STRATEGIC VERSUS NONSTRATEGIC TERRORISTS

Although the National Strategy for Homeland Security emphasizes the dominating fact that terrorists are strategic, early spending decisions on homeland security and, especially, on critical infrastructure protection have been widely criticized for being dominated by pork-barrel politics. As the 9/11 Commission Report delicately puts it, “In a free-for-all over money, it is understandable that representatives will work to protect the interests of their home states and districts. But this issue is too important for politics as usual to prevail. Resources must be allocated according to vulnerabilities” (2003, 396). ⁴

In keeping with this, the cornerstone of the National Infrastructure Protection Plan is “its risk management framework” (DHS 2006a, 35). This framework does take an adversary's intentions into account, but it does not treat adversaries as fully strategic actors who try to counter a defender's decisions, who “modify their tactics and targets to exploit perceived vulnerabilities and observed strengths” (White House 2003, viii). Not treating an adversary as fully strategic can lead to a significant misallocation of resources. This section illustrates the potential for misallocation with a simple model in which the defender only has two sites to protect. The next section extends the model to an arbitrary number of sites and develops a more general framework for analyzing the allocation problem.⁵

In the two-site model, the defender has a total of R resources to allocate and suffers a loss of L₁ if site 1 is successfully attacked and L₂ if site 2 is successfully attacked. Let r₁ represent the resources dedicated to protecting the first site and r₂ be the resources allocated to the second where r₁+r₂=R. Take v_j(r_j) to be the vulnerability of site j; that is, v_j(r_j) is the probability that an attack on site j succeeds if the defender allocates r_j to protecting it. Finally, the probability of an attack on site 1 is α₁ and the probability of an attack on site 2 is α₂ where α₁+α₂= 1. All of this implies that the defender's expected loss to allocating r and R−r to sites 1 and 2, respectively, is L(r)=α₁L₁v₁(r)+α₂L₂v₂(R−r). The first term on the right in this equality is the expected loss from an attack on the first site, L₁v₁(r), weighted by the probability of an attack on that site. The second term is the expected loss from an attack on the second site weighted by the probability of an attack on that site.

The basic elements of this model parallel the three key components of DHS's approach to risk management which are vulnerability, threat, and consequence. Vulnerability “is the probability that a particular attack will succeed against a particular target” (GAO 2005, 25), and this is precisely what v_j is. Threat “is the probability that a specific target is attacked in a specific way” (Willis et al. 2005, 8) and is formalized as the probability of attack α_j. Finally, the loss L_j corresponds to the consequence which “is the range of loss or damage that can be expected from a successful attack” (2006a, 41).⁶

Because resources are limited, investing more in one site means investing less in another. Figure 1(a) plots the defender's expected loss and the trade off it faces when the value of the first site relative to the second is 3/2 (i.e., L₁=3 and L₂=2), the probability of a successful attack on a site is

, and total resources are R=10. The relative odds of an attack on a site are also taken to be the same as that of the sites' relative value (i.e., α₁/α₂=3/2, which means α₁=3/5 and α₂=2/5).

Strategic versus Nonstrategic Terrorists

The defender's optimal allocation r*≈8.3 minimizes its expected loss. At this allocation, the marginal gain from spending more on protecting the first site is just offset by the marginal loss of having less to spend on the second site. More formally, the optimal risk-management allocation r* given the threat assessment α₁ satisfies the first-order condition dL(r*)/dr=0 or, equivalently, r* solves

.

But, equating the marginal costs and benefits in this way gives the wrong allocation against a strategic adversary. Figure 1(b) illustrates the misallocation. At r*, a terrorist group can impose a higher expected loss on the defender by attacking the second site rather than the first. Therefore a strategic adversary seeking to impose the highest expected cost on the defender will attack the second site.⁷

Put another way, the allocation r* and attack probability α=3/5 are not a Nash equilibrium of the underlying game. In a Nash equilibrium each actor plays optimally against the other; that is, each chooses a strategy which maximizes its payoff given the other actor's strategy. In this sense, r* and α are only “half” an equilibrium: the defender is playing optimally against the attacker's strategy α, but the attacker is not playing optimally against the defender. The allocation r* fails to reflect the strategic nature of the threat.

What is the optimal strategic allocation in this example? If the attacker is trying to impose the highest expected loss on the defender, the defender wants to minimize this maximum expected loss. Allocation

in Figure 1(b) does this. If the defender allocates less than this to protecting site 1, then that site becomes the most attractive target because it offers the attacker its highest expected payoff. If the defender spends more than

on site 1, say r*, site 2 becomes the most appealing target as striking that site now gives the attacker its highest expected payoff.

In brief, the risk-management allocation r* does take the adversary's intentions into account (through α). But this approach does not treat the adversary as fully strategic: if the terrorists anticipated the allocation r*, they would not want to behave in a way consistent with the threat assessment on which the allocation r* was based. Failing to take the strategic nature of the threat into account in this example leads to a significant misallocation of resources as the defender over spends on defending site 1 by more than 40% of the optimal allocation (

).

A BASIC FRAMEWORK

This section presents a basic game-theoretic framework for analyzing the resource-allocation problem. The solution to this baseline model also provides a simple algorithm for finding the unique optimal allocation of scarce resources against a strategic adversary. As will be seen, many sites may not receive any resources in the optimal allocation.

A defender must decide how to allocate R resources across S sites; a terrorist group then chooses which target to attack. In the static version of the game discussed here, the attacker is unable to observe the defender's allocation. (The dynamic version in which the attacker can see how the defender allocates its resources before deciding which site to hit is discussed below.) A strategy for the defender is simply a resource allocation (r₁, …, r_S) such that r₁+[sdot ][sdot ][sdot ]+r_S≤R; a strategy for the terrorist group is a set of probabilities (α₁, …, α_S), where α_j is the probability the terrorist group attacks site j with α₁+[sdot ][sdot ][sdot ]+α_S=1.

To specify the player's payoffs, suppose that the defender suffers a loss L_j>0 if site j is successfully attacked. Allocating more resources to a site lowers the probability of a successful attack but with diminishing effect. In symbols, the probability that an attack on site j succeeds is v_j(r_j) with

and

as long as this site is imperfectly defended (i.e., as long as v_j(r_j)>0).⁸

. As for the attacker's payoffs, let G_j>0 be the attacker's gain from successfully striking site j. Then the terrorist group's expected gain is

.

Finally resources are scarce in the sense that the defender cannot perfectly defend every target. That is, any allocation of R leaves one or more sites imperfectly defended.

ASSUMPTION 1. (SCARCE RESOURCES). For any allocation of R, v_j(r_j)>0 for some j.}

A Nash equilibrium is a pair of strategies (r, α) such that each player's strategy maximizes its payoff against the other player's strategy. If, therefore, the defender's allocation is (r₁, …, r_S), the terrorist group will attack the site offering the highest expected gain. That is, the attacker strikes a site j such that G_jv_j(r_j)=M_A(r) where M_A(r)≡max{G₁v₁(r₁), …, G_Sv_S(r_S)}.

A simple algorithm yields the defender's optimal allocation. Assume that the defender has not allocated any of its resources and that the sites are indexed so that if the defender allocates no resources to any site, the attacker's expected gain from striking site 1 is larger than its expected gain to striking site 2, which is larger than its expected gain to striking site 3, and so on. That is, G₁v₁(0)>G₂v₂(0)>[sdot ][sdot ][sdot ]>G_Sv_S(0).⁹

in Figure 2, where

, G₂v₂(0), …, G_Sv_S(0)}. From this point on, the defender must allocate resources to both sites 1 and 2 in order to keep G₁v₁(r₁)=G₂v₂(r₂). Otherwise the attacker would strike either site 1 or 2 depending on which offered the highest gain, and the resources allocated to the other site would effectively have been wasted.

The Optimal Allocation

When a Site Becomes Harder to Defend

The defender continues allocating resources to sites 1 and 2 keeping the attacker's payoffs to striking these sites equal until these two sites are no more attractive than site 3, that is, until

,

, G₃v₃(0), …, G_Sv_S(0)}. Now the defender allocates resources to sites 1, 2, and 3 so as to keep G₁v₁(r₁)=G₂v₂(r₂)=G₃v₃(r₃). The defender continues dividing its resources in this way reducing the attacker's maximum expected gain and having to spread its resources across more and more sites until it fully allocates its resources.

This allocation and the algorithm leading to it produce what may at first seem to be counterintuitive results. Suppose a site becomes uniformly more difficult to defend at the margin, that is, the marginal return to investing in protecting the site decreases. Formally, v_j(r) shifts to w_j(r), where w_j(0)=v_j(0) and

for all r. Intuitively, one might expect that this change would induce the defender to spend less on the site that is harder to secure, because this investment now does less to reduce the expected loss if the site is attacked. But this is not the case. As a site becomes more difficult to protect at the margin, the defender allocates more to that site and necessarily, less to the other sites. The attacker also becomes more likely to attack that site as shown below.

Figure 3 illustrates these results for the case in which site 3 becomes harder to protect and the original equilibrium allocation is

. If the original allocation remains unchanged, striking site 3 offers the attacker its highest payoff. But if the attacker is sure to go after this site, protecting any other site is a waste of resources and

is not an equilibrium allocation. Relative to

, the defender needs to spend more on site 3, which necessarily means spending less on the other sites. Equalizing the attacker's payoffs across the defended sties leads to allocation r* in Figure 3.

Stating the results formally, assume more generally the strict inequalities, mentioned previously, are weak so that G₁v₁(0)≥G₂v₂(0)≥[sdot ][sdot ][sdot ]≥G_Sv_S(0). Then the defender's optimal allocation minmaxes the attacker by minimizing the attacker's maximum payoff. That is, the defender's unique equilibrium allocation

solves min{M_A(r): r₁+[sdot ][sdot ][sdot ]+r_S≤R. Still more formally,

LEMMA 1. The defender has a unique minmax allocation.

Proposition 1. Nash equilibria of the static baseline game exist, and the defender plays the minmax allocation

in every Nash equilibrium.¹⁰

Proofs of these claims and the propositions below are in the Appendix. In related work, Powell (2006) analyzes the dynamic version of the baseline model in which the attacker decides which site to strike after seeing how the defender allocates its resources. He shows that subgame perfect equilibria exist and that the defender plays the minmax allocation

in all of them. The minmax allocation

thus turns out to be a remarkably robust equilibrium allocation, remaining optimal in both zero-sum and nonzero-sum versions of the game as well as in both static and dynamic versions.

These results are surprising in two ways. As is well known, minmax strategies are equilibrium strategies in two-actor, zero-sum games. The players' payoffs are diametrically opposed in these games so that anything that increases one actor's payoff decreases the other's. As a result, holding an adversary's payoff down as low as possible by minmaxing it also serves to maximize a player's own payoff and mutual minmaxing is an equilibrium. But the allocation problem described earlier is generally not zero sum, and minmax strategies are usually not equilibrium strategies in nonzero-sum games.

To see that the baseline allocation game is typically not zero sum and appreciate the significance of the zero-sum issue, it is useful to begin by noting the close relationship between the baseline model and “Colonel Blotto” games which have long been used to study force allocation issues (e.g., Blackett 1958; Shubik and Weber 1981; Tukey 1949). The classic blotto game is a two-actor, zero-sum game in which players A and B compete on S independent battlefields. Player A has F_A units of force to distribute among these battlefields, and player B has F_B. If A allocates

to battlefield k and B allocates

, A receives a payoff of

in the ensuing battle and B gets

. The players' payoffs in the game are the sums of their battlefield payoffs.

When the basic allocation game is zero sum, it is also a blotto game. If, for example, the defender's loss when a site is destroyed equals the attacker's gain; that is, if L_j=G_j for all sites j, then the defender's loss at j if it allocates r_j and the attacker “allocates” α_j is α_jL_jv_j(r_j) and exactly equal to the attacker's gain α_jG_jv_j(r_j).¹¹

But the basic allocation game is generally not zero sum and therefore not a blotto game. The defender and attacker's payoffs are not diametrically opposed if, for example, they rank the sites differently. Suppose more concretely G₁v₁(0)>G₂v₂(0) but L₁v₁(0)<L₂v₂(0), then attacking site 1 maximizes the attacker's payoff but does not minimize the defender's. Indeed, even if the the attacker and defender rank the sites in the same order (i.e., L_i>L_j [hArr ] G_i>G_j), the game will not be zero sum unless the defender and attacker put precisely the same relative weights on the sites (i.e., L_i/L_j=G_i/G_j for all i and j).

Relaxing the zero-sum assumption is important for two reasons. First, assuming that the terrorist's gain equals the defender's loss may not sound very demanding intuitively. But its formal equivalent—that the actors' put the same relative weights on the sites—seems much more demanding and less substantively appealing. Still more fundamentally, some situations simply cannot be modeled as zero-sum games. Indeed, neither the case in which threats have nonstrategic components nor the case in which the defender is uncertain of the attacker's payoffs are zero sum.

Why does the defender minmax the attacker in the basic allocation game even though it is not zero sum? Observe that regardless of how the defender ranks the sites, the optimal use of the defender's resources is to protect only those sites that will be attacked. Spending on sites that will not be attacked brings no gain and means that there is less to be spent on sites that will be attacked with positive probability. But only the sites offering the attacker the highest expected payoff will be attacked with positive probability and allocating resources solely to these sites has the effect of minimizing the attacker's maximum payoff.

The second surprising aspect of the robustness of the minmax allocation is that the defender's equilibrium allocation remains the same in both the static and dynamic versions of the game. These different settings typically lead to different equilibrium behavior (see Powell 2006 for an analysis of the dynamic model).

The emphasis in this analysis is on the defender's allocation. But it is useful to specify the attacker's equilibrium strategy

given the optimal minmax allocation

. Let T_A(r) be the set of sites giving the attacker its highest payoff at r. These are the sites the attacker might hit if the allocation were r. More formally, T_A(r) is the set of pure-strategy, best replies to

:

If the defender plays

, the attacker's payoff to striking any site in

is the same as its payoff to striking any other site in

and higher than its payoff to going after any site not in

. This implies that the probability the attacker strikes any site

is zero:

. As for the probabilities of striking sites in T_A,

must minimize the defender's expected loss

against the attacker's strategy

if

is to be an equilibrium. This requires the defender's marginal returns to investing in these sites be the same: ∂L/∂r_j=∂L/∂r_i for all

whenever r_j>0 and r_i>0.¹²

for j∈T_A. Adding the

's for j∈T_A and using the fact that these probabilities sum to one yields

. Corollary 1 summarizes these results:

COROLLARY The attacker's strategy in the generically unique equilibrium is to attack site

with probability

.

It follows that the probability of an attack on a site increases as that site becomes harder to defend. As shown previously, the defender spends more on a site (and necessarily less on the other sites) as that site becomes harder to defend at the margin. Because this site is both harder to defend and more is being spent on it, the marginal effect any additional investment has on reducing the probability that an attack on this site will succeed is lower. Given this lower marginal effect, the defender's larger investment in this site will only be optimal if the site is more likely to be attacked which raises the marginal expected gain from trying to protect this site enough to rationalize the defender's higher level of spending.¹³

BORDER DEFENSE, COUNTER-TERRORIST OPERATIONS, AND INTELLIGENCE

In addition to defending individual sites, the United States is attempting to “harden” its borders by making it more difficult for terrorists to enter and operate in the country. The United States also engages in significant counterterrorist operations throughout the world. And, good intelligence is seen to be critically important to disrupting terrorist attacks. Indeed, “intelligence and warning,” “border and transportation security,” and “domestic counter-terrorism” are three of the six critical mission areas specified in the National Strategy for Homeland Security (White House 2002). This section examines the trade off between defending individual sites as in the basic framework above and allocating resources to border defense, intelligence, or counterterrorist operations.

The key to introducing this trade off into the baseline model is that investing in these activities lowers the probability of a successful attack on all sites whereas investing in defending some sites only helps protect those sites. These activities “tend to reduce the overall level of risk without having to know in advance what the targets are, while also complementing site defenses” (O'Hanlon et al. 2003, 2).¹⁴

Consider border defense. Given the difficulty of securing the borders, it might seem that the marginal gain from investing more in border defense is likely to be very small and less than the marginal gain from spending that money on hardening, say, a nuclear power plant or a chemical facility. If so, why allocate anything to border defense? Would it not be better to devote everything to site defense? But surely spending nothing on border defense cannot be the right thing to do.

The analysis below explains why spending on activities that protect all of the sites but have a very low marginal return makes sense. The defender's optimal allocation does equate the marginal gain from investing in border defense with the marginal gain from site defense. But as the analysis of the baseline case shows, any additional spending on site defense must be spread across multiple sites in order to keep the attacker's payoff to striking any one of these sites equal to its payoff to hitting any other site. Consequently, the marginal gain to site defense depends on the number of sites being defended and decreases as that number increases. Indeed, if the number of defended sites is quite large, the marginal gain to additional spending on site defense is very small and eventually equals the marginal gain to border defense even if the latter is also quite small.

The allocation algorithm leading to these results is also closely related to the algorithm that yields the optimal allocation in the baseline case. Suppose that the defender allocates all of its resources to border defense and none to site defense. (To ease the exposition, “border defense” will be used broadly to include counterterrorist and intelligence activities as well.) Then the attacker would strike the site that offers the highest expected payoff given this allocation. If, therefore, the defender wants to hedge by dedicating some of its resources to site defense, it should spend them on that site. But the more the defender takes out of border defense and spends on that site, the harder that site becomes. Eventually attacking that site will be no more appealing than going after what was originally the second most attractive site. If at this point the defender wants to invest still more in site defense, it must devote those additional resources to protecting both of those sites. At some point the payoff to attacking either of those sites will be the same as that of attacking what was originally the third most attractive site. The defender must divide any further spending on site defense among those three targets. As the defender shifts more and more resources from border defense to site defense, it has to spread the latter over more and more sites. The optimal division of resources between border and site defense occurs when the marginal gain from allocating more to site defense is just offset by the marginal loss of spending less on border defense.

To incorporate border defense in the basic framework formally, suppose that the defender has to decide how much to direct to the S individual sites and to border defense which has a low marginal return but protects all of the sites. The probability that an attack on a site succeeds now depends on the resources allocated to that individual site and to border defense. More specifically, the probability an attack on j succeeds is the probability that the terrorist group penetrates the border times the probability of a successful attack on j given that the terrorists reach that site. In symbols, the probability that an attack on j succeeds is v_j(r_j)β(b), where β(b) is the probability that the terrorists penetrate the border if the defender allocates b to border defense. This probability decreases as more resources are spent on border defense with diminishing returns (β′<0 and β′′>0). To capture the idea that it is much easier to protect one site than secure the entire border, assume that if the defender only had one site to protect, it would always do better by spending on that site rather than on border defense, but the marginal returns to such spending are decreasing. Formally,

ASSUMPTION 2. (BORDERS ARE HARD TO DEFEND). ∂β(b−r_j)v_j(r_j)/∂r_j<0 and

for all sites j, any

, and any r≤b.

The defender's and attacker's payoffs are now

and

. As before, the terrorist group attacks sites offering the highest expected gain; that is, those j such that G_jv_j(r_j)β(b)=max \{G₁v₁(r₁)β(b), …, G_Sv_S(r_S)β(b)} = β(b)M_A(r), where r=(r₁, …, r_S) is the defender's allocation to site defense.

As will be seen, the same fundamental formal approach can be used to characterize the defender's optimal allocation when the defender can invest in border defense, faces threats with nonstrategic components, or is uncertain of the terrorist's priorities. The intuition underlying this approach is also straightforward. The algorithm derived in the baseline case can be used to define a curve through the space of possible allocations where movement along the curve defines the fundamental trade off the defender faces in the particular substantive case: spending more on border defense and less on site defense, more on the strategic component of the threat and less on the nonstrategic component, or more to protect against one type of terrorist and less on any other. The equilibrium allocations lie along the curve where the marginal gain of spending more is just offset by the marginal loss of spending less.

In the case of border defense, take allocation r=(r₁, …, r_S) with

to be a point in

at which the defender is spending

on site defense and b=R−σ on border defense. Now define the curve P_B through

parameterized by

. At P_B(σ), the defender spends σ on site defense with those resources optimally allocated against the attacker. That is, r(σ) is the unique allocation that minimizes M_A(r) subject to r₁+[sdot ][sdot ][sdot ]+r_S≤σ and can be found with the algorithm used to solve the baseline model.

We can now think of moving along P_B from P_B(0) to P_B(R) as the allocations that result from the defender's spending less on border defense in order to spend more on site defense while being sure to allocate however much it does spend on site defense optimally. It follows that any equilibrium allocations must lie on P_B. Otherwise the resources dedicated to site defense would not be allocated optimally. Moreover, the equilibria must lie at critical points along P_B, where the marginal gain from dedicating slightly more to site defense is just offset by the marginal loss of spending slightly less on border defense.

To formalize these marginal conditions, define a pseudo-equilibrium attack strategy

to be a best response to r that also rationalizes the defender's relative spending on the sites in T_A(r), where, recall, T_A(r) is set of attacker's pure-strategy, best replies to r. In the case of border defense, this means that

for all j∉T_A(r);

whenever j, k∈T_A(r) with r_j>0 and r_k>0; and

whenever j, k∈T_A(r) with r_j>0 and r_k=0.

Now observe that the defender's expected loss at point

can be written as Π_D(σ, α)β(R−σ), where

. It follows that the equilibrium allocations lie on P_B, where

. The expression on the left is the marginal reduction in the defender's expected loss from spending slightly more on site defense, whereas the right side is the marginal reduction due to spending slightly more on border defense.

This marginal condition is quite intuitive if all of the sites are identical; that is, L_j=L₁, G_j=G₁, and v_j=v₁ for 1≤j≤S. In these circumstances, the defender simply divides the resources spent on site defensive evenly across all of the sites, and the attacker strikes each site with equal probability. The defender's payoff therefore can be written as

. The first-order condition is

. Hence, the marginal benefit of spending more on site defense (the left side of the previous equation) is the marginal gain of spending more on a site divided by the number of sites. If the number of sites is large, this marginal gain to site defense is small.

Proposition 2 states the results in the more general case when the sites are not identical.

Proposition 2. Nash equilibria exist, and the defender always plays a pure strategy lying on P_B where the marginal loss of investing less on border defense just offsets the marginal gain of spending more on site defense. This condition generically satisfies

which is equivalent to:¹⁵

THREATS WITH STRATEGIC AND NONSTRATEGIC COMPONENTS

Some threats entail both a strategic and nonstrategic component and defending against one kind of threat also protects against the other. Chemical facilities in densely populated areas may pose a risk because of accidents as well as terrorism. For example, the EPA report on the risks posed by chemical accidents (Belke 2000) is now frequently cited in connection with the dangers of terrorist attacks (e.g., Flynn 2004). Many public health measures taken to detect and contain contagious diseases defend against natural outbreaks as well as deliberate bioterror attacks (Chyba 2001, 2002). Improved building codes for tall towers can help people evacuate more quickly in the event of a fire as well as a bomb (Dwyer and Lipton 2005). Better emergency communications and interoperability facilitate coordination and response to natural disasters and deliberate attacks (Mayer-Schönberger 2003).

Recognizing these complementarities, the Department of Homeland Security intends to take the overlap between natural disasters and terrorist attacks into account. Announcing the 2006 Urban Areas Security Initiative grants, Secretary Chertoff explained

The answer and the way of finding the optimal allocation turn out to be quite similar to the problem of border defense and draw heavily on the analysis of the baseline case. Suppose that the defender completely discounts the possibility of a terrorist attack and allocates all of its resources optimally against the nonstrategic threat. This allocation equates the marginal gain from spending a little more on site j with the marginal gain of spending a bit more on k, where these margins depend on the chances of an accident at j and k, but not on the prospects of a deliberate attack. Spending a bit more or less on either site does not affect the nonstrategic threat to the other site.

But equating the margins to optimize spending against a nonstrategic threat means that a terrorists' payoffs will generally not be equal across the sites. Suppose that the attacker's payoff to striking site t₁ is higher than its payoff to going after any other site. Then this site is sure to be attacked at the optimal allocation against the nonstrategic threat. If therefore the defender wants to hedge against a terrorist attack, it has to shift resources to t₁ and necessarily away from the other sites, doing so in a way that keeps the marginal gains of protecting those other sites against the nonstrategic threat equal. As the defender allocates more resources to protecting t₁, the attacker's payoff to striking that site decreases until some other site, say t₂, now becomes an equally attractive target given the defender's allocation. If the defender shifts still more resources away from protecting against the nonstrategic threat, it must begin to distribute those resources across t₁ and t₂ so as to keep the terrorists' payoff to attacking either of the those sites the same. The defender continues shifting resources in this way, having to spread them across more and more sites, until the marginal gain of spending more on the strategic threat is just offset by the marginal loss of spending less on the nonstrategic threat.

To introduce the non-strategic component, let η_j be the probability of an “accidental,” that is, non-strategic, attack on site j. Then the overall probability of an attack on j is η_j+α_j(1−η_j).¹⁶

. As always, the attacker strikes sites offering the highest expected payoff, i.e., sites in T_A(r).

As for the comparative statics, anything that increases the marginal gain of defending sites not subject to deliberate attack (i.e., those j∉T_A(r)) leads (weakly) to greater spending on those sites and vice versa. ¹⁷

increases), then the marginal cost of protecting against the nonstrategic threat rises and the defender (weakly) shifts resources toward the strategic threat by spending more on the sites in T_A(r). Finally, suppose a site subject to deliberate attack becomes more difficult to guard at the margin. Then the marginal cost of strategic defense goes up, and the defender shifts resources away from it by spending less on the sites in T_A(r).

As with border defense, the equilibrium allocations can be characterized formally by describing a curve through the space of possible allocations. In the case of border defense, the defender moved along the curve as it shifted resources from border defense to site defense, optimally allocating the latter among the sites. In the present case, the defender moves along the curve by shifting resources from defending against the nonstrategic threat to protecting against the strategic threat where the resources dedicated to each type of threat are optimally allocated against that threat. An equilibrium allocation lies on this curve at the point where the marginal gain from spending more to defend against the strategic threat is just offset by the marginal loss of having less to spend on defending against the nonstrategic threat.

To describe this curve analytically, suppose that the defender completely discounts the strategic threat. The resulting allocation δ equates the marginal benefit to protecting a given site against the nonstrategic threat with the marginal benefits of protecting any other site. That is, δ is the unique allocation minmizing

and, therefore, satisfying three conditions: (i)

whenever δ_j>0 and δ_k>0, (ii)

whenever δ_j>0 and δ_m=0, and (iii) δ₁+[sdot ][sdot ][sdot ]+δ_S=R.

The path P_η begins at δ, and a point is on P_η if and only if the resources guarding against the nonstrategic threat are optimally allocated against that threat and those devoted to the strategic threat minimize the attacker's expected payoff. In symbols, r∈P_η if and only if conditions (i) and (ii) hold for all sites not in T_A(r). These conditions guarantee that resources devoted to the nonstrategic threat, that is, to j∉T_A(r), cannot be profitably reallocated among those sites. As for the resources dedicated to the strategic threat, the definition of T_A(r) ensures that the resources spent on those sites minimize the attacker's payoff over these sites.¹⁸

, where σ is defined over the interval

with

and

. As the defender moves along P_η, it is effectively changing the mix between the strategic and the nonstrategic threat but optimally allocating however much it dedicates to each of these separate threats.

To identify an equilibrium allocation where the marginal gain from allocating more to the strategic threat is just offset by the marginal loss from dedicated less to the nonstrategic threat, define the defender's expected loss from guarding against the strategic threat at r=P_η(σ) to be

and from the nonstrategic threat to be

. The defender's expected loss can now be written as is L(r(σ), α)= L_η(σ, α)+L_D(σ, α). Then an equilibrium allocation lies on P_η and generically satisfies

, where

is the attacker's pseudo-equlibrium strategy at r.¹⁹

Before stating the results in the general case, consider the simpler case in which there are H identical high-value sites for the attacker and S−H identical low-value sites. The latter are so much less attractive that the attacker would never go after them.²⁰

. The left side of this equation is the marginal gain to spending more against strategic threats; the right side is the marginal gain to spending more on protecting against nonstrategic threats.

Observe that if the number of potential targets subject to strategic threat (H) is very large, the probability of a strategic attack on any of them (1/H) is very small. The probability of a attack on this site is swamped by the nonstrategic component, and the equilibrium condition approximates

. But this is just what the defender would do if it completely ignored the terrorist threat and simply allocated its resources optimally against the non-strategic threat.

Turning to the general case:

Proposition 3. Nash equilibria exist, and the defender always plays a pure strategy lying on P_η where the marginal loss of spending less on the non-strategic threat just offsets the marginal gain of spending more on the strategic threat. This condition generically satisfies

which is equivalent to

for any

for which

. ²¹

UNCERTAINTY ABOUT THE ATTACKER'S PREFERRED TARGETS

The previous discussion assumed that the defender knows how the attacker ranks potential targets. But what if the defender is uncertain about the targets that a terrorist group would most like to strike? How should a defender allocate its resources in the face of this uncertainty?

We simplify matters here by assuming that the defender is unsure whether it is facing one of two possible types of attacker, γ and τ. These types may get different payoffs from attacking a particular site and may even rank the sites in a different order. Attacker γ might, for example, rank a successful attack on a nuclear facility higher than on a chemical site, whereas τ might rank the latter higher than the former. ²²

The answer to the resource allocation question and the approach to finding the optimal allocation are quite similar to the border-defense and nonstrategic threat problems. Suppose the defender were certain it was facing γ and allocated all of its resources accordingly. (The baseline algorithm can be used to find this allocation.) This allocation equalizes γ's payoff to attacking across the defended sites. But this allocation generally is not optimal against a terrorist like τ with different priorities. The optimal allocation against γ will usually not minmax τ by equalizing its payoffs across all of the defended sites.

Suppose attacking site t₁ maximizes τ's payoff at the optimal allocation against γ. Then the defender, if it wants to hedge against the prospect of facing τ, should shift resources to t₁ and necessarily away from the other sites while keeping γ's payoff to attacking those sites equal. As the defender dedicates more resources to defending against τ by hardening t₁, τ's payoff to striking this site decreases until some other site, say t₂, becomes equally attractive. At this point allocating still more resources to protect against τ means that the defender has to spend those resources on both t₁ and t₂ so as to keep τ's payoff to attacking them equal while at the same time keeping γ's payoff to attacking the other sites equal. The defender continues shifting resources in this way, having to spread them across more and more sites, until the marginal loss of spending less on defending against γ just offsets the marginal gain from allocating more to defending against τ. These marginal conditions define an equilibrium allocation.

To formalize the defender's uncertainty and the argument as a whole, suppose that the defender believes that it is facing γ with probability μ and τ with probability 1−μ. Type γ's payoff to a successful attack on site j is

and τ's payoff is

. Then the attackers' expected payoffs are

and

, where α and ϕ are γ's and τ's strategies. The defender's expected loss is

The comparative static results parallel those in the case of threats with nonstrategic components. Anything that increases the marginal gain to defending against one type induces the defender to spend (weakly) more guarding against that type. If, for example, the defender is more confident it is facing γ (μ increases), the defender spends (weakly) more on the sites γ threatens (the j∈T_γ(r) which are the sites offering γ its highest payoff). Similarly, if a site γ threatens becomes harder to protect at the margin, the defender shifts resources away from defending against γ and invests more in protecting against τ by reallocating from sites in T_γ(r) to sites in T_τ(r).

As before, we characterize the defender's equilibrium allocation by describing a curve through the space of possible allocations. In the case of border defense, the defender moved along the curve by shifting resources between border defense and site defense while optimally allocating the latter among the sites. In the case of threats with nonstrategic components, the defender moved along the curve by shifting resources between defending against the nonstrategic and strategic threats where the resources dedicated to each type of threat are optimally allocated against that threat. In the present case, the defender moves along the curve by shifting resources between the optimal allocation against γ and against τ. An equilibrium allocation lies on this curve where the marginal loss of having less to spend on defending against γ just offsets the marginal gain from dedicating more to defending against τ.

To specify the path, let g and t be the defender's optimal allocation if it were sure that it was facing γ or τ, respectively. Then a point is on the path P_g from g to t if the defender is allocating all of its resources against the sites that offer γ or τ their highest payoffs. That is,

if and only if

. Parameterize movement along P_g in terms of σ, the resources the defender allocates against τ, by defining P_g(σ) to be the point r∈P_g, where

for

with

and

.

Clearly any equilibrium allocation must lie on P_g. If r′∉P_g, then

. Consequently, there exists a site k such that

and k∉T_γ(r′)∪T_τ(r′). Thus, neither γ nor τ attack k with positive probability, so the defender can profitably deviate from r′ by reallocating

across the sites in T_γ(r′)∪T_τ(r′).

An equilibrium allocation equates the marginal loss of spending less on guarding against γ by increasing σ with the marginal gain to spending more defending against τ. To characterize these allocations, take the defender's expected loss from defending against γ and τ at any

to be

and

. Combining these payoffs gives the defender's total payoff

. Then the equilibrium allocation lies on P_g and generically satisfies

, where

and

are γ's and τ's pseudo-equilibrium strategies. This leaves:

Proposition 4. Bayesian Nash equilibria exist, and the defender always plays a pure strategy lying on P_g where the marginal loss of spending less on defending against γ just offsets the marginal gain of spending more protecting against τ. This condition generically satisfies

which is equivalent to: