Deterrence, preemption, and panic: a common-enemy problem of terrorism.

I. INTRODUCTION

Terrorism has become a global phenomenon and a matter of concern for both developed and developing countries. While there is a large list of terrorist organizations, many of which are "local" and target one particular country, over the last two decades some terrorist organizations have been targeting multiple countries. Al-Qaeda is the prime example at present. (1) In the 1990s, the Abu Nadal Organization (ANO) carried out activities against 20 countries, including the United States, France, Israel, and various Arab countries (U.S. Department of State 1997). (2)

Multicountry targeting not only involves strategic interdependence between a terrorist organization and individual countries, but it also implies such interdependence among the target countries themselves. In this article, we adopt a game-theoretic approach to study this problem, by explicitly formulating the choice problem facing a terrorist organization and its target countries. In the game-theory terminology it can be called a common-enemy problem. (3) We consider a two-stage game where, in the first stage, the countries simultaneously decide their levels of both preemption and deterrence, followed in the next stage by the organization deciding its levels of attack.

Attacks are conceptualized through the notion of a production function relating "terror input" to damage. Damage from terror not only includes direct effects like loss of property and human tragedy in the form of casualties and fatalities, but is also meant to encompass the fear it engenders among the public. That is, terrorism is viewed as production of terror. As recognized since Hobbes, human life is governed by a social contract, which presupposes social stability. Terror is aimed at undermining this very stability, leading to significant distortions in social contracts. Recently, Becker and Rubinstein (2010) have argued that small-probability events such as an incident of terror can, via producing fear, have major effects on people's utility and well-being. According to Richman, Cloninger, and Rospenda (2008), 9/11 has had a lingering effect on mental health of Americans in the form of depression, anxiety, drinking, and so forth. Additional empirical evidence that terrorism typically has large effects on economic outcomes is provided by Abadie and Gardeazabal (2003). (4)

A main innovation of this article is to characterize fear and its implications. More specifically, we differentiate between weak-marginal-fear and strong-marginal-fear, referring to whether an increase in the deployment of the terror input leads to a decrease or an increase in its marginal effect on expected damage. We consider equilibrium in each of these two cases--as well as in a situation, where, depending on the magnitude of the attacks, there could be a "regime" switch from weak-marginal-fear to strong-marginal-fear. In the presence of such a regime switch there could be an equilibrium, which we shall call brink-of-panic (BOP), in which the terrorist organization is indifferent between operating in the weak- and the strong-marginal-fear regimes (to be made precise later). Interestingly, the resultant situation is knife-edge in the aggregate in that an arbitrarily small reduction in preemption by any target country would lead to a drastic increase in the scale of attack by the organization with the countries "plunging" into panic. But (a) it is "individually stable" in the sense that it can be sustained as a (self-enforcing) Nash-equilibrium and (b) it can occur over a parameter range, having a positive measure. Furthermore, multiple equilibria may arise.

One of the main insights from our model is that the equilibrium nature of counter-terrorism policies and how they respond to changes in parameters facing the terrorist organization, depend on the characteristics of fear, in particular whether fear is "strong" or "weak." We provide a tractable formalization of these two kinds of fear, modeling them through parametric restrictions. In particular, the model sheds light on the merit of the dictum, "offense is the best defense," that is, whether more offense is the optimal response to a higher threat of terrorism. We find that as a response to an increase in the potency of the organization--interpreted as an increase in terrorism--the target countries scale up their security-deterrence measures, but may either decrease or increase preemption, depending on whether the marginal-fear effect is weak or strong. Along the BOP equilibrium "an increase in terrorism" necessarily leads to an increase in preemption, despite the fact that the equilibrium occurs in the weak-marginal-fear region. Thus we find that the above-mentioned dictum holds in the strong-marginal-fear, or what we call in Section III, the panic equilibrium, as well as in the BOP equilibrium, while it does not hold in the weak-marginal-fear or the no-panic equilibrium.

Explicit modeling of a terrorist organization's behavior yields some unusual or unexpected insights too. In comparison to Sandier and Sequeira (2006), for example, who adopt a reduced-form approach and directly postulate that (a) an increase in security-deterrence by one target country induces the terrorist organization to focus more on other target countries and (b) preemption measures undertaken by target countries that reduce the overall strike capacity of the organization are strategic substitutes, our analysis shows that these may not hold. That is, (1) a terrorist organization may not step up its activity in country j in response to greater security-deterrence measures undertaken by another target country i and (2) while preemption measures undertaken by any target country is a "good" public good for all, it may not imply strategic substitutability. Furthermore, we are able to qualitatively identify these situations where (1) and (2) hold. Hence, imposing properties (a) and (b) by fiat is, to some extent, misleading.

In terms of welfare implications for target countries we find that there is a role for cooperation irrespective of whether the marginal fear effect is weak or strong, and whether preemption measures are strategic substitutes or complements. Under cooperation the aggregate level of preemption is higher compared to the non-cooperative equilibrium, while security-deterrence levels by individual countries are lower. These results hold as long as there are no possible regime switches. If a regime switch from weak- to strong-marginal-fear is possible, however, additional implications follow. In the BOP equilibrium explicit collusion may not be required at all--because the countries coordinate on the appropriate levels of preemption anyway so as to avoid panic. Furthermore, the presence of multiple equilibria imply that, even in the absence of explicit collusion, coordination alone can be welfare improving.

We now explicitly relate our article to the literature, in particular the game theoretic ones. (5) Das and Lahiri (2006) have formulated a non-cooperative game between a terrorist organization and one target country, in which the former selects its scale of attack and the latter its security-deterrence level. In an interesting paper, Hausken (2008) extends the Das and Lahiri (2006) framework by considering a case where the government uses preemptive strategies, and the terrorist organization can defend its assets. Sandier and Siqueira (2006) and Bandopadhyay and Sandier (2011) extend the literature by allowing for multiple target countries. Further, they examine the interactions between the target countries in terms of choosing their strategic variables: preemption in Bandopadhyay and Sandier (2011), and both preemption and deterrence in Sandier and Sequiera (2006). The terrorist organization is modeled implicitly through a probability-of-attack function facing a target country. Another stream of the literature, which includes the present article as well as Carceles-Poveda and Tauman (2011), not only allows for interactions among target countries, but also models the terrorist organization's behavior explicitly. In so doing, these papers move away from "reduced-form" formalization of the externalities between target countries implicit in much of the earlier literature (which is subject to a "Lucas-type critique").

The paper closest to this article is the one by Carceles-Poveda and Tauman (2011). They break new ground in focusing on the issue of endogenous coalition formation, analyzing conditions under which the coalition against terror is likely to be large (or otherwise). Moreover, they show that among the countries that do cooperate, while all incur deterrence costs, only a subset incur preemption costs.

This article differs from the literature, including Carceles-Poveda and Tauman (2011), in many respects, most significantly in its emphasis on analyzing the effects of fear induced by terror attacks, and on whether the induced fear is "weak" or "strong." Our analysis shows that many of the results critically depend on whether the equilibrium is in the "strong-fear," or the "weak-fear" region. We find, for example, that the dictum that "offense is the best defense," goes through in the strong-fear region, but not in the weak-fear region. In line with our findings (and as mentioned earlier), one finds less support for this dictum in Carceles-Poveda and Tauman (2011), Proposition 2, whose analysis can be interpreted as focusing on the weak-fear region. (6) Moreover, in case there is the possibility of regime switch from weak fear to strong fear, one can have equilibria where the countries are at the brink of panic. We find that the nature of fear is critical in determining the nature of coalition formation as well.

This article adds to the analysis of coalition formation in Carceles-Poveda and Tauman (2011) in two respects. First, we show that under the strong-fear regime, there is a critical minimum mass required for a new country to join an existing anti-terror coalition. Thus, while the grand coalition forms once this critical mass is reached, no coalition may form at all since joining small coalitions are not profitable. Second, we find that in the BOP equilibrium, explicit coalition may not be required. Just the possibility that panic may develop may be enough to sustain a collusive level of outcome.

Finally, it is the explicit modeling of the terrorist organization, which allows a clear understanding of the importance of the "fear factor." This is because the terrorist organization internalizes the "fear factor." For example, it will strive for larger attacks in case the fear effect is strong, which in turn demands a greater response from the target countries. Unless the terrorist is modeled explicitly, these effects will not be brought about in a transparent fashion.

Before getting into formal analysis, it is important to state the context in which we analyze the problem. First, with regard to the terrorist organization, we do not consider what their "fundamental" or long-term objective is. It is supported by a group of people who presumably want some sort of a public good that is costly to provide on the part of target countries. The merit or demerit of the "cause" is immaterial for the analysis. For example, in his classic study, Pape (2005) concludes that religious fundamentalism has little to do with suicide terrorism; nationalism seems to be the root cause. We do not get into the political aspects surrounding terrorism. The terrorist organization is seen as an entity deriving utility from imposing damage on some target countries. Second, the analysis completely abstracts from various problems of asymmetric information. (7)

The rest of the article proceeds as follows. In Section II we set up the basic model, and analyze the case where the marginal-fear effect is either uniformly strong or uniformly weak. In Section III, we examine an integrated model that allows for a regime switch from weak- to strong-marginal-fear. Section IV discusses some extensions of our model and concludes the paper.

II. THE MODEL

Our analysis dealing with a terrorist organization and target countries is built on two main hypotheses. (1) Like the target countries, the terrorist organization is a rational entity, taking into account benefits and opportunity costs of its actions. (2) Reducing damage from terror is costly for a target country and these costs are taken into consideration while setting counter-terrorism policies; in other words, countries do not aim to reduce or eliminate terror at any arbitrarily high cost.

There is one terrorist organization--hence forth called the Organization--and at least two target countries, indexed by i [member of] {1, 2 ..., I}. We view terrorist acts in terms of a production function. Let [x.sub.i] denote the level of a composite input being used against country i. We will call it the terror input supplied to country i. The strategy vector of the Organization is ([x.sub.1], ..., [x.sub.I]), where [x.sub.i] [greater than or equal to] 0.

The target countries choose security-deterrence as well as preemptive measures, each represented by a univariate index, namely, [s.sub.i] and [z.sub.i], respectively. The strategy vector of country i is thus a non-negative vector ([s.sub.i], [z.sub.i]).

The target countries and the Organization play a two-stage game, with the former moving first and simultaneously choosing levels of their counter-terrorism measures, followed by the latter choosing its terror input levels in stage 2. The nature of timing amounts to commitment on the part of the target countries with respect to counter-terrorism policies. (8) We ignore repeated interactions, or negotiations between the Organization and the target countries.

The expected damage to a country, [D.sub.i], is a function of [x.sub.i] and [s.sub.i], that is, [D.sub.i] = F([s.sub.i], [x.sub.i]). We assume F([s.sub.i], [x.sub.i]) : [[0, [infinity]).sup.2] [right arrow] [0, [infinity]) to be adequately differentiable, satisfying

A1: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

That is, the level of deterrence has a negative and decreasing marginal effect on expected damage. The expected damage increases with the terror input and its marginal effect decreases with security-deterrence. As discussed earlier, there are two components of damage: material and psychological. It is reasonable to suppose that the marginal effect of terror input on material damage is subject to decreasing returns and that on psychological damage follows increasing returns. Consequently, [F.sub.xx] [??] 0.

We do not model the long-term goal of the Organization; neither do we take a position on how reasonable it is. Our primitive is that there is a group of sympathizers behind the "cause" and the Organization is an outgrowth of that. It derives its utility from the expected damage it is able to directly inflict on the target countries, equal to [[sigma].sub.i] [D.sub.i]. It is presumed that all countries are targeted equally. (10)

The Organization also cares about the cost of its "production." These include the cost of recruiting, financing, and training terrorists, cost of equipment and that of building and maintaining infrastructure, and so forth. They are specified by

(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The functions C(x) and c(x) respectively are the "common-resource" and country-specific costs. The former would include, for example, the costs of maintaining infrastructure to train terrorists. (In other words, a terrorist must be trained in some common facility before being assigned to a country-specific "mission.") Note that Z is the aggregate level of preemptive measures undertaken by all target countries. An increase in Z (in terms of bombing terrorist bases, freezing suspect funds, etc.) shifts up the Organization's cost function. (11)

A decrease in ct reflects either a technology improvement or an exogenous infusion of resources for the Organization. It can be interpreted as the Organization becoming more potent or simply "an increase in terrorism."

The positive sign of the cross partial in the C(x) function means that an increase in aggregate preemption or a decrease in the potency of the Organization increases its common-resource marginal cost.

We assume increasing marginal country-specific costs, that is,

A2: c"(x) > 0.

At this point we do not impose any restriction on the sign of [C.sub.XX], that is, whether the common-resource marginal cost function is increasing, constant, or decreasing, We shall see below that the sign of [C.sub.XX] will determine the nature of the cross effect of security-deterrence choice by one country on the Organization's choice of terror input toward another.

A. The Organization's Problem: Stage 2

It seeks to maximize [[summation].sub.i] [D.sub.i] - T with respect to ([x.sub.1], ..., [x.sub.I]), given [s.sub.i] and [z.sub.i]. The first-order conditions are:

(2) [F.sub.x]([s.sub.i], [x.sub.i]) = [C.sub.x](Z + [alpha], X) + c'([x.sub.i]), [for all] i.

The left-hand and right-hand side expressions, respectively, indicate the marginal benefit and marginal cost facing the Organization. We impose that

A3: (i) [absolute value of c"] is sufficiently large, such that [F.sub.xx](x) - I [C.sub.XX](x) - c" (x) < 0, [F.sub.xx](x) - c"(x) < 0 and (ii) [absolute value of [F.sub.x]x(x) - [C.sub.XX](x)] - c"(x) > (I - 1) [absolute value of [C.sub.XX](x)],

which ensures that the second-order conditions for the Organization's problem are met.

The following proposition summarizes the response of the Organization to changes in deterrence and preemptive measures.

PROPOSITION 1. Let assumptions A1-A3 hold. An increase in the aggregate preemption forces the Organization to lower its scale of activities in all target countries, while an increase in security-deterrence by country i induces the Organization to (a) decrease [x.sub.i] but (b) increase, leave unchanged or decrease [x.sub.j] (j [not equal to] i) as [C.sub.XX] [??]. 0.

Proof. See Supporting information, Appendix S1.

The first part of Proposition 1 is apparent. It is also intuitive that if country i enhances its security-deterrence measure, the Organization scales [x.sub.i] down. Aggregate level of terror input, X, tends to fall. This decreases or increases the marginal cost of using common resources as [C.sub.XX] [??] 0. In turn, it decreases or increases the "total" marginal cost of [x.sub.j]--and therefore induces the Organization to step up or reduce [x.sub.j]--according as [C.sub.XX] [??] 00. Hence, unlike Sandier and Sequeira (2006), it is not necessary that the Organization would redirect its activities more toward a particular country if it faces more stringent security-deterrence measures in another.

Having noted Proposition 1, we however assume henceforth, for the sake of analytical tractability, the intermediate case of

A4: [C.sub.XX] = 0.

This assumption essentially implies no externalities between target countries originating from their choice with regard to security-deterrence. Given A4, we can specify, without loss of generality,

(3) T = (Z + [alpha])([[sigma].sub.i] [x.sub.i]) + [[sigma].sub.i]c([x.sub.i]).

Accordingly, Equation (2) simplifies to

(4) [F.sub.x]([s.sub.i], [x.sub.i]) = Z + [alpha] + c'([x.sub.i]).

In keeping with Proposition 1, Equation (4) implicitly yields [x.sub.i] = g([s.sub.i], Z + [alpha]), where

(5) [partial derivative]g/[partial derivative][s.sub.i] = [F.sub.sx]/(c" - [F.sub.xx]) < 0;

[partial derivative]g/[partial derivative]Z = -1/(c" - [F.sub.xx]) < 0.

B. Counter-Terrorism Measures by Target Countries: Stage 1

The target countries incur the damage from terror, as well as the costs of security-deterrence and preemptive measures. Let the latter two be indicated by [[beta].sub.i]u([s.sub.i]) and [[lambda].sub.i] v([z.sub.i]), respectively. The total cost function is given by [[OMEGA].sub.i]([x.sub.i], [s.sub.i], [z.sub.i]) = F([s.sub.i], [x.sub.i]) + [[beta].sub.i]u([s.sub.i]) + [[lambda].sub.i]v([z.sub.i]). We assume

A5: u', v', u", v" > 0,

that is, the respective cost functions are convex. A target country's objective is to minimize this cost. Note that the benefit to country i from preemption enters through the Organization's choice of [x.sub.i]. Thus, had we assumed a simultaneous game, it would have meant minimizing [[OMEGA].sub.i] at given [x.sub.i], implying optimal [z.sub.i] to be zero.

Because of the two-stage nature of the game, the qualitative nature of equilibrium would, in general, depend partly on third-order derivatives. For analytical simplicity as well as ease of interpretation, we assume however that

A6: [F.sub.ss], [F.sub.sx], [F.sub.xx], c", u" and v" are all constant.

In what follows we compare non-cooperative equilibrium with cooperative equilibrium in which the target countries coordinate their counter-terrorism policies. The optimization problems facing the countries in the two scenarios would require various second-order conditions to be met. These are collectively captured in:

A7:

(R1) c" > [F.sub.xx].

(R2) [[lambda].sub.i]v" [(c" - [F.sub.xx]).sup.2] [A.sub.i] - [B.sub.i] > 0, where

[A.sub.i] [equivalent to] ([F.sub.ss] + [[beta].sub.i]u")[(c" - [F.sub.xx]).sup.2] + (2c" - [F.sub.xx])[F.sup.2.sub.sx]

(6) [B.sub.i] = [(c").sup.2][F.sup.2.sub.sx] - [A.sub.i][F.sub.xx].

(R3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

positive semi-definite,

(7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In this section, we assume that [F.sub.xx] is either positive or negative for all x > 0; given this, A7 is met if c", [[beta].sub.i] and [[lambda].sub.i] are large enough. (12)

"We now analyze the non-cooperative Nash behavior of target countries in Stage 1. Country i aims to:

(8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

subject to Equation (5). Under our assumption of [C.sub.XX] = 0, country i's cost is not affected by any other country's security-deterrence measures. But it is affected (partly) by others' choice of preemption levels. This is the basis of externality between the choices of preemption, which is not internalized in the non-cooperative equilibrium. (13)

The following first-order conditions hold for country i:

[partial derivative][[OMEGA].sub.i]/[partial derivative][s.sub.i] = [F.sub.s](x) + [F.sub.x](x)[partial derivative]g/[partial derivative][s.sub.i] + [[beta].sub.i]u'([s.sub.i]) = 0

(9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(10) [??] [[lambda].sub.i]v'([z.sub.i]) = [F.sub.x](x)/(c" - [F.sub.xx).

Remarks

1. In Equations (9) and (10), the left-hand-side terms are respectively the marginal costs of deterrence and preemption, which are increasing in their respective arguments.

2. The marginal benefits are indicated by the terms in the right-hand side. For security-deterrence, the marginal benefit is obtained directly as well as through its impact of the Organization's choice of [x.sub.i]. From Equation (9), it decreases with deterrence. (14) However, the marginal benefit from preemption works through its impact on [x.sub.i] only.

In view of Equation (10),

LEMMA 1. The marginal benefit of preemption increases or decreases with preemption as [F.sub.xx] [??] 0.

Suppose [F.sub.xx] < 0, that is, the marginal damage from the terror input is diminishing. It implies that the lower the level of this input, the higher is the marginal damage. Because a higher [z.sub.i] implies, ceteris paribus, a higher Z and hence a lower level of [x.sub.i], it follows that at a higher level of preemption, the marginal benefit from preemption is higher. Similarly, if [F.sub.xx] > 0, the marginal benefit decreases with preemption.

Lemma 1 is illustrated in Figure 1 for the case of [F.sub.xx] < 0. At given [z.sub.j]'s, for country i, the marginal benefit from preemption is reflected by the magnitude of the slope of the [F.sub.x] (x) function. As can be seen, a higher [z.sub.i] is associated with a lower level of [x.sub.i] and hence a greater marginal benefit.

Earlier, we have discussed that the greater the fear effect, the higher is the value of [F.sub.xx]. Hence, a corollary of Lemma 1 is that the marginal benefit of preemption increases or decreases with preemption, as the marginal fear effect is sufficiently large or small. This will serve as the key in understanding the role of preemptive measures.

Equations (9) and (10) imply the following rankings of counter-terrorism policies across countries.

PROPOSITION 2. If [[beta].sub.i] = [[beta].sub.j] and [[lambda].sub.i] < [[lambda].sub.j] then [s.sub.i] = [s.sub.j] and [z.sub.i] > [z.sub.j]. Whereas if [[beta].sub.i] < [[beta].sub.j] and [[lambda].sub.i] = [[lambda].sub.j], then [s.sub.i] and [z.sub.i] < [z.sub.j].

The next proposition relates the levels of two policies employed by a target country, as well as whether preemption levels chosen across countries are strategic substitutes or complements.

PROPOSITION 3. (A) At any given vector of preemption levels chosen by other countries, security-deterrence ([s.sub.i]) and preemption ([z.sub.i]) chosen by any target country are gross substitutes in that a higher [s.sub.i] is associated with a lower [z.sub.i] and vice versa.

(B) Preemption levels chosen across the target countries are strategic complements or substitutes as the marginal fear effect is weak or strong enough, that is, as [B.sub.i] [??] 0 (where Bi is defined in (R2)).

(C) Given that preemption levels by other countries are adjusted optimally, [s.sub.i] and [z.sub.i] are net substitutes or complements, as the marginal fear effect is weak or strong enough, that is, as [B.sub.i] [??] 0. (15)

Proof (A) Equation (9) implicitly defines [s.sub.i] as a function of Z + [alpha], with

(11) d[s.sub.i]/d (Z + [alpha]) = [partial derivative][s.sub.i]/[partial derivative][z.sub.i] = c" [F.sub.sx]/[A.sub.i] < 0. (16)

Thus, at any given Z' [equivalent to] Z - [z.sub.i], [s.sub.i] and [z.sub.i] are negatively related.

(B) By definition, [B.sub.i] [??] 0 as [F.sub.xx] [??] [(c").sup.2] [F.sup.2.sub.sx]/[A.sub.i]. If the marginal fear effect is high enough, [F.sub.xx] would be positive and large enough to outweigh [(c").sup.2] [F.sup.2.sub.sx]/[A.sub.i]; otherwise, [F.sub.xx] is positive and small enough or negative such that [F.sub.xx] < [(c").sup.2] [F.sup.2.sub.sx]/[A.sub.i]. This is how the marginal fear effect is related to the sign of [B.sub.i].

We substitute Equation (11) into Equation (10) to implicitly express [z.sub.i] as a function of Z + [alpha] with

(12)

d[z.sub.i]/d(Z + [alpha]) = d[z.sub.i]/dZ = [B.sub.i]/[[lambda].sub.i]v"[(c" - [F.sub.xx]).sup.2] [A.sub.i]] [??] 0 as [B.sub.i] [??] 0.

Further, by virtue of (R2), d[z.sub.i]/dZ < 1. Since Z is the sum of all z's,

d[z.sub.i]/d[z.sub.j] = d[z.sub.i]/dZ x (1 + d[z.sub.i]/d[z.sub.j]) [??] d[z.sub.i]/d[z.sub.j] = d[z.sub.i]/dZ/(1 - d[z.sub.i]/dZ).

It follows that d[z.sub.i]/d[z.sub.j] [??] 0 as [B.sub.i] [??] 0. (C) Combining Equations (11) and (12),

(13) d[s.sub.i]/d[z.sub.i] = [[lambda].sub.i]v"[(c" - [F.sub.xx]).sup.2]c"[F.sub.sx]/ [B.sub.i] [??] 0 as [B.sub.i] [??] 0.

It is important to understand why, unlike the case of public-good provision, preemption levels may not be strategic substitutes, and why security-deterrence and preemption may be net substitutes or complements.

Considering the first initially, there are two effects at work, Suppose [z.sub.j] increases. It tends to increase the aggregate level of preemption, thereby shift up the marginal cost function facing the Organization. The Organization reduces the supply of terror input to any particular country i, which is induced to reduce its security-deterrence. Since security-deterrence and preemptive measures are gross substitutes, a reduction in the former tends to imply a higher level of preemption by country i. In brief, because of substitutability between security-deterrence and preemption as alternative forms of counter-terrorism policies, preemption levels across target countries tend to be strategic complements. (17)

The other effect relates to the interdependence between [z.sub.i] and [z.sub.j] at any given vector of [s.sub.i]'s. ff the marginal-fear effect is weak enough to the extent that [F.sub.xx] < 0, then by Lemma 1, the marginal benefit from preemption increases with preemption. As an increase in [z.sub.j] tends to increase aggregate preemption, the marginal benefit from [z.sub.i] increases. Country i responds by increasing [z.sub.i]. On this account also preemption levels are strategic complements. (18) But, by analogous reasoning, if the marginal fear effect is strong enough so that [F.sub.xx] > 0, preemption choices across countries would be strategic substitutes at given [s.sub.i]'s.

Combining these two effects yields that preemption levels may be strategic complements or substitutes. If [F.sub.xx] < 0, the fear effect is weak and works in the same direction as the gross substitutability effect. If 0 < [F.sub.xx] < [(c").sup.2][F.sup.2.sub.sx], the fear effect works in the opposite direction but is outweighed by the gross substitutability effect. The net result is that preemption choices are strategic complements. It is only when the fear effect is strong enough to outweigh the substitutability effect between deterrence and preemption, preemption choices are strategic substitutes. (19)

Finally, as detailed above, an increase in aggregate preemption, on the one hand, implies a decline in security-deterrence, and, on the other, may induce a particular country to increase or reduce its level of preemption measures depending on the strength of the marginal fear effect. This explains why security-deterrence and preemption may be net substitutes or complements.

The nature of the overall equilibrium thus depends on whether the marginal fear effect is weak or strong.

C. Weak-Marginal-Fear Effect: [B.sub.i] > 0

The solution is illustrated in Figure 2. Given that [B.sub.i] > 0, Equation (12) implies that d[z.sub.i]/dZ > 0. Hence the [z.sub.i] = [[phi].sub.i](Z) schedule in the right quadrant is upward sloping, The left quadrant shows the negative relationship between [s.sub.i] and [z.sub.i] in view of Equation (13).

Summing up [[phi].sub.i] (Z) curves vertically yields the [[sigma].sub.i][[phi].sub.i](Z) [equivalent to] [bar.[phi]](Z) curve. As shown in Appendix S1, [bar.[phi]]'(Z) < 1. Thus the [bar.[phi]](Z) curve intersects the 45[degrees] line. An equilibrium exists and it is unique. The point [Z.sup.n] marks the equilibrium level of aggregate preemption, where superscript n stands for non-cooperation. The preemption level chosen by country i is read off the [[phi].sub.i] (Z) curve, indicated by [z.sup.n.sub.i]. The corresponding solution of sn is obtained in the second quadrant.

We now consider the comparative-statics effects of a decrease in the parameter [alpha], which poses the question of how target countries react when the Organization becomes more potent. From Equation (12), as ct decreases, the [[phi].sub.i](Z) and [bar.[phi]](Z) curves shift down. The implication is that each target country lowers its level of preemptive activity. Furthermore, given that [B.sub.i] > 0 and thus security-deterrence and preemption are net substitutes, a lower [z.sub.i] is associated with a higher [s.sub.i]. In summary, the target countries "beef up" their security-deterrence measures, but reduce preemptive ones.

Intuitively, as [alpha] falls, the Organization tends to step up the terror input in each country. To counter this the target countries enhance their security-deterrence levels. Gross substitutability between security-deterrence and preemption implies that each country tends to lower the level of preemption. In addition, a weak-fear effect implies either a decline in the marginal benefit from preemption as the Organization increases its terror input, inducing a target country to lower its preemptive measure or a weak increase in marginal benefit from preemption and hence a small increase in the level of preemption, which is outweighed by the decrease in preemption due to the gross substitutability effect. The net effect on preemption is negative.

D. Strong-Marginal-Fear Effect: [B.sub.i] < 0

In this case, from Equations (12) and (13), respectively, d[z.sub.i]/dZ < 0 and d[s.sub.i]/d[z.sub.i] > O. Figure 3 is the analog of Figure 2. Both the [[phi].sub.i](Z) and [bar.[phi]](Z) curves are downward sloping. The intersection of the latter with the 45[degrees] line defines equilibrium, which is unique. The locus between [s.sub.i] and [z.sub.i] is upward sloping.

Consider now a decrease in [alpha]. Both curves in the first quadrant shift to the right, implying an increase in [z.sub.i] for all i. From the left quadrant, [s.sub.i] increases. In this case, the Organization chooses a higher input level in response to a decline in [alpha]. This sufficiently improves the marginal benefit from preemption, so that preemption levels increase too.

Combining the cases where the marginal fear effect is weak and where it is strong, we have

PROPOSITION 4. Irrespective of whether the marginal fear effect is weak or strong, there is a unique non-cooperative equilibrium. An increase in the potency of the Organization leads each target country to step up its security-deterrence level. Preemptive measures are scaled down or up as the marginal fear effect is weak or strong.

It is interesting that Carceles-Poveda and Tauman (2011, Proposition 2) find that with an increase in potency of the terror organization, preemption may in fact decrease. This can be traced to the fact that the damage to the country function in Carceles-Poveda and Tauman is linear in the level of terror input, so that their framework can be interpreted as modeling a situation with weak fear.

The reason as to why we obtain a different result when the marginal fear effect is strong is quite intuitive. Note that as the terror potency of the organization increases, terror input is likely to increase. Consequently, preemption will tend to increase. However, given that deterrence will also tend to increase, and that deterrence and preemption are gross substitutes (Proposition 1), this will have a negative impact on the level of deterrence. As Proposition 4 shows, however, for countries with a strong fear effect it is the first effect that dominates. This is because of two reasons. First, countries with a strong-fear effect have more to lose in case there is a terrorist attack. Thus, for such countries, an offensive strategy may be more attractive. Whereas for countries with a weak-fear effect, the free-riding intrinsic to an preemptive strategy makes it less attractive. Second, with a strong-fear effect, the preemption levels of the different countries are strategic substitutes (Proposition 3(B)). This weakens the second (negative) effect, since a reduction in the preemption level of one country will tend to increase preemption by the other countries.

E. Cooperation Among All Countries

Countries facing a common terrorist organization may wish to coordinate their policies to deal with the "common enemy." What is the nature of the cooperative solution and how does it compare with that under non-cooperation? The focus here is on examining the implications of cooperation. This is in contrast to Carceles-Poveda and Tauman (2011) who endogenize the formation of anti-terror coalitions. Even so, our analysis has some implications for endogenous coalition formation.

Suppose that in Stage 1 all target countries collectively decide [s.sub.i] and [z.sub.i] by minimizing their joint cost. Further let there be compensatory side payments such that participation in such a grand coalition is not an issue.

In Stage 2 the Organization faces the same optimization problem, however, and thus it has the same input supply function as in Equation (5). In Stage 1, [[summation].sub.i][[OMEGA].sub.i] = [[summation].sub.i][F([s.sub.i], g([s.sub.i], Z)) + [[beta].sub.i]u([s.sub.i]) + [[lambda].sub.i]v([z.sub.i])] is minimized with respect to [s.sub.i] and [z.sub.i]. Using the partials of the g (x) function, the first-order conditions with respect to [s.sub.i] and [z.sub.i] are:

(14) [F.sub.s](x) + [F.sub.x](x)[F.sub.sx]/(c" - [F.sub.xx]) [[beta].sub.i]u'([s.sub.i]) = 0,

(15) - [I.summation over (i=1)][F.sub.x](x)/(c" - [F.sub.xx]) + [[lambda].sub.i]v'([z.sub.i]) = 0.

Equation (15) captures the internalization of spill-over effects of preemption among the target countries. But notice that Equation (14) is same as Equation (9). It is because given that [C.sub.XX] = 0, there is no direct cross effect of [s.sub.i] on country j's cost. (20) This leads us to

PROPOSITION 5. Full cooperation in terms of jointly setting deterrence and preemptive measures is equivalent to partial cooperation in terms of coordinating preemptive measures only.

To compare the cooperative to the non-cooperative solution, substitute the implicit function of [s.sub.i] in terms of Z derived from Equation (14) into Equation (15) and then differentiate Equation (15) to obtain

(16) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [??] and [??] are defined earlier in Equation (8). (21) This equation implies

(17) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

On the basis of the function [[phi].sub.i](Z), we obtain the following proposition.

PROPOSITION 6. Compared to non-cooperation, at the cooperative equilibrium, each country chooses a lower level of security-deterrence ([s.sup.c.sub.i] < [s.sup.n.sub.i]) and the aggregate level of preemptive measures is higher ([Z.sup.c] > [Z.sup.n]). If countries are symmetric, or they are asymmetric and [B.sub.i] > 0, then [z.sup.c.sub.i] > [z.sup.n.sub.i] [for all]i. Otherwise, [z.sup.c.sub.i] > [z.sup.n.sub.i] for some i.

Proof Let [bar.[phi]](Z) [equivalent to] [[sigma].sub.i] [[phi].sub.i](Z)" Suppose [B.sub.i] > 0 [for all]i. Then, as shown in Appendix S1, 0 < [bar.[phi]]'(Z) = [[sigma].sub.i] [phi]'(Z) < 1. If [B.sub.i] < 0 for all i, [bar.[phi]]'(Z) < 0. Hence, in general, [bar.[phi]]'(Z) < 1. In turn, this implies that, if [Z.sup.c] solves Z = [bar.[phi]](Z), then, for any particular Z = [Z.sub.0], [Z.sup.c] [??] [Z.sub.0] if and only if [bar.[phi]]([Z.sub.0]) [??] [Z.sub.0].

Now, from the deterrence-setting rule, which is same under both non-cooperation and cooperation, for any given Z, [s.sub.i] is the same between the two regimes, and, comparing Equations (10) and (15), this implies that [z.sub.i] is higher under cooperation. That is, [[phi].sub.i](Z) > [[phi].sub.i](Z). This implies that for any Z, [phi](Z) > [phi](Z). Hence [bar.[phi]]([Z.sup.n]) > [bar.[phi]]([Z.sup.n]) = [Z.sup.n], implying [Z.sup.c] > [Z.sup.n] (since, as already proved, [Z.sup.c] > Z if and only if [bar.[phi]](Z) > Z), Because the same negative locus between [s.sub.i] and Z holds both under non-cooperation and cooperation, [Z.sup.c] > [Z.sup.n] [??] [s.sup.c.sub.i] < [s.sup.n.sub.i].

Since [Z.sup.c] > [Z.sup.n], [z.sup.c.sub.i] > [z.sup.n.sub.i] for some i. If all target countries are symmetric, then, trivially, [z.sup.c.sub.i] > [z.sup.n.sub.i] [for all]i. Or if Bi > 0, as shown in Appendix S1, [[phi]'.sub.i](Z) > 0. Then [for all]i, [z.sup.c.sub.i] = [[phi].sub.i]([Z.sup.c]) > [[phi].sub.i]([Z.sup.n]) > [[phi].sub.i]([Z.sup.n]) = [z.sup.n.sub.i].

F. Partial Coalition

By cooperation we have so far meant cooperation by all target countries. Obviously there are transactions costs in forming such a "grand coalition"--even if side payments are promised. What if only a proper subset of countries form a coalition?

By definition, such a coalition leaves out other countries who would adjust their strategies non-cooperatively in response to the formation of the coalition. These adjustments may imply qualitatively asymmetric implications toward the member countries in the coalition and the "outside" countries--even if all countries are symmetric otherwise.

In what follows we consider a comparative-statics analysis of the effects of M out of I target countries forming a coalition, where all countries face symmetric parameters with respect to their cost functions. (We do not undertake a full analysis of alternative coalition formation, deviation from coalition, etc.)

While the formal analysis is laid out in Appendix S1, the implications are intuitive. At given level of preemption chosen by outside countries, those in the M-member coalition internalize the externalities from preemption and choose a higher level of preemption. Consider the response by outside countries. In case the marginal fear effect is weak (respectively strong), from Proposition 3(B), preemption levels are strategic complements (respectively substitutes), which implies that these countries increase (respectively decrease) their preemption levels.

Interestingly, regardless of which way the outside countries adjust their preemption levels, their response, per se, constitutes an additional positive effect on the choice of preemption by member countries. When the marginal fear effect is weak and outside countries increase preemption, strategic complementarity also implies that the coalition would increase its preemption. When the marginal effect is strong and outside countries decrease preemption, strategic substitutability implies the same for the coalition.

The overall effect on preemption chosen by the coalition is thus unambiguously positive. Further, this effect either supplements the increase in preemption by outside countries or outweighs the negative adjustment by these countries. Hence aggregate preemption is higher vis-a-vis the non-cooperative situation. Finally, given that the level of security-deterrence chosen by any particular country, inside or outside the coalition, is negatively related to aggregate preemption, it follows that deterrence levels by all countries decrease. The following proposition summarizes these results.

PROPOSITION 7. A coalition by a proper subset of target countries leads to

(a) a higher level of preemption by member countries in the coalition,

(b) a higher or lower level of preemption by outside countries according as the marginal fear effect is weak or strong,

(c) a higher aggregate level of preemption, and

(d) a lower level of security deterrence by all countries,

While it is evident that under a grand coalition each country benefits, in the case of a coalition by some countries, a member country may or may not benefit. In case of weak-marginal-fear effect, outside countries increase their preemption levels which benefit the coalition. But if the marginal fear effect is strong, the negative adjustment in preemption by outside countries imposes a negative externality on the coalition. It is then possible that this adverse effect outweighs the direct gains from cooperation among the member countries. All else the same, the smaller the size of the coalition, the greater is the number of outside countries and hence the greater is the magnitude of the negative externality effect.

In other words, if the marginal fear effect is strong, then a (partial) coalition may not be sustainable. A sustainable coalition would require a "critical mass" of countries joining hands together. In what follows, this point is illustrated through a numerical example.

An Example. Consider a simple preemption game, in which the target countries choose their preemptive activity levels, while security-deterrence measures are given for exogenous reasons. Let c([x.sub.i]) and v([z.sub.i]) functions be [x.sup.2.sub.i]/2 and [z.sup.2.sub.i]/2, respectively, and F([x.sub.i]) = [epsilon][x.sub.i] + [x.sup.2.sub.i]/4. Hence the total cost facing country i has the expression [[OMEGA].sub.i] = [epsilon][x.sub.i] + [x.sup.2.sub.i]/4 + [[lambda][z.sup.2.sub.i]/2. As [F.sub.xx] > 0, there is a possibility of negative welfare effect of coalition. Let [[OMEGA].sup.l] denote the total cost of a country within the M-member coalition, and [[OMEGA].sup.n] its total cost in the precoalition state. It is worked out in Appendix S 1 that if [epsilon] = 20, [alpha] = 2, [lambda] = 65, I = 28, and M = 3, then [[OMEGA].sup.l]/ [[OMEGA].sup.n] = 1.37, that is, the total cost under coalition is higher. Hence this coalition does not benefit (even if the damage from terror is less). The same holds if M = 4. But for M = 5 or higher, [[OMEGA].sup.l] / [[OMEGA].sup.n] < 1. Hence, in this example, "5" is the critical number.

On the other hand, an outside country always benefits, as it enjoys the positive externality from the enhanced level of preemption undertaken by the countries in the coalition. In summary,

PROPOSITION 8. A coalition always benefits the outside countries, whereas it may not benefit itself if the size of the coalition is sufficiently small. (22)

III. INTEGRATING WEAK- AND STRONG-MARGINAL-FEAR EFFECTS

Thus far we have assumed that the marginal fear effect is either throughout weak or throughout strong irrespective of the level of terrorist activity. It may be more realistic to suppose that the society is resilient and can repair small tears in the social fabric. It is only when the scale of the terror activity crosses a critical level does the strong fear effects "take over."

In this scenario it seems apt to interpret the strong-marginal-fear effect as panic and the weak-marginal-fear effect as no-panic, and, these terms will, in fact, be used in the current section. A richer set of possibilities arise when no-panic gives away to panic after a threshold. As discussed in Section I, apart from no-panic or panic equilibrium, there may arise BOP equilibria. Also, multiple equilibria of two different kinds can arise. First, some parameter configurations support both BOP and panic equilibria. Second, for the same parameter values that permit a symmetric BOP equilibrium, there exist an infinite number of asymmetric BOP equilibria.

All else the same, the analysis integrating the two cases, no-panic and panic, becomes quite complex however. To keep analytical tractability, we restrict ourselves to a preemption game only, and further assume symmetry of the cost function of preemption, that is, [[lambda].sub.i] = [lambda], [for all]i. (23) As we shall see, cost symmetry does not necessarily imply symmetry of preemption choices across countries.

"No panic turning into panic" is modeled by postulating an expected damage function that is initially concave and then convex in [x.sub.i]--as shown in Figure 4. This is equivalent to the marginal expected damage being initially decreasing, and then increasing in the terror input. With some abuse of notation, we let F(x) denote the expected damage function. (24)

A. Different Possibilities and Heuristic Arguments

How various possibilities and equilibria may emerge can be understood by focusing on the problem facing the Organization in Stage 2. Recall its first-order condition in Equation (4). The right-hand side is the marginal cost of [x.sub.i] and the left-hand side is the marginal benefit, which, in our revised notation, is [F.sub.x](x). Figure 5 shows various possibilities, supposing that Z, the aggregate preemption, is already chosen by countries in Stage 1.

Clearly, in panel (A), the Organization chooses [x.sup.1] along the no-panic portion of the [F.sub.x](x) curve. It represents a no-panic equilibrium. In panel (B), the two curves intersect thrice, at [x.sup.1], [x.sup.2], and [x.sup.3], but the second-order condition is fulfilled only at [x.sup.1] and [x.sup.3]. (Multiple intersection does not mean multiple equilibria.) Given that area A exceeds area B, the Organization chooses [x.sup.1]. This corresponds to a no-panic equilibrium. Panel (C) is opposite of panel (B), Area A is less than area B, the Organization chooses [x.sup.3] and it corresponds to a panic equilibrium. In panel (D), area A equals area B and the Organization is indifferent between [x.sup.1] and [x.sup.3]. We make an intuitive tie-breaking rule that in case of indifference the Organization does not take trouble of mounting a high level of activity [x.sup.3] and selects [x.sup.1] instead. This corresponds to a BOP equilibrium. For now we ignore panels (E) and (F).

It follows that, for the same [F.sub.x] schedule, no-panic, panic, or BOP equilibrium will occur in Stage 1 depending on whether the aggregate preemption, Z, is high enough, low enough, or assumes a critical (intermediate) value.

Turning now to Stage 1, the choice of Z would critically depend upon the shift parameter k of the marginal cost function of preemption. If it is smaller than a critical value, say [[lambda].sub.1], the target countries would all choose high enough [z.sub.i] such that the no-panic equilibrium will prevail. Similarly, there would exist another critical value, say [[lambda].sub.3], so that for [lambda] > [[lambda].sub.3] there is a panic equilibrium.

The BOP equilibrium is associated with a unique level of aggregate preemption, which we will denote as [??]. What may not be apparent is that this equilibrium would exist for a range of parameter values, in particular an interval of intermediate values of [lambda], not just at a single value. This is because the marginal damage from terror to country i, at given [z.sub.j], j [not equal to] i, is discontinuous with respect to [z.sub.i] and the BOP equilibrium occurs at this point of discontinuity. Note that, as long as the sum total of preemption by other countries (j [not equal to] i) is not high enough, so that at [z.sub.i] = 0 the Organization selects [x.sup.3], a positive but sufficiently small [z.sub.i] induces the Organization to reduce [x.sub.i] along the panic portion of the [F.sub.x] curve. But once [z.sub.i] exceeds a critical value, the Organization "jumps down" to a point such as [x.sup.1] and accordingly there is a discrete decline in the marginal damage from terror. The upshot is that the BOP equilibrium, which is "knife-edge" in terms of the scale of terrorist activity and scale of fear, would hold for a parameter range having a positive measure.

It is also noteworthy that a downward shift of the marginal cost curve facing the Organization (via a decrease in [alpha]) implies a higher [??]. This means that along the BOP equilibria, an increase in the potency of the Organization invites a higher level of aggregate preemption--opposite of the effect of a decrease in ct on preemption in the no-panic case. This is because averting panic is the primary objective of target countries in the BOP equilibrium.

B. Formal Analysis

We shall work with specific functional forms of the cost and the expected damage functions. Let

A8: c([x.sub.i]) = [x.sup.2.sub.i]/2 and v([z.sub.i]) = [z.sup.2.sub.i]/2.

The next assumption formalizes the idea that panic effect is small to begin with, becoming large after a critical level of attack.

A9: F(x) satisfies

(a) F(x) : [0, [infinity]) [right arrow] [0, [infinity]) is thrice differentiable.

(b) [there exists] X > 0, such that F(x) = f(x) [for all]x [less than or equal to] X, where f(x)=ax - b[x.sup.2]/2, a > [alpha], b > 0 and X = a/b.

(c) For x [greater than or equal to] X, F(x) = [gamma]h(x) such that (i) f'(X) = [gamma]h'(X) = 0, (ii) for x > X, h"(x) > 0 > h'" (x) and (iii) [lim.sub.x [right arrow] [infinity]] h"(x) = 0. (25)

(d) (1 - [gamma]h"([x.sup.3](Z))h"([x.sup.3](Z)) + [gamma]h'([x.sup.3](Z)) h'"([x.sup.3](Z)) > 0 [for all]Z.

Thus f(x) and [gamma]h(x), respectively, capture the damage before and after panic takes effect, with [gamma] parametrizing the strength of the panic effect. We call [gamma] the "panic coefficient." Figure 6 illustrates the [F.sub.x](x) function and the marginal cost function facing the Organization.

In view of A8, the Organization's first-order condition for choosing [x.sub.i] is:

(18) [F.sub.x]([x.sub.i]) = Z + [alpha] + [x.sub.i], [for all] i.

Thus [x.sub.i] depends on aggregate preemption, not its distribution, implying that the Organization chooses the same level of attack against all countries, that is, [x.sub.i] = x. (26) The area A (respectively B) is that between the [F.sub.x](x) and Z + [alpha] + [x.sub.i] curves over the interval ([x.sup.1], [x.sup.2)] (respectively ([x.sup.2], [x.sup.3])).

The analysis of various cases begins now. For a given Z, if [gamma] is small enough, there is no intersection of the Organization's marginal cost curve with the [gamma]h' (x) function. For a single, critical value of [gamma], the former will be tangential to the latter and for higher values, there will be two intersections (since h'"(x) < 0, there cannot be more than two intersections with the [gamma]h'(x) curve).

To begin with, suppose Z = 0. Consider that value of [gamma], say [[gamma].sub.0], such that the area A equals area B. Let [x.sup.1.sub.0], [x.sup.2.sub.0], and [x.sup.3.sub.0] denote three levels of [x.sub.i] at which the [F.sub.x] curve intersects the marginal cost line Z + [alpha] + [x.sub.i]. The following equation defines [[gamma].sub.0]:

(19) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Clearly, for [gamma] < [[gamma].sub.0], area A exceeds area B for any Z > 0. Consequently, a no-panic equilibrium is the only possibility for any [lambda] > 0.

Now consider the case where [gamma] > [[gamma].sub.0]. There must exist a critical, positive Z, say [??], such that area A = area B, so that the Organization is indifferent between choosing [x.sup.1] and [x.sup.3]. The equation below solves [??]:

(20) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

It follows that no-panic, BOP, or panic equilibrium occurs respectively as Z >/< [??]. Further, since a change in [lambda] affects the choice of [x.sub.i] only through its impact on Z, and, [gamma]h'(x) is increasing in [gamma] without bound at any given x, we have

LEMMA 2. [??] is independent of [lambda], decreases with [alpha], and increases without bound with respect to [gamma].

We now turn to Stage 1 of the game. Because the Organization chooses either [x.sup.1] (Z) or [x.sup.3](Z), the total cost function facing any target country can take either of two forms:

(21) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where Z' is the sum total of preemption used by other countries. (27)

Clearly [[OMEGA].sup.1] ([z.sub.i]; x) < [[OMEGA].sup.3]([z.sub.i]; x). Also, [z.sup.1](Z', [lambda]) < [z.sup.3](Z', [lambda]), where [z.sup.k](Z', [lambda]) minimizes [[OMEGA].sup.k]([z.sub.i]; Z'), k = 1, 3 at a given Z'. (28) Figure 7 illustrates this. The total cost incurred by country i is read off the [[OMEGA].sup.1] (x) or the [[OMEGA].sup.3] (x) function according as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We are in a position to formally define various equilibria.

Definition. A preemption vector ([z.sup.1.sub.1], ..., [z.sup.1.sub.I]) is said to constitute a no-panic equilibrium if, for Z' = [[summation].sub.j [not equal to] i] [z.sup.1.aub.j],

(a) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

(b) [[summation].sub.i] [z.sup.1.sub.i] > [??],

(c) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Definition. A preemption vector ([[??].sub.1], ..., [[??].sub.I]) is said to constitute a brink-of-panic equilibrium if for Z" = [[summation].sub.j [not equal to] i] [[??].sub.j]

(a) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

(b) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

(C) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Definition. A preemption vector ([z.sup.3.sub.1], ..., [z.sup.3.sub.I]) constitutes a panic equilibrium if for Z'"= [[summation].sub.j [not equal to] i] [z.sup.3.sub.j],

(a) [z.sup.3.sub.i] = argmin [[OMEGA].sup.3] ([z.sub.i], Z'", [lambda]) [for all]i,

(b) [E.sub.j][z.sup.3.sub.j] < [??],

(C) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

These definitions are general, independent of the functional forms we assume. Part (a) says that the equilibrium [z.sub.i] is the optimal point on the respective [[OMEGA].sup.k] function in its domain. Part (b) specifies whether Z</> [??]. Part (c) is an incentive-compatibility constraint saying that it does not pay to a target country to change [z.sub.i] so as to jump to a point on the alternative cost function. The BOP equilibrium is shown in both panels of Figure 8. (29) The other two kinds of equilibria can be shown in similar diagrams, having the feature that, at [??]/I the total cost is not minimized when relevant portions of [[omega].sup.1] (x) and [OMEGA]3 (x) functions are combined.

Various equilibria in the ([gamma], [lambda]) space are characterized in Figure 9 and we have

PROPOSITION 9. Let assumptions A1, A3, A8, and A9 hold. As illustrated in Figure 9, depending on the parameters [gamma] and [lambda], various equilibria arise.

Proof." See Appendix S1.

Remarks

1. As discussed earlier, BOP or panic equilibria arise only when [gamma] > [[gamma].sub.0].

2. For a given [gamma] [member of] ([[gamma].sub.0], [[gamma].sub.1]), no panic (respectively panic) equilibrium obtains if [lambda] is relatively small (respectively large). For intermediate values of [lambda], the BOP equilibrium obtains.

3. If, for any given [gamma], [lambda] lies in-between [[lambda].sub.3] and [[lambda].sub.2] or in-between [[lambda]'.sub.3] and [[lambda]'.sub.2] there an be two kinds of equilibria: panic and BOP.

Multiple equilibria essentially arise out of a coordination failure. (30) In the BOP equilibrium, every country expects the others to supply [??]/I, so that supplying the same amount is optimal for it. In the panic equilibrium, however, every country expects the other to supply a lower level of preemption at [z.sup.3] ([lambda]). Thus panic-prevention is too costly, and all countries prefer to supply [z.sup.3] ([lambda]) instead.

4. We see that if [gamma] > [[gamma].sub.1], there is no no-panic equilibrium (which is intuitive), and at any BOP equilibrium, the scale of attack is zero. This will be explained below.

Notice further that, if it exists, a no-panic equilibrium is unique and symmetric. This follows from the cost function of preemption across countries being symmetric and the marginal benefit and the marginal cost of preemption being respectively decreasing and increasing in the level of preemption. The same holds for panic equilibrium by virtue of A9(d), ensuring that the second-order condition is met.

Given that a panic equilibrium may co-exist with a BOP equilibrium for some parameter values, it is natural to ask why the same is not true for the no-panic equilibrium. The reason is that both panic and BOP equilibria entail relatively "little" aggregate preemption. For a panic equilibrium this statement is almost tautological. Even for a BOP equilibrium, preemption is "just sufficient" to prevent panic from setting in. However, no-panic equilibrium is associated with a relatively high level of preemption. Therefore, if there is a no-panic equilibrium for some parameter configuration, other equilibria cannot arise for those parameter values.

We now focus on the BOP equilibria, which is novel. For each symmetric BOP equilibrium (given that it exists), we find that there are an infinite number of asymmetric BOP equilibria. (31) To see this, start with a symmetric BOP equilibrium in which each country "supplies" preemption equal to [??]/I. Corresponding to this set of strategies, consider another in which country i' s level of preemption is equal to [z'.sub.i] = [??]/I + [[epsilon].sub.i], where (i) [[epsilon].sub.i] is small and (ii) [summation][[epsilon].sub.i] = 0. Step (ii) implies that [summation] [z'.sub.i] = [??]. Notice from Figure 8 that at the symmetric BOP equilibrium, [[OMEGA].sup.1] is increasing in [z.sub.i]. Hence for [[epsilon].sub.i] sufficiently small, [[OMEGA].sup.1] is increasing in [z'.sub.i] whether [epsilon] [??] 0. Choosing [z'.sub.i] = [??]/I + [[epsilon].sub.i] is thus preferred to any [z.sub.i] > [z'.sub.i]. In conjunction with Figure 9, it is also seen that as long as [lambda] is either in the interior of ([[lambda].sub.1], [[lambda].sub.2]) or strictly less than [[lambda]'.sub.2], and [[epsilon].sub.i] is small enough, we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Hence [z'.sub.i] is also preferred to any [z.sub.i] < [z'.sub.i]. Thus any country i supplying the preemption level of [z'.sub.i] constitutes a BOP equilibrium.

Intuitively, a BOP equilibrium is intrinsically tied to the panic threat. Averting this threat is of paramount importance as long as the cost of preemption is not too high. If one of the countries lowers the level of preemption by a small amount compared to an existing symmetric BOP equilibrium, the other countries have an incentive to make up this short-fall.

We next turn to some comparative statics. An increase in the potency of the Organization (a decrease in a) leads to a decline or an increase in aggregate preemption depending on whether it is a no-panic or panic equilibrium. This is consistent with our analysis in the previous section. Interestingly, if it is a BOP equilibrium, a higher [alpha] implies a higher [??], meaning a greater level of aggregate preemption--which is opposite of the response along the no-panic equilibrium--even though in this equilibrium too the Organization operates on the no-panic segment of the marginal damage curve. It is because the sole objective of the countries in the BOP equilibrium is to avert panic.

Consider an increase in [gamma], the panic coefficient. Along the BOP equilibrium, an increase [gamma] implies a decrease in the level of terror input employed by the Organization and hence a decline in the expected damage from terror. This may appear somewhat surprising. However, the reason is that the more prone the countries are to panic, the more aggressive will they be to prevent panic and thus the smaller is the scale of attack or terror input used by the Organization. Hence if [gamma] is sufficiently large, the marginal cost line [??] + [alpha] + x will intersect the f'(x) line at point a or above it in Figure 6. This explains why in Figure 9, [x.sub.i] = 0 in the BOP equilibrium when [gamma] > [[gamma].sub.1]. (32)

PROPOSITION 10. Let assumptions A1, A3, A8, and A9 hold.

(a) Whenever it exists, the no-panic equilibrium is unique and symmetric. A similar statement holds for the panic equilibrium.

(b) Associated with each symmetric BOP equilibrium are infinitely many asymmetric BOP equilibria.

(c) In a BOP equilibrium an increase in the potency of the Organization leads to a higher level of aggregate preemption. Moreover, an increase in the panic coefficient implies a higher level aggregate preemption, as well as a lower scale of attack by the Organization.

C. Cooperation

The analysis of cooperative behavior for the no-panic (respectively panic) equilibrium is quite similar to that for the weak-marginal-fear (respectively strong-marginal-fear) case analyzed earlier. Consequently, we focus on the BOP equilibrium, and, in this case, several new insights on the nature of cooperation emerge.

First, there exist parameter values (namely, [lambda] not too small) such that the aggregate preemption level, [??], will be the first best anyway. Thus, the standard argument for cooperation--that it helps internalize some externality--does not apply. The intuition hinges on the fact that in the first-best outcome the countries are constrained to supply "enough" preemption, so as to avoid panic. But this is exactly the symmetric BOP equilibrium. Second, there is a rationale for cooperation however, if the BOP equilibrium involves an asymmetric allocation of preemption among the target countries. Third, there is a further ease for coordination if there is a panic equilibrium, even when a BOP equilibrium co-exists. This is because in this case a symmetric BOP equilibrium Pareto dominates the panic one.

The last two arguments, which favor cooperation, are, interestingly, based on coordination on the appropriate non-cooperative equilibrium, not on explicit collusion. Even in the absence of explicit collusion, the presence of multiple equilibria implies that cooperation can help to coordinate on the welfare maximizing equilibrium.

We then turn to the formal analysis. In order to focus on the case of interest let [[gamma].sub.1] > [gamma] > [[gamma].sub.0], so that there is some role for preemption. (33) Under cooperation the countries jointly solve [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [x.sup.*] (x) solves the Organization's optimization problem. We first look for solution assuming that it operates in the no-panic region, that is, [x.sup.*]([[summation].sub.i][z.sub.i]) = [x.sup.1]([[summation].sub.i][z.sub.i]). From the first-order conditions it follows that any solution must be symmetric. Consequently, the symmetric solution is:

(22) [z.sup.C]([lambda]) = I(a + b[alpha]/[[lambda][(1 + b).sup.2] - b[I.sup.2]].

It is easy to see that [z.sup.C]([lambda]) > [z.sup.*] ([lambda]), where recall that [z.sup.*]([lambda]) is the symmetric solution to minimizing [[OMEGA].sup.1]([z.sub.i], [[summation].sub.j [not equal to] i] [z.sub.j]). This is intuitive as the cooperation internalizes the externalities generated by preemption.

However, note that the outcome where every country supplies [z.sup.C]([lambda]) may not be the first best, because if I [z.sup.C]([lambda]) < [??] panic will be triggered. In that case the first best, after taking the possibility of panic into account, involves every country supplying [??]/I, so as to avoid panic. If, however, I [z.sup.C] ([lambda]) [greater than or equal to] [??], then the first best involves every country supplying [z.sup.C]([lambda]). Given that [z.sup.C]([lambda]) is decreasing in k, the first best involves every country supplying [??]/I if and only if [lambda] > [[lambda].sub.c], where [[lambda].sup.c] solves [z.sup.C]([lambda]) = [??]/I. Given that [[lambda].sub.c] > [[lambda].sub.1] (see Equations (23) and (A.10)), we focus on two cases:

Case 1. [[lambda].sub.0] [less than or equal to] [lambda] [less than or equal to] min {[[lambda].sub.2], [[lambda].sub.c]}, so that the first-best outcome involves every country supplying a preemption level of [z.sup.C]([lambda]), ensuring that panic is prevented. In this case collusion, internalizing the externalities, would increase welfare irrespective of the nature of equilibrium.

Case 2. [[lambda].sub.c] < [lambda] [less than or equal to] [[lambda].sub.2]. (34) In this case the first best involves all the countries supplying [??]/I, so that the first best outcome coincides with the symmetric BOP equilibrium. If, moreover, the non-cooperative equilibrium is symmetric, then the cooperative and non-cooperative outcome is the same, so that there is no role for cooperation. In case it is asymmetric, however, cooperation can improve aggregate utility of the target countries by improving the allocation of preemption. This follows since, with convex and symmetric costs of preemption, efficiency involves a symmetric allocation of preemption among countries.

Continuing with Case (2), if [[lambda].sub.3] < [lambda] [less than or equal to] [[lambda].sub.2], then there are both panic and BOP equilibria. It is straightforward to show that the countries prefer the symmetric BOP equilibrium over the panic equilibrium. This follows since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where the first inequality follows because the outcome where every country supplies [??]/I constitutes a BOP equilibrium, and the last inequality from fact that for all [lambda] [greater than or equal to] [[lambda].sub.3], [??]/I > [z.sup.3] [(I - 1)[??] / I, [lambda]]. Consequently, coordination may help the countries reach the symmetric BOP equilibrium, which welfare dominates the panic equilibrium.

Summarizing the above discussion we have our next result.

PROPOSITION 11. Let assumptions A1, A3, A8, and A9 hold. Further, suppose that [[gamma].sub.0] < [gamma] < [[gamma].sub.1] and [[lambda].sub.c] < [lambda] [less than or equal to] [[lambda].sub.2].

(i) A symmetric BOP equilibrium exists, which, moreover, leads to the first best outcome.

(ii) If the equilibrium is an asymmetric BOP one, then coordinating to obtain the symmetric BOP equilibrium is welfare improving.

(iii) If [[lambda].sub.3] < [lambda] [less than or equal to] [[lambda].sub.2] and there is a panic equilibrium, then coordinating to obtain symmetric BOP equilibrium is welfare improving.

For [gamma] [greater than or equal to] [[gamma].sub.1], very similar results go through.

D. Generalizations and Other Possibilities

While the preceding analysis has assumed specific forms of the preemption cost function, the cost function of producing terrorism, as well as the damage function, many of them were used for the sake of technical convenience only. For instance, the linear segment of F(x) in the no panic zone, combined with a linear cost function of the Organization, yielded a closed-form solution for [x.sub.i] in this zone--as well as the possibility that a sufficiently high level of aggregate preemption totally eliminate terrorist attacks. Any general downward sloping [F.sub.x](x) function over x [less than or equal to] X would have served the purpose. The assumptions on [F.sub.x](x) over the panic zone implied strict concavity of the marginal damage function and hence a unique panic equilibria. The main insights and particularly how the BOP equilibrium emerges would hold as long as the marginal damage function is downward sloping initially, and strictly concave when upward sloping.

Consider now panel (E) in Figure 5. The marginal damage function is not concave in its upward sloping range and thus, in the panic region, the Organization would choose any arbitrarily high level of [x.sub.i]. This is obviously not a plausible outcome. A natural assumption which eliminates this possibility would be the existence of an exogenous capacity constraint on the terrorist organization, say at [bar.x] (as shown in Figure 5). (35) Indeed, the analysis of this case is simpler--and it does not hamper any of the results we have obtained--because the panic equilibrium holds at [bar.x] and it is robust to a marginal change in Z as long as Z < [??].

Finally, in a scenario such as panel (F) there are many intersections of the marginal damage function (in its upward range) with the marginal cost function. Multiple panic equilibria is an additional possibility. But, as long as the Organization's optimal [x.sub.i] declines with aggregate preemption (which is quite plausible), the BOP and no-panic equilibria would arise in the same way as before.

IV. CONCLUDING REMARKS AND EXTENSIONS

Some terrorist organizations operate globally and target many countries. We have developed a game-theoretic model incorporating the behavior of target countries and the terrorist organization explicitly. The damage from terror is defined in terms of the panic or fear it engenders and we have modeled the implications of a regime switch from no-panic to panic. Both preemption and deterrence are considered as counter-terrorism policy instruments. Our analysis allows us to ask several policy questions of interest--in particular, if more offense is the best response to more potential terror. Different kinds of equilibria emerge, including one which is rather non-standard, namely, the BOP equilibrium.

One can consider many relevant extensions of the problem we have taken up, such as more than one terrorist organization, various forms of informational asymmetries, learning and reputation building, strategic negotiations, and so forth. In closing, we briefly comment on two direct extensions of our model, both relating to asymmetry in targeting by the Organization.

This article has assumed that the Organization targets all countries equally. What happens when there is differential targeting for extraneous reasons? This can be modeled by attaching differential weights, say [[mu].sub.i], in the objective function of the Organization to the expected damage caused to a country. (These weights may be seen as hatred parameters.) Its objective function can be set up as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

To focus on difference in targeting, let us assume that the countries are symmetric otherwise, that is, [[beta].sub.i] = [beta] and [[lambda].sub.i] = [lambda].

Consider first the case where the marginal fear effect is either uniformly strong, or uniformly weak. Straightforward calculations show that if [[mu].sub.i] > [[mu].sub.j], then [s.sub.i] > [s.sub.j], that is, a relatively high-target country will choose a relatively high security-deterrence levels. How do the preemptive measures compare? A relatively more targeted country chooses a relatively lower (respectively higher) level of preemption, if the marginal fear effect is weak (respectively strong). This is because preemption measures are strategic substitutes (complements) if the marginal fear effect is weak (strong). Such asymmetric targeting may have interesting implications for the case of no panic yielding to panic. In reference to the BOP equilibrium in particular, we conjecture that countries facing relatively greater targeting supply the total level of preemption that averts panic, while the other countries free-ride.

Instead of targeting being exogenous, it may be a reaction to preemption levels chosen by target countries. Broadly speaking, this would have the following implications. First, all else the same, such "endogenous targeting" implies a higher marginal cost of preemption activity and this would tend to reduce the level of preemption. Second, insofar as targeting is a function of the relative magnitude of preemption, it generates complementarity in preemptive activity: an increase in preemption by country i makes the Organization target country j less than before, inducing the latter to increase its preemptive measures. Third, cooperation among target countries would contain the element of minimizing the relative effect on targeting through a choice of relatively more symmetric levels of preemption than the intrinsic source of asymmetry among target countries would have otherwise implied.

ABBREVIATIONS

ANO: Abu Nadal Organization

BOP: Brink-of-Panic

doi: 10.1111/ecin.12034

Online Early publication July 5, 2013

REFERENCES

Abadie, A., and J. Gardeazabal. "The Economic Cost of Conflict: A Case Study of the Basque Country." American Economic Review, 93(1), 2003, 113-32.

--. "Terrorism and the World Economy." European Economic Review, 52(1), 2008, 1-27.

Bandopadhyay, S., and T. Sandier. "The Interplay between Preemptive and Defensive Counter-Terrorism Measures: A Two Stage Game." Economica, 78(311), 2011, 546-64.

Becker, G., and Y. Rubinstein. "Fear and Response to Terrorism. An Economic Analysis." Mimeo, Brown University, 2010.

Carcels-Poveda, E., and Y. Tauman. "A Strategic Analysis of the War against Transnational Terrorism." Games and Economic Behavior, 71(1), 2011, 49-65.

Das, S. P., and S. Lahiri. "A Strategic Analysis of Terrorist Activity and Counter-Terrorism Policies." B. E. Journals in Theoretical Economics (Topics), 6(1), 2006, 1-28.

Hausken, K. "Whether to Attack a Terrorist's Resource Stock Today or Tomorrow." Games and Economic Behavior, 64(2), 2008, 548-64.

Mirza, D., and T. Verdier. "International Trade, Security, and Transnational Terrorism: Theory and Empirics." World Bank Policy Research Working Paper 4093, December 2006.

Pape, R. A. Dying to Win: The Strategic Logic of Suicide Terrorism. New York: Random House, 2005.

Richman, J., L. Cloninger, and K. M. Rospenda. "Macrolevel Stressors, Terrorism, and Mental Health Outcomes: Broadening the Stress Paradigm." American Journal of Public Health, 98(2), 2008, 323-29.

Sandler, T., and K. Siqueira. "Global Terrorism: Deterrence Versus Pre-Emption." Canadian Journal of Economics, 2006, 39(4), 1370-1387.

U.S. Department of State. Patterns of Terrorism, 1996, 1997.

SUPPORTING INFORMATION

Additional Supporting Information may be found in the online version of this article:

Appendix S1. Various Proofs

Figure S1. Coalition. (A) Weak Marginal Fear. (B) Strong Marginal Fear

Figure S2. Definition of [[lambda].sub.2]

SATYA P. DAS AND PRABAL ROY CHOWDHURY *

* The paper has benefited from a reviewer's comments and seminar presentations at the Universidad Carlos III de Madrid, the City University of Hong Kong, the Hong Kong University of Science and Technology, and the Delhi School of Economics. Remaining errors are ours.

Das: Professor, Indian Statistical Institute-Delhi Centre, 7 S J S Sansanwal Marg, New Delhi 110016, India. Phone 11-4149-3936, Fax 11-4149-3981, E-mail das@isid.ac.in

Roy Chowdhury: Professor, Indian Statistical Institute-Delhi Centre, 7 S J S Sansanwal Marg, New Delhi 110016, India. Phone 11-4149-3930, Fax 11-4149-3981, E-mail prabalrc@isid.ac.in

(1.) Not only does Al-Qaeda itself target many countries, it purportedly supports local or regional terrorist organizations, thus creating inter-linkages among the activities of the latter.

(2.) Since Abu Nadal's death in 2002 the ANO has remained largely passive.

(3.) Mirza and Verdier (2006) also stress the necessity of such a formulation, but they stop short of setting up a formal analytical model.

(4.) In their follow-up work, Abadie and Gardeazabal (2008) argue that mobility of productive capital in an open economy may account for much of the impact of terrorism.

(5.) There is an earlier theoretical literature dealing with hostage scenarios and subsequent negotiations between terrorists and governments.

(6.) Given that our results depend on the nature of fear induced by terror, that is, whether it is weak or strong, something that does not depend on the sequencing of moves, the differences in results are not driven by the fact that in the present framework the counter-terrorism efforts are all in the first stage, while in Carlceles-Poveda and Tanman (2011), these are chosen in the second stage. Also, none of our results are driven by issues related to first-mover advantages or disadvantages, something that would of course he affected by the sequencing.

(7.) Last but not least of all, our analysis should not be construed as potentially helping terrorist acts in any way; We say this, given the response received by Steven Levitt to his writings on terrorism in the "Freakonomics" blog.

(8.) It will be noted later that in a simultaneous game, optimal preemption is always zero.

(9.) For the cases of [F.sub.x]x < 0 and [F.sub.xx] > 0 the respective examples will be: F([s.sub.i], [x.sub.i]) = A[x.sub.i] - [x.sup.2.sub.i] + [x.sub.i][(K - [s.sub.i]).sup.2] and F([s.sub.i], [x.sub.i]) = [x.sup.2.sub.i] + [x.sub.i][(K - [s.sub.i]).sup.2], where A and K are positive constants that are sufficiently large.

(10.) In Section IV we discuss some implications of allowing for asymmetric targeting.

(11.) This specification ignores country-specific preemptive actions, and thereby enables us to differentiate this policy sharply from security-deterrence which is largely a country-specific policy.

(12.) All conditions in A7 need not be binding, for example, if [F.sub.xx] [less than or equal to] 00, (R1) is met for any c" > 0.

(13.) Such externality would not arise if all preemptive actions were country-specific.

(14.) The derivative of the term in the square bracket of Equation (9) with respect to [s.sub.i] equals -[F.sub.ss] - [F.sup.2.sub.sx]/(c" - [F.sub.xx]), which is negative.

(15.) Here, "net" refers to that levels of preemption by other countries are not given.

(16.) Given A1, A5, and (R1) in A7, [A.sub.i] > 0.

(17.) This is captured by the positive term [(c").sup.2] [F.sup.2.sub.sx] in the definition of Bi. Indeed, if countries were to keep their security-deterrence levels unchanged, this term would not appear in d[z.sub.i]/d[z.sub.j].

(18.) Algebraically it is verified by the fact that if [F.sub.xx] < 0, the term [B.sub.i] is unambiguously positive.

(19.) In a simpler way it can be seen that in the standard public-good case, the good itself contributes directly to private utility and thus implies strategic substitutability. Here, preemption is akin to a public input, which may reduce or enhance the marginal effect of one's own provision of this input toward own welfare.

(20.) Hence if preemption levels were constant there is no over- or under-provision of deterrence.

(21.) Check that, in view of Equation (16), our assumption (R3) in A7 implies that the second-order conditions are being met.

(22.) We do not analyze the issue of coalition-proofness, that is, the problem of incentive of a member country or a group of member countries to leave an already successful ("profitable") coalition.

(23.) Looking at preemption alone is of independent interest, when, for example, security-deterrence is handled by a different government wing, with deterrence levels impacting on aspects other than terrorist threat. For instance, too high a level of deterrence would adversely affect tourism industry and the trade sector of a country. Thus factors other than security concerns may be more decisive in choosing security-deterrence levels.

(24.) A counter argument to a fear function such as in Figure 4 may be that people get used to terror as time passes. Incidents that used to cause panic would be taken on a stride after several occurrences. Our position is that fear is a relative concept--just as much as utility from consuming the same good. (Cell phones that used to be a novelty a decade and half ago are not so any more.) However, relative to a bench-mark (initial) range of terror activity, a jump in it will surely cause panic for some time period to come, and this is what our static analysis is designed to capture.

(25.) An example of h(x) satisfying 4(c) is h(x)= 2/3 [(x - X).sup.1.5]

(26.) This holds even if the cost functions of preemptive measures are asymmetric.

(27.) We explicitly write the parameter [lambda] in the (OMEGA) function, since, as we shall see, various cases depend on the value of this parameter in a systematic way. Other parameters matter too, but we avoid making them explicit for the sake of notational ease.

(28.) The proof follows from the fact that f" < 0 < h" and [gamma]h'([x.sup.3](z)) > f'([x.sup.1] (z)).

(29.) There are two panels, depending on the magnitude of [??]/I relative to [z.sup.3] that minimizes the [[OMEGA].sup.3] function.

(30.) If there were a single target country then [[lamba].sub.2] = [[lambda].sub.3] and multiple equilibria cannot arise.

(31.) This is a situation of multiple equilibria in a different dimension compared to the coexistence of BOP and panic equilibria.

(32.) This possibility would not have risen if f(x) satisfied the Inada condition at x = 0.

(33.) All the results go through for [gamma] [less than or equal to] [[gamma].sub.1] if in Proposition 11 we replace [[lambda].sub.i] by [[lambda]'.sub.i].

(34.) The cases where [[lambda].sub.2] < [lambda] < [[lambda].sub.c], or where [lambda] > max {[[lambda].sub.c], [[lambda].sub.2]} yield no new insights and hence are ignored.

(35.) We can think of this constraint as arising from an absolute limit on the funds, or volunteers available to the terrorist organization.

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Author:	Das, Satya P.; Chowdhury, Prabal Roy
Publication:	Economic Inquiry
Article Type:	Abstract
Geographic Code:	1USA
Date:	Jan 1, 2014
Words:	13780
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