different regions or groups to equalize detection probabilities, maximize the number of attacks prevented, or maximize the total benefits of preventing attacks. While several numerical examples informed by available unclassified data were presented to illustrate these models, no claim is being advanced that these models have been implemented in the field. Nonetheless, by highlighting the link between the detection and prevention of attacks and the deployment of undercover agents, our hope is that in view of these models, counterterrorism professionals and researchers will be motivated to think more carefully about counterterrorism staffing decisions.
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Edward H. Kaplan is the William N. and Marie A. Beach Professor of Management Sciences, Professor of Public Health, and Professor of Engineering at Yale University Professor Kaplan’s research has been reported on the front pages of the New York Times and the Jerusalem Post, editorialized in the Wall Street Journal, recognized by the New York Times Magazine’s Year in Ideas, and discussed in many other major media outlets. The author of more than 125 research articles, Professor Kaplan received both the Lanchester Prize and the Edelman Award, two top honors in the operations research field, among many other awards. An elected member of both the National Academy of Engineering and the Institute of Medicine of the US National Academies, he has also twice received the prestigious Lady Davis Visiting Professorship at the Hebrew University of Jerusalem, where he has investigated AIDS policy issues facing theState of Israel. Kaplan’s current research focuses on the application of operations research to problems in counterterrorism and homeland security.