WHY DIRECT COUNTER-TERROR (CT) MEASURES ONLY MAY

FAIL: AN ANALYSIS OF DIRECT AND PREVENTIVE CT MEASURES∗

Satya P. DasVisiting, University of Illinois

at Urbana-Champaignsatyadas@illinois.edu

Sajal LahiriSouthern Illinois University Carbondale

lahiri@siu.edu

This draft: March 12, 2018

Preliminary: Comments welcome

Abstract

We present a model of three-period game between a terrorist organization (Org) and a defend-ing state (State), where the Org chooses aggregate terror input and the State chooses (a) the levelof preemptive measures which shift the marginal cost curve facing the Org and hence affect thesupply curve of terror, and (b) the level of redressal of grievances of the Org’s population whichaffects the derived demand for terror as a means to an ‘end’ (demand). (a) and (b) are interpretedrespectively as direct and preventive counter-terrorism (CT) measures (equivalently as stick andcarrot policies). There is two-sided asymmetric information: the Org and the State do not fullyknow each other’s preferences.

We define the War on Terror (WoT) to be ‘lost’ or ‘won’ if, in response to an exogenousincrease in militancy or terrorism, the State steps up it CT measures so that the new equilibriumlevel of terror increases or does not increase. A central result is an ‘impossibility theorem’ that theState cannot win WoT with preemptive measures only as long as the marginal cost of preemptivemeasures is increasing. In general, the optimal response of the State to an exogenous increasein terrorism or militancy is to increase preemption as well as grant more grievance redressal.Hence, direct and preventive CT measures complement each other: they are not substitutes. Thecore model is extended to consider an increase in terrorism due to anger and frustration arisingfrom collateral damage associated with preemptive strikes, the presence of heterogeneous groupswithin a terrorist organization such as moderates and hardliners, a piecemeal conciliatory offer ofless preemption and more grievance redressal by the State, and Bayesian learning by the Org ofthe preference of the State if the State grants any further grievance redressal.

JEL Classification: C72, D74, F51, F52Keywords: Terrorism, Counter-Terror Measures, Preemption, Grievance Redressal

∗We’re thankful to Rajat Deb, Shale Horowitz and seminar participants at Jadavpur University, Indian Statistical In-stitute, 16th CRETE held at Milos, Greece, and Southern Methodist University for useful comments. The usual caveatapplies.

1 Introduction

Most theoretical literature on counter-terrorism (CT) policies or strategies confine themselves to

security-deterrence measures and preemption. Among many others, it includes Sandler (2005), Das

and Lahiri (2006), Sandler and Siqueira (2006), Bueno De Mesquita and Dickson (2007), Hauksen

(2008), Jindapon and Neilson (2009), Bandopadhyay and Sandler (2011), Carcels-Poveda and Tau-

man (2011), Sandler et al. (2011) and Das and Roy Chowdhury (2014). This approach falls in line

with the idea of war on terror (WoT) or global war on terror (GWoT), coined by the former U.S. Pres-

ident George Bush shortly after 9/11. A decade and half later, a fair and general assessment would be

that WoT is far from being won. On one hand, the likelihood of spectacular attacks such as 9/11 has

receded - thanks to the deployment of comprehensive security measures, strikes on terrorist havens,

scrutiny of movement of funds, larger volume of intelligence gathering and the like. On the other

hand, the number of terror attacks and human lives lost by these attacks have increased dramatically.

According to Institute for Economics and Peace (2015) report, there were less than 2,000 terrorist

incidences globally in the year 2000, killing less than 4,000, and in 2014 nearly 13,000 terrorist in-

cidents were reported which killed over 30,000 people. While most terrorist activities are confined

to “SANIP” — Syria, Afghanistan, Nigeria, Iraq and Pakistan, the rest of the world witnessed 54%

increase in terrorist incidents in 2013, compared to 2012. In 2016, the number of terror attacks in the

rest of the world was between 75 to 80% of attacks in SANIP (Global Terrorism Dataset, University

of Maryland).1 The threat of terrorism is global and persistent. In this sense victory in WoT remains

a far cry.

There is an increasing understanding now that WoT is not just about raising security, killing or

capturing terrorists, squeezing the flow of funds, destroying their bases, infiltrating their organiza-

tions, etc. Furthermore, poverty and lack of education may be contributing factors toward terrorism

but are not the primary causes. It is imperative to understand what the terrorist organizations really

want or what their ultimate goal may be. The U.S. Department of Defense defines terrorism as “the

calculated use of unlawful violence or threat of unlawful violence to inculcate fear; intended to coerce

or to intimidate governments or societies in the pursuit of goals that are generally political, religious,

or ideological.” Painstaking collection of quality data by Pape and Feldman (2010) suggests that di-

rect and indirect control or occupation by foreign armed forces is a primary cause of suicide terrorism.

Lack of political and religious freedom and failed states provide the breeding ground of terrorists and

terrorism (Krueger (2007), Berman and Laitin (2008) and Berman (2011)). Krueger (2007) capsules

these factors as “geopolitical grievances.” A broader list of geopolitical grievances or fears of the

population from which terrorist organizations emerge is provided by Hegghammer (2009). In order

to ‘combat’ terrorism, increasing references are being made towards redressing such grievances.

In her speech delivered at the OSCE (Organization for Security and Cooperation in Europe) Ex-

pert Conference in Astana, Kazakhstan, the U.S. Deputy Coordinator for Homeland Security and

1The number of deaths in SANIP due to terror attacks far outweighs that in the rest of the world however.

1

Multilateral Affairs, Anne Witkowsky discussed the need for international cooperation at various lev-

els to counter terrorism and at the same time said the following:

“To make progress on this front, we must resolve legitimate grievances peacefully and

strive to foster good governance, reduce poverty and corruption, and improve education,

health and basic services. In particular, we need to focus our efforts on locations where the

risk for radicalization is exceptionally high. Once we understand the unique combination

of grievances and needs that are driving recruitment in these “hot spots,” we can set about

creating tailor-made programs and strategies in response.” (Witkowsky (2010))

David Sedney, the Deputy Assistant Secretary of Defense for Afghanistan,Pakistan and Central

Asia from 2009-2013 and the Deputy Assistant Secretary of Defense for East Asia from 2007-2009,

writes

“Our singular focus on killing, without any serious attempt to ameliorate basic societal

problems – and the absence of a moral core for our actions – have led huge swathes of the

world to see us as the evil doers. Extremists today seek revenge for those we have killed,

to punish us for abuses they suffer, and to end our support for abusive, corrupt rulers.”

(Sedney (2015))

In the White House Summit on Countering Violent Extremism in 2015, President Obama had voiced

that “we have to address grievances terrorists exploit.”

From an analytical perspective we may view security-deterrence, intelligence gathering and pre-

emption as direct, short-run measures to ‘fight’ terrorism, whereas measures to mitigate or redress the

fundamental grievances of communities from which terrorist organizations emerge can be seen as a

long-term preventive strategy towards tackling terrorism - akin to medicines or procedures to fight or

cure a disease as opposed to preventive measures to counter the origin of a disease.2

One of the notable success stories in more or less eliminating terrorism and conflict is the Northern

Ireland case. The main grievance of the nationalists (Sinn Fein and the Irish Republican Army) was

the lack of representation of the Catholics in actual policymaking, and the main tenet of the peace

agreement has been power-sharing by all the groups in the Northern Ireland administration. Recent

peace accord between the Colombian government and FARC that has led to a cessation of more

than fifty years of violence and terrorism includes redressing longstanding grievances of the rebels

in terms of government’s promise of large-scale investments in rural areas, which, historically, have

been neglected in Colombia.3 The main objective of Hamas and Hezbollah is an establishment of a

Palestinian Islamic state and Taliban’s aim is to create an Islamic theocracy within Afghanistan; these

are geopolitical objectives that are potentially negotiable.

2Sometimes, preventive counter-terror measures refer also to adequate security measures, conciliation, mediation etc;see Eijkman and Schuurman (2011). But our definition of preventive strategies is different.

3In Sri Lanka, while the military force dealt a defeat for LTTE in 2009, the civil war bordered on terrorism in that themain target of LTTE was the government; killing of civilians by LTTE occurred toward the end of the conflict.

2

The paper takes the position that while direct measures to curb terrorism are necessary, they are

hardly sufficient to contain or reduce terrorism; it is critical to actively pursue preventive measures in

terms of redressing grievances that are legitimate and reasonable, although they carry their own costs

for the target countries. More specifically, this paper develops an analytical model with double-sided

asymmetric information which integrates these direct and preventive measures. Together, they may

be called the “DP” approach to terrorism. Indeed, a key result of our analysis that, as long as the

marginal cost of direct measures is increasing, it is impossible to win WoT by direct measures only.

This is in the sense that in response to an exogenous increase in militancy or terrorism, a rational

target state would not intensify its CT measures to eventuate a decline in the equilibrium level of

terrorism. Put differently, the target state would be only chasing the problem without being able to

overcome it. This underscores the importance of the DP approach to terrorism.

Furthermore, in sharp contrast to the existing literature (e.g., Sandler and Siqueira (2006), Bueno

De Mesquita (2005b), Das and Lahiri (2006), Carcels-Poveda and Tauman (2011) and Das and

Roy Chowdhury (2014)), utility enhancement is not the basis of terror attacks by a terror organi-

zation in our model. Terror is used by the terror organization in the pursuit of inducing a target state

to grant redressal. Put differently, in our model preventive measures reduce the incentive to engage in

terrorist attempts, and, thus, commitment to some redressal of terrorists’ grievances is critical.

In order to keep our analysis in perspective, it must be noted however that most recent, major

instances of terrorism perpetrated by ISIS and modern Al Qaeda seem to be driven more by a global

agenda of spreading extreme ideology. Such objectives do not come under the purview of geopolitical

grievances. They are illegitimate and unacceptable, even partially. Authors such as Kagan et al.

(2016) strongly warn that these objectives pose an existential threat — effectively a WoT of a very

complex kind — to liberal democracies, and, they would require a very different kind of intervention

— perhaps fostering political and religious competition and freedom, as recommended by Iannacone

and Berman (2006). The analysis of such preventive CT measures is, however, beyond the scope of

the current paper.

Section 2 lays out the formal structure of our model, whereas Section 3 analyzes the optimal

choice of of direct and preventive counter-terror measures. Comparative statics with respect to an

exogenous increase in terrorism or militancy and an exogenous change in cost of preemption are

defined and analyzed in Section 6. This section demonstrates and delineates a situation where an

increase in militancy or terrorism cannot be adequately countered by direct counter-terror measures

only. Three extensions are presented in Section 5, namely, a collateral-damage-induced increase in

terrorism and militancy, the presence of hardliners along with moderates in the terrorist organization,

and piecemeal concessions by the State towards a negotiated settlement. Section 6 presents a model

of learning by the terrorist organization about the preference of the target state from its attempt to

offer redressal of grievances and its impact on the state’s choice of preventive measures in the first

place. Section 7 concludes the paper.

3

2 Environment

There is one terrorist organization, the “Org,” and one target state, the “State.” If the State can pre-

commit itself to any level of crackdown on terrorists regardless of the cost of crackdown, the problem

of terrorism may not surface and preventive measures may become redundant. This is a plausible

scenario where the State is autocratic. But it is not a realistic situation where democratic nations

are the potential targets. Democracies would typically care about costs of preemption (crackdown)

including those of resources used and collateral damage, and, this is why terror organizations would

use terror against the state in the first place to achieve their objective. The State in our model is a

non-autocratic entity.

The population which the Org represents lives in an environment that is partly dictated by the

State, towards whom the population and the Org feel resentful. It could be occupation by armed

forces of the State, a regime abhorred by the population but supported by the State or a failed state

due to prior policy of the State. Redressing the grievance of the Org’s population by the State is

the preventive measure towards terrorism. Grievance redressal is, of course, not the only type of

preventive measures. They would include, for instance, promoting political and religious freedom and

separation between religion and the state; the present paper abstracts from these types of preventive

measures towards — or root causes of — terrorism.

The Org is a homogeneous entity. There is no mixture of moderates and extremists and thus

no differential reactions from within the Org to grievance redressal. (This will relaxed later.) Let

grievance redressal be captured by a single (composite) dimension, namely the unit interval g ∈ [0, 1]:

an increase in g means more grievance redressal. It is a unilateral “concession,” not an outcome of

any bargaining process. The upper limit on g, equal to g, is normalized to one, and achieving it is the

primary goal of the Org, whereas the initial g is normalized to zero. It is worth stressing again that

g is intended to capture terrorist demands that have some justification or legitimacy — not goals like

political, cultural or religious subjugation.

An increase in g is utility-enhancing for the Org but utility-reducing for the State. The Org’s and

State’s utility functions are u∗U(g) and z∗C(g) respectively, where u∗ and z∗ are positive and constant.

(The role of the multiplicative parameters will be discussed shortly.) Both are mappings [0, 1] → R,

which are continuous and twice differentiable.

Assumption 1. U′ > 0 > U′′; C′ < 0, C′′ < 0,C(1) = 0.

Observations.

[a] First and foremost, the damage from terror does not enter the Org’s utility function. The function

u∗U(·) reflects the long-term, “ultimate” objective of a terrorist organization. This is in sharp con-

trast to the existing analytical literature on terror and CT measures in which a terrorist organization

is explicitly or implicitly modeled as an entity who gains direct utility from inflicting damage to de-

fending states, e.g. Sandler and Siqueira (2006), Bueno De Mesquita (2005b), Das and Lahiri (2006),

4

Carcels-Poveda and Tauman (2011) and Das and Roy Chowdhury (2014). In our model here, terror

is a means used by the Org against the State in order to elicit an increase in g.

g g0

zC(g)

uV (g)

State

Org

Figure 1: V(·) and C(·) Functions

[b] Granting redressal is costly to the State in terms of utility loss; C′′ < 0 means increasing marginal

cost of grievance redressal which will be necessary for an interior equilibrium solution of g ∈ (0, 1).

How far the State will be willing to provide redressal to the Org’s population depends crucially on its

utility function z∗C(g).

[c] Compared to a given z, a higher z can be interpreted as a State which is more unwilling to grant

grievance redressal — a hardliner State. We may think of z∗ as an outcome of interaction between

hardliners and softliners within the State.4

From the Org’s utility function we derive another function u∗[U(1) − U(g)] ≡ u∗V(g), which

represents a shortfall of utility from the full redressal utility and thus measures a utility deficit. Clearly,

V ′(g) < 0 < V ′′(g). Figure 1 illustrates v∗V(·) and z∗C(·) functions.

We will later use specific functions satisfying Assumption 1, which will serve as examples.

Example A: U(g) = vgλ, v > 0, λ ∈ (0, 1); V(g) = v(1 − gλ);

C(g) = c(1 − gβ), c > 0, β > 1. (1)

Example B: U(g) = A − v(1 − g)λ, v > 0, A > v; V(g) = v(1 − g)λ, λ > 1;

C(g) = c(1 − gβ), β ≥ λ.5 (2)

4However, a variation in u does not reflect well how extremist or moderate a terrorist organization is insofar as theultimate objective of both is to obtain g = g. One way to distinguish between extremists and moderates with a terrororganization would be via incorporating a bargaining process with the State where extremists are identified as those whoare less willing to bargain and more willing to engage in violence to achieve their objective. Still another way of thinkingabout extremists within the broad purview of our framework is to define those who receive direct utility from the damageto target countries caused by terror as extremists, whereas moderates are the ones use terror only as a means to an end interms of grievance redressal.

5

In addition to grievance redressal, the State incurs utility loss due to damage from terror.

Assumption 2. Utility losses to the State from granting grievance redressal and receiving damage from

terror are additively separable.

Apart from grievance redressal (preventive counter-terror (CT) measures), another set of control

variables is available to the State — namely, direct CT measures in the form of preemptive strikes

against the Org and curbing its financial resources which tend to reduce the terror-producing capa-

bility of the Org or equivalently increase the marginal cost of producing terror. For instance, drone

strikes are frequently used by the U.S. to kill leading terrorists and damage terrorist camps and fa-

cilities. The U.S. Office of Terrorism and Financial Intelligence (TFI) devises strategies to combat

financing of domestic and international terrorist organizations. Such measures are lumped together

as preemption, which is measured along R+. The total cost of preemption is given by the function

H(m, κ) where m represents the scale of preemptive measures and κ ∈ R+ is a shift parameter. The

function H: R2+ → R+ is continuous and twice differentiable, and, satisfies

Assumption 3. Hm > 0, Hmm ≥ 0,Hκ > 0, and Hmκ > 0.

Thus the marginal cost of preemption is non-decreasing.6 As we shall see below, whether the marginal

cost of preemption is constant or increasing will be central in understanding the role of both preemp-

tive and preventive CT measures. The last two restrictions on the H(·) function signify that an increase

in the parameter κ increases total as well as marginal cost of preemption.

The following will be an example of the H(·) function:

H(m, κ) =κ

2m2. (3)

The Org “produces” terror by using terrorists, capital, technology etc. Let terror be measured

by a single index x (e.g. the number of planned terror attacks within a time period), which is com-

monly observed. Let it be called an aggregate terror ‘input,’ which leads to material damage and

psychological damage (fear) to the State.7 Such damages are intended to pressurize the State to grant

redressal.8 Let D ∈ R+ denote the total damage from terror, the magnitude of which depends on

security-deterrence measures including intelligence and is subject to randomness. Let

Assumption 4.

D = αξx, α > 0, ξ > 0, (4)5In this example, β > 1 is sufficient to ensure Assumption 1. The stronger assumption β > λ satisfies Assumption 11,

which will be introduced later.6Our analysis abstracts from the optimal strategic choice of security-deterrence or intelligence gathering.7Indeed, the fear factor is the hall-mark of terrorism, distinguishing it from other forms of violence.8While most terrorist incidents and fatalities occur in regions outside the developed countries, such occurrences pose a

threat to the interests of developed countries in those regions and the same organizations also carry out attacks on the soilof developed countries. Terrorism threat is thus global - a concern for the developed and developing countries alike. In thissense the Org inflicts damage from terror to the State, a developed country presumably.

6

where α is non-stochastic and inversely reflect the effectiveness of security-deterrence and ξ the

stochastic component. Security-deterrence measures can thus be seen as a negative technology shock

in the production of damage. The distribution of ξ is common knowledge, iid and has mean equal to

one.

Producing terror input is costly for the Org, which faces a linear total cost function f (·), and,

Assumption 5. The marginal cost of producing x is given by

n = n(a + m), n ∈ R+, n′ > 0 ≥ n′′, a > 0. (5)

where recall that m is the level of preemptive measures.

Observations.

[a] An increase in preemption shifts up the Org’s marginal cost at a non-increasing rate.

[b] The parameter a is exogenous. A decrease in a captures the notion of a technological improvement

or a reduction in input prices in producing terror. Importantly, we interpret it as an exogenous increase

in terrorism or militancy.

[c] We shall later use a linear n function:

n = a + m. (6)

Let L(m) with L′ > 0 denote the disutility to the Org from the collateral damage associated with

preemption.

A central feature of our model is double-sided asymmetric information. The Org wouldn’t know

for sure some of the parameters facing the State and vice versa.

It is common knowledge that the ultimate objective of Hamas is to obtain a Palestine-Islamic

state. But how strong or weak is the preference of Israel and U.S. towards having an independent

Palestine-Islamic state can only be guessed by Hamas.9 Likewise, the State would also be facing

informational problems regarding the Org. It may not be able gauze the value the Org attaches to its

goal — how strongly the Org feels about its grievances. 9/11 led the Western countries to learn for

9Another source of informational problems facing the Org relating to parameters pertaining to the State would be thefollowing. Terrorism distinguishes itself from other forms of organized violence in terms of creating a sense of fear andanxiety among civilians. For example, after the 9/11 attack in the USA, air travel within the US dropped dramatically. De-spite public announcements about anti-terrorist measures, many Americans were far from being reassured. According to astudy in the New England Journal of Medicine, over a longer run a majority of Americans reported symptoms of stressand depression after the attacks. See http://www.ptsd.va.gov/professional/trauma/disaster-terrorism/research-findings-traumatic-stress-terrorism.asp; last accessed on 9/12/2016. However, the level of thisfear differs from case to case and the terrorist have no way of knowing the exact extent of this effect. In sharp contrast tothe reactions of 9/11, life in London came back to near normality the day after the 7/7 attack in 2005. It seems reasonable tothink that while terrorism is meant to cause damage to civilians, the extent of material and psychological damage is moreaccurately assessed by the target country than by the terrorist organization.

7

the first time what the terrorists “really wanted” and how strongly they felt about it.10

In our model, there is asymmetric information about each other’s preferences: the true utility

function is privy to the respective parties.11 The sequential nature of our game model (to be intro-

duced in a moment) together with asymmetric information do not however permit much analytical

tractability without having to assume specific distributions of u and g.

Assumption 6. The State doesn’t know the (true) value u∗ in Org’s utility function and has a proba-

bilistic belief of it given by a uniform distribution over (0, u]. The Org doesn’t know the (true) value

z∗ in the State’s utility function and has a probabilistic belief of it with a log-uniform distribution in

the interval [z, z].12,13

As will be seen later, asymmetric information — particularly on the part of the terrorist organiza-

tion regarding parameters facing the State — is critical in how terrorism is used as a means to an end.

There is however no learning of each other’s preferences. (This will be relaxed in Section 6.)

2.1 The Stages/Periods

Figure 2: Timeline of Decisions

We envisage a three-period, sequential game with asymmetric information. The timeline of deci-

sions is exhibited in Figure 2. In the beginning of period 1 the State and the Org jointly inherit g0 = 0

and consequently the problem of terrorism.

Assumption 7. Grievance redressal, g, is irreversible.

It reflects commitment on the part of the State, or, rolling back on concessions is too costly for

the State in terms of reputation and credibility. If g relates to military occupation with supporting

military hardware and logistics, the resource cost in reversing a pull back may be too high.

10Terror organizations are always innovating on their methods, and the target governments have no way of exactlyknowing the form of the new attacks or the cost function of organizing an attack. The true cost of organizing a terroristattack is better known to terrorist organizations than to others. This will be another source of asymmetric informationproblem facing the State.

11Previous versions of the paper show that our results continue to hold when there is information asymmetry over eachother’s costs.

12Thus z and z are the lower limit and upper limit of ln z.13For use of log-uniform distribution, see Frieden (1998). Another version of the paper assumes log-uniform distribution

for u as well; the results remain unaffected however.

8

During period 1 the State chooses simultaneously the level of g1 = g in the closed interval [0, 1]

as well as the scale of preemptive measures m > 0.

Definition. The choice in period 1 said to be full initial redressal or full peace if g = g = 1. It is noinitial redressal if g = 0 and partial initial redressal if 0 < g < 1.

If it is full peace, the State accedes to the terrorists’ grievances or ‘demands,’ grievance-redressal

deficit is zero and the game ends. There is no further threat of terrorism. In event of no full peace at

the end of period 1, during the first part of period 2, say 2a the Org produces the level of aggregate

terror input for periods 2 and 3 together, equal to x2 + x3. Further,

Assumption 8. Technology of producing composite terror input is such that a variation of it across the

periods 2 and 3 is prohibitively costly, so that the Org chooses x2 = x3 = x.14

Total production of terror input is thus 2x. After the execution of x2 = x by the Org,

Assumption 9. In the end or second part of period 2, say 2b, the State faces a binary choice: not to

increase g any further or increase it to its limit. That is, g2 = g or 1.

Definition. The choice in period 2b is said to be delayed peace or no peace according as g2 = 1 or

g2 = g < 1.

Assumption 10. If g2 = 1 the Org aborts its planned attack for period 3 and disposes x3 off costlessly.

If g2 = g < 1, the Org carries out its already planned terror mission x3 = x in period 3 at no further

cost. There is no fresh decision making and the game ends in period 3.

Observations.

[a] This way of modeling grievance redressal essentially reduces a dynamic problem to a static one.

[b] Choice of x3 back in period 2a signals the Org’s commitment to use terror if grievances are not

fully addressed. Otherwise, the State would never choose g2 = 1 at the end period 2 and there will be

no full peace with probability one.

3 Analysis

The equilibrium concept is the sub-game-perfect Bayes-Nash equilibrium of a sequential game. It is

equivalent to the Stackelberg solution. We thus work backwards.

3.1 Period 3 Payoffs

After observing x in period 2a, the State chooses at the at the end of period 2,

g2 =

g if z∗C(g) − αx > 0

1 otherwise.(7)

14More generally, the ratio between x2 and x3 is constant.

9

This is illustrated in Figure 3. The payoffs or net utilities in period 3 are

State: z∗C(g) − αx, or 0

Org: u∗U(g) − L(m), or u∗U(1) − L(m),

according as g2 = g or 1.

x

gg = 10

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Full redressal<latexit sha1_base64="R67APHHkHz+I35c8wTuJmwHQYFE=">AAAB9HicbVBNS8NAEJ3Ur1q/qh69LBbBU0l6UW8FQTxWMFZoY9lsJu3SzSbsbpRS+j+8eFDx6o/x5r9x2+agrQ8GHu/NMDMvzATXxnW/ndLK6tr6RnmzsrW9s7tX3T+402muGPosFam6D6lGwSX6hhuB95lCmoQC2+Hwcuq3H1FpnspbM8owSGhf8pgzaqz0cJULQRRGCrWmoletuXV3BrJMvILUoECrV/3qRinLE5SGCap1x3MzE4ypMpwJnFS6ucaMsiHtY8dSSRPUwXh29YScWCUicapsSUNm6u+JMU20HiWh7UyoGehFbyr+53VyE58HYy6z3KBk80VxLohJyTQCEnGFzIiRJZQpbm8lbEAVZcYGVbEheIsvLxO/Ub+oezeNWtMt0ijDERzDKXhwBk24hhb4wEDBM7zCm/PkvDjvzse8teQUM4fwB87nD+Bekj4=</latexit><latexit sha1_base64="R67APHHkHz+I35c8wTuJmwHQYFE=">AAAB9HicbVBNS8NAEJ3Ur1q/qh69LBbBU0l6UW8FQTxWMFZoY9lsJu3SzSbsbpRS+j+8eFDx6o/x5r9x2+agrQ8GHu/NMDMvzATXxnW/ndLK6tr6RnmzsrW9s7tX3T+402muGPosFam6D6lGwSX6hhuB95lCmoQC2+Hwcuq3H1FpnspbM8owSGhf8pgzaqz0cJULQRRGCrWmoletuXV3BrJMvILUoECrV/3qRinLE5SGCap1x3MzE4ypMpwJnFS6ucaMsiHtY8dSSRPUwXh29YScWCUicapsSUNm6u+JMU20HiWh7UyoGehFbyr+53VyE58HYy6z3KBk80VxLohJyTQCEnGFzIiRJZQpbm8lbEAVZcYGVbEheIsvLxO/Ub+oezeNWtMt0ijDERzDKXhwBk24hhb4wEDBM7zCm/PkvDjvzse8teQUM4fwB87nD+Bekj4=</latexit><latexit sha1_base64="R67APHHkHz+I35c8wTuJmwHQYFE=">AAAB9HicbVBNS8NAEJ3Ur1q/qh69LBbBU0l6UW8FQTxWMFZoY9lsJu3SzSbsbpRS+j+8eFDx6o/x5r9x2+agrQ8GHu/NMDMvzATXxnW/ndLK6tr6RnmzsrW9s7tX3T+402muGPosFam6D6lGwSX6hhuB95lCmoQC2+Hwcuq3H1FpnspbM8owSGhf8pgzaqz0cJULQRRGCrWmoletuXV3BrJMvILUoECrV/3qRinLE5SGCap1x3MzE4ypMpwJnFS6ucaMsiHtY8dSSRPUwXh29YScWCUicapsSUNm6u+JMU20HiWh7UyoGehFbyr+53VyE58HYy6z3KBk80VxLohJyTQCEnGFzIiRJZQpbm8lbEAVZcYGVbEheIsvLxO/Ub+oezeNWtMt0ijDERzDKXhwBk24hhb4wEDBM7zCm/PkvDjvzse8teQUM4fwB87nD+Bekj4=</latexit><latexit sha1_base64="R67APHHkHz+I35c8wTuJmwHQYFE=">AAAB9HicbVBNS8NAEJ3Ur1q/qh69LBbBU0l6UW8FQTxWMFZoY9lsJu3SzSbsbpRS+j+8eFDx6o/x5r9x2+agrQ8GHu/NMDMvzATXxnW/ndLK6tr6RnmzsrW9s7tX3T+402muGPosFam6D6lGwSX6hhuB95lCmoQC2+Hwcuq3H1FpnspbM8owSGhf8pgzaqz0cJULQRRGCrWmoletuXV3BrJMvILUoECrV/3qRinLE5SGCap1x3MzE4ypMpwJnFS6ucaMsiHtY8dSSRPUwXh29YScWCUicapsSUNm6u+JMU20HiWh7UyoGehFbyr+53VyE58HYy6z3KBk80VxLohJyTQCEnGFzIiRJZQpbm8lbEAVZcYGVbEheIsvLxO/Ub+oezeNWtMt0ijDERzDKXhwBk24hhb4wEDBM7zCm/PkvDjvzse8teQUM4fwB87nD+Bekj4=</latexit>

Partial orno redressal

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Figure 3: Partial Versus Full Redressal in Period 2b

3.2 Output of Terror Decision by the Org in Period 2

In period 2a, the Org’s choice of x is based partly - yet critically - on its own belief about z and the

probability that the State will or will not fully redress its grievances. Let pO denote the probability

that the State will not fully redress the grievances. Since z follows a log-uniform distribution over the

interval [z, z] ∈ R, denoting r = z − z,

pO(x, g) = Prob. [zC(g) − αx > 0]

=z + ln C(g) − lnα − ln x

r. (8)

Lemma 1:∂pO

∂x= −

1rx

< 0 and∂pO

∂g=

C′(g)rC(g)

< 0. (9)

Hence, an increase in terror increases the probability that the State will fully redress the Org’s

grievances. This is the raison d’etre of terrorism in our model. There are indeed examples of terror

attacks that have induced target countries to grant concessions. Hezbollah suicide attacks prior to May

2000 against Israeli soldiers were followed by withdraw of Israel army from Lebanon in May 2000.

Israeli withdrawal of occupation of Gaza and West Bank in 2005 was preceded by suicide attacks

from Hamas during the second Intifada. Most recently, Taliban’s brutal terror attacks in January 2018

10

led President Ashraf Ghani to offer political concessions to Taliban in March 2018.15 The second

part of (9) says that the higher the initial redressal of grievance the less is the utility loss from full

redressal and hence the higher will be the probability that full redressal be granted.

Without loss of generality let the time discount factor for both the State and the Org be unity.

Accounting for the disutility from preemptive measures undertaken by the State to the Org, the dis-

tribution of the discounted value of Org’s utility as of period 2 is:

2u∗U(g) − L(m) with probability pO

2u∗U(1) − L(m) with probability 1 − pO.

Its expected payoff (net of the cost of producing terror) has the expression: 2u∗U(1) − L(m) −

pO(x, g)2u∗V(g) − f (2x), where, recall that V(g) = U(1) − U(g). The optimal choice of terror by

the Org satisfies the first-order condition

−∂pO(x, g)

∂x· 2u∗V(g) = 2 f ′(·)

⇔u∗V(g)

rx= n (10)

⇒ x = x(n, g) · u∗, where (11)

x ≡V(g)rn

;∂x∂n

= −V(g)rn2 < 0,

∂x∂g

=V ′(g)

rn< 0. (12)

Observations.

[a] The function x(·) reflects that terrorism used by the Org is a means to an end. It varies directly

with the population’s deprivation of some good or state of the world which is ‘controlled’ by the

state. It can also be viewed as a derived demand for terror to elicit redressal. If g = 1, then V = 0 and

therefore x = 0; terrorism ceases if grievances are fully redressed.

[b] Because of the asymmetric information regarding the utility function of the State, the benefit of

engaging in terror lies in raising the probability facing the Org that full redressal will be granted by

the State. The optimal production of terror input depends on the sensitivity of the ex post probability

of no further redressal or full redressal with respect to terror, not on the probability itself.

[c] The disutility from collateral damage, L(m), doesn’t affect the choice of terror input by the Org -

as it is a sunk cost from its perspective.

15The actions by the Irish Republican Army (IRA) also possibly led the former British Prime Minster Margaret Thatcherto initiate secret talks with the IRA, which ultimately led to a lasting peace in Northern Ireland.

11

3.3 Payoffs of the State and Its Objective Function

While the amount produced of the terror input is non-stochastic from the viewpoint of the Org, it is

ex ante stochastic from the perspective of the State, equal to:

x = x(g, n) · u. (13)

Keeping in view the expected damage from terror in period 2a based on (13) and the binary choice

problem it would face in period 2b, the State, assumed to be risk-neutral, chooses m and g in period

1. Its period 1 payoff is z∗C(g) − H(m, κ), while its period 2 payoff has the expression

z∗C(g) − αE(x), where E(x) =x(n, g)u

2=

V(g)u2rn

. (14)

Although the State’s payoffs in period 3 are z∗C(g) − αx if g2 = g, or 0 if g2 = g = 1, obtaining

the expression of its expected payoff is not straightforward as we must evaluate the probabilities of

no further redressal and full redressal from the State’s perspective. Let p denote the probability that

it will not grant any further redressal — which is different from the ex post probability of no full

redressal facing the Org, p0. Viewed in period 1, the future decision rule in period 2b will be: no

further redressal if z∗C(g) − αx(n, g)u > 0. Defining

Υ(g) ≡C(g)V(g)

, (15)

p(n, g) ≡ Prob [z∗C(g) − αx(n, g)u > 0] = Prob [u < uS ], where uS ≡rz∗nα· Υ(g) (16)

=rz∗nαu· Υ(g) ∵ u is distributed uniformly, Assumption 6 (17)

⇒∂p∂n

=rz∗

αu· Υ(g) > 0, and

∂p∂g

=rz∗nαu· Υ′(g).

An increase in preemptive measures tends to reduce expected damage from terror and thus im-

prove the probability that full redressal will not be granted; thus ∂p/∂n > 0. As for ∂p/∂g, two

opposing effects are at play. On one hand, a higher partial redressal reduces further gains to the Org

from full redressal, thereby reducing the Org’s use of terror; it tends to decrease the probability of

full redressal (i.e. increase p). On the other, it reduces further losses to the State from granting full

redressal and hence tends to increase the probability of full redressal (i.e. decrease p). The sign of

∂p/∂g is ambiguous in general. We impose

Assumption 11. Υ′(g) > 0⇔ V(g)C′(g) −C(g)V ′(g) > 0,

so that ∂p/∂g > 0. It is shown in the Appendix that the Υ(g) functions in Example A and Example B

indeed satisfy this condition. Assumption 11 underscores that partial redressal is a ‘hindrance’ to full

redressal: a higher g implies lesser future gains to the Org from full redressal, lesser future production

12

of terror and hence lesser incentive on the part of the State to grant full redressal.

We may now state the State’s expected payoff in period 3:

B(n, g) =1u

∫ uS (n,g)

0

[z∗C(g) − αx(n, g)u

]du =

pz∗C(g)2

,

where the expressions of p and x have been made use of and recall that n = n(a + m).

Summing up the State’s discounted values of payoffs in three periods, its objective function is

expressed as

Ω(m, g) = z∗C(g) − H(m, κ) + z∗C(g) − αE(x) + B(n, g),

where the Org’s best response function in period 2, given by eq. (13), is embedded in Ω(m, g). The

State maximizes Ω with respect to m and g. The optimal solutions constitute the sub-game-perfect

and Stackelberg equilibrium.

It will however be simpler to analyze CT measures if the objective function is stated in the (n, g)

space. We invert the n(·) function to obtain

m = m(n, a), where∂m∂n

=1n′> 0;

∂2m∂n2 = −

n′′

n′3≥ 0;

∂m∂a

= −1, (18)

and re-state Ω(m, g) as

Ω(n, g) = z∗C(g) − H(m(n, a), κ) + z∗C(g) −αx(n, g)u

2+ B(n, g). (19)

3.4 Choice of Direct and Preventive CT Measures

Equilibrium strategies are defined by the vector (n∗, g∗, x∗) such that (n∗, g∗) = arg maxn, g

Ω(n, g) and

x∗ = x(n∗, g∗)u∗. Assuming interior solutions, the first-order conditions with direct and preventive CT

measures are:

∂Ω

∂n≡ Ωn = −

αu2

(1 + p2

) ∂x∂n︸ ︷︷ ︸

Marginal benefit in terms ofreduced terror

−Hm [m(n, a), κ]∂m∂n︸ ︷︷ ︸

Marginal cost ofengaging in preemption

= 0 (20)

∂Ω

∂g≡ Ωg = −

αu2

(1 + p2

) ∂x∂g︸ ︷︷ ︸

Marginal benefit in terms ofreduced terror

+(2 + p2

)z∗C′(g)︸ ︷︷ ︸

Marginal utility costfrom grievance redressal

= 0.16 (21)

The expressions of the second partials of the Ω(n, g) functions are worked out in the Appendix,

where it is shown that Ωnn < 0. Note that the magnitude of Ωnn increases with (a) the preemption

cost function parameter κ and (b) the magnitude of marginal effect of preemption on the marginal

16An alternative expression of Ω(n, g) and the first-order conditions are given in the Appendix.

13

cost function of producing terror |n′′|. The sign of Ωgg is ambiguous in general but negative as long

as |C′′| and |V ′′| are large enough relative to each other — and we assume so. We further let κ and the

parameters which govern the magnitudes of n′′, C′′ and V ′′ be large enough relative to one another

so that the remaining second-order condition, namely, ∆ ≡ ΩnnΩgg − Ω2ng > 0, is met. A numerical

example using Example A such that the second-order conditions are satisfied is provided in Section

6.17

Reverting back to the first-order conditions

Observations.

[a] In (20) the first term is the marginal benefit from preemptive measures. They shift up the marginal

cost of producing the terror input and tend to reduce terror. The second term is the marginal cost of

undertaking preemptive measures. Likewise, in (21), the first term represents the marginal benefit

from grievance redressal: it diminishes the Org’s benefit from using terror and hence the production

of terror. The second term is the marginal cost of grievance redressal in terms of reduced (direct)

utility to the State.

[b] The difference between the two measures is that preemption increases the Org’s cost of producing

terror, while grievance redresses reduces the Org’s benefit from using terror. The former acts as a

stick, and the latter a carrot.

Given the solutions, say (n∗ and g∗), the optimal preemption level is solved: m∗ = m(n∗, a) and

the level of expected terror production E(x) is determined from (14).

How does an exogenous change in one type of CT measures affect the incentive to use the other?

Definition. Preemption and grievance redressal are said to be individually substitutes or comple-

ments of each other if an exogenous increase in one tends to reduce or enhance the optimal use of

the other.

In a direct way, an increase in preemption tends to reduce terror and thereby weaken the State’s

incentive to redress the grievances of the Org’s population. Likewise, grievance redressal will reduce

the need to use preemptive measures. Hence direct and preventive CT measures are individually

substitutes of each other. However, there is an element of complementarity. By tending to reduce the

Org’s production of terror, partial grievance redressal improves the likelihood that no full redressal

will be the preferred choice. This enhances the marginal benefit from and incentive to use preemptive

measures. Through this channel of interdependence, they tend to be individually complements of

each other. However, it is proved in the Appendix that Ωng < 0, i.e., the direct effect outweighs the

complementarity effect and the two measures are individually substitutes of each other. Each first-

order condition spells a a negative locus between n and g. Optimal choice of the CT measures, n∗ and

g∗, is displayed in Figure 4.

At this point we introduce a geometric construct and name it an iso-terror curve in the (n, g)

17It can also be verified that, at the equilibrium defined by (20) and (21), Assumption 11 is satisfied if and only if1 − 4p − p3 < 0, i.e., p > 0.246. This is verified in the numerical example in section 6.

14

@

@g= 0

g (Grievance Redressal)g

gg

nn

@

@n= 0

n

n

nx(n, g) = constant

E

tt

Figure 4: Optimal Choice of CT Measures

space. While from the Org’s perspective, the equilibrium level of terror input is xu∗, from the State’s

viewpoint, the ex ante expected level of it is xu. The parameter u being given, an iso-terror curve is

defined by x(n−, g−

) = constant. It is obviously downward sloping, as shown by the tt curve in Figure

4. Starting with a given iso-terror curve, any point to the left (or right) of it represents higher (lower)

level of terror. As proved in the Appendix, the tt curve is steeper than the nn curve. It may be steeper

or flatter than the gg curve, although the former is less likely than the latter.18

Comparing Direct CT Measures Only to Direct and Preventive CT Measures

We now utilize the model to evaluate counter-terror measures. To begin with, if the State were not

open to preventive measures, n in Figure 4 will be its optimal choice of preemptive measures. Com-

pared to this situation, if preventive measures are chosen rationally, the optimal choice of grievance

redressal is accompanied by less intense preemptive measures, reflected by n∗ < n. Does it mean

that the State is willing to accept more terror at (n∗, g∗) than at (n, 0)? Since the tt curve lies to the

right of the point n, the answer is negative. The State would not want utility loss in granting redressal

18The tt curve would be steeper than the gg curve if the marginal cost of grievance redressal is large enough to ensure aninterior solution of g and yet not too large, such that a given change in n leads to a large adjustment in g along the gg curve.Formally

dndg

∣∣∣∣∣x−

dndg

∣∣∣∣∣gg

=A

V pn

(C′ +

1−p2

2p2 ·CV′

V

) ,where

A ≡ V(2 + p)C′′ −

1 + p2

2p·

CV ′′

V+

pV2 p2g

C

+ npnV ′(C′ +

1 − p2

2p2 ·CV ′

V

).

Hence, the iso-terror curve is steeper than the gg curve if and only if dn/dg|x < dn/dg|gg

⇔ A > 0.

15

and at the same time exposing itself to a greater expected damage from terror. This leads to our first

proposition.

Proposition 1: Equilibrium level of terror production and expected damage from terror are less when

both types of CT measures are undertaken optimally than when only the direct preemptive measures

undertaken.

4 Comparative Statics

A parametric change in our model can be interpreted in a cross-sectional sense across a cross-section

of terrorist organizations and target countries. Alternatively, we can imagine that in the beginning of

period 1 when the State and the Org inherit the terrorism problem, they know how the game is going

to play out, period 1 equilibrium choice of the State and their subsequent contingent choices, based

on original parameter values, and, at that point before the State actually undertakes the CT measures

during period 1, there is an unanticipated permanent parameter change.

4.1 The State’s Type

How does the choice of CT measures vary across the State’s type, z. As shown in the Appendix, at

given n and g, an increase in z∗ implies (a) an increase in marginal gain from the preemptive measures

and no change in the marginal cost of such CT measure, and, (b) a decrease in the ratio of marginal

gain to marginal loss from preventive measures. Thus, the nn curve shifts to the right and the gg curve

to the left, implying

Proposition 2: A hardliner State will choose a higher level of preemption and less grievance redressal

than a softliner State.

By virtue of being eminently intuitive, Propositions 1 and 2 lend credence to the plausibility of

our model’s underlying structure.

4.2 An Exogenous Increase in Terrorism or Militancy

The central question we seek to answer is: what should be the nature of response with regard to direct

and preventive CT measures to “an increase in terrorism” or an increase in militancy? A natural

way to think about an exogenous increase in terrorism is in terms of technology improvement or a

decrease in the cost of inputs in producing terror, which would exogenously lower the marginal cost

of producing terror. In the model this amounts to a decline in the parameter a.

We are now in a position to frame the “ultimate” question on winning the war on terror via the

following definition.

Definition. Winning or not winning the War on Terror The State wins the war on terror in a weak

or strong sense if in the face of an increase in terrorism or militancy defined as above and the optimal

16

response in terms of CT measures, the equilibrium level of terror production remains unchanged or

declines.

A central point of this paper is that whether the State would win WoT is shown to depend on the

nature of the marginal cost of preemption.

4.3 Constant Marginal Cost of Preemption

Irrespective of whether marginal cost of preemption is constant or increasing, a change in a does not

affect the marginal condition with respect to preventive measures, that is, the first-order condition

(21). But, if the marginal cost of preemption is constant, the first-order condition with respect to n is

unaffected too. Hence, there is no change in n or g, and, ∂m/∂(−a) = 1. An increase in preemption

is called for, which exactly offsets the downward change in a. As a result, there is no change in the

expected terror output. Terror is contained and WoT is won in the weak sense by preemptive measures

only.

Proposition 3: If the marginal cost of preemption is constant, the optimal response to an increase

in militancy is to increase preemptive measures with no change in g such that there is no change in

terror production or in expected terror output.

4.4 Increasing Marginal Cost of Preemption

Suppose momentarily that grievance redressal is not an option for extraneous reasons, i.e., g is ex-

ogenously given and thus eq. (21) does not apply. The nn curve shifts inward as the parameter a falls.

For any given g, n falls. Thus the ‘equilibrium’ marginal cost of terror production declines. How does

the State adjust its preemptive measures? Using the relationship between m and n and the first-order

condition (20),

dmd(−a)

= 1 +∂m∂n+

∂n∂(−a)

+

=

Hmm

(∂m∂n

)2

+ Ωnn

Ωnn> 0 in view of (A.5).

That is, the State steps up preemptive measures. But, this does not fully offset the decline in a (as n

falls). The implication is that expected terror output increases.

The upshot is that, in response to an increase in terrorism, the State steps up its preemptive

measures but not sufficiently enough. Higher levels of preemptive measures are associated with higher

levels of expected damage from terror. Provocatively put, war on terror cannot be won. It does not

however mean that preemptive actions are futile: damage from terror would be greater in the absence

of more preemptive actions.

Inadequacy of direct CT measures — if the marginal cost of preemption is increasing — is illus-

trated in Figure 5. For simplicity of exposition, let n = a + m be the marginal cost of producing terror.

17

x0

x0

x

a + ma0a10

State’s Choice of m

Org’s

Choice

ofx

a1 + m1

bc

a0 + m0

HmHm

h

Marginal Benefitfrom Preemption

in terms of reduced damagefrom terror

constantmarginal cost

increasingmarginal cost

Marginal Gainsto the Org from

full grievance redressal

Figure 5: An Exogenous Increase in Militancy or Terrorism and Whether WoT Can be Won byPreemptive Measures Only

Because preemptive measures work through the Org’s marginal cost of producing terror, the total and

marginal benefits to the State from preemption is a function of a + m. Let the initial value of a be a0

and let it decline from a0 to a1. The downward sloping line in the top quadrant shows the marginal

benefit to the State from preemption. It depicts the choice of the preemption level by the State based

on its objective function Ω.

If the marginal cost of preemption were constant, equal to h, the State chooses m = a0b when

a = a0. The bottom quadrant shows the Org’s choice of x. Its marginal benefit from using terror is

shown by the downward sloping line in the lower quadrant, based on its choice rule for terror, eq.

(10). The optimal choice of the terror is x0. When a falls from a0 to a1, under constant marginal cost

h, the State increases preemption one-to-one relative to the decline in a. The new level of preemption

is a1b. The marginal cost facing the Org does not change and the production of the terror remains

unchanged at x0. There is no change in the expected damage from terror. WoT is won in the weak

sense.

However, if the marginal cost of preemption is increasing and thus the Hm function is upward

sloping, the optimal level of preemption increases from m0 go m1 as a falls from a0 to a1. But notice

that a1 + m1 = c < a0 + m0 = b. There is a decline in the equilibrium marginal cost facing the Org,

even after the upward adjustment in preemption. The optimal terror input level is now x′, higher than

x0. The expected damage from terror is higher. WoT is lost so to speak. As terrorism increases, the

State responds no doubt but not enough to fully counter the problem. It ends up chasing the problem

without being able to overcome it.

18

Hawk or Dove Response?

We now bring preventative measures (g) into the analysis. In Figure 4, because a decrease in a shifts

in the nn curve in, a decline in n is accompanied by an increase in g. In other words, in the face

of an increase in terrorism or militancy, the rationale behind a proactive grievance redressal rests

on increasing marginal cost of preemption. In turn, an increase in g tends to reduce the need for

preemption since the two CT measures are individually substitutes of each other. The net impact on

preemption is thus generally ambiguous. There is however a strong presumption is that it is positive.

Algebraically,

dmd(−a)

=∂m∂(−a)

+∂m∂n·∂n

∂(−a)= 1 +

HmmΩgg

(n′)2∆

=

−

r2[z∗C(g)]3

p3[αuV(g)]2 +Hmn′′

(n′)3

Ωgg − Ω

2ng

∆(22)

> 0

as long as |n′′|, |C′′| and |V ′′| are sufficiently large vis-a-vis each other. In summary, an increase in

militancy induces the State to engage in more preemptive and preventive measures.

To better understand the interplay between the two types of CT measures, we use the following

definition.

Definition. Preemption and grievance redressal are jointly substitutes or complements of eachother if, as equilibrium response to a change in the environment, both measures change in opposite

directions or in the same direction.

Thus, while the two forms of CT measures are individually substitutes of each other, they are

jointly complementary in response to an exogenous increase in the terrorism. It is not whether the

optimal response to an increase in terrorism ought to be more hawkish and less dove-like or less

hawkish and more dove-like. It calls for more of hawkish and dove-like measures — a carrot and

stick response.

Effect on Terror

Turn to Figure 6. The tt curve is drawn flatter than the gg curve in panel (a) and steeper in panel (b).

Comparing the original and the new equilibrium points E∗ and E1 in panel (a) we see that the new

iso-terror line tt1 lies to the left of original iso-terror line tt. Hence more terror is accompanied with a

decrease in a. However, if preventive measures g were unchanged, the new equilibrium point would

be E2 and since this point lies to the left of tt1, it is associated with a higher level of terror than at E1.

Thus, as long as the iso-terror curve is flatter than the gg curve, an exogenous increase in terrorism

leads to a higher level of expected terror, i.e., WoT is still not won, but, the new equilibrium level of

terror is less than what it would be in the absence of response in terms of preventive measures.

19

g (Grievance Redressal)g

gg

nn

n

n

nx(n, g) = constant

E1E2

n2

n1

g1

E

nn1

tt

tt1

(a) Iso-Terror Curve Flatter than the gg Curve

g (Grievance Redressal)g

gg

nn

n

n

n

E1E2

n2

n1

g1

E

nn1

tt tt1

(b) Iso-Terror Curve Steeper than the gg Curve

Figure 6: Effects of an Exogenous Increase in Militancy or Terrorism on CT Measures and Equilib-rium Terror

In panel (b) of Figure 6 where the iso-terror curve is steeper than the gg curve, the new iso-terror

curve tt2 lies to the right of the old iso-terror curve tt and hence is associated with less terror. That is,

on the face of an exogenous increase in terrorism, the equilibrium level of terror is less. While the pre-

sumption of this case is smaller, it underscores the critical role of reducing terror through preventive

CT measures and opens the possibility that WoT despite increasing marginal cost of preemption.

We can turn to Figure 7, an analog of Figure 5, to further understand how preemptive and preven-

tive measures together constitute a better response. We simply notice in the lower quadrant that as

preventive measures reduce the utility deficit for the Org, they shift the Org’s marginal benefit curve

in. If the dashed red line is the new marginal benefit curve from terror, the equilibrium terror pro-

duction after the State’s response is at the point x′′, which is less than the original equilibrium terror

production x0. WoT is won. Of course, whether WoT is won or not depends on the magnitude of the

shift of the marginal benefit curve. The general point is that a two-pronged response leads to a better

outcome than using preemption only.

We now work out numerical examples to further establish this point.

20

x0

x0

x

a + ma0a10

State’s Choice of m

Org’s

Choice

ofx

a1 + m1

bc

a0 + m0

HmHm

h

Marginal Benefitfrom Preemption

in terms of reduced damagefrom terror

increasingmarginal cost

Marginal Gainsto the Org from

full grievance redressal

Preventivemeasures

x00

Figure 7: Preemptive and Preventive Measures: Response to an Increase in Terrorism

An Illustrative Example

Choose C(·) and V(·) functions in Example A, and, H(·) and n(·) functions as defined in (3) and

(6). Let the parameter values be c = v = 0.1, λ = 0.5, β = 2.5, κ = 0.0125, r = 25, u = 20,

z∗ = 0.0125, α = 0.1 and a = 1. The domain of m is restricted to [0, 0.5]. At these parameter values,

the equilibrium p is always less than unity for all values of g ∈ [0, 1] and m ∈ [0, 1].

The resulting Ω function is depicted in Figure 8. The optimal levels of preemption and grievance

redressal are m∗ = 0.09 and g∗ = 0.54 respectively. The equilibrium level of terror is 9.7304e − 04.

The probability of no full grievance redressal is 50%. The maximized value of the State’s payoff is

positive, equal to 0.0012. Thus partial redressal is preferred to no redressal and full redressal.

Note that a = 1 is the initial intercept of the marginal cost function of producing terror. If there

is an exogenous decline in a by 20%, i.e., if a = 0.8, then m∗, g∗ and x∗ are equal to 0.11, 0.58 and

0.0010 respectively. The probability of no further redressal is 44% and maximized Ω is 9.4209e− 04.

Thus, in response to an exogenous decline in the marginal cost of producing terror, the State steps

up both direct and preventive CT measures, while the equilibrium level of terror is higher. This is an

example where the iso-terror curve is flatter than the gg curve.

Consider the same set of parameters except that β = 1.5 instead of 2.5. When a = 1, the equi-

librium levels of CT measures and terror are: m∗ = 0.10, g∗ = 0.46 and x∗ = 0.0012. Suppose the

parameter a declines to 0.8; new equilibrium solutions are m∗ = 0.11, g∗ = 0.56 and x∗ = 0.0011.

Notice that the equilibrium level of terror is smaller in the face of a decline in a. This is an example

where the iso-terror curve is steeper than the gg curve and WoT is won in the strong sense. Simulation

exercises for various sets of parameters confirm that such an outcome results in very selective ranges

21

Figure 8: An Example of Ω(m, g) Function

of parameter values.

We now state our next proposition.

Proposition 4: If the marginal cost of preemption is increasing, an exogenous increase in militancy

or terrorism in terms of a decrease in a has the following effects:

a. the State scales up preemptive measures as long as |n′′|, |C′′| and |V ′′| are large enough relative

to one another;

b. the marginal cost of producing terror by the Org goes down even though the State may increase

preemptive measures;

c. the State undertakes more preventative measures;

d. the equilibrium level of terror increases or decreases as the iso-terror curve is flatter or steeper

than the gg curve;

e. irrespective of whether terror increases or decreases, the equilibrium level of terror when both

forms of CT measures are undertaken is less compared to when only direct or preemptive CT

measures are undertaken.

Proposition 4 implies that combining preemptive and preventive measures are likely to yield a

22

better outcome than reliance on preemptive measures only and has the potential of winning the war

on terror.

4.5 A Decrease in the Cost of Preemption

Drones are often used for preemptive strikes against terrorists and terrorist bases in the Middle East.

Over time, the technology of drones has improved in terms of their combative capacity as well as

accuracy. The Predator fleet has been upgraded over time. The ‘Reapers’ are now the recent and

effective versions of ‘Predators.’ Despite the negative connotation associated with drone strikes in

terms of innocent lives lost - which also happens in other forms of conflict - civilian casualties rates

have declined due to technology improvements and stricter rules for releasing weapons; see Brooks

(2014). In the financial front, Ryder (2015) contains a detailed account of financial controls and

strategies since 2001 which are aimed to counter terrorist activities. These are suggestive of a decline

in the cost of preemptive measures to curb terrorism and let it be captured by a decrease in the

parameter κ in our model. In view of its effect on the nn curve and how equilibrium terror varies

along the gg curve,

Proposition 5: A decrease in the marginal cost of preemption calls for a higher level of preemption

and a lower level of preventive measures. The production of terror input and expected damage from

terror may decrease or increase however.

That is, an improvement in preemptive capability may or may not help the State win WoT. The

latter possibility arises as a higher level of preemptive measures is combined with less grievance

redressal for the Org’s population.

5 Extensions

5.1 Collateral Damage Induced Increase in Terrorism

Military strikes against terrorists and their bases cause collateral damage to civilian population. By

engendering anger and resentment it can worsen the terrorism problem in two ways. First, anger

and resentment breed revenge, implying a shift of input supply to produce terror.19 It causes a de-

crease in the marginal cost of producing terror, a supply-side effect on terrorism.20 We will term it as

preemption-induced militancy. Second, anger and resentment may harden the demand of the popula-

tion and the Org for their long-run goal of full grievance redressal. This is a preference shift toward

19See Rosendorff and Sandler (2010) for a game-theoretic analysis of the State’s decision to engage in preemption inanticipation of such backlash.

20In an interview with Huffpost Live in 2014, the former President Jimmy Carter said that “I believe it [drone strike]creates more additional terrorists, with the fervor of killing Americans, than we would be if we were not using the dronesto kill people.” In 2015, the retired Army General, Mike Flynn submitted in an interview with Al Jazeera that drones havecreated terrorists, perhaps more terrorist than they killed. Available empirical work on drone strikes till date, e.g., Johnstonand Sarbahi (2016) does not find robust link between such strikes and increase in terrorism.

23

the long-run goal, not revenge. In turn it increases the need for terror as a means towards achieving

this end. Both channels imply a higher equilibrium level of terror, compared to their absence.

What are the implications for CT measures?

5.1.1 Preemption Induced Militancy

Let preemption-induced militancy in the model be introduced through an additional parameter in the

marginal cost function facing the Org. Let n = n(a + (1 − δ)m), where δ represents such militancy.

If δ ≥ 1, preemptive strikes are totally ineffective. This is rather implausible. It seems reasonable to

assume that 0 < δ < 1. If so, the effectiveness of preemption is partly offset by the anger-revenge-

militancy factor δ. We thus have a more general m(·) function and Ω(·) function:

m = m(n, a, δ), where∂m∂n

=1

(1 − δ)n′> 0;

∂m∂a

= −1

1 − δ< 0;

∂m∂δ

=m

1 − δ> 0

Ω(n, g, δ) = z∗C(g) − H(m(n, a, δ), κ) + z∗C(g) − αx(n, g)u

2+ B(n, g).

But the nature of tradeoffs associated with preemptive and preventive measures remains the same

qualitatively. Akin to the a decrease in a, an increase in δ tends to lower the Org’s marginal cost of

producing terror. The nn curve shifts in.21 As a result, n falls and g rises. The implications towards

preemptive and preventive CT measures and equilibrium terror are qualitatively similar to that of a

decline in a.

Proposition 6: The effects of an increase in preemption induced militancy are qualitatively same as

those of an exogenous increase in militancy or terrorism, as described in Proposition 4.

5.1.2 Preemption Induced Preference Shift

Preemption induced preference shift of the Org towards its long-term goal can be thought in terms

of a non-additive utility function of the Org with respect to grievance redressal and collateral dam-

age. We assume collateral damage to be proportional to m and specify the Org’s utility function as

u∗U(L(m), g) ≡ U(m, g), where Um < 0 and Ug > 0. In addition, let Ugm > 0 capturing that collateral

damage amplifies the Org’s marginal desire to achieve its long-term objective of complete redressal.

An example will be U(m, g) = A+ vφ(m)(gλ−1), φ′ > 0, which implies V = V(m, g) = vφ(m)(1−gλ).

The Org’s decision rule (first-order condition) with respect to the choice of terror x is now

u∗V(m, g)rx

= n, (10’)

21We have

Ωnδ = −

[Hmm

1 − δ+

Hm

(1 − δ)2n′

]< 0,

implying that n falls with δ at given g and hence the nn curve shifts in.

24

which is the analog of (10). While it is hard to analytically characterize the implications for a general

U(·) function, for the specific example above, we obtain a closed-form solution of x:

x =u∗v(1 − gλ)

rn(a + m)/φ(m). (23)

Hence preemption induced preference shift acts as a decrease in the marginal cost of terror — sim-

ilar to the effect of revenge and collateral damage induced militancy. It is rather unlikely (although

possible) that an increase in preemption leads to an increase in the production of terror; this happens

when n(a + m)/φ(m) falls with m. Otherwise, preemption induced shift in preferences weakens the

effectiveness of preemption. But qualitative implications similar to our earlier baseline model.

5.2 Moderates and Hardliners within the Organization

So far, the terrorist organization is assumed to be a homogeneous entity whose members may be

interpreted as being ‘moderates’ in that they do not derive direct pleasure from terror attacks inflicted

on the State. We now recognize that the Org includes ‘hardliners’ who are more aggressive. They

may be thought of as those having a utility function which is increasing in grievance redressal, and,

terror inflicted on the State unless and until the State redresses their grievances fully. Specifically, let

them be identified by a utility function: u∗U(g) + bx, b > 0 if g < 1 and u∗U(1) if g = 1. The term

bx differentiates them from the moderates. Hardliners are thus ‘extremists’ in that they have such

extreme preferences.22 While for a moderate using violence is a means to an end, for an hardliner the

means partly serves as the end.

Assume that θ ∈ [0, 1) fraction of Org’s members are hardliners. We follow the utilitarian ap-

proach and express the Org’s collective preference by additively aggregating the utility functions.

The Org’s utility prospects are then

2u∗U(g) + θbx − L(m) with probability pO

2u∗U(1) − L(m) with probability 1 − pO.

It may be recalled that 1 − p0 is the ex post probability that the State would fully redress the Org’s

grievances. The Org’s expected utility has the expression:

X ≡ 2u∗U(1) − 2p0u∗V(g) + 2θbp0x. − L(m). (24)

The third term captures the presence of hardliners in the Org. This apart, it is important to note

that an increase in terror tends to raise the probability of full redressal, which, on one hand, benefits

all members (indicated by the second term), but, on the other, undermines the gains to the hardliners

in terms of expected damage from terror. In other words, an increase in terror reduces the utility

22This is however different from religious extremism, the study of which is outside the scope of our analysis.

25

of the hardliners insofar as it decreases the probability of no full redressal. We’ve ∂(p0x)/∂x =

p0 − x∂p0/∂x = p0 − x/(rx) = p0 − 1/r, which in general is ambiguous in sign. If p0 − 1/r < 0,

it is (paradoxically) possible that as long as the door for grievance redressal is open, an increase in

the proportion of hardliners in the Org leads to less terror at a given levels of direct and preventive

counter-terror measures. While such a possibility cannot be ruled out it is rather implausible. We

assume r to be large enough such that in equilibrium p0 − 1/r > 0.

The pertinent question is: what are the impacts of an increase in presence of hardliners — in

terms of an increase in θ — on the choice of CT measures and equilibrium level of terror? In the light

of (24), the marginal benefit to the Org from terror has the expression

2u∗V(g)rx

+ 2θb(p0 −

1r

).

The marginal cost of terror equals 2n as earlier. Hence the first-order condition with respect to the

Org’s optimal choice of terror reads as:

V(g)u∗

rx= n − θb

(p0 −

1r

)≡ n(n

+, p0−

, θ−), 23 (25)

Note that, unlike when θ = 0, not only the sensitivity of p0 with respect to terror (= 1/r) but also the

magnitude of p0 affects the optimal choice of terror. Condition (25) leads to

x =V(g)u∗

rn(n, p0, θ). (26)

However, because of the dependence of marginal benefit from terror on the ex post probability

p0 itself, the model is considerably more complex. Terror output and the ex post probability of (no)

full redressal are determined jointly. Eq. (8), which defines p0, and (26) simultaneously solve them

in terms of n, g and u∗.24 Let these implicit functions be x(n, g, u; θ) and p0(n, g, u; θ), where u = u∗.

From the State’s perspective then,

x = x(n, g, u; θ) =V(g)u

rn(n, p0((n, g, u; θ), θ). (27)

As shown in the Appendix,

∂x∂n

< 0;∂x∂g

< 0;∂x∂u

> 0;∂x∂θ

> 0;

∂p0

∂n> 0;

∂p0

∂g> 0; 25 ∂p0

∂u< 0;

∂p0

∂θ< 0.

(28)

23We ignore preemption induced militancy and assume δ = 0.24In the (x, p0) space eqs. (8) and (26) imply respectively a downward sloping curve and an upward sloping schedule.

Their intersection point defines the solution.

26

The directions of change in x and p0 with respect to CT measures and the random term u are same

as in our base model. In addition, an increase in the presence of hardliners leads to more terror and a

higher probability of full redressal.

In particular, since terror output increases with u there exists a critical u ≡ u, such that the State

would grant full or no further redressal according as u ≷ u, where u is defined by

z∗C(g) = αx(n, g, θ, u), implying

u = u(n, g, θ), implying

p = p(n, g, θ). (29)

Given the implicit functions x(n, g, u; θ) and p(n, g; θ), the objective function of the State Ω is defined

accordingly:

Ω(n, g; θ) = z∗C(g) − H(m(n, a), κ) + z∗C(g) − αE(x(n, g, u; θ) + B(n, g; θ). (30)

While the structure of the State’s problem is similar to our baseline case, the model is no more

analytically tractable. There are two reasons. First, the terror output expression is not multiplicatively

separable in non-stochastic and stochastic components. Second, The marginal effects of CT measures

on the State’s discounted sum of payoff must include their marginal impact on the ex post probability

p0. Since a higher p0 tends to imply a higher level of terror production and ∂p0/∂n and ∂p0/∂g are

both positive, (a) such marginal effects constitute a decline in the marginal benefits associated with

either CT measure and (b) the decline in the marginal benefits increases with θ.26

Absent an analytical characterization, we will be content with a heuristic assessment of likely

impacts of the presence of hardliners in the Org. Since, per se, the presence of hardliners implies

less marginal benefits from CT measures, the net impact on CT measures is in general ambiguous

and likely be small. This would imply a higher level of expected terror.27 In other words, preemptive

measures and grievance redressal both are less effective in dealing with terror in the presence of

hardliners in the Org. The implication is that extreme preferences are harder to tackle.

5.3 Piecemeal Concessions

The sub-game perfect equilibrium - which, in this game, is the Stackelberg solution, is inefficient due

to two reasons. First, the collateral damage L(m) to the Org is not internalized by the State’s choice

of CT measures, since, from the viewpoint of the Org, it is a sunk cost and hence does not affect the

25This holds under Assumption 11.26In the core model, ∂p0/∂n > 0 and ∂p0/∂g > 0 did not affect the marginal benefits of CT measures.27In the light of (25), the direct effect of θ on terror output and the indirect (negative) effect through a change p0 depend

on the direction of change in the term θ(p0 − 1/r). As shown in the Appendix, ∂[θ(p0 − 1/r)]/∂θ > 0. Hence, ∂Ex/∂θ =

E(∂x/∂θ) > 0.

27

Org’s choice of terror input in period 2a. In this sense there is an over-provision of preemption at the

sub-game perfect equilibrium. Second, with respect to grievance redressal the State’s behavior takes

into account its effect on the Org’s incentive to engage in terror, but does not internalize its positive

effect on the Org’s surplus. This implies an under-provision of grievance redressal.

Formally, returning to our original model without hardliners in the Org, a super-national planner

would maximize the joint welfare, say W, of the two players, equal to the sum of their individual

levels of welfare, where

W = 2u∗U(1) − L(m) − 2p0(x, g)u∗V(g) − f (2x) + Ω(m, g),

The first-order conditions of the planner’s problem are:

∂W∂m

= Ωm − L′(m) = 0, (31)

∂W∂g

= Ωg − 2∂p0

∂gu∗V(g) − 2p0u∗V ′(g) = 0. (32)

Observe that at the solution of the sub-game perfect equilibrium

∂W∂m

∣∣∣∣∣(m∗, g∗)

= −L′(m) < 0,∂W∂g

∣∣∣∣∣(m∗, g∗)

= −2∂p0

∂gu∗V(g) − 2p0u∗V ′(g) > 0,

which imply an over-provision of preemption and under-provision of grievance redressal.

It follows that there is a scope for Pareto improving negotiation. Even if in the non-cooperative

situation, the State is the Stackelberg leader it would benefit from a precommitment of the Org to

reduce terror. While characterizing negotiation outcomes is beyond the scope of this paper, we ask

whether, starting from the Stackelberg solution defined by eqs. (20) and (21), the State, on its own

volition to signal a willingness for negotiated settlement, or yielding to pressure by superpowers or

super-national bodies like U.N., can reduce terror by taking measures, so that it does not incur first-

order loss of surplus and yet the welfare of the Org improves.

In the parlance of the literature in public finance on piecemeal tax reform, we may call this

a piecemeal concession or peace proposal, which will move towards the first best by “offering” less

preemption and providing more redressal. But any such arbitrary move would not reduce the expected

damage from terror, as the two changes have opposite effects. Consider a proposal that relates dm

and dg by the following rule:28

pnz∗C′dg = Hm(·)dn, (33)

28In designing this rule we follow the literature on the piecemeal tax reforms, which consider many types of reforms.These reforms typically involve simultaneous changes in at least two taxes, and often a rule is specified linking the changesin the different taxes. See for example, Atkinson and Stiglitz (1976), Hatta (1977), Michael et al. (1993), Diewert et al.(1989), Keen and Ligthart (2002), Emran and Stiglitz (2005), Naito and Abe (2008), and Hatzipanayotou et al. (2011).

28

where the respective coefficients are evaluated at (m∗, g∗). In view of (14),

d lnE(x) =V ′

Vdg −

n′

ndn. (34)

Substituting (20), (21) and (33) — or, equivalently, (A.3), (A.4) and (33) — in the above equation

and using that p = (rz∗n/αu) · (C/V), we find (proved in the Appendix)

d lnE(x) =2n′z∗C′

Hm(·)dg < 0. (35)

That is, starting from the initial equilibrium, a piecemeal concession of reducing preemption and

further redressing the grievances satisfying the rule (33) will unambiguously reduce expected damage

from terror and increase the utility of the Org. At the same time, by virtue of the envelope property

the effect on the net utility of the State will be negligible as long as the piecemeal changes are small

enough. Formally,

Proposition 7: Starting from the equilibrium defined by (20) and (21), there exists a rule such that

a piecemeal, small enough concession by the State involving a reduction in preemption and further

redressal of the grievance according to that rule will be Pareto-improving and at the same unambigu-

ously reduce the use of terror.

6 Revision of Beliefs by the Org: the False-Expectations Problem

Direct utility loss aside, the State may be reluctant to offer partial redressal lest it should lead the Org

to believe that it is “soft” and thereby encourage the Org to use more terror to achieve its end. One

may even speculate that the net effect of partial redressal on terror attacks would be positive. This

would cast doubt on the effectiveness of grievance redressal in addressing terrorism.

Formally put, the Org may update its belief about the State’s preference and attach a higher

probability that the State is soft if g chosen by it lies in the open interval (0, 1). This however is false

expectation, since there is no change in the true preference of the State. But it may influence the Org’s

choice of terror and, therefore, the State’s choice of CT measures in the first place.

There are two caveats to this reasoning however. First, a false expectation may not necessarily

arise. If the choice environment facing the State is probabilistic in period 1 which includes full

redressal alongside no or partial redressal, then the choice of partial redressal does not necessarily

strengthen the Org’s probabilistic belief that the State is a softliner — insofar as partial redressal is

revealed preferred to full redressal.

Second, suppose a partial redressal by the State does lead the Org to believe with a higher prob-

ability that the State is a softliner. Even then, such a false expectation may not necessarily pose a

problem for the State — because the Org’s choice of terror does not depend per se on its probabilis-

tic belief on whether the State would grant full redressal. Rather, it is a function of the change in this

29

probability due to a change in the choice of the terror input — that is, ∂p0/∂x. If this marginal effect

is independent of p0,29 the Org’s updating of belief does not affect its choice of terror and thus does

not impact on the State’s choice of CT measures.

In the event that false expectations arise and they enhance terror and influence the State’s decision

on CT measures, there may be a feedback effect: while the State’s decision is influenced by the Org’s

posterior belief a la Bayes, the latter depends on the State’s decision. The current section considers a

Bayesian updating of beliefs by the Org if the State grants partial redressal, and, presents a situation

where the false-expectations problem arises and affects the Org’s choice of terror. Tractability, how-

ever, requires us to consider a scenario that departs from our core model in many ways. For example,

the State’s choice problem is discrete, and, ex ante, probabilistic.

Our main result is that while the scope of the State’s choice of partial redressal may be reduced

when there is a false expectation problem, grievance redressal still remains a rational response in the

face of an increase in militancy or terrorism.

It is important to note that an obvious solution to this problem, which is not featured in our model,

will be to combine grievance redressal with a threat of more preemption in the future conditional upon

terror chosen by the Org.

6.1 The Environment

We deviate from our baseline model as follows.

• We abstract from the State’s choice of preemptive measures and assume that the level of such

measures is exogenous.

• There is one-sided asymmetric information. While the Org does not know for sure the true pref-

erences of the State, the State knows u∗ and thus the true preference of the Org. For simplicity,

we let u∗ = 1.

• The marginal cost of producing terror is strictly increasing. More specifically, let the marginal

cost be linear in terror production, x2 + x3 and given by 12 · ω · (x2 + x3), where a decrease in ω

represents an exogenous increase in terrorism.30,31

• There are two possible values of z or the State’s type zJ where J = H, L and zH > zL > 0. The

Org’s prior on the State’s type is given by:

z =

zH with Prob π0 ∈ (0, 1)

zL with Prob 1 − π0,(36)

29In the Appendix we provide an example where it is indeed the case.30Increasing marginal cost ensures an interior solution of terror production.31The parameter n here is different from the n function used in previous sections.

30

where π0 is exogenous. The State knows which type it is: a hardliner and relatively uncompro-

mising (z∗ = zH) or a moderate (z∗ = zL). We will call the State as an H-state or an L-state as its

true preference parameter is zH or zL.

• Partial grievance redressal can assume one value only, g, in the open interval (0, 1).

• Whether or not the State grants g is probabilistic due to an extrinsic source of uncertainty which

impacts on the State’s choice. Let us assume a specific source. Recall that in our baseline model

there is a commonly known random iid flow of damage from terror which was denoted by ξ

with mean unity. Instead, assume that there is some persistence of such shocks over time:

ξt = ρξt−1 · εt, ρ > 0, (37)

where εt has an iid distribution, with a range ⊂ R+ and E(εt) = µε . This stochastic process is

common knowledge. Put differently, an ‘innovation’ by the Org in causing damage from the

same amount of aggregate terror input has a spillover effect into future. The parameter ρ would

depend on the efficacy of specific types of security measures to counter such technology of

attack, and, learning to deal with it would imply ρ < 1. However, in our finite-period model it

is immaterial whether this process is stationary. Hence, let us assume, instead, that ln ξ follows

a random walk, i.e., ρ = 1. In the beginning of period 1, the State inherits a realization of

ξ1 = ξ0ε1 (as a result of terror inputs engaged against it in period 0), where ξ0 is given and

normalized to one so that ξ1 = ε1. The State decides g in period 1 after ε1 realizes; this makes

the choice of grievance redressal probabilistic.

• If the State chooses g = 1, it “solves” the terrorism problem in period 1. Thus, updating of

beliefs assumes relevance only when 0 < g < 1. In order to focus on the false expectations

problem, we impose additionally that the choice of full redressal is not available in period 1 —

due to exogenous reasons like binding political constraints, while the State remains open to full

redressal in future (period 2). Otherwise, the choice of partial redressal would reveal preference

over full redressal as well and thus may not enhance the Org’s probabilistic belief that the State

is an L-type.

• Let θJ ≡ Prob(g = g|zJ) denote the probability that the State prefers and thus chooses g = g

instead of 0 if its type is J. Let πc be the Org’s posterior or conditional probability that the State

is a hardliner, given the information set, I2, available in period 2. It is defined by the Bayes rule:

πc =θHπ0

θHπ0 + θL(1 − π0). (38)

Remarks.

[a] It is expected in view of Proposition 2 that a hardliner will be less inclined to grant full

redressal, i.e., θH < θL. This implies πc < π0, i.e., granting any redressal bolsters the Org’s

31

belief that the State is a softliner regardless of its true type.

[b] The probabilities θJ’s are endogenous however: if in period 2 the Org’s output is influenced

by πc, both θH and θL would depend on πc. This exerts a feedback effect on πc via the Bayes

rule as long as the θH/θL ratio is a function of πc. In order solve πc, we invoke perfect foresight

by the State.

• In the Appendix we first provide an example where the false expectations problem does not

arise. This happens if the productivity shock εt obeys a log-uniform distribution. Next we show

that the false-expectations problem does arise if the realizations of εt are governed by a Pareto

distribution and this is the case which we work out fully. However, the dependence of the Org’s

optimal choice of terror production on grievance redressal is more complicated than it is in

our core model (see (12)). As a consequence, for analytical tractability, we need, apart from

Assumptions 1 and 11, a further restriction on the preferences of the State and the Org with

respect to grievance redressal.

Assumption 12.

limg→1

Υ(g)→ ∞, (39)

where Υ(g) ≡ C(g)/V(g); see (15). As will be seen, it implies that limg→1 x = 0. This as-

sumption, thus, places the model within our maintained broad premises that terror production

ceases with full redressal. It is easy to check that Example A does not satisfy Assumption 12,

but Example B does and hence serves as an illustrative example.

• In the environment outlined above we will derive the false-expectations result under Pareto

distribution of terror damage shock with the help of a further assumption:

Assumption 13. The level of partial redressal, g, is above a threshold defined by

C(0)C(g)

> max[1.5,

zH

(1 − π0)zH + π0zL

](40a)

Υ(g)Υ(0)

>zH

(1 − π0)zH + π0zL.32 (40b)

Note that, (40a) is consistent with Assumption 1, and (40b) with Assumptions 11 and 12.

6.2 Main Results

The situation depicted above implies that in period 1 the State faces a binary discrete choice problem,

which is, ex ante, probabilistic, depending upon the realization of ξ1. However, the full characteriza-

tion of this scenario is rather long and technical. We thus relegate it to the Appendix and present here

the two main results when εt is governed by a Pareto distribution.32As a concrete example, suppose zH/zL = 3 and π0 = 0.5. Thus, inequalities (40a) and (40a) boil down to both

C(0)/C(g) and Υ(g)/Υ(0) exceeding 1.5. In Example B, if β = λ = 1.2, both (40a) and (40b) are met as long as g ≥ 0.4.The values of parameters c and v do not matter. These are overly sufficient conditions however.

32

Proposition 8: Relative to no revision of beliefs by the Org, if the Org revises its beliefs via the Bayes

rule, both θH ≡ Prob(g = g|zH) and θL ≡ Prob(g = g|zL) are smaller, i.e., the State, irrespective of its

type, is less likely to grant partial redressal.

This is the false expectations result, implying a reduced scope for partial redressal.

However, as long as there is a positive probability that partial redressal is preferred to no redressal,

Proposition 9 below implies that the rationale behind partial redressal as a counter terror measure

remains in tact, albeit less strongly than otherwise. Let a higher level of militancy or terrorism be

captured by a decline in the parameter ω which lowers the marginal cost function of the Org in

producing terror. In keeping with the spirit of our core model, we obtain that

Proposition 9: Irrespective of the State’s type, a higher level of militancy implies a higher probability

that partial redressal is preferred to no redressal.

Hence, the upshot is that (a) the false-expectations problem does not eliminate the role of preven-

tive measures and (b) a higher level of terrorism or militancy implies a greater likelihood of grievance

redressal. Although we have assumed a specific source of probabilistic choice by the State, this

conclusion is general.

7 Concluding Remarks

Unfortunately, the world map is currently filled with conflicts of various types: inter-state threat

of wars, intra-country civil wars and ethnic conflicts and terrorism. Each type of conflict has its

own characteristics. This paper is concerned with terrorism. A key postulate of our analysis is that

a fundamental grievance constitutes a root cause of terrorism. We postulate a three-period game

between a State and a terrorist organization. In our core model the terrorist organization does not

receive any direct satisfaction from causing damages through terror; it uses terrorism as a means

towards inducing the State to redress their grievances. Beside addressing grievances, the State also

has another instrument in its possession, preemption, which increases the Organization’s marginal

costs of producing terror. Asymmetric information is an essential ingredient of our model.

An increase in militancy which reduces the marginal cost of terrorist attacks calls for more

grievance redressal or conciliation as well as more intensive preemption — a carrot and stick response.

This combination is shown to be more effective in countering terror than just direct, preemptive mea-

sures. There is a possibility that it can win WoT in the face of an increase in militancy or terrorism in

the sense that equilibrium terror may decrease, whereas preemptive measures alone cannot achieve it

as long as the marginal cost of such measures is increasing. These implications highlight the inade-

quacy or limits of direct CT measures — which have been voiced in recent times. In order to contain

and possibly reverse the rise of terrorism it is important on the part of target countries to engage in

33

grievance redressal.33

We have analyzed various extensions of our core model. In particular, it is shown that increased

militancy in terms of a rise in extreme preferences to cause harm as a direct objective is harder to

deal with by using preemptive measures or grievance redressal. A strong case arises for neutralizing

hardliners by diplomatic — and other means such as fostering political and religious competition

and freedom (Abadie (2006), Krueger (2007) and Berman (2011)). An analysis of these measures is

however outside the purview of this paper, but constitute an important future research agenda.

The paper has assumed a unilateral action on the part of the State to accord grievance redressal. In

Section 5 we have considered a piecemeal concession or proposal by the State as a prelude towards a

negotiated settlement. In any negotiation the State would grant some redressal or engage in less pre-

emptive strikes in return for a commitment by the Org to be less engaged in terror through surrender

of arms, cession of certain kinds of activities etc. Bueno De Mesquita (2005a) considers heterogene-

ity within the terrorist organization and examines how negotiations with moderates within the Org

may be subverted by extremists. But the State would still make concessions as the moderates would

then cooperate with the State improving counter-terror capabilities. It will be fruitful to examine CT

measures in a bargaining framework without and with heterogeneity within a terrorist organization.

33There are exceptions however, where terrorism in a democratic country has apparently been eliminated without re-course to preventative measure, e.g., the localized, religion-based Khalistan movement in India or the non-religious move-ment by the Tamil Tigers in Sri Lanka.

34

Appendix

Section 3

Examples A and B Satisfy Assumption 11

Without any loss of generality, let c = v. In Example A, Υ(g) =(1 − gβ

)/(1 − gλ

). Check that

Υ(0) = 1 and, using L’Hopital’s rule, Υ(1) = β/λ > 1. Hence Υ(1) > Υ(0) and thus Υ′(g) > 0 for

some interval in [0, 1]. We have

Υ′(g) = λgλ−1(1 − gβ) − βgβ−1(1 − gλ).

Suppose, to the contrary, Υ(·) is not increasing monotonically. Then ∃ a g in the open interval (0, 1)

where it attains an extremum value and g is defined by the first-order condition

Υ′(g) = gλ−1(1 − gλ)[λΥ(g) − βgβ−γ

]= 0. (A.1)

At g = g, Υ′′ = −βgλ−1(1−gα)gβ−λ−1 < 0. Hence g is unique and Υ(·) attains its maximum value

at g ∈ (0, 1), and Υ(g) > Υ(1). But, substituting (A.1) in the expression for Υ(g), we get

Υ(g) =1 − gβ

1 − gλ=β

λ· g(β−λ) <

β

λ= Υ(1),

since g ∈ (0, 1) and β > λ. That is, Υ(g) < Γ(1), a contradiction. Hence, Υ(g) is monotonically

increasing in g.

In Example B, Υ(g) =(1 − gβ

)/ (1 − g)λ and

Υ′(g) =λ − k(g)

(1 − g)λ+1 ,

where k(g) ≡ λgβ + β(1 − g)gβ−1. Since β ≥ λ > 1, we’ve k′(g) = βgβ−2[β − 1 − (β − λ)g] > 0 for any

g ∈ [0, 1], and thus max k(g) = k(1) = λ⇒ Υ′(g) > 0 for any g ∈ [0, 1).

Alternative Expressions forΩ(n, g) and the First-Order Conditions

Using the expressions of p and x, (19) can stated as

Ω(n, g) =

(2 +

p2−

12p

)z∗C(g) − H(m(n, a), κ). (A.2)

35

The first-order conditions (20) and (21) can thus be expressed alternatively as

Ωn =(1 + p2)z∗C(g)

2p2

∂p∂n−

Hm(m(n, a), κ)n′

,

=1 + p2

2p2 ·r[z∗C(g)]2

αuV(g)−

Hm(m(n, a), κ)n′

= 0. (A.3)

Ωg =

(2 +

p2−

12p

)z∗C′(g) +

(12

+1

2p2

)z∗C(g)

∂p∂g

= (2 + p)z∗C′(g) −1 + p2

2p·

z∗C(g)V ′(g)V(g)

= 0. (A.4)

Second Partials of theΩ(n, g) Function

Differentiating Ωn in (A.3),

Ωnn = −1p3 ·

r2[z∗C(g)]3

[αuV(g)]2 −Hmmn′ − Hmn′′

n′3< 0. (A.5)

In view of (A.4),

Ωg

z∗= (2 + p)C′(g) −

αu(1 + p2)2rz∗n

V ′(g)

⇒Ωgg

z∗= (2 + p)C′′(g) −

1 + p2

2p·

C(g)V ′′(g)V(g)

+p[V(g)]2

C(g)p2g. (A.6)

The first two terms are positive as C′′(g) < 0 < V ′′(g), whereas the last term is positive. It follows

that Ωgg < 0 as long as |C′′| and |V ′′| are large enough relative to each other

Differentiating Ωg in (A.4) with respect to n,

Ωng

z∗=

[C′(g) +

1 − p2

2p2 ·C(g)V ′(g)

V(g)

]pn < 0. (A.7)

Furthermore,

Ωna =Hmm

n′> 0 (A.8)

Ωga = 0. (A.9)

36

Iso-Terror Curve is Steeper than the nn Curve

The slopes of the iso-damage curve and the nn curve are respectively

dndg

∣∣∣∣∣x

=nV ′

V;

dndg

∣∣∣∣∣nn

= −Ωng

Ωnn=

[C′(g) +

1−p2

2p2 ·C(g)V′(g)

V(g)

]z∗pn

1p3 ·

r2[z∗C(g)]3

[αuV(g)]2 +Hmmn′−Hmn′′

n′3

.

The iso-terror curve is steeper than the nn curve if and only if

dndg

∣∣∣∣∣nn>

dndg

∣∣∣∣∣x⇔

[C′(g) +

1−p2

2p2 ·C(g)V′(g)

V(g)

]r(z∗)2

αu ·CV

1p3 ·

r2[z∗C(g)]3

[αuV(g)]2 +Hmmn′−Hmn′′

n′3

>nV ′

V

⇔pz∗

n

[V(g)C′(g) −C(g)V ′(g) −

1 − p2

2p2 C(g)V ′(g)]−

(Hmmn′ − Hmn′′) nV ′(g)n′3

> 0,

which is met in view of Assumptions 3, 5 and 11.

The Effect of an Increase in z∗ on nn and gg Curves

By definition, at given n and g, p(·) is proportional to z∗. In view of (A.3), we can write Ωn as

Ωn =1 + p2

2·αuV(g)

rn2 −Hm(m(n, a), κ)

n′

which increases with p and thus increases with z∗ at given n and g. Hence, the nn curve shifts to the

right.

In view of (A.4) the ratio of marginal gains from grievance redressal to its marginal cost has the

expression:1 + p2

2p(2 + p)·

C(g)V ′(g)C′(g)V(g)

which is decreasing in p and therefore decreases with z∗ at given n and g. The gg curve shifts to the

left.

Section 5

Partials of x(n, g,u;θ) and p(n, g,u;θ) Functions

Totally differentiating (8) and (26),

rdp0 +dxx

=C′

Cdg

−θbn

dp0 +dxx

=V ′

Vdg −

1n

dn +1u

du +b(p0 −

1r

)n

dθ

37

dxx

= −r

rn + θbdn +

(θb

rn + θbC′

C+

rnrn + θb

V ′

V

)dg +

rnu(rn + θb)

du +rb

(p0 −

1r

)rn + θb

dθ

dp0 =1

rn + θbdn +

nrn + θb

(C′

C−

V ′

V

)dg −

nu(rn + θb)

du −b(p0 −

1r

)rn + θb

dθ

The last two expressions imply the signs in (28). Furthermore,

∂[θ(p0 −

1r

)]∂θ

=rn

(p0 −

1r

)rn + θb

> 0.

Inequality (35)

Starting from (34),

d lnE(x) =V ′

V· dg −

n′pz∗C′

Hm· dg, using (33)

=2p(2 + p)C′

(1 + p2))C· dg −

n′pz∗C′

Hm· dg, using (A.4)

=[(2 + p)C′z∗n′ − pn′z∗C′

]·

dgHm

, using (A.3) and p =rz∗nαu·

CV

=2C′z∗n′

Hm· dg < 0.

Section 6

We begin with the State’s choice in the latter half of period 2. It may revise its decision on g after observing xas well as the realization of ξ2. For any given g chosen in period 1 it would maintain status quo if

zJC(g) − αxE2(ξ3) > 0⇐⇒ zJC(g) − αxE2(ξ2ε3) > 0

⇐⇒ zJC(g) − αxξ2µε > 0⇐⇒ ξ2 <zJC(g)

x, by normalizing αµε = 1,

where the subscript 2 indicates that the expectation is conditional on the information set I2.Turning now to the Org’s choice of terror, ξ2 is not known to it in the first half of period 2. Based on

realized ξ1 = ε1 it faces a probability distribution of ξ2.

Prob(ξ2 <

zJC(g)x

)= Prob

(ε2 <

zJC(g)xξ1

)= F

(zJC(g)

xε1

)(A.10)

⇒ p0 = πF(

zHC(g)xε1

)+ (1 − π)F

(zLC(g)

xε1

)≡ p0(x; ·), where

π =

π0 if g = 0πc if g = g ∈ (0, 1).

(A.11)

The Org’s choice of terror production is guided by the first-order condition −∂p0/∂x · V(g) = ωx, where∂p0/∂x depends on the distribution of ε. As mentioned earlier, we consider two specific density functions of

38

ε, namely, log-uniform and Pareto, which respectfully illustrate situations where a false expectation does notcause a problem in that it does not affect terror output and therefore CT measures, and where it is a problem.

If ε has a log-uniform distribution and its upper bound and lower bound are ε > 0 and ε > 0, then

F(·) =ln(zJC(g)) − ln(xε1) − l

l − l, (A.12)

where l ≡ ln ε. However, substituting (A.12) into (A.11) and differentiating yield ∂p0∂x = − 1

x(l−l) , which isindependent of π. Hence terror output is unaffected any update of beliefs by the Org.

Pareto Distribution

Henceforth, we assume that ε has a Pareto distribution:

F = 1 −(εmin

ε

)θ, ε ≥ εmin, (A.13)

and, for simplicity, let εmin = θ = 1.34 We have

p0 = 1 −xε1φ(π)

C(g), where φ(π) ≡

zH − π(zH − zL)zHzL

=⇒ −∂p0

∂x=ε1φ(π)C(g)

=⇒ x =φ(π)ε1

ωΥ(g), (A.14)

where we have used the first-order condition of producing terror.Remarks.[a] Unlike in the case of the log-uniform distribution, here ∂p0/∂x is dependent of π, which makes terror outputsensitive to the probability π.[b] Terror output increases with productivity shock and decreases with marginal cost of producing terror. Itdoes not however depend on the State’s type per se, but is dependent on the Org’s belief about the State’s type.[c] It follows from (A.14) that, if πc < π0, the Org produces more terror than otherwise. Hence the falseexpectation problem arises under Pareto distribution of terror productivity shocks.[d] A higher g already granted in period 1 reduces the utility loss to the State from full redressal and hencemakes terror marginally more effective in increasing the odds facing the State in favor of full redressal. Thiselement was absent under the specific distribution of u assumed in our main model but it is present under Paretodistribution of damage technology shock. The implication is that in the presence of this marginal effect, a highergrievance redressal tends to increase terror irrespective of whether or not there is a false expectations problem.However, Assumption 11, Υ′(g) > 0, ensures that this effect is outweighed by the reduced utility deficit effectfacing the Org from a higher level of partial redressal. Hence the net effect is that terror production declineswith grievance redressal. Furthermore, in view of (A.14), under Assumption 12, terror production is zero if fullredressal is accorded.

Choice between No and Partial Redressal in the Absence of Expectations Revision

It will be convenient and instructive to first analyze the State’s probabilistic choice of preventive measures inthe absence of expectations revision by Org. In view of (A.14), the Org’s choice of terror input production is:

x =φ(π0)ε1

ωΥ(g). (A.14′)

34If θ > 1, we have to assume strongly increasing marginal cost of producing terror.

39

Turning now to the State’s decision in period 1, we need to piece together its payoffs in all three periods.Letting H1 denote the exogenous cost of preemptive measures, the State’s period 1 payoff is zJC(g)− H1. Fromthe State’s perspective, given I1, terror production is non-stochastic as it knows the true preference of the Org;but the damage from it is stochastic and the expected damage from terror in period 2 is αxE1(ξ2) = xε1 (recallthat αµε is normalized to unity). The expected period-2 surplus has the expression

zJC(g) − xε1 = zJC(g) −φ(π0)ε2

1

ωΥ(g). (A.15)

In period 3 there may or may not be any damage from terror depending on whether the State doesn’t ordoes grant full redressal. Given I1, the expression for the expected period 3 surplus of the State is constructed asfollows. Starting with period 2, given I2, the expected utility loss from terror damage in period 3 is αxE2(ξ3) =

xξ2αµε = xε1ε2. The State maintains status quo as long as αxE2(ξ3) < zJC(g), i.e., if

ε2 ≤zJC(g)

xε1=

zJωΓ(g)φ(π0)ε2

1

≡ ζ(ε1−

; zJ+, g±

, π0+, ω

+), Γ(g) ≡ C(g)Υ(g).35 (A.16)

A higher ε1 implies a higher level of damage from terror, and a smaller likelihood of the State not willing togrant full redressal irrespective of its type. A type-H State is more likely not to grant full redressal than type-L.A higher π0 or a higher ω implies less terror and hence a higher likelihood that the State wouldn’t grant fullredressal. A higher g exerts two opposite effects on the likelihood that the State does not accord full redressal.On one hand, it leads to less terror production and thus the State is more likely not to grant full redressal;but, on the other, it reduces the utility cost to the State from granting full redressal and thereby decreases thelikelihood that the State would maintain status quo. Accordingly, the signs of Γ′(g) and ∂ζ/∂g are ambiguous.

Based on I1, the expected payoff to the State in period 3 has the expression∫ ζ(ε1;zJ ,g,π0,ω)

1

[zJC(g) − xε1ε2

]dF(ε2) =

∫ ζ(ε1;zJ ,g,π0,ω)

1

zJC(g) −φ(π0)ε2

1ε2

ωΥ(g)

dF(ε2). (A.17)

In view of (A.15) and (A.17), the discounted value of State’s surplus, given I1, has the expression:

Ω(ε1; zJ , g, π0, ω) = −H1 + 2zJC(g) −φ(π0)ε2

1

ωΥ(g)

+

∫ ζ(ε1;zJ ,g,π0,ω)

1

zJC(g) −φ(π0)ε2

1ε2

ωΥ(g)

dF(ε2)

= −H1 + 3zJC(g) −φ(π0)ε2

1

ωΥ(g)·[2 + ln ζ(ε1; zJ , g, π0, ω)

]. (A.18)

This function is the basis of the State’s choice between g = 0 or g in period 1, depending upon therealization of ε1. The State’s surplus, irrespective of whether it is a hardliner or a softliner, decreases withpositive technology shock from terror in period 1, and, the surplus enjoyed by a hardliner is higher (since itsutility from any given level of grievance redressal, short of full redressal, is greater).36 Figure A1 depicts the

35Since ε2 ≥ 1, the probability that the State does not grant full redressal in period 2 is 0, or equivalently, it grants fullredressal with probability 1 if ζ ≤ 1. This happens when ε1 exceeds a threshold,

√zJωΓ(g)/φ(π0). We assume that zJ and/or

the marginal cost of terror production is high enough such that this threshold is large enough so as to imply (in view of thedownward sloping Pareto density function) negligible probability of such occurrence in period 1. Effectively, we ignore theabove constraint and presume ε2 = ζ ≥ 1.

36We have

∂Ω

∂ε1= −

2φ(π0)ε1

ωΥ(g)[1 + ln ζ(ε1; zJ , g, π0, ω)

]< 0;

∂Ω

∂zJ= [2 + F(ζ(ε1; zJ , g, π0, ω)]C(g) > 0.

40

respective surplus Ω as a function ε1 at any given g < 1.

Figure A1: The State’s Expected Surplus as a Function of ε1

We next develop the gains function from partial redressal, defined as the difference between the State’ssurplus from granting partial redressal g and that from no redressal: Y(ε1; zJ , π0, ω) ≡ Ω(ε1; zJ , g, π0, ω) −Ω(ε1; zJ , 0, π0, ω) R 0.

Y(ε1; zJ , π0, ω) =φ(π0)ε2

1

ω

[2 + ln ζ0

Υ(0)−

2 + ln ζΥ(g)

]− 3zJ[C(0) −C(g)], (A.19)

where ζ0 ≡ ζ(ε1; zJ , 0, π0, ω); ζ ≡ ζ(ε1; zJ , ,π0, ω).

This leads to a simple decision rule: choose g = 0 or g according as Y(ε1; zJ , π0, ω) ≶ 0, depending on therealization of ε1. Properties of the gains function defined in (A.19) are summarized in the following proposition.

Proposition A1: As long as Γ(g) satisfies (40a), (a) Y(ε1; zJ , π0, ω) is monotonically increasing in ε1, (b)Y(ε1; zJ , π0, ω) > 0 if ε1 exceeds a threshold and (c) Y(ε1; zH , π0, ω) < Y(ε1; zL, π0, ω).

Proof. We have∂Y∂ε1

=2ε1φ(π0)Λ

ω, where Λ ≡

1 + ln ζ0

Υ(0)−

1 + ln ζΥ(g)

. (A.20)

We show that for any ε1, Λ > 0, so that ∂Y/∂ε1 > 0. Define a function

K(g) ≡1 + ln ζ(ε1; zJ , π0, ω)

Υ(g)(A.21)

and note that Λ = K(0) − K(g). Using the definition of the ζ function and Assumption 11,

K ′(g) =C(g)V ′(g) − V(g)C′(g)

[C(g)]2 [1 + ln ζ(·)] +2V(g)C′(g) −C(g)V ′(g)

[C(g)]2

=C(g)V ′(g) − V(g)C′(g)

[C(g)]2 ln ζ(g, zJ , ε1, πc;ω) +V(g)C′(g)

[C(g]2

< 0, since ζ ≥ 1

⇒ Λ > 0, proving part (a).

From the definition of the Y function,

limε1→∞

Y(·) = limε1→∞

φ(π0)ε21

ω

(2

Υ(0)−

2Υ(g)

)− 3zJ[C(0) −C(g)] = ∞,

41

where we have used ∂ζ/∂ε1 < 0, and, hence, limε1→∞ ζ = min ζ = 1. This proves part (b). Next,

∂Y∂zJ

= −[(2 + F(ζ0)C(0) − (2 + F(ζ)C(g)

]< −[(2 + 0)C(0) − (2 + 1)C(g)] since 0 ≤ F(·) ≤ 1= −2[C(0) − 1.5C(g)]< 0 in view of (40a),

which proves part (c).Part (a) of Proposition A1 says that the greater the damage technology shock, the higher are the gains from

partial redressal whether the State is type-L or type-H. It implies, in contrast to the surplus functions, that thegains function is upward sloping. Part (b) establishes that gains from partial redressal are positive if the damagetechnology shock is sufficiently is high enough. Part (c) indicates that gains from partial redressal are less forthe H-State than for the L-State.

In view of parts (a) and (b) of Proposition A1, if Y(1; zJ , π0, ω) > 0, then Y(ε1; zJ , π0, ω) > 0 for any ε1 ≥ 1and there is zero scope for no redressal. Partial redressal will be granted with probability one. Proposition A2articulates a more general environment where Y(1; zJ , π0, ω) < 0, so that no redressal arises as the preferredchoice over relatively low realizations of the technology shock. Hence, there would a threshold value of ε1,below or above which the preferred choice will be no redressal or partial redressal. 37

It is seen from (A.19) that Y(1; zJ , π0, ω) < 0 if, ω, the height of the marginal cost function facing the Orgin producing the aggregate terror input exceeds a threshold. Formally,

Proposition A2: Let the marginal cost facing the Org exceed a threshold such that for both J = H, L, Y(1; zJ , π0, ω) <0. Then there exists a threshold ε1J such that no or partial redressal is the preferred choice by the J-State ac-cording as ε1 ≶ ε1J . The threshold ε1J solves

Y(ε1; zJ , π0, ω) = 0. (A.22)

Proof. The first term in (A.19) is relevant.

limω→∞

Y(1; zJ , π0, ω)

= limω→∞

φ(π0)[2 + ln ζ(1; zJ , g, π0, ω)]ωΥ(0)

[limω→∞

2 + ln ζ(1; zJ , 0, π0, ω)2 + ln ζ(1; zJ , g, π0, ω)

−Υ(0)Υ(g)

]− 3zJ[C(0) −C(g)]

= limω→∞

1/ωΥ(0)

[1 −

Υ(0)Υ(g)

]− 3zJ[C(0) −C(g)]

= −3zJ[C(0) −C(g)]< 0.

This implies Proposition A2.Figure A2 depicts Propositions A1 and A2. The J-State would grant partial redressal if ε1 > ε1J . An

important implication follows immediately:

Proposition A3: The probability that the hardliner will grant partial redressal (as opposed no redressal) isless than that for the moderate, i.e., θH ≡ Prob[g = g|zH] < Prob[g = g|zL] ≡ θL.

37Further, the respective threshold would differs between an H-State and an L-State.

42

Figure A2: Ω(ε1) and Y(ε1) Functions

Choice under Bayesian Revision of Beliefs

We now introduce Bayesian updating of beliefs by the Org. The surplus function associated with no redressalis unchanged, while that for partial redressal is influenced by the revision of beliefs by the Org:

Ωc(ε1; zJ , g, πc, ω) = −H1 + 3zJC(g) −φ(πc)ε2

1

ωΥ(g)· (2 + ln ζc), where ζc ≡ ζ(ε1; zJ , g, πc, ω), (A.23)

where πc is posterior probability facing the Org that the State is type H. A priori, πc R π0, although it will be de-duced that πc < π0. As in case of no revision of beliefs, ∂Ωc(ε1; zJ , g, πc, ω)/∂ε1 < 0 < ∂Ωc(ε1; zJ , g, πc, ω)/∂zJ ,i.e., the respective Ωc(·) functions are shaped and positions similar to the respective Ω(·) as shown in FigureA1.

Next, we define the gains functions under Bayesian learning, which are dependent on both prior and pos-terior probabilities π0 and πc:

Yc(ε1; zJ , πc, π0, ω) ≡ Ωc(ε1; zJ , g, πc, ω) − Ω(ε1; zJ , 0, π0, ω). (A.24)

The following proposition states the properties of the Yc(·) function.

Proposition A4:∂Yc

∂ε1> 0;

∂Yc

∂zJ< 0;

∂Yc

∂πc> 0;

∂Yc

∂ω< 0.

Proof. We’ve

∂Yc

∂zJ= −

[(2 + F(ζ0))C(0) − (2 + F(ζ))C(g)

]< 0 in the light of (40a) (A.25)

∂Yc

∂πc=∂Ω

∂πc= −

(1 + ln ζc)ε21φ′(πc)

ωΥ(g)> 0. (A.26)

Furthermore,

∂Yc

∂ε1=

2ε1

ωΛc;

∂Yc

∂ω= −

ε21

ω2Λc = −

ε1

2ω·∂Yc

∂ε1, (A.27)

where Λc ≡φ(π0)(1 + ln ζ0)

Υ(0)−φ(πc)(1 + ln ζc)

Υ(g).

We prove now that Λc > 0, which implies ∂Yc/∂ε1 > 0 > ∂Yc/∂ω. Suppose πc ≥ π0. Noting that

43

φ(π)[1 + ln ζ(·)] is a decreasing function of π,

Λc > φ(π0)[1 + ln ζ0

Υ(0)−

1 + ln ζΥ(g)

]= φ(π0)[K(0) − K(g)] = φ(π0)Λ > 0.

Now suppose πc < π0. It implies φ(π0) < φ(πc). Rewriting Λc,

Λc =φ(π0)(1 + ln ζ0)

Υ(0)−φ(πc)(1 + ln ζ + ln[φ(π0)/φ(πc)])

Υ(g)

>φ(π0)(1 + ln ζ0)

Υ(0)−φ(πc)(1 + ln ζ)

Υ(g)

=φ(π0)(1 + ln ζ)

Υ(0)

[1 + ln ζ0

1 + ln ζ−Υ(0)Υ(g)

·φ(πc)φ(π0)

]. (A.28)

Note that, given (40b), Υ(0)Υ(g) ·

φ(πc)φ(π0) < 1. If ζ0 ≥ ζ, then (A.28) and thus Λc are positive. If ζ0 < ζ, we note

that (1 + ln ζ)/ζ is a decreasing function of ζ and thus

1 + ln ζ0

1 + ln ζ>ζ0

ζ=

C(0)Υ(0)C(g)Υ(g)

⇒ Λc >φ(π0)(1 + ln ζ)

Υ(g)

[C(0)C(g)

−φ(πc)φ(π0)

]in view of (A.28)

> 0, when (40a) holds.

Hence, under (40a)-(40b), the properties of Yc(·) function with respect to ε1, zJ and n are analogous tothose of the Y(·), defined in (A.19). In addition and importantly, ∂Yc/∂πc > 0 means that the higher the Org’sposterior belief that the State is a hardliner, the greater are the gains from partial redressal.

To allow for the possibility of no redressal as the preferred choice, we impose Yc(1; zJ , 1, π0, ω) < 0.Similar to the case of no updating of beliefs by the Org, the last inequality holds if the marginal cost facing theOrg exceeds a threshold.

Proposition A3 that the hardliner State is less inclined to grant partial redressal than the softliner holdsunder Bayesian updating of beliefs too. The Bayes rule, eq. (38), then immediately implies that partial redressallowers the Org’s probabilistic belief that the State is a hardliner.

Corollary 1: πc < π0.

Next, since ∂Yc/∂πc > 0, Corollary 1 implies that Yc(ε1; ·) function lies to the right of the Y(ε1; ·) for bothtypes, as shown in Figure A3. Compared to Y(·), Yc(·) > 0 holds and therefore partial redressal is the preferredchoice over a smaller range or ε1 (to the right of εc

1L and right of εc1H for the L-type and the H-type respectively).

The false expectations result, that is, Proposition 8 in the text follows.Note that εc

1J solves Yc(ε1; zJ , πc, π0, ω) = 0, which implicitly defines εc1J ≡ h(zJ , πc, π0, ω). Hence, under

Pareto distribution,

θJ = Prob(ε1 ≥ εc1J) =

∫ ∞

εc1J

dF(ε) =1εc

1J. (A.29)

Solution of πc

The posterior probability πc is endogenous, and, has not been solved yet. However, once we express the Bayes’rule (38) as

πc =θH(πc; ·)π0

θH(πc; ·)π0 + θL(πc; ·)(1 − π0)≡ ∆(πc; ·), (A.30)

the equilibrium πc is a fixed point of the mapping ∆(πc; ·).

44

Figure A3: Impact of Expectations Revision by the Org

Lemma 2: |∆′(πc)| < 1.

Proof. Differentiating (A.29) with respect to πc,

∂ ln θJ

∂πc= −

∂ ln εc1J

∂πc= −

∂h/∂πc

εc1J

=1εc

1J·∂Yc/∂πc

∂Yc/∂ε1

= −φ′(πc)

2·

(1 + ln ζc)/Υ(g)Λc (< 0)

= −φ′(πc)

2·

1φ(π0)Υ(g)Υ(0) ·

1+ln ζ0

1+ln ζc − φ(πc). (A.31)

Both ζ0 and ζc are functions of zJ and from their respective definitions, ∂ ln ζ0/∂zJ = ∂ ln ζc/∂zJ = 1/zJ . Hence

∂(

1+ln ζ0

1+ln ζc

)∂zJ

= −ln ζ0 − ln ζc

zJ(1 + ln ζ)2

If ζ0 < ζc,

∂(

1+ln ζ0

1+ln ζc

)∂zJ

> 0⇒∂2 ln θJ

∂πc∂zJ< 0⇒

∂ ln θH

∂πc<∂ ln θL

∂πc⇒ ∆′(πc) < 0,

so that the equilibrium πc is unique.If ζ0 > ζc, then

∂ ln θH

∂πc>∂ ln θL

∂πc

⇒ ∆′(πc) = πc(1 − πc)(∂ ln θH

∂πc−∂ ln θL

∂πc

)> 0.

It is however sufficient to prove πc(1 − πc)∂ ln θH/∂πc < 1, which would imply ∆′(πc) < 1. In view of (A.31),

πc(1 − πc)∂ ln θH

∂πc<−πc(1 − πc)φ′(πc)/2Υ(g)Υ(0)φ(π0) − φ(πc)

< 1, if

−πc(1 − πc)φ′(πc)

2<Υ(g)Υ(0)

φ(π0) − φ(πc)

45

⇔Υ(g)Υ(0)

>2zH − πc(1 + πc)(zH − zL)

2[zH − π0(zH − zL)], which holds if

Υ(g)Υ(0)

>zH

(1 − π0)zH + π0zL.

The last inequality is same as Assumption (40b). There if Assumption (40b) is met, then πc(1−πc)∂ ln θH/∂πc <1, which implies that ∆′(πc) < 1.

Lemma 2 implies that the solution of πc in eq. (A.30) is unique.

An Exogenous Increase in Terrorism or Militancy

As stated in the text, this is captured by a decline inω that lowers the marginal cost function of Org in producingterror. Differentiating (A.29) with respect to ω and using (A.27), (∂θJ/∂ω)/θ j = −1/2ω < 0, which is indepen-dent of zJ . Thus, for any π0, there is no change in the posterior probability πc. However, although the Org’sbeliefs on the preference of State are unaltered, the Org produces more terror in the face of lower marginal costof producing it. This changes the gains from partial redressal. In view of Proposition A4, ∂Yc/∂ω < 0, i.e., alower ω implies higher gains from partial redressal. This is equivalent to an leftward shift of the Yc(ε1; ·) func-tions in Figure A3. Both θH and θL increase, although their ratio and thus πc remain unchanged. Proposition 9follows.

46

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