Atrocity in Identity: Dynamics of Extremist Groups ∗

Xiaoli Guo†

Abstract

I formalize how an identity of ethnicity, religion or other attributes is manipu-

lated to influence intra- and intergroup relations. As the extremist groups commit-

ted atrocities in the name of Islam, Muslim backlash soared globally. Aggravated

discrimination against Muslims slows down their pace of assimilation, which in turn

may aid the growth of the extremist groups. The model suggests that the public’s

negative perception of the Islam identity and the Muslim community’s instinct to

preserve their identity in the face of threats are essential for successful provocation.

Exclusion of Muslims exacerbates the situation, unless they are productive and ex-

panding. Surprisingly, a less extreme or a more heterogenous Muslim community is

conducive to the extremists’ recruitment by provocation. Provocation efficacy de-

creases in the Muslims’ productivity and increases in their consumption, and most

often has an indeterminate relation with their group size.

∗I thank Robert Carroll, Jens Grosser, Mark Isaac, Will Moore, Dmitry Ryvkin, Am-mar Shamaileh, David A. Siegel and Mark Souva for helpful comments. All the remainingerrors are mine.†Department of Political Science, Florida State University, Tallahassee, FL 32306-2180,

USA; E-mail: xlguo.fsu@gmail.com

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1 Introduction

Though the notion of “Islamophobia” has existed for centuries, Muslim backlash soared in

the U.S. and across Europe after Islamic extremist groups like al-Qeada and IS launched

terror attacks in their homelands (e.g. Aslund and Rooth, 2005; Hanes and Machin, 2014;

Schuller, 2016). In the face of looming threats of terrorism, the public and the government

in these countries most often become increasingly hostile to the Muslim communities (e.g.

Bakalian and Bozorgmehr, 2009). Aggravated hostility contributes to a renewed sense of

solidarity and a revival of their identity among Muslims, and slows down their pace of

assimilation (Abdo, 2006; Barrett, 2007; Bakalian and Bozorgmehr, 2009).

A widespread concern is that the more cohesive and traditional Muslim communities

would in turn facilitate the Islamic extremists’ recruitment and growth. From a strategic

perspective, the extremists may have deliberately committed atrocities to spur discrimi-

nation against and exclusion of Muslims, and create intergroup antagonism, polarization,

and radicalization in society.

In this paper, I presents a game-theoretical model to formalize the mechanisms in

this vicious circle. The model addresses the following questions: how do atrocities in

the name of Islam affect intergroup relation and the dynamics of assimilation in the

Muslim community? Under what conditions do the extremists have incentives to apply

the provocation strategy? What policies of the public and the government, and what

attributes of the Muslim community exacerbate the situation and benefit the extremists?

The findings indicate that the public’s negative perception of the Islam identity and

the Muslim community’s instinct to preserve their identity in the face of threats are

essential for successful provocation. Only when the Muslim community, in response to

worsening out-group discrimination, reaches a certain threshold of reaction to preserve

in-group cohesion, the extremists find it profitable to commit atrocities for recruitment

at all.

When the majority exclude Muslims, the threshold becomes easier to meet. The

higher the degree of exclusion, the larger the increase of loyal Muslims who stop assimilate.

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However, for a productive and expanding Muslim community, the decrease in provocation

efficacy is more salient when they are excluded and discriminated, compared with when

they are only discriminated.

Surprisingly, for the sake of promoting recruitment, the extremists benefit from a less

extreme or a more heterogenous Muslim community. Moreover, the efficacy of provoca-

tion decreases in the Muslim community’s productivity and increases in their consumption

level, but often has an indeterminate relation with their group size. Yet, if the Muslim

community is unproductive, and the everyday discrimination against them is already se-

vere even before the extremists commit atrocities, then the efficacy of provocation strictly

increases in the Muslims’ group size.

Though the model is motivated by the Islamic extremists’ manipulation of the Islam

identity, it potentially can be applied to other scenarios where an identity of ethnicity,

religion, class or other attributes is used to influence intra- and intergroup cooperation

and conflict. Before presenting the model and explaining the findings, I elaborate the

concepts and critical assumptions made in the model, and walk through the process

intuitively below.

1.1 Identity, In-group Favoritism and Assimilation

The minority identity is defined by the combination of descent-based and non-descent-

based attributes (Lee et al., 2004; Chandra, 2012; Mele and Siegel, 2016, 2017), including

an individual’s ethnicity, religion, language, accent, habit of diet and dress, etc. A minor-

ity group member can “dilute” his identity through assimilation, given that an identity to

some extent is a social construct (e.g. Posner, 2004; Dickson and Scheve, 2006; Penn, 2008;

Chandra, 2012). For example, in the case of Muslims, behavior of assimilation includes

anglicising their names, intermarrying with non-Muslims, speaking Arabic or visiting a

mosque less frequently (Goldwasser, 1998; Kulczycki and Lobo, 2002).

A minority group member’s incentives of assimilation arise from the high benefits of

interacting with outsiders. There is a significant body of scholarly evidence that shows

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the ubiquitousness of in-group favoritism, no matter in a minority or majority group.

Realistic group conflict theory (Sherif, 1966) argues that this inclination is caused by real

intergroup conflict, while social identity theory (Tajfel and Turner, 1979) postulates that it

is driven by a psychological need to enhance one’s group status and consequently promote

one’s self-image. The downside of in-group favoritism is the differentiation, ignorance,

discrimination, or even malicious hostility towards out-group individuals. Assimilation to

alleviate out-group discrimination at the expense of in-group favoritism can be beneficial

to a minority group member, since the majority group most often is economically and

politically dominant.

Yet, assimilation is costly. In this model, I consider the psychological cost1 that in-

creases in the minority group member’s identity consciousness and the strength of the

identity. Assume that the minority are heterogenous regarding their identity conscious-

ness, and as a result have different psychological cost. When the cost is sufficiently high,

an in-group member forgoes the benefits of assimilation and chooses to differentiate. The

outcome of individual choices could be what is usually observed in society that part of

the minority assimilate while the others do not. I also assume that the strength of the

identity increases in the scale of minority group members that assimilate. The proportion

of in-group members who assimilate and the strength of the identity thus are mutually

dependent and endogenous.

1.2 Atrocities and Discrimination

Despite the prevalence of in-group-out-group distinctions, most people reveal unconscious,

subtle, indirect and ambiguous biases. Conscious, explicit, direct, and unambiguous bi-

ases that underlie aggression and even hate crime are often harbored by extremists and

triggered by perceived threats (Fiske, 2002). These dangerous biases and discrimination

1 It will not be problematic to interpret the cost of assimilation in another way, forexample, the time and effort an individual needs to input depending on his ability ofpassing (Caselli and Coleman, 2013; Mele and Siegel, 2016, 2017) or learning (Eguia,2017).

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can give rise to polarization and radicalization in society (Isenberg, 1986), which are what

the extremists seek for.

The extremists conduct specular atrocities like suicide bombings, mass shootings, or

kidnapping and beheading hostages, in the name of the minority identity.2 The anti-

humanity and anti-civilization nature of atrocities provokes strong negative sentiments

like nausea, fear, anger and hatred among the audience. Negative stereotypes of the

minority, such as brutality and violence associated with Muslims, are reinforced, and the

majority create negative expectations of the minority members’ behavior (Stephan and

Stephan, 1996), especially of those who do not assimilate.

Perceived threats to the majority can be symbolic, such as destruction of their cher-

ished values and beliefs like democracy and freedom, which contributes to prejudice and

biases against the minority (Sears, 1988; Biernat, Vescio and Theno, 1996; Dunbar et al.,

2000; Esses, Haddock and Zanna, 1993; McLaren, 2003; Riek, Mania and Gaertner, 2006).

The threats can also be realistic in that the majority’s safety, economy, politics, and overall

well-being are at risk (Riek, Mania and Gaertner, 2006). If the minority are of relatively

lower socioeconomic status, which implies that their productivity tends to be low, but

of a large or growing size so that they consume more resources, the threats looms even

more salient (Quillian, 1995). As a result, incompatible goals and competition between

the groups are placed under the spotlight, which turns subtle and “cool” discrimination

into explicit and “hot” discrimination.

Muslim backlash following terror attacks happened in the U.S., England, France,

Germany, Belgium, Italy and even traditionally neutral Switzerland and liberal Holland,

among others. In the week after 9/11, more than 600 bias incidents against Middle

Eastern and Muslim Americans, in the form of anti-Arab activities, property damage,

violence, harassment and threats, were reported in metropolitan newspapers across the

country.3 A few weeks after 9/11, the U.S. government generated a series of decrees,

2 The extremists can conduct terrorism out of different purposes, such as provocation,intimidation, bargaining, etc. Here I focus on provocation to smooth the argument.

3Source: http://saalt.org/policy-change/post-9-11-backlash/

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administrative rules, executive orders, and laws that targeted Middle Eastern and Muslim

immigrant populations (Bakalian and Bozorgmehr, 2009). The post-9/11 terrorism policy

has radically reshaped immigration policy in the U.S., which “promotes the notion that

immigrants are suspects first and welcome newcomers second, if at all.” (Tumlin, 2004)

Anti-Islam and anti-immigration politicians also rise and gain important grounds in recent

elections in the U.S. and across Europe.

1.3 Minority Reaction and Differentiation

From the perspective of the already–disadvantaged minority, intensified biases, discrim-

ination and hostility from the majority, especially as they are legalized and established,

pose both symbolic threats to their beliefs and realistic threats to their well-being. For

example, after the terror attacks on Charlie Hebdo in Paris in January 2015, a poll for

BBC by ComRes showed that almost half of the British Muslims in the survey felt that

being a Muslim in Britain was difficult due to prejudice against Islam. About a quarter

said they had some sympathy for the motives behind the attacks, or disagreed that acts

of violence against those who publish images of the Prophet can never be justified. More-

over, 8% claimed to know Muslims who felt strongly sympathetic towards people fighting

for IS and Al-Qaeda.4

In response to rising out-group threats, the minority tends to promote in-group fa-

voritism to avoid betraying in-groups (Tajfel and Turner, 2004), maintain in-group soli-

darity (Bulbulia and Sosis, 2011), or protect in-group sacred values (Atran and Ginges,

2012). The in-group favoritism can be material interests, or moral support—providing

consolation, reassurance, and a sense of belonging and security, etc.

Group leaders and entrepreneurs often play an important role in influencing the minor-

ity’s behavior (Posner, 2004; Chandra and Wilkinson, 2008; Mele and Siegel, 2016, 2017),

such as preserving the identity or promoting in-group cohesion. In the 9/11 backlash, ad-

vocacy organizations representing Middle Eastern and Muslim Americans appealed their

4Source: http://www.comresglobal.com/polls/bbc-radio-4-today-muslim-poll/

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people to “claim their rights as Americans, to raise their voices, and to fight back against

hate crimes, bias incidents, prejudice and discrimination, and governmental abuses of

power” (Bakalian and Bozorgmehr, 2009).

After 9/11, a less-assimilated trend among Muslims in the U.S. are observed. Quan-

titive evidence shows that Muslim immigrants living in states which experienced the

sharpest increase in hate crimes after 9/11 also have greater chances of “intra-marriage”,

higher fertility, lower female labor force participation, and lower English proficiency

(Gould and Klor, 2016). Their interest in Quran and traditional Islamic practices re-

vived (Bryan, 2005; Abdo, 2006). Traditional women’s groups began to flourish, and

more young women began to wear the hijab (Haddad, 2007). Islamic day schools and

Sunday schools proliferated across the country (Schmidt, 2004). Their frequency of at-

tending mosque services also increased (Haddad, 2007). Mosques and Islamic community

centers have strengthened not only their religious role but also their social and politi-

cal function, such as becoming more engaged in every election since 2001 (Bakalian and

Bozorgmehr, 2009).5

1.4 Radicalization and Recruitment

Radicalization here refers to a process of gradually converting to the extremists’ violent

ideology espousing terrorism. Extant studies have explored the effect of various factors—

religious, socioeconomic, psychological, etc.—on radicalization, yet have no consensus on

why and how it happens. In several major explanations of radicalization (Moghaddam,

2005; Silber and Bhatt, 2007; Sageman, 2008), personal crisis relating to one’s identity

management is critical to the conversion (King and Taylor, 2011).

In social psychological research, these identity-related issues include discrimination,

lack of integration into mainstream society (Jenkins, 2007), and managing a dual identity

5Civic and religious organizations at time work to promote interfaith dialogue andcooperation, and are viewed as part of the solution to the problem of intergroup conflict.The point I emphasize here is that these organizations’ effort in a hostile environmentfacilitates the in-groups to seek refuge in their community and religion.

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(Stroink, 2007). In my model, the extremists’ atrocities exacerbate out-group discrimina-

tion against the minority, especially those who do not assimilate, which creates barriers

for them to integrate into the mainstream society. Moreover, minority group members

who find difficulties in managing both a Western identity and an inherited identity such

as being a Muslim, tend to have high consciousness regarding the inherited identity,

which makes them less likely to assimilate. Overall, minority group members who dif-

ferentiate rather than assimilate are more likely to experience the identity-related issues

contribute to radicalization. Therefore, I make the simple assumption that they have a

higher probability of being radicalized, compared with the in-groups who assimilate and

the out-groups, though this probability in absolute value is very small.

The extremists benefit from a larger pool of radicalized individuals that potentially

would get engaged in terrorism. In my model, under certain conditions, the extremists

can enlarge this pool through committing atrocities to induce more loyal minority group

members that do not assimilate. Terror activities, especially the specular ones, are costly.

Hence, the extremists are incentivized to apply the provocation strategy only if their gains

from recruitment outweigh the costs of implementing atrocities. This happens when they

are able to induce a sufficiently large increase in loyal in-group members.

The efficacy of the provocation strategy, for one thing, hinges on the majority group’s

policy towards the minority. In particular, if the majority not only discriminate but also

exclude the minority, in the sense of reducing intergroup interactions, then the extremists

can garner a larger increase of loyal in-groups. For another, the efficacy is also affected by

the minority’s attributes, including their consumption level and productivity, and their

extremism and heterogeneity regarding the identity consciousness. The results of these

effects are not always intuitive.

2 Model

Based on Fershtman, Gneezy and Hoffman (2011)’s framework about taboos and stable

identity, I first characterize the dynamics of assimilation in a world before the extremists’

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commit atrocities. Then I specify the extremists’ incentives regarding the provocation

strategy, and detail how it spurs discrimination from the majority and reaction from the

minority. I also analyze how the efficacy of provocation is affected by the majority’s policy

and the minority’s attributes.6

2.1 Pre-atrocities Assimilation

A society consists of two groups, a minority and a majority. Normalize the population

size of the society to be 1, and let the proportion of the minority be ρ ∈ (0, 12). Minority

group members, labeled by i, have the identity I. They are heterogenous regarding their

identity consciousness, denoted by φ, which is uniformly distributed over [0, 1].

In-group favoritism exists in both the minority and majority groups. Yet minority

group members can alleviate the majority’s out-group discrimination through assimila-

tion. Assume that it is observable to both the in-groups and outsiders whether i chooses

to assimilate (denoted by A) or differentiate (denoted by ∼ A).7

Let a minority group member i’s payoff of interacting with a majority group member

be b ∈ [0, 1]. In particular, i’s payoff is b when he assimilates, and b when he does not,

where b > b. By contrast, when i interacts with an in-group member, his payoff is ω ∈

[0, 1], with ω for assimilation and ω otherwise, where ω > ω. When i assimilates, he also

assumes some self-enforced cost, φσ, generated by betraying his identity I, where σ ∈ [0, 1]

represents the strength of the identity. Overall, depending on whether i assimilates or

not, his expected payoffs are

ui =

(1− ρ)b+ ρω − φσ if A

(1− ρ)b+ ρω if ∼ A

(1)

6 All details of the model that are not presented in the paper are in the appendix.7 In reality, assimilation probably is not an all or nothing option. An alternative is

to model assimilation as a continuous variable, and use a cut point to differentiate thein-groups who assimilate more and those who assimilate less. Since this will not causeany substantial changes in the results, I use the simpler way of a binary choice.

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The minority group member chooses to assimilate if it yields a higher payoff, which means

∆ui = (1− ρ)(b− b)− ρ(ω − ω)− φσ > 0 (2)

For simplicity, let ∆b = b − b, which represents i’s gains from alleviating out-group

discrimination by assimilation; and ∆ω = ω−ω, which represents his gains from in-group

favoritism due to being loyal to his identity.

I consider two factors that affect the value of ∆b: the minority group’s consumption

level of scarce resources and its productivity. Let e ∈ [0, 1] measures the minority’s average

consumption of resources, and π ∈ [0, 1] be their average productivity. Assume that an in-

group gains more from alleviating discrimination when the minority’s consumption level

is higher (∂∆b∂e

> 0), and less when the minority produce more (∂∆b∂π

< 0).

Let k = (e− π)ρ denote the total difference between the minority’s consumption and

productivity. If e − π > 0, then I call the minority group “unproductive,” in the sense

that its consumption exceeds the productivity; otherwise, it is “productive.” Intuitively,

out-group discrimination rises if the minority are less productive, no matter the in-groups

assimilate or not (i.e. ∂b∂k< 0, ∂b

∂k< 0). Assume that the situation is comparatively worse

for the in-groups who do not assimilate (i.e. ∂∆b∂k

> 0). Consequently, when the minority

group expands, assimilation is more beneficial if the minority are unproductive (∂∆b∂ρ

> 0

if k > 0), but less so when they are productive (∂∆b∂ρ

< 0 if k < 0).

The minority group member who is indifferent between assimilating or not is defined

as

φo ≡ min{max{0,

g(·)︷ ︸︸ ︷(1− ρ)∆b− ρ∆ω

σ}, 1} (3)

Every minority group member with φ > φo differentiates, while those with φ < φo assimi-

lates. g(·) represents the difference between i’s gains from assimilation and differentiation.

If g(·) ≤ 0, which means the gains from assimilation do not exceed those from differen-

tiation, then φo = 0. Namely, every i with identity consciousness higher than zero does

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not assimilate. Otherwise, if g(·) > 0, then φo ∈ (0, 1], and at least some minority

group members assimilate for the higher benefits of alleviating out-group discrimination,

at the expense of acquiring more in-group favoritism. More minority members differen-

tiate when the gains from in-group favoritism is higher ( ∂φo

∂∆ω< 0), the strength of the

identity is stronger (∂φo

∂σ< 0), and the size of the minority group is larger (∂φ

o

∂ρ< 0). More

minority members assimilate when the gains from alleviating out-group discrimination is

higher ( ∂φo

∂∆b> 0).

The proportion of minority group members who differentiate is 1− φo. Assume that

the strength of the identity increases in the scale of loyal in-groups,

σ = α(1− φo) (4)

where α ∈ [0, 1] captures factors other than (1 − φo) that affect σ. As a result, the

strength of the identity and the proportion of minority group members who assimilate

are endogenous.

In Figure 1, the horizontal axis measures the indifferent in-group’s identity conscious-

ness φo, and the vertical axis measures the strength of the identity σ. The curve represents

condition (3), which captures the proportion of minority group members who assimilate

given the strength of the identity σ; and the line represents condition (4), which describes

the strength of the identity determined by the proportion of minority group members who

assimilate.

To solve for φo∗, plug condition (4) into (3), which gives

φo∗ =α±

√α2 − 4αg(·)2α

(5)

When α2 − 4αg(·) < 0, we are in Case 1 of Figure 1. There are no intersections between

the curve and the line. In equilibrium, φo∗ = 1, located at point C in the curve; σ∗ = 0,

located at point D in the line. The implication is that even for the minority group

member of the highest identity consciousness, when he differentiates, the losses from out-

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φo φo*σ*

α

σ

C

D

(a) Case 1

φ1o*

φo φ2o*

σ2*

σ1*

α

σ

A

B C

D

(b) Case 2

Figure 1: Stable and unstable pool of in-group members

group discrimination exceed the gains from in-group favoritism and his psychological cost.

Therefore, all minority group members assimilate, and the identity has zero strength in

this situation.

If α2 − 4αg(·) > 0 and g(·) > 0,8 we are in Case 2, where there are two intersections

between condition (3) and (4).9 Point A represents a stable couple (φo∗1 , σ∗1), while point

B represents an unstable one. The dashed lines with arrows illustrate the adjustment

process, as a result of which, every φo and σ to the left of point B converge to (φo∗1 , σ∗1)

located at point A, while every φo and σ to the right of point B end up at φo∗2 = 1 located

at point C and σ∗2 = 0 located at point D.

From now on, I focus on the stable equilibrium at point A, where φo∗ =α−√α2−4αg(·)2α

.

As g(·) increases, which means the benefit from assimilation exceeds that from differ-

entiation to a greater extent, more minority group members assimilate (∂φo∗

∂g(·) > 0), and

8 As analyzed above, if g(·) ≤ 0, all minority in-groups differentiate, and the extremistshave no incentive to apply the provocation strategy.

9 When α2 − 4αg(·) = 0, there is only one intersection between (3) and (4), and thecurve and the line are tangent to each other. Analysis of this special case does not yieldmore insights.

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the strength of the identity diminishes ( ∂σ∂φo∗

< 0). This process is not monotonic—once

α2 − 4αg(·) < 0, then we are in Case 1.

2.2 Atrocities and Recruitment

The extremists, labeled by r, choose to commit (denoted by T ) costly atrocities in the

name of the minority identity to provoke polarization and radicalization in society, or not

(denoted by ∼ T ). Assume that the minority group members who do not assimilate tend

to encounter more negative identity-related experience, and have probability θ1 of being

radicalized. By contrast, this probability of the minority group members who differentiate

and the majority is θ2, where θ2 < θ1.

The extremists’ payoffs from recruitment, depending on whether they commit atroc-

ities or not, are given by

ur =

(1− φo)ρθ1 + [1− (1− φo)ρ]θ2 − cηv if T

(1− φo)ρθ1 + [1− (1− φo)ρ]θ2 if ∼ T

(6)

where φo represents the minority group member who is indifferent between assimilating

or not after atrocities are committed, and c, η and v denote the unit cost, salience and

amount of atrocities, respectively. The extremists benefit from a larger pool of loyal

in-groups, before or after atrocities are committed (∂ur∂φo

< 0; ∂ur∂φo

< 0).

The extremists only commit atrocities if it yields a higher payoff, which means

∆ur = ρ(θ1 − θ2)(φo − φo)− cηv > 0 (7)

Namely, the provocation strategy is profitable to the extremists only when it manages to

induce a sufficient proportion of minority group members to stop assimilating, i.e.

∆φo = φo − φo > ∆φo ≡ cηv

ρ(θ1 − θ2)(8)

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Δφo

φ~o φo

α

σ

Figure 2: Shift in loyal minority in-groups caused by provocation

Figure 2 depicts the direction of the shift that the extremists aim for, where φo > φo.

This condition equals to (1−φo) < (1− φo), which means a larger proportion of minority

group members differentiate after atrocities in the name of their identity are committed.

∆φo is the threshold of shift that incentivizes the extremists to engage in atrocities for

recruitment.

2.3 Discrimination, Antagonism, and Assimilation

For simplicity, I assume that after the extremists committed atrocities, the majority’s

biases and discrimination against the minority group members who differentiate are ag-

gravated, but their attitude towards thoese who differentiate do not change.10

As in the world before atrocities are committed, assume that the aggravation of dis-

crimination increases in the minority group’s consumption of scarce resources and de-

creases in its productivity. In addition, assume that the aggravation increases in the

10 If discrimination against the minority who assimilate is also aggravated but to a lessextent, all implications of the model and all propositions still hold, though the magnitudeof the results changes somewhat.

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magnitude of the extremists’ atrocities, which is determined by both the salience η and

the quantity v of violence.

Backlash from the majority group provokes reactions from the minority, which leads to

a higher level of in-group favoritism towards the members who differentiate and stay loyal

to the identity.11 Let λ ≥ 0 be the degree of the minority’s reactions to the aggravated

out-group discrimination. The minority group member i’s post-atrocities payoffs are given

by

ui =

(1− ρ)b+ ρω − φσ if A

(1− ρ)[b− γ1(e− π)ρ− γ2ηv] + ρ[ω + λ(γ1(e− π)ρ+ γ2ηv)] if ∼ A

(9)

where γ1 and γ2 are the majority’s weights of the problems of resource scarcity and

atrocities, respectively.

A minority group member differentiates if the second line of equation (9) is larger

than the first line, which means

∆ui = (1− ρ)∆b− ρ∆ω + (1− ρ)[γ1(e− π)ρ+ γ2ηv]− ρλ[γ1(e− π)ρ+ γ2ηv]− φσ > 0 (10)

The in-group member who is indifferent between assimilating or not is defined as

φo ≡ min{max{0,

everyday difference, g(·)︷ ︸︸ ︷(1− ρ)∆b− ρ∆ω +

difference caused by atrocities, h(·)︷ ︸︸ ︷{(1− ρ)[γ1(e− π)ρ+ γ2ηv]− ρλ[γ1(e− π)ρ+ γ2ηv]}

σ}, 1}

(11)

where g(·) captures the gains from assimilation due to the existing everyday out-group

discrimination, and h(·) captures the difference in payoffs caused by the extremists’ atroci-

ties. Based on the previous result, ∂φo∗

∂g(·) > 0 when g(·) > 0, it can be inferred that φo < φo

only if h(·) < 0. Namely, provocation only works if, for the difference in i’s payoffs

caused by atrocities, the gains from differentiation exceeds those from assimilation. For

11 Again, if the post-atrocities in-group favoritism is overall higher but especially so forthe loyal minority group members, all the results still hold.

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convenience, let l(·) = −h(·), then

φo∗ =α−

√α2 − 4αg(·) + 4αl(·)

2α(12)

According to the previous analyses, the extremists use the provocation strategy for

recruitment if it induces a sufficient amount of minority group members to stop assimi-

lating

∆φo = φo∗ − φo∗ > ∆φo (13)

Condition (13) equals to

λ > λ ≡ 1− ρρ

+cηv

ρ2(θ1 − θ2)[γ1(e− π)ρ+ γ2ηv][

αcηv

ρ(θ1 − θ2)+√α2 − 4αg(·)] (14)

with the assumption that γ1(e − π)ρ + γ2ηv > 0.12 It implies that the extremists only

commit atrocities when the minority group’s reaction to the majority’s backlash reaches

a certain threshold λ. Intuitively, if the extremists’ atrocities only aggravate out-group

biases against the minority who differentiate, more minority group members would as-

similate to mitigate the discrimination. Yet, if the minority group, under the influence

of its leaders and advocacy organizations, provides more in-group favoritism to the loyal

members to counteract the backlash, the trend of assimilation can be brought to a halt

or even reversed.

Proposition 1 The extremists only apply the provocation strategy if the minority group’s

reaction to the majority’s backlash reaches a certain threshold λ.

Next, consider the effect of the minority’s group size on the decrease in assimilation

12 The substantial meaning of this assumption is that atrocities aggravate the majority’sdiscrimination against the minority group members who do not assimilate. This assump-tion does not hold if π−e > γ2ηv

γ1ρ, namely the minority’s productivity highly exceeds their

consumption of resources, which totally compensates for the negative effect of atrocities.For the interest of this model, I consider the situation where the minority’s productivitydoes not reach this high level.

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induced by the atrocities.

∂∆φo

∂ρ=

∂φo

∂ρ− ∂φo

∂ρ

=∂g(·)∂ρ

1√α2 − 4αg(·)

− [∂g(·)∂ρ− ∂l(·)

∂ρ]

1√α2 − 4αg(·) + 4αl(·)

(15)

In particular,

∂g(·)∂ρ

= −∆b−∆ω + (1− ρ)∂∆b

∂ρ(16)

When the minority group is unproductive (i.e. k > 0, ∂∆b∂ρ

> 0), we have ∂g(·)∂ρ≥ 0 if

∂∆b∂ρ≥ ∆b+∆ω

1−ρ . In this case, the everyday discrimination against the minority, especially

the loyal in-groups, is very severe due to their high consumption. As the minority group

expands and consumes more, it becomes more beneficial for i to assimilate to alleviate

this severe discrimination, despite the higher chance of interacting with an in-group.

The pre-atrocities scale of loyal in-groups thus is smaller when the minority group is

larger.13 On the other hand, as the minority group grows, the atrocities-induced in-group

favoritism increases (∂l(·)∂ρ

> 0), and consequently the post-atrocities scale of loyal in-groups

is enlarged. Overall, the provocation strategy is more effective for a larger minority group

(∂∆φo

∂ρ> 0).

If ∂∆b∂ρ

< ∆b+∆ω1−ρ , then the majority’s everyday discrimination against the minority is

less severe, and the sign of ∂∆φo

∂ρis indeterminate.14 In this case, i’s gains from assimilation

decrease as the minority group grows, since everyday discrimination is not severe while

the chance of interacting with an in-group goes up. As a result, both the pre- and post-

atrocities scales of loyal in-groups are enlarged as the minority group expands (∂φo

∂ρ< 0,

∂φo

∂ρ< 0). Yet, provocation is more effective when the increase of the scale of loyal in-

groups is bigger, which is more likely to occur if the pre-atrocities scale (i.e. 1 − φo)

13 It is worthy noting that everyday discrimination is also severe after atrocities arecommitted, which to some extent shrinks the post-atrocities pool of loyal in-groups. Butthe effect of everyday discrimination is stronger before than after atrocities are committed.

14Specifically, ∂g(·)∂ρ

< 0 and ∂l(·)∂ρ

> 0, therefore ∂g(·)∂ρ− ∂l(·)

∂ρ< 0.

17

is smaller, or the post-atrocities scale (i.e. 1 − φo) is larger. Depending on the values

of the parameters, the extremists do not necessarily prefer a smaller or larger minority

group, since the former enlarges the pre-atrocities scale less while the latter enlarges the

post-atrocities scale more, and both may contribute to a bigger increase of loyal in-groups.

Similarly, if the minority is productive (i.e. k < 0, ∂∆b∂ρ

< 0), then we also have

∂g(·)∂ρ

< 0. In this scenario, the relation between the minority’s group size and the effi-

cacy of the provocation strategy is also indeterminate. Overall, only when the minority

are unproductive and the everyday discrimination against them is severe, provocation

determinately becomes more effective as the minority group expands.

Related to this, the minority’s group size has indeterminate effect on the extremists’

payoff.

∂∆ur∂ρ

= (θ1 − θ2)∆φo + ρ(θ1 − θ2)∂∆φo

∂ρ(17)

Since (θ1−θ2)∆φo > 0, when the efficacy of provocation increases in the minority’s group

size (∂∆φo

∂ρ> 0), the extremists’ payoff also rises for a larger minority (∂∆ur

∂ρ> 0). Yet,

it is possible to have ∂∆ur∂ρ

> 0 even when ∂∆φo

∂ρ< 0. That is, the extremists may still

benefit from a larger minority group even when it leads to a smaller increase in the scale

of loyal in-groups. The reason lies in that when a larger minority is weighted by a smaller

increase, it may generate a bigger pool of loyal in-groups, compared with when a smaller

minority is weighted by a larger increase.

Proposition 2 1. If the minority are unproductive and the everyday discrimination

against them is severe, provocation determinately becomes more effective as the mi-

nority group expands. Otherwise, the minority’s group size has indeterminate effect

on the efficacy of provocation and on the extremists’ payoff.

2. The extremist group may obtain a higher payoff from a larger minority group even

when it leads to a smaller increase in the scale of loyal in-groups.

When the minority group’s consumption level is higher, it is easier to induce a larger

18

increase of loyal in-groups by provocation, which yields the extremists a higher payoff

(∂∆φo

∂e> 0; ∂∆ur

∂e> 0). If the minority is more productive, then the effect is the oppo-

site (∂∆φo

∂π< 0; ∂∆ur

∂π< 0). The salience and the quantity of atrocities both boost the

extremists’ recruitment and payoff (∂∆φo

∂η> 0, ∂∆φo

∂v> 0; ∂∆ur

∂η> 0, ∂∆ur

∂v> 0).

Proposition 3 The efficacy of provocation and the extremists’ payoff increase in the mi-

nority group’s consumption level, the salience and the quantity of atrocities, and decrease

in the minority group’s productivity.

2.4 Exclusion and Assimilation

In this section, I explore the effect of another policy of the majority towards the minority—

they not only discriminate but also to some extent exclude the minority, for example, in the

form of occupational exclusions, residential and school segregation, and social ostracism.

Denote β ∈ (0, 1] as the probability of a minority member interacting with an in-group.

Assume that β > ρ, which means i’s chance of in-group interaction is higher than the

proportion of the minority in the population. A larger value of β indicates a higher degree

of exclusion.

The minority group member’s post-atrocities payoffs, depending on whether he as-

similates or not, are

ui =

(1− β)b+ βω − φσ if A

(1− β)[b− γ1(e− π)ρ− γ2ηv] + β[ω + λ(γ1(e− π)ρ+ γ2ηv)] if ∼ A

(18)

Note that provocation affects all minority members’ payoffs by reducing their chances of

out-group interactions. In addition, for the loyal in-groups, out-group discrimination is

aggravated.

When the second line of equation (18) equals the first line, a post-atrocities minority

19

group member is indifferent between assimilating or not, who is defined as

φo ≡ min{max{0,

m(·)︷ ︸︸ ︷(1− β)∆b− β∆ω−

n(·)︷ ︸︸ ︷{βλ[γ1(e− π)ρ+ γ2ηv]− (1− β)[γ1(e− π)ρ+ γ2ηv]}

σ}, 1}

(19)

where m(·) represents the post-atrocities gains from assimilation due to the existing ev-

eryday out-group discrimination, and n(·) represents the post-atrocities gains from differ-

entiation caused by atrocities.

The extremists apply the provocation strategy if the increase of loyal in-groups is

sufficiently large, that is, ∆φo = φo∗− φo∗ > ∆φo. To fulfill this condition, the minority’s

reaction to the aggravated out-group discrimination needs to reach a certain threshold,

denoted as λ. Based on the assumption β > ρ, λ > λ. The threshold is lower and

easier to satisfy when the majority both exclude and discriminate against the minority,

compared with when they only discriminate. Moreover, as the degree of exclusion rises,

the extremists are able to induce a larger increase in loyal in-groups and obtain a higher

payoff ( ∂∆φo

∂∆β> 0; ∂∆ur

∂∆β> 0).

Proposition 4 The threshold of the minority group’s reaction that incentivizes the ex-

tremists to adopt the provocation strategy is lower when the majority both exclude and

discriminate the minority, compared with when they only discriminate. As the degree of

exclusion rises, the extremists are able to induce a larger increase of loyal in-groups and

obtain a higher payoff.

Next, consider the effect of the minority’s group size on the efficacy of provocation. If

the minority group is productive (i.e. k < 0, ∂∆b∂ρ

< 0), then provocation is less effective

as the group expands. The larger the minority, the more they produce and mitigate both

everyday and atrocities-induced discrimination, and the higher chance they interact with

in-groups. Hence, i’s gains from assimilation decrease (∂g(·)∂ρ

< 0, ∂m(·)∂ρ

< 0), and the pre-

atrocities scale of loyal in-group is enlarged. Moreover, i’s gains from atrocities-induced

in-group favoritism also decreases (∂n(·)∂ρ

< 0), since less severe out-group discrimination

20

generates lower in-group favoritism as reaction from the minority. The post-atrocities

scale of loyal in-groups accordingly shrinks. Overall, for a larger minority group, the

increase in the scale of loyal in-groups due to provocation is smaller (∂∆φo

∂ρ< 0), and the

extremists’ payoff from recruitment decreases (∂∆ur∂ρ

< 0). If the minority is unproductive

(i.e. k > 0, ∂∆b∂ρ

> 0), the relation between the minority’s group size and the efficacy of

provocation is indeterminate.

Interestingly, when the minority are productive, as the group expands, the decrease

in provocation efficacy is lager when they are excluded, compared with when they are

only discriminated (∂∆φo

∂ρ> ∂∆φo

∂ρ, if k < 0).15 Namely, a lower chance of post-atrocities

out-group interaction decreases the efficacy of provocation as the productive minority

expand.

The reason of this counter-intuitive result is twofold. The productive monority allevi-

ates out-group discrimination and reduces the gains from assimilation as it grows. With

exclusion, first, i’s chance of interacting with an out-group and incurring the decrease

in gains is lower, which makes the expected losses from assimilation lower. And second,

the post-atrocities in-group favoritism as reaction to the reduced discrimination also de-

creases. i’s higher chance of in-group interaction then makes the expected losses from

differentiation larger, compared with when he is only discriminated. Both effects tend to

make provocation less effective for a larger minority group that is excluded.

Proposition 5 1. With exclusion, if the minority group is productive, provocation is

less effective as the group expands; otherwise, the relation is indeterminate.

2. As a productive minority group expands, the decrease in provocation efficacy is lager

when the minority are excluded and discriminated, compared with when they are

only discriminated.

As in the scenario where the majority only discriminate, the minority group’s pro-

ductivity diminishes the increase of loyal in-group members and the extremists’ payoff of

15 Since the sign of ∂∆φo

∂ρis indeterminate, it is possible that the provocation efficacy

actually increases when they are only discriminated.

21

recruitment (∂∆φo

∂π< 0, ∂∆ur

∂π< 0), while the minority group’s consumption level, and the

salience and quantity of atrocities enlarge both ( ∂∆φo

∂e> 0, ∂∆ur

∂e> 0; ∂∆φo

∂η> 0, ∂∆ur

∂η> 0;

∂∆φo

∂v> 0, ∂∆ur

∂v> 0).

2.5 Extremism, Heterogeneity, and Assimilation

This section explores how the minority’s average level of extremism and heterogeneity of

their identity consciousness affect the efficacy of provocation and the extremists’ payoffs.

Denote the minority group’s mean of identity consciousness by x ∈ [0, 1], where a higher

value of x indicates a more extreme minority group on average. Let φ ∈ [x − µ, x + µ],

assuming that x − µ ≥ 0 and x + µ ≤ 1, where µ ∈ [0, 12] measures the minority’s het-

erogeneity of identity consciousness.16 A higher value of µ indicates a more heterogenous

minority group.

Consider the scenario where the majority only discriminate but do not exclude the

minority. The pre-atrocities scale of loyal in-groups is x+µ−φo2µ

, and the strength of the

identity is given by σ = α(x+µ−φo)2µ

. Accordingly, the pre- and post-atrocities minority

group members who are indifferent between assimilating or not, φo and φo, can be defined.

Following the process in the previous sections, solve for the threshold of the minority’s

reaction that incentives the extremists to commit atrocities, denoted as λ. Surprisingly,

when the minority is on average more extreme or more homogenous, λ becomes higher and

more difficult to satisfy (∂λ∂x> 0, ∂λ

∂µ< 0). Intuitively, when µ is fixed and x increases, every

minority in-group has higher identity consciousness, and is more likely to differentiate even

before atrocities are committed. To make the high-consciousness member not differentiate,

assimilation needs to be more rewarding, which implies that the minority’s post-atrocities

in-group favoritism also needs to reach a higher level to counteract the temptation. By

contrast, when x is fixed and µ increases, the minority simultaneously has more high- and

low-consciousness members. The members of low identity consciousness tend to assimilate

even if it is not particularly profitable. Therefore, relatively low post-atrocities in-group

16 Note that when x is of extreme values closed to 0 or 1, the range of φ can be small.This does not affect the results, since I look at the marginal effects of x and µ.

22

favoritism would be sufficient to induce them to stop assimilating.

From another perspective, the efficacy of the extremists’ provocation strategy de-

creases in the minority’s extremism.

∂∆φo

∂x= −α(x+ µ)

2[

1√α2(x+ µ)2 − 8αµg(·)

− 1√α2(x+ µ)2 − 8αµg(·) + 8αµl(·)

] < 0 (20)

As analyzed above, a higher x means the minority has a larger proportion of high-

consciousness members, which generates more loyal in-groups in response to everyday

discrimination and provocation. Equation (20) shows that this effect is stronger before

than after atrocities are committed, which hinges on the condition that i’s post-atrocities

gains from assimilation decreases due to the minority’s reaction.

Figure 3: Effect of extremism on increase of loyal in-groups

φ~Ho

φ~Lo

Post-atrocities effect of x

φHo φL

o

Pre-atrocities effect of x

ΔφHo ΔφL

o

Figure 3 illustrates this finding. φoL and φoH represent the pre-atrocities minority

group member who is indifferent between assimilating or not when the minority’s average

level of extremism x is low and high, respectively. The effect of an increase in x on the

efficacy of provocation is (φoL − φoH). Similarly, after atrocities are committed, the effect

of x is (φoL − φoH). Since the effect is stronger before atrocities are committed, that is

φoL − φoH > φoL − φoH , we have φoL − φoL > φoH − φoH . Therefore, ∆φoL > ∆φoH , which means

23

that the efficacy of provocation is lower when the minority group is of higher extremism

(∂∆φo

∂∆x< 0).

This finding is also counter-intuitive. The lesson is that attributes of the minor-

ity, which are traditionally viewed as favorable for the extremists to garner more loyal

in-groups, do not necessarily incentive them to commit atrocities. After all, it is the in-

crease of loyal in-groups that matters. If the minority’s higher level of extremism already

generates a good increase of loyal in-groups before atrocities are committed, there may

not be much room to induce more by provocation.

At last, whether the minority’s heterogeneity mitigates the efficacy of atrocities in

recruitment is indeterminate. As the minority become more heterogenous, the proportions

of in-groups of higher and lower identity consciousness both increase. For the increase of

high-consciousness members, the effect on the efficacy of provocation is similar to what

is analyzed above about the effect of the minority’s extremism, represented in equation

(20). While for the increase of lower-consciousness members, the effect is in the opposition

direction. Depending on the values of the parameters, the overall effect, namely the sign

of ∂∆φo

∂µ, can be positive, zero or negative.

Proposition 6 1. The threshold of the minority’s reaction that motivates the extrem-

ists to conduct atrocities is higher as the minority becomes more extreme on average,

and is lower as the minority becomes more heterogenous on average.

2. The efficacy of provocation decreases in the minority’s extremism, and has an inde-

terminate relation with the minority’s heterogeneity.

3 Conclusion

As the Islamic extremist groups committed series of atrocities in the past decades, polar-

ization along ethnic and religious cleavages has exacerbated in the Western societies. In

the vicious circle, the extremists use terror attacks to intimidate and infuriate the public

and the government and spur Muslim backlash. Intergroup antagonism leads to a more

24

cohesive Muslim community, which potentially generates more radicalized individuals and

feeds the extremist group’s need of recruitment. There is a substantial body of anecdotes,

observations, and empirical evidence which suggest that the audience’s perception and

Muslims’ consciousness of their identity are critical for the success of the extremists’

provocation strategy.

In this paper, I specify and formalize the elements and links in this process, and

identify the conditions under which a vicious circle can emerge. I make several critical

assumptions in the model. First, a minority identity, such as being a Muslim, matters to

an in-group member’s utility, and therefore assimilation is psychologically costly. Second,

in-group favoritism and out-group discrimination are universal. A minority group mem-

ber chooses to assimilate or not partially based on her gains from alleviating out-group

discrimination versus acquiring more in-group rewards. Third, intergroup biases and dis-

crimination are subtle and “cool” in everyday life, but they can become explicit and “hot”

due to sentiments like fear and hatred provoked by atrocities. Fourth, in response to rising

out-group discrimination and threats, a minority group provides more in-group favoritism

to preserve its identity and sacred values.

Analyses of the model reveal that, for one thing, the minority group’s reaction to a

hostile environment is an integral part of the vicious circle. Unless the reaction reaches a

certain level, the extremists cannot make a profit out of the provocation. The threshold

is lower when the majority does not only discriminate but also exclude the minority.

This finding highlights the importance of mitigating the impact of atrocities in both the

majority and minority communities, and maintaining smooth inter-group communication

and interaction, since it is the intergroup action-reaction that contributes to polarization.

Surprisingly, when the minority group is on average more extreme in their identity

consciousness, the threshold decreases, but when they are more heterogenous, it increases.

The reason lies in that, to the extremists, it is the increase of minority group members

who differentiate that matters. If the minority’s higher level of extremism or homogeneity

already generates a large increase of loyal in-groups before the extremists take action,

there is not much space to induce more by provocation. Therefore, some attributes

25

of the minority group that are traditionally considered as nutrition for terrorism may

not motivate the extremists to apply the provocation strategy, though they are indeed

dangerous from a pre-atrocities perspective.

If the minority is highly productive, to the extend that it exceeds the minority’s

consumption and counteracts the negative influence of provocation, the extremists have

no incentive to commit atrocities for recruitment. Yet, since the minority oftentimes

are of relatively low socioeconomic status, their productivity probably would not reach

such a high level. Still, the efficacy of provocation decreases in the productivity of the

minority, and increases in their consumption level. In addition, a larger minority group

boosts the efficacy of provocation, if the the in-groups are unproductive and the everyday

discrimination against them is severe. Yet, most often the relation between the minority

group size and the effect of provocation is indeterminate.

This model is an attempt to diagnose the polarization problem before seeking for

a prescription for restoring social solidarity. The findings indicate that efforts should

be made to promote intergroup interaction and cooperation, which helps to mitigate

antagonistic action-reaction in the face of violent atrocities. A secured environment to

both the majority and the minority is helpful to reduce the perception of out-group threats

that trigger in-group cohesion to defend an identity. More fundamentally, a deeper and

better understanding of the role of identity in intergroup relation is needed, since a more

compatible society that is able to accommodate multiple identities is essential to resist

the extremists’ provocation.

26

Appendices

Appendix .A Discrimination, Antagonism, and Recruitment

.A.1 Threshold of reaction of the minority group

In equilibrium, the pre-atrocities minority group member who is indifferent between as-

similating or not is

φo∗ =α−

√α2 − 4αg(·)2α

(21)

while the post-atrocities indifferent minority group member is

φo∗ =α−

√α2 − 4αg(·) + 4αl(·)

2α(22)

The extremists commit atrocities if

∆φo = φo∗ − φo∗ > ∆φo (23)

which equals to √α2 − 4αg(·) + 4αl(·)−

√α2 − 4αg(·)

2α>

cηv

ρ(θ1 − θ2)(24)

Plug in the parameters in g(·) and l(·), then the condition equals to

λ > λ ≡ 1− ρρ

+cηv

ρ2(θ1 − θ2)[γ1(e− π)ρ+ γ2ηv][

αcηv

ρ(θ1 − θ2)+√α2 − 4αg(·)] (25)

when γ1(e− π)ρ+ γ2ηv > 0.

27

.A.2 Effect of minority group size on scale of change in loyal minority group

members due to atrocities

∂∆φo

∂ρ=

∂φo

∂ρ− ∂φo

∂ρ

=∂g(·)∂ρ

1√α2 − 4αg(·)

− [∂g(·)∂ρ− ∂l(·)

∂ρ]

1√α2 − 4αg(·) + 4αl(·)

(26)

In particular,

∂g(·)∂ρ

= −∆b−∆ω + (1− ρ)∂∆b

∂ρ(27)

and

∂l(·)∂ρ

= (1 + λ)[2ργ1(e− π) + γ2ηv]− γ1(e− π) (28)

Since λ > 1−ρρ ,

∂l(·)∂ρ

> γ1(e− π) + (1 + λ)γ2ηv > γ1(e− π) +γ2ηv

ρ(29)

Given the assumption ργ1(e−π)+γ2ηv > 0, we have γ1(e−π)+ γ2ηvρ > 0, which means ∂l(·)

∂ρ > 0.

Appendix .B Exclusion and Assimilation

.B.1 Threshold of reaction of the minority group, with exclusion

The minority group member assimilates if

∆ui = (1− β)∆b− β∆ω + (1− ρ)[γ1(e− π)ρ+ γ2ηv]− ρλ[γ1(e− π)ρ+ γ2ηv]− φσ > 0 (30)

28

Accordingly, the minority group member who is indifferent between assimilating or not is

defined as

φo ≡ min{max{0,

m(·)︷ ︸︸ ︷(1− β)∆b− β∆ω−

n(·)︷ ︸︸ ︷{βλ[γ1(e− π)ρ+ γ2ηv]− (1− β)[γ1(e− π)ρ+ γ2ηv]}

σ}, 1}

(31)

where m(·) represents the post-atrocities gains from assimilation due to the existing ev-

eryday out-group discrimination, and n(·) represents the post-atrocities gains from differ-

entiation caused by atrocities.

The pre-atrocities minority group member who is indifferent between assimilating or

not is

φo∗ =α−

√α2 − 4αg(·)2α

(32)

and the post-atrocities indifferent minority group member under exclusion is

φo∗ =α−

√α2 − 4αm(·) + 4αn(·)

2α(33)

The extremists use violent identity recruitment only if The extremists commit atrocities

for recruitment only if

∆φo = φo∗ − φo∗ > ∆φo (34)

which equals to √α2 − 4αm(·) + 4αn(·)−

√α2 − 4αg(·)

2α>

cηv

ρ(θ1 − θ2)(35)

29

Plug in the parameters in m(·) and n(·), then the condition equals to

λ > λ ≡ 1− ββ

+cηv

ρ2(θ1 − θ2)[γ1(e− π)ρ+ γ2ηv][

αcηv

ρ(θ1 − θ2)+√α2 − 4αg(·)]− 4α(β − ρ)(∆b+ ∆ω)

γ1(e− π)ρ+ γ2ηv

(36)

Given that β > ρ,

λ− λ =1− ρρ− 1− β

β+

4α(β − ρ)(∆b+ ∆ω)

γ1(e− π)ρ+ γ2ηv> 0 (37)

.B.2 Effect of exclusion on provocation efficacy and extremists’ payoff

∂∆φo

∂β=

∆b+ ∆ω + (λ+ 1)[γ1(e− π)ρ+ γ2ηv]√α2 − 4αm(·) + 4αn(·)

> 0 (38)

and

∂∆ur∂β

= ρ(θ1 − θ2)∂∆φo

∂β> 0 (39)

.B.3 Effect of minority group size on provocation efficacy and extremists’

payoff

∂∆φo

∂ρ=

∂φo

∂ρ− ∂φo

∂ρ

=∂g(·)∂ρ

1√α2 − 4αm(·)

− [∂m(·)∂ρ

− ∂n(·)∂ρ

]1√

α2 − 4αm(·) + 4αn(·)(40)

where

∂g(·)∂ρ

= −∆b−∆ω + (1− ρ)∂∆b

∂ρ(41)

∂m(·)∂ρ

= (1− β)∂∆b

∂ρ(42)

30

and

∂n(·)∂ρ

= [β(1 + λ)− 1]γ1(e− π) (43)

If the minority group is productive (i.e. k < 0, ∂∆b∂ρ

< 0), then provocation is less

effective as the group expands. In this case, ∂g(·)∂ρ

< 0 and ∂m(·)∂ρ

< 0. Since λ > 1−ββ

,

if k < 0, which means e − π < 0, then ∂n(·)∂ρ

< 0. Moreover, since ∂g(·)∂ρ

< ∂m(·)∂ρ

, overall

∂∆φo

∂ρ< 0.

If the minority is unproductive (i.e. k > 0, ∂∆b∂ρ

> 0), the relation between the

minority’s group size and the efficacy of provocation is indeterminate. In this case, we

have ∂m(·)∂ρ

> 0 and ∂n(·)∂ρ

> 0. No matter ∂g(·)∂ρ

is bigger or smaller than zero, the sign of

∂∆φo

∂ρdepends on the values of the parameters.

The extremists’ payoff from recruitment decreases in the minority’s group size,

∂∆ur∂ρ

= −(1 + λ)[γ1(e− π)ρ+ γ2ηv] < 0 (44)

.B.4 Difference in effect of productive minority group size on provocation

efficacy, with and without exclusion

When the minority are productive, as the minority group expands, the efficacy of provo-

cation is lower when they are excluded, compared with when they are only discriminated.

∂∆φo

∂ρ− ∂∆φo

∂ρ=

∂g(·)∂ρ

1√α2 − 4αg(·)

− [∂g(·)∂ρ− ∂l(·)

∂ρ]

1√α2 − 4αg(·) + 4αl(·)

− ∂g(·)∂ρ

1√α2 − 4αg(·)

+ [∂m(·)∂ρ

− ∂n(·)∂ρ

]1√

α2 − 4αm(·) + 4αn(·)(45)

Since g(·) > m(·) and l(·) > n(·), then 1√α2−4αg(·)+4αl(·)

> 1√α2−4αm(·)+4αn(·)

. From the

analyses above, when k < 0, ∂g(·)∂ρ

< 0, ∂m(·)∂ρ

< 0, and ∂g(·)∂ρ

< ∂m(·)∂ρ

; ∂l(·)∂ρ

> 0, and ∂n(·)∂ρ

< 0.

Given all these conditions, there is ∂∆φo

∂ρ> ∂∆φo

∂ρ.

31

Appendix .C Heterogeneity, Extremism, and Recruitment

.C.1 Threshold of reaction of the minority group

The pre-atrocities scale of loyal in-groups is x+µ−φo2µ

, and the strength of the identity is

given by

σ =α(x+ µ− φo)

2µ(46)

Accordingly, the pre-atrocities minority group member who is indifferent between assim-

ilating or not is defined by

φo ≡ min{max{x− µ, 2µg(·)α(x+ µ− φo)

}, x+ µ} (47)

while the post-atrocities indifferent in-group member is

φo ≡ min{max{x− µ, 2µ[g(·)− l(·)]α(x+ µ− φo)

}, x+ µ} (48)

solve for

φo∗ =α(x+ µ)−

√α2(x+ µ)2 − 8αµg(·)

2α(49)

and

φo∗ =α(x+ µ)−

√α2(x+ µ)2 − 8αµg(·) + 8αµl(·)

2α(50)

The extremists use atrocities to promote recruitment if

∆φo = φo∗ − φo∗ > ∆φo (51)

32

that is,√α2(x+ µ)2 − 8αµg(·) + 8αµl(·)−

√α2(x+ µ)2 − 8αµg(·)

2α>

cηv

ρ(θ1 − θ2)(52)

which equals to

λ > λ ≡ 1− ρρ

+cηv

2µρ2(θ1 − θ2)[γ1(e− π)ρ+ γ2ηv][

αcηv

ρ(θ1 − θ2)+√α2(x+ µ)2 − 8αµg(·)] (53)

Regarding the effect of heterogeneity on the threshold, We have ∂λ∂µ> 0 only if

α2(µ2 − x2) >cηv

√α2(x+ µ)2 − 8αµg(·)ρ(θ1 − θ2)

(54)

which by the assumption (x− µ) ≥ 0 cannot be true. Therefore, ∂λ∂µ< 0.

Regarding the effect of extremism on the threshold,

∂λ

∂x=

cηv

2µρ2(θ1 − θ2)[γ1(e− π)ρ+ γ2ηv]

2α2(x+ µ)√α2(x+ µ)2 − 8αµg(·)

> 0 (55)

.C.2 Effect of heterogeneity on provocation efficacy

Regarding the effect of heterogeneity on the efficacy of provocation, LetW = 1√α2(x+µ)2−8αµg(·)

and Y = 1√α2(x+µ)2−8αµg(·)+8αµl(·)

, then

∂∆φo

∂µ= 2g(·)W − [2g(·)− 2l(·)]Y − α(x+ µ)

2(W − Y ) (56)

The sign of ∂∆φo

∂µis indeterminate.

33