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Online Appendix for “Managing Terror”
Peter Schram
October 14, 2015
Contents
I Historical Appendix and Other Examples 2
1 Background on the Haqqani Network and AQI 21.1 The Islamic State of Iraq and Al-Qaeda in Iraq: 2003-2010 . . . . . . . . . . . . . . . . . . . 21.2 Haqqani Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Other Examples of the Model 6
II Model Appendix 6
3 Folk Theorem Proof 63.1 Folk Theorem Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
4 Principal’s Strategies 84.1 No Contracting Strategy: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.2 Transfer Contracting Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.3 Worst-Case Organizational Contracting Strategy . . . . . . . . . . . . . . . . . . . . . . . . . 9
5 Proving Proposition 0 95.1 Part 1: Define the Vertices of the Convex Hull . . . . . . . . . . . . . . . . . . . . . . . . . . 105.2 Part 2: Define the Efficiency Frontier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.3 Part 3: Determine the feasible and efficient set: . . . . . . . . . . . . . . . . . . . . . . . . . . 13
6 Calculating “Worst Payoff” for the Principal 15
III Estimation Appendix 16
7 Structural Estimation Example 16
8 Recovering Number of Subversive Acts 17
IV Model Extension 17
9 Motivation 18
1
10 Auditing Model 1810.1 Actors and Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1810.2 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1910.3 Utilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
10.3.1 Agent Stage Game Utilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2110.3.2 Principal Stage Game Utilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
10.4 Supergame Utilities and Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
11 Principal’s Strategies 2311.1 Equilibria and Parameter Space Refinements . . . . . . . . . . . . . . . . . . . . . . . . . . . 2311.2 No Contracting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2311.3 Transfer Contracting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2411.4 Organizational Contracting with Auditing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2511.5 Principal’s Decision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
12 Mechanics of Organizational Contracting with Auditing 2812.1 First Deviation: Cell member deviates from xm,t = ρt to xm,t = 1 for one round . . . . . . . 2912.2 Second Deviation: Cell leader deviates from xl,t = ρt to xl,t = 0 for one round . . . . . . . . . 3012.3 Third Deviation: Cell leader forms homogeneous cell for one round . . . . . . . . . . . . . . . 31
12.3.1 Preventing single round deviation . . . . . . . . . . . . . . . . . . . . . . . . . . 3112.3.2 Preventing multiple round deviation . . . . . . . . . . . . . . . . . . . . . . . . . 3212.3.3 Preventing single and multiple round deviation: constructing the auditing
function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3312.4 Forth Deviation: Defecting and avoiding punishment . . . . . . . . . . . . . . . . . . . . . . . 33
V Leadership Targeting Analysis: Data, Alternative Explanations and Ro-bustness 33
A Summary of Targeting Data, May 2007-April 2008 33
B Alternative Explanations 34B.1 Alternative Explanation: Opportunistic Violence, Infighting, and Retaliation . . . . . . . . . 35B.2 Ruling out Alternative Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
C Robustness Checks 35C.1 Population Normalized . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35C.2 Dropping Governorates and Months . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
Part I
Historical Appendix and Other Examples
1 Background on the Haqqani Network and AQI
1.1 The Islamic State of Iraq and Al-Qaeda in Iraq: 2003-2010
The group that became al-Qaeda in Iraq was started by Abu Musab al-Zarqawi, a Jordanian
militant with personal and financial ties to Osama bin Laden. After fleeing Afghanistan
following the 2001 US invasion, Zarqawi began supervising camps run by of Ansar al-Islam,
a Kurdish-Sunni terror organization operating in northeastern Iraq. In early 2004, Zarqawi
2
established the militant group Jama’at al-Tawhid wal-Jihad to resist US military forces in
Iraq. On October 2004, he publicly declared allegiance to Osama bin Laden and renamed the
group Tanzim Qaidat al-Jihad fi Bilad al-Rafidayn (which translates to “the Organization
of Jihad in the Country of Two Rivers”), and the group became known as al-Qaeda in Iraq.
Though Zarqawi was killed in June 2006, the organization outlived its founder and provided
the foundation for the Islamic State of Iraq and Syria (ISIS) today.
From the beginning, AQI possessed ambitious goals. In addition to attempting to incite
a civil war through attacks on Shi’a civilians, AQI sought to expel American troops from
Iraq, overthrow the established Iraqi government, and establish an Islamic state operating
under Sharia law. To meet these goals, AQI was particularly brutal. Whereas most Sunni
insurgent groups used conventional and guerrilla tactics against US and coalition forces, AQI
used these tactics as well as suicide attacks targeting members of the Iraqi Security Force,
UN troops, humanitarian workers, and Shi’a Iraqi civilians. AQI also controversially engaged
in the beheading of Iraqi and foreign hostages, which led to al-Qaeda Central (referring to the
primary network of al-Qaeda operatives in Afghanistan and Pakistan, hereafter referred to
as AQC) to warn Zarqawi in November 2005 that such tactics might weaken Iraqi civilians’
support of the group (CTC, 2005).
Over the course of 2006, tensions rose between AQI and local Sunni insurgent groups due
to AQI’s AQI’s move into racketeering and smuggling (which had previously been a major
source of revenue for the other Sunni groups), and AQI’s killing of members of other Sunni
insurgent groups (Fishman, 2009). This tension was made public on September 17, 2006,
when several Sunni tribes in the Anbar Province announced an alliance with US forces and
pledged to fight AQI. This event, known as the “Day of Awakening” or the “Anbar Awaken-
ing,” is viewed as a major turning point for the US military and Iraqi government (Biddle
et al., 2012). Support for the Awakening movement grew as AQI continued to attack Sunni
groups and as AQI on 15 October 2006 created their own political entity, the Islamic State
of Iraq (ISI), which claimed to be the legitimate government across much of Iraq.
The declaration of the ISI openly demonstrated AQI’s political intent, but also had a tactical
purposes. The ISI was an institution that AQI leadership hoped would absorb other tribal
and insurgent groups; if the ISI was indeed the “true” Islamic rulers of Iraq as they claimed,
rebelling against their dictates, as those in the Anbar Awakening were doing, was a violation
of Islamic ideology (Fishman, 2009). However, the declaration of the ISI did little to change
the attitudes of the tribes that had already turned against AQI, and it created additional
tension among the militant groups that were allied with AQI but were not willing to accept
3
the ISI as the true Islamic government of Iraq.
The violent and criminal behavior of the members of AQI continued to alienate other Sunni
militant groups. In late 2007, the Islamic Army in Iraq, an umbrella group of Sunni insurgent
and tribal organizations, denounced the ISI. While AQI attempted to reach out after the
statement, this progress was hindered by AQI’s dysfunction and the actions of its members.
Around this time, AQI began turning away foreign fighters, but this did not seem to stop
the group’s decline. By 2010, the US military reported 80% of AQI leadership had been
removed, and in the first five months of 2010 the number of attacks and casualties on US
troops were at the lowest level since 2003 (BBC, 2010).
A great deal of insight into AQI’s decline can be gathered by the “Analysis of the State
of ISI,” a leaked internal document which circulated to al Qaeda leadership. Beginning in
2006, AQC became concerned with what was being done in Iraq, and requested information
on the inner workings on the group. The document that was reported back (which was later
intercepted by the United States in 2008) portrayed AQI as a highly dysfunctional organi-
zation that, amongst other things, was never able to properly integrate foreign and local
fighters into a cohesive fighting force. In addition to the quote introducing the paper, the
group discusses the counter-productive behavior of local Iraqi fighters:
“It reached a stage where one of the brothers who was stationed close to one of
the IEDs that was originally set to target one of the renegades, saw one of the
American’s convoy passing by and did not set off the IED off, and he was later
asked about his behavior and he answered, and we wish he did not, that he did not
have orders to attack the Americans ???!!!!... The apostates and the Americans, as
a result, started launching their attacks to destroy us. We lost cities and afterward
villages, and the desert became a dangerous refuge.”(CTC, 2007)
Because of the failure to integrate foreign mujahideen into the group, in 2007 AQI began
turning foreign fighters away. This action highlights the extent of the personnel problems
that the group faced; as AQI was losing ground in Iraq, leadership determined it would be
better to turn manpower away than to try to take on additional soldiers.
4
1.2 Haqqani Network
Since its inception in the 1970s, the Haqqani network has developed into the most lethal
terrorist network operating in Afghanistan today (Dressler, 2012). Geographically, the group
has a number of operational and training bases in the North Waziristan region of the Fed-
erally Administered Tribal Areas in Pakistan and has a strong insurgent presence in the
Loya-Paktiya region in Afghanistan. While technically the group was subsumed by the
Afghan Taliban in 1995 and group members have played a strong role in assisting the Tal-
iban’s takeover of Kabul and other areas in Afghanistan pre-2001, the leaders of the network
still maintain control over actions taken in the Loya-Paktiya region and still have indepen-
dent funding sources.
The Haqqani network possesses two non-exclusive goals: to control the security environment
in the greater Loya-Paktiya region in Afghanistan, and to help foster the global Jihadist
movement (Dressler, 2012; Brown and Rassler, 2013). To meet these goals, the Haqqanis
welcomed Arab jihadists to their group.
After the Soviet Union withdrew from Afghanistan, the Haqqani network as well as a num-
ber of other mujahideen groups fought to topple the communist Democratic Republic of
Afghanistan. Arab jihadis were involved in the Battles of Jalalabad (1989), Khost (1990-1)
and Gardez (1991), but as the Battle of Jalalabad demonstrated, the Arab fighters were not
necessarily well utilized. After several commanders at the Battle of Jalabad refused to coop-
erate or lead the Arab mujahideen into battle, one of Jalaluddin Haqqani’s colleagues– Haji
Khalid– commanded these forces into what was generally agreed upon as a poorly planned
and executed battle. However after the Battle of Jalabad, Ma’afi Khan, a Haqqani comman-
der, noted that ”the lack of a unified, structured command [at the battle] created a huge
obstacle on our way to success,” and the Haqqanis adapted their organization. The next
year, in the Battle of Khost, for the first time Arab mujahideen fought under the command
of other Arabs who were integrated into the Haqqani chain of command. The new command
structure appeared effective as both Khost and Gardez fell to the Haqqanis, and in time
these victories proved instrumental in the decline of the communist government (Brown and
Rassler, 2013). The integration of foreigners continued throughout the history of the group,
and to further ensure local and foreign fighters continue to work together, the Haqqanis
today are known to have assigned facilitators who act as liaisons between indigenous and
foreign fighters (Dressler, 2010).
5
2 Other Examples of the Model
• Lawyers and Economists in the FTC (covered in Wilson (1989)). The US Federal
Trade Commission employes both lawyers and economists. Wilson discusses how these
two types of actors possess different preferences over what types of cases they work
on. For example, Wilson claims lawyers are happiest with pursuing allegations that two
companies have conspired to fix prices, while economists are happiest with the discovery
that the market power of a company may reduce industry competition. By integrating
both parties, the FTC becomes a more effective organization.
• Actors in the Forest Service: Wilson (1989) discusses how, between 1950-1980, the US
Forest Service changed from being composed of almost entirely foresters to a group
including engineers, biologists and economists. Wilson claims this diversity of actors
made the Forest Service an agency that was better able to represent the interests of the
public.
Part II
Model Appendix
3 Folk Theorem Proof
Folk Theorem: For any (vl, vm) ∈ V with vl > EUl[SNE] and vm > EUm[SNE], there ex-
ists a δ such that for all δ ∈ (δ, 1) there is a subgame-perfect equilibria with payoffs (vl, vm).
Proof: Let X (ω) refer to a mixed action profile that assigns a fixed probability weight
to a set of action profiles.1 Assume there is an action profile X (ω) with EUl[X (ω)] = vl
and EUm[X (ω)] = vm and consider the following strategy profile: In period 1 each player
plays the action profile defined by X (ω). Each player continues to play X (ω) so long as the
realized actions were those defined by X (ω) in all previous periods. If at least one player did
not play according to X (ω), then each player plays SNE for the rest of the game.
Letting X ∗l (ρt) denote the terror cell leader’s optimal deviation from X (ω) given ρt, this
strategy profile is a Nash equilibrium when:
Ul,1(X ∗l (ρt)) +δEUl[SNE]
(1− δ)≤ Ul,1(X (ω), ρt) +
δEUl[X (ω)]
(1− δ)1For simplicity, it is assumed this cannot be correlated with type.
6
Which can be re-written as:
(1− δ)(Ul,1(X ∗l (ρt))− Ul,1(X (ω), ρt)
)≤ δEUl[X (ω)]− δEUl[SNE]
This inequality is satisfied for a range of δ < 1 because it holds strictly in the δ = 1 limit.
This profile is subgame perfect because in every subgame off the equilibrium path, the strate-
gies are to play SNE.
Additionally, the assumption that there exists a EUl[X (ω)] = vl and EUm[X (ω)] = vm
for all (vl, vm) ∈ V is proven in the first part of the proof for Proposition 0.
3.1 Folk Theorem Example
In the example in the paper, it is assumed δ → 1, and then the principal’s worst possibleoutcome is selected. The example below considers the “best case” for the principal, when theagents both match action to ρt for all t.
First, for the best-case to be feasible, it is assumed that (1− ρ)(1− 32β) + βρ
2> 0. Without
this assumption, the foreign agent does strictly better playing the Nash stage game equilib-ria (setting xm,t = 0). This assumption could be relaxed, but it would require considering amixed strategy on the Pareto frontier.
For matching action to the state of the world to be sustainable in a infinite horizon gamewith a grim-trigger strategy, it must be that the two inequalities hold:
−1 + 32β
(1− ρ)(2β − 1)≤ δ
−1 + 3β2
ρ(2β − 1)≤ δ
These are the requisite conditions for a domestic fifer (above) and a foreign fighter (below).
In the top condition, as ρ moves towards 1, the condition is more difficult to support (re-quires a larger discount rate). In the bottom condition, as ρ moves towards 0, the conditionbecomes more difficult to support. This is fairly intuitive. If ρ = 0, then the foreign typemember would perform poorly matching action to the state of the world, and could be muchbetter selecting xm,t = 1. The reverse holds for the domestic type fighter. This impliesthat δ can be lowest for values of ρ closer to 1/2. While this condition is influenced by theassumption that agents match action to the state of the world in organizational contracting,generally equilibria on the efficiency frontier are easier to be sustained with values of ρ thatare not very close to 0 or 1.
For both conditions below, as β increases, δ must increase to support the cooperative action.This is also intuitive- if β was low, then inefficiency costs are low, which means matchingaction to the state of the world only has small benefit.
7
4 Principal’s Strategies
4.1 No Contracting Strategy:
Information Decision: i = 0
Partnering Decisions: a = d
Breaking Ties Decision: b = nr
Action Strategies: xl,t = xm,t = 0 for all ρt, h(t), and t.
Expected Payoffs:
EUl = EUm = −(1− β)ρ/(1− δ)EUp = −2ρ/(1− δ)
4.2 Transfer Contracting Strategy
Information Decision: i = 0
Transfer Contract: For all t, set gl,t = (3β2− 1)xl,t + β
2xm and gm,t = (3β
2− 1)xm,t + β
2xl
Partnering Decisions: a = d
Breaking Ties Decision: b = nr
Action Strategies: xl,t = xm,t = ρt for all ρt, h(t), and t.
Expected Payoffs:
EUl = EUm = −(ρ− βρ)/(1− δ)EUp = −θ(4β − 2)ρ/(1− δ)
8
4.3 Worst-Case Organizational Contracting Strategy
Information Decision: i = 1
Organizational Contract: If a = d, then b = r
Partnering Decisions: a = f
Breaking Ties Decision: b = nr
Cell Leader Strategies: In t = 1, play Action Profile 1 with probability α = −4βρ+β+2ρ(β−2)(ρ−1)
,
and Action Profile 2 with probability 1− α. If in period t− 1 the cell member does not do
the same, then play xl,t = 0 for all remaining t. If not, then continue mixing over action
profiles 1 and 2.
Cell Member Strategies: In t = 1, play Action Profile 1 with probability α, and
Action Profile 2 with probability 1 − α. If in period t − 1 the cell leader does not do the
same, then play xm,t = 1 for all remaining t. If not, then continue mixing over action profiles
1 and 2.
Expected Payoffs:
Ul = −[α(β(1+ρ
2) + (1− β)(1− ρ)) + (1− α)βρ
]/(1− δ)
Um = −[(1− α)β(1− ρ) + αβ(1−ρ
2)]/(1− δ)
Up = − [ψ + α(1− ρ)] /(1− δ)
5 Proving Proposition 0
Definition: This paper will refer to the three action profiles by their specified number:
• Action Profile 1 : xl,t = 1, xm,t = ρt for all t.
• Action Profile 2 : xl,t = ρt, xm,t = ρt for all t.
• Action Profile 3 : xl,t = ρt, xm,t = 0 for all t.
Proposition 0: If δ → 1, the set of efficient SPNE strategies are constructed by mixingover Action Profiles 1 and 2, and over Action Profiles 2 and 3.
9
The roadmap for the proof is as follows. First, I will demonstrate that the payoffs of thethree action profiles above are three of nine vertices to the convex hull of possible payoffs.Second, I will identify the efficiency frontier of action profiles. This will be accomplished byidentifying the action profiles not on the efficiency frontier as Pareto inferior to the convexcombination of action profiles on the frontier. Finally, I will identify the feasible set of actionprofiles as defined by the Folk Theorem.
5.1 Part 1: Define the Vertices of the Convex Hull
First it must be shown that the expected value of any single-stage payoff can be generated
through some combination of selecting xl,t ∈ {0, ρt, 1} and xm,t ∈ {0, ρt, 1}.
Abusing notation, let xm ∈ [0, 1] and xl ∈ [0, 1] denote two arbitrary action profiles in
time t. Abusing utility notation , the payoff for the terror cell leader (domestic type) for this
arbitrary action profile is:
Ul,t(fixed) = −βxl + xm2
− (1− β) ∗ |ρt − xl|
And the payoff for the terror cell member is:
Um,t(fixed) = −β(1− xl + xm2
)− (1− β) ∗ |ρt − xm|
With these profiles fixed, it is necessary to define the probability weight each agent places
on the pure strategies. These are:
η = Pr(xl,t = 0), κ = Pr(xl,t = ρt), 1− η − κ = Pr(xl,t = 1)
λ = Pr(xm,t = 0), µ = Pr(xm,t = ρt), 1− λ− µ = Pr(xm,t = 1)
Without loss of generality, assume λ and µ are fixed so that xm = µρt + (1 − λ − µ).
When this holds, then the expected payoff for the terror cell leader from mixing over 0, ρt,
and 1 is:
EUl,t(mixing) = η[−βxm
2− (1− β) ∗ ρt
]+ κ
[−βρt + xm
2
]+ (1)
+(1− η − κ)
[−β 1 + xm
2− (1− β) ∗ (1− ρt)
]
10
If EUl,t(mixing) can be equivalent to Ul,t(fixed), then it has been shown that through a
convex combination of xl,t ∈ {0, ρt, 1}, any payoff from an arbitrary action profile can be
expressed. For the two utility terms to hold with equality, it must be that:
xl = κρt + (1− η − κ)(1)
&
|ρt − xl| = ηρt + (1− η − κ)(1− ρt)
If ρt 6= 0, these are two independent equations defined by two variables (κ and η). Therefore,
a closed form solution exists.2 If ρt = 0, then the system of equations is dependent, and any
η and κ such that xl = 1− η − κ will solve the equation above.
By fixing η and κ such that xl = κρt + (1− η − κ)(1), it can also be shown EUm,t(mixing)
can be made equivalent to Um,t(fixed).
Therefore, the vertices below define the convex hull of payoffs in stage t for a fixed ρt:
Action ordered pair (xl,t, xm,t)(Asterisk Denote Efficiency
Frontier)
Payoffs (Ul,t, Um,t)
(0, 0)∗ (−(1− β)(ρt), −β − (1− β)(ρt))
(0, ρt) (−β ρt2− (1− β)(ρt), −β(1− ρt
2))
(0, 1) (−12β − (1− β)(ρt), −1
2β − (1− β)(1− ρt))
(ρt, 0)∗ (−β ρt2, −β(1− ρt
2)− (1− β)(ρt))
(ρt, ρt)∗ (−βρt, −β(1− ρt))
(ρt, 1) (−β ρt+12, −β(1− ρt+1
2)− (1− β)(1− ρt))
(1, 0) (−12β − (1− β)(1− ρt), −1
2β − (1− β)(ρt))
(1, ρt)∗ (−β ρt+1
2− (1− β)(1− ρt), −β(1− ρt+1
2))
(1, 1)∗ (−β − (1− β)(1− ρt), −(1− β)(1− ρt)
Note that the action ordered pair (xl,t, xm,t) notation will be used throughout the rest of
this and the following section.
5.2 Part 2: Define the Efficiency Frontier
It remains to be shown that the following convex combinations (in the table below) define
the efficiency frontier for any realization of ρt. For the line equation in the table below, I
2As an example, if ρt > xl, then κ = xlρt
and η = 1− xlρt
.
11
abuse notation further and let the “x” term denote terror cell leader’s expected utility, and
the “y” term denote the terror cell member’s expected utility.
Connecting Points Line equation
(1, 1) to (1, ρt) y = −4βρt + 2−2ρtβ
+(
2β− 3)x+ β + 6ρt − 4
(1, ρt) to (ρt, ρt) y = β∗(β(2ρt−1)−2ρt+2)β−2
+ ββ−2
x
(ρt, ρt) to (ρt, 0) y = β(2ρt − 1)− 2ρt + (β−2)βx
(ρt, 0) to (0, 0) y = b2(4p−3)+b(2−6p)+2p3b−2
+ x b2−3b
Note that moving from the (1, 1) to (1, ρt) line to the (1, ρt) to (ρt, ρt) line has the slopes
strictly decreasing. This similarly holds for moving from the (1, ρt) to (ρt, ρt) line to the
(ρt, ρt) to (ρt, 0) line and so on. This implies the the function defined by these line segments
is concave, which means that any convex combination outside of the ones defined above
would be Pareto inferior. For example, any point on the (1, 1) to (ρt, ρt) line is weakly worse
than a point defined on the frontier above.
It remains to be shown that the action profiles (0, ρt), (ρt, 1), (0, 1), and (1, 0) are Pareto
inferior to the efficiency frontier for all ρt and β. This is accomplished as follows. First,
for each Pareto inferior action profile, I identify the terror leader’s utility from that action
profile as falling within a certain segment (or multiple segments, depending on parameter
values) of the efficiency frontier. For example, for Pareto inferior action profile (0, ρt), the
terror cell leader does better under (0, ρt) than under (ρt, ρt), and worse under (0, ρt) than
under (ρt, 0). Therefore, through some convex combination of strategies on the efficiency
frontier (mixing strategies (ρt, ρt) and (ρt, 0)), it is possible to give the terror cell leader a
payoff equivalent to his payoff from (0, ρt).
With the first part identified, I then take the terror cell leader’s utility from the inferior
strategy (from the (0, ρt) example this is −β ρt2− (1− β)(ρt)), and I plug it into the x value
of the respective segment of the efficiency frontier. This will identify the utility value for
the terror cell member when the terror cell leader is receiving his equivalent payoff on the
efficiency frontier. If this calculated utility value for the terror cell member is greater than
the utility of the terror cell member from action profile (0, ρt), then the point on the frontier
is Pareto improving.
The following table identifies the Pareto inferior action profiles, the segment(s) of the ef-
ficiency frontier that the inferior action profiles lies below, and the Pareto improving point
that lies on the frontier. It also defines the “outside condition;” the outside condition defines
12
under what conditions the inefficient point falls under the designated segment of the Pareto
frontier. This is:
InefficientPoint
Falls on followingsegment of
Pareto-frontier
Outsidecondition?
Pareto improving point
(0, ρt) (ρt, ρt) to (ρt, 0) Always(−β ρt
2− (1− β)(−ρt), 5β2ρt−2β2−8βρt+4ρt
2β
)(ρt, 1) (1, ρt) to (ρt, ρt) Always
(−β ρt+1
2, (3β2−4β)(ρt−1)
2(β−2)
)(0, 1) (1, ρt) to (ρt, ρt)
β2(2β−1)
≥ ρt
(−β ρt+1
2, 6β2ρt−3β2−6βρt+4β
2(β−2)
)(0, 1) (ρt, ρt) to (ρt, 0) β
2(2β−1)< ρt
(−1
2β − (1− β)(ρt),
(3β−2)(2βρt−β−2ρt)2β
)(1, 0) (1, 1) to (1, ρt)
2−2β2−β > ρt
(−1
2β − (1− β)(1− ρt), −2βρt−7β+2ρt+4
2
)(1, 0) (1, ρt) to (ρt, ρt)
2β−2β−2≤ ρt, &0 ≤
1− ρt − β2
(−1
2β − (1− β)(1− ρt), β(2βρt−β−2ρt+2)
2(β−2)
)(1, 0) (ρt, ρt) to (ρt, 0) 0 > 1− ρt − β
2
(−1
2β − (1− β)(1− ρt), 2β2ρt−β2+2βρt−4β−4ρt+4
2β
)Through algebra, it can be seen that the inefficient action profiles are in fact Pareto in-
ferior to the defined points in the right most column. Thus, the combinations of the five
action profiles above define the efficient set.
5.3 Part 3: Determine the feasible and efficient set:
It remains to be shown where the feasible set lies. By the Folk Theorem, this is all points
that exceed the minmax payoffs for the two agents in the terror cell. Based on the table
immediately above, in expectation the terror cell leader’s minmax utility falls within the
(1, ρt) to (ρt, ρt) range. This can be seen because β2(2β−1)
≥ ρ for all possible β and ρ.3
The minmax payoff for the terror cell member can fall within two possible segments on
the efficiency frontier. When 3β−2(4β−2)
≤ ρ then it falls within the (ρt, ρt) to (ρt, 0) segment.
When 3β−2(4β−2)
> ρ holds, it falls within the (1, ρt) to (ρt, ρt). An example of this is shown
below in Figures A1 and A2 show this for a fixed β = 0.75 and either ρ = 0.15 or ρ = 0.55.
Note the blue polygon represents the convex full of possible payoffs.
3This is because β ∈ (2/3, 1] and 2ρ ∈ ( 12, 1].
13
In either case, the set of efficient and feasible outcomes can be defined by over mixing
Action Profiles 1 and 2 and by mixing over Action Profiles 2 and 3.
14
6 Calculating “Worst Payoff” for the Principal
It is assumed state-contingent mixing is not possible. This implies that the agents cannot
select one mixing profile for some values of ρt and a different mixing profile for a different set
of values for ρt. Because the principal does best when agents select (ρt, ρt) any movement
along the Pareto frontier away from this action profile makes the principal strictly worse. it
can be seen from Figure A1 and A2 that the principal will do worst when one of the agents
is receiving his minmax payoff.4
First consider the minmax for the terror cell leader. In expectation, the terror cell leader
would receive −β2− ρ+ βρ when agents play their stage-game Nash equilibrium action pro-
file. Because in expectation this utility falls between the utility from action profiles (1, ρt)
and (ρt, ρt), I let λ denote the probability of matching action to (1, ρt), 1 − λ denote the
probability of matching action to (ρt, ρt), and solve:(−β
2− ρ+ βρ
)< α
(−β ρ+ 1
2− (1− β)(1− ρ)
)+ (1− α) (−βρ)
This gives:
α <−4βρ+ β + 2ρ
(β − 2)(ρ− 1)
The Folk Theorems imply that this value of α is an upper bound on how much the (1, ρt)
action profile can be played.
I also need to consider the minmax payoff for the terror cell member. It should be noted that
if the terror cell member’s minmax utility falls within the (ρt, ρt) to (ρt, 0) segment, then the
principal does worse when the minmax for the terror cell leader is achieved (defined by α
above). To reach the terror cell member’s minmax when the cell members are mixing over
the (ρt, ρt) to (ρt, 0) strategies, I let γ denote the probability of playing the (ρt, 0) action
profile:
−1
2β − (1− β)(1− ρ) < γ
(−β(1− ρ
2)− (1− β)(ρ)
)+ (1− γ) (−β(1− ρ))
4In the first part of this proof, it was shown that any payoff in the convex hull of payoffs forthe agents could be achieved by mixing over 0, 1, and ρt. Along the Pareto frontier, becausethe principal’s payoffs are linear, the full set of payoffs for the principal can be achieved bymixing over Action Profiles 1, 2, and 3. This implies that this section is, in fact, calculatinga “worst case.”
15
This gives:
γ <−4βρ+ 3β + 2ρ− 2
(β − 2)ρ
Comparing the upper bounds of α and γ, αupper > γupper. For ease, the paper will simply
set α with equality to the upper bound for calculating utilities. Therefore, the worst the
principal can do is when the terror cell leader achieves his minmax payoff while mixing over
(1, ρt) and (ρt, ρt) with probability weights by α and 1− α (respectively) as defined above.
Part III
Estimation Appendix
7 Structural Estimation Example
(Local Kill)i,t(Fired upon)i,t + (Local Kill)i,t
= (Zarqawi)tβ1 + FEi + ωi,t
Consider what this model means for a single district (district i) for two periods (one before
the Zarqawi targeting, and one after). By Assumption 1 β1 is equal to:
β1 =(Not subversion)i,2 + (Subversion)i,2
(Not subversion)i,2 + (Subversion)i,2 + (Fired upon)i,2−
− (Not subversion)i,1 + (Subversion)i,1(Not subversion)i,1 + (Subversion)i,1 + (Fired upon)i,1
+ ψi,2 − ψi,1
Decomposing the ψi,t terms and expressing the above (whenever possible) in terms of the
principal’s preferred levels gives:
β1 =(Not subversion)∗i,2 + (Subversion)i,2
(Not subversion)∗i,2 + (Fired upon)∗i,2−
−(Not subversion)∗i,1 + (Subversion)i,1
(Not subversion)∗i,1 + (Fired upon)∗i,1+ µi,2 − µi,1
By the definition of ρi,t (using (7)) and taking expectations gives:
16
E [β1] = −E [ρi + εi,2] + E [ρi + εi,1] + E [µi,2 − µi,1] +
+E[
(Subversion)i,2(Fired upon)∗i,2 + (Not subversion)∗i,2
]−
−E[
(Subversion)i,1(Fired upon)∗i,1 + (Not subversion)∗i,1
]Which, by Assumption 1, 2, and 3, can be re-written:
E [β1] = E[
(Subversion)i,2Ki,2
]− E
[(Subversion)i,1
Ki,1
]
8 Recovering Number of Subversive Acts
The following establishes the technique for recovering a lower bound:
minimize [(Subversion)i,post−Zarqawi − (Subversion)i,pre−Zarqawi]
Such that:
β1 =(Subversion)i,post−Zarqawi
Ki,post−Zarqawi− (Subversion)i,pre−Zarqawi
Ki,pre−Zarqawi
Using the condition on β1 above and substituting this into the top generates:
minimize
[Ki,post−Zarqawi ∗ β1 +
Ki,post−Zarqawi ∗ (Subversion)i,pre−ZarqawiKi,pre−Zarqawi
− (Subversion)i,pre−Zarqawi
]Because violence in Iraq was continuing to grow, it is assumed Ki,post−Zarqawi > Ki,pre−Zarqawiin this setting. Therefore, to minimize the difference, the level of subversion pre-Zarqawishould be fixed at 0. Therefore, the lower-bound of post-Zarqawi subversion can be measuredas: β1 ∗Ki,post−Zarqawi.
17
Part IV
Model Extension
9 Motivation
In the existing “Managing Terror” model, there are two concerns. First, it might be expected
the principal could develop a sense of if a terror cell was subverting, investigate to see if
this was the case, and then punish or disband the cell if subversion was occurring (in other
words, the principal might possess the ability to audit and punish). Second, once the agent’s
terror cell is organized, the cell stays with that organizational structure for the remainder of
the infinite horizon game (in other words, there is no changing from a homogeneous to het-
erogeneous formation, or vice versa). This extension addresses these two concerns, though
is less concise and more technical.
The following is written as a stand-alone model. Much of this is redundant with the model
that currently exists in “Managing Terror.”
10 Auditing Model
10.1 Actors and Organization
In the game, the principal offers contracts to agents within a terror cell. Feminine pronouns
refer to the principal, and masculine pronouns refer to the agents.5 Within the cell, there
are two classifications of agents. One agent is classified as the “terror cell leader.” The terror
cell leader will select another agent, the “terror cell member,” to act as a partner, and these
two agents will repeatedly conduct operations.
Each agent has a type, denoted y ∈ {d, f}, which corresponds to if an agent is domestic
(y = d) or foreign (y = f).6 It is assumed the terror cell leader is a domestic type.7 Thus,
when the cell leader selects a partner, either he forms a homogenous cell by selecting a type
y = d agent as partner, or he forms a heterogeneous cell by selecting a type y = f agent as
partner. The terror cell leader’s choice is denoted by a ∈ {d, f}, where a = d is forming a
homogeneous cell and a = f is forming a heterogeneous cell. For simplicity, it is assumed the
terror cell leader has the option to work with either a single foreign type agent or a single5This assignment was determined by a randomizing procedure and is done entirely for convenience.6This framing should not only be considered for the foreign-local fighter dichotomy. It can also be though, more generally,
of two types of agents possessing different and offsetting preferences.7While this model assumes this is not a strategic choice, even if the principal could select this, she would be weakly worse
off by letting an agent of type y = f act as terror cell leader.
18
domestic type agent– in other words, the foreign agent and domestic agent that the terror
cell leader selects from is the same throughout the game.
Crucially, while the principal knows the type of the terror cell leader, the principal does
not costlessly observe the type of the terror cell member (in other words, she does not cost-
lessly observe a). If the principal chooses to learn the type of the terror cell member, she
must incur a one-shot cost, denoted by Ψ.
10.2 Structure
The games structure is presented below, then expanded upon.
1. The principal offers contracts.
2. The terror cell leader forms a terror cell (sets partner type a ∈ {d, f}).
3. Nature sets the state of the world ρt.
4. The agents in the terror cell select actions xl,t ∈ [0, 1] and xm,t ∈ [0, 1].
5. The principal’s action utilities are realized.
6. The principal has the “audit” option to pay cost Ψ and assess the previous cell compo-
sition.
7. All utilities are realized.
8. The principal has the option to reject (or “break”) the terror cell by setting b ∈ {r, nr}.If disbanding occurs, all players receive reservation utility for the remainder of the game.
9. The game repeats infinitely, starting at Step (1) .
Each period of the game can be broken into three components. Stages (1) and (2) comprise
the organizational component, where the terror group is formed. Stages (3)-(4) comprise
the operational component, where agents observe the situation on the ground and conduct
operations. Stages (5)-(8) comprise the auditing component, where the principal gains in-
formation, chooses to audit (or not), and then chooses to disband the terror cell (or not).
It should be noted that this final component is most relevant when the principal chooses to
audit the agents.
In (1), the principal specifies a contract for the stage game. One form of contracting has
19
the principal offer a utility transfer to the agents in the terror cell. In period t, a transfer to
the terror cell leader is denoted by gl,t ≥ 0, and a transfer to the cell member is denoted by
gm,t ≥ 0. The set of transfers given to the terror cell leader and terror cell member in period
t is gt.
In (2), the terror cell leader selects his partner’s type by setting a ∈ {d, f}.
In (3), the state of the world is realized. The state of the world is denoted by random
variable ρt ∈ [0, 2ρ], which identifies the action profile that the principal most prefers at pe-
riod t. It is assumed 2ρ ∈ (12, 1], realizations of ρt are independently drawn from a uniform
distribution, and the agents observe the state of the world but the principal does not. When
ρt is low, the conditions on the ground are such that AQI leadership would prefer its agents
take more actions directed against domestic actors to secure local power; if ρt+1 < ρt, AQI
leadership would prefer its terror cells to take more actions against domestic actors in period
t+1 than in period t. The reverse holds for ρt being high and attacks against Coalition forces.
In (4), each agent in the terror cell selects an action. The terror cell leader’s action is
denoted by xl,t ∈ [0, 1] and the terror cell member’s action is denoted by xm,t ∈ [0, 1]. The
principal observes these selections. When agents select low values of xm,t and xl,t, this rep-
resents the agents selecting operations targeting domestic actors. When agents select high
values of xm,t and xl,t, this represents the agents selecting operations targeting coalition
forces. The actions are aggregated to determine xt, which denotes the profile of violence
being carried out by the terror cell. The following equation determines xt:
xt =xm,t + xl,t
2(2)
In (5), the principal’s “action utilities” are realized. The “action utilities” are the utili-
ties the principal receives from the actions the agents are undertaking. Formally, this is
UaP,t − |xl,t − ρt| − |xm,t − ρt| + εt, where εt is a i.i.d. random variable (assumed uniform
for ease with εt ∈ [−ε, ε]). This will be described below, but essentially here the principal
receives a noisy signal from the terror cell if the agents are subverting (or not).
In (6), the principal has the ability to monitor the composition of the terror cell (setting
It ∈ {0, 1}). Doing so incurs cost Ψ. This strategy can be conditional on terms of a contract
instituted in (1).
In (7) all utilities (agents and principal) are realized.
20
In (8), the principal can disband the terror cell. The principal’s choice is denoted by
b ∈ {r, nr}, where b = r corresponds to the principal rejecting the cell and b = nr cor-
responds to the principal not rejecting the cell. If the principal rejects the cell, all agents
and the principal receive their reservation utility (denoted U0a and U0
p respectively) in each
period.
In (9), the game repeats starting with step (1).
10.3 Utilities
10.3.1 Agent Stage Game Utilities
The utility functions for the stage game are presented then described below. For the
terror cell member, the utility function of a domestic type agents is:
Um,t(xm,t, ρt, gm, y = d) = −βxt − (1− β) ∗ |xm,t − ρt|+ gm,t (3)
And for a foreign type terror cell member:
Um,t(xm,t, ρt, gm, y = f) = −β(1− xt)− (1− β) ∗ |xm,t − ρt|+ gm,t (4)
First consider the −βxt term in the utility function of the domestic type agent. The domes-
tic agent prefers taking actions that secure local power and prefers other agents within their
terror cell do the same. Therefore, as the profile of violence xt moves further away from the
domestic fighter’s ideal point of xt = 0, he incurs a disutility. The β term can be thought of
as a utility weight the agent places on the value of actions he prefers relative to the cost of
conducting actions. This similarly holds for the foreign fighters, whose ideal violence profile
is xt = 1.
The −(1− β) ∗ |xm,t− ρt| term captures the inefficiency from deviating from the state of the
world. Because ρt represents the most efficient action profile for conducting an insurgency,
any deviation from ρt generates a cost borne by the agent.8
The gm,t term denotes transfers.
8This assumption could be relaxed and the cost of one agent deviating from ρt could also be borne by both members of thecell; changing this assumption does not radically change the model and will therefore not be considered.
21
The terror cell leader (who is always a domestic type) has utility:
Ul,t(xl,t, ρt, gl) = −βxt − (1− β) ∗ |xl,t − ρt|+ gl,t (5)
It is assumed β ∈ (23, 1). This assumption implies an important feature: for a single iteration
of the stage game, the Nash equilibrium outcome has agents match action to their ideal point
rather than the state of the world.
10.3.2 Principal Stage Game Utilities
The principal’s stage game utility is:
Up,t(xl,t, xm,t, ρt, gt) = −|xl,t − ρt| − |xm,t − ρt| − θgt − iΨ
The −|xl,t − ρt| and −|xm,t − ρt| terms captures the disutility the principal receives when
agents select actions that differ from the state of the world. The θ term (with θ > 0) is a
constant capturing the relative cost of transfers to the agents. The ψ term (with ψ > 0) is
a constant capturing the information cost of learning the terror cell member’s type.
10.4 Supergame Utilities and Strategies
The variable X denotes the sequence of actions played in the stage game, or X =
{(xl,1, xm,1) (xl,2, xm,2), ...}. Similarly, ρ denotes the sequence of realized values of ρt, or
ρ = {ρ1, ρ2, ...}. The variable G denotes the sequence of realized transfers, or G = {g1, g2, ...}.Finally, the variable δ denotes the discount rate of the agents and principal. Conditional on
the principal not selecting b = r, the utilities of the agents in the terror cell are:
Ul(X , ρ, G, d) =∞∑t=1
δt−1Ul,t(xl,t, ρt, g, d)
Um(X , ρ, G, y) =∞∑t=1
δt−1Um,t(xm,t, ρt, g, y)
And the principal’s utility is:
Up(X , ρ, G, i) =∞∑t=1
δt−1Up,t(xl,t, xm,t, ρt, gt)
The agent’s and principal’s strategies are mappings from all information sets to strategy
profiles. Because this is an iterated game, this is a mapping from the period number t, the
state of the world ρt, and the t-history (denoted h(t)) into an action profile for period t. The
mapping Sl will denote the strategy for the terror cell leader, Sm will denote the strategy for
the terror cell member, and Sp will denote the strategy for the principal.
22
11 Principal’s Strategies
11.1 Equilibria and Parameter Space Refinements
This paper will only consider subgame perfect Nash equilibria. Additionally, this paper
will introduce an efficiency refinement:
Efficiency Refinement: Strategy S is efficient if there does not exist strategy S ′ such
that for i, j ∈ {l,m}, i 6= j denoting the agents in the terror cell:
Ui(S ′, ρ, G, y) > Ui(S, ρ, G, y) and Uj(S ′, ρ, G, y) ≥ Uj(S, ρ, G, y).
This is Pareto-efficiency applied to the agents in the terror cell. The refinement implies
if two agents are interacting repeatedly, the outcome will not be such that one player could
be made better off without making the other player worse off.
This paper will also introduce Condition 1:
Condition 1: When Condition 1 is in place, U0a = U0
p = −1.
This condition ensures all actors are better off when the principal does not walk-away from
the terror cell (b = nr). While relaxing Condition 1 could speak to the agents’ and prin-
cipal’s decision to join into the terror group, this is outside of the scope of the question
motivating this paper.9
The following subsections will describe the strategies the principal selects from. It is as-
sumed that the principal will select one of these three strategies, and implement it in the
first stage of each iteration.
11.2 No Contracting
In this setting, the principal offers no contracts for each iteration of the stage game.
When this occurs, the terror cell leader will form a homogeneous terror cell and agents will
set xl,t = xm,t = 0 for all periods. The principal will neither audit nor terminate the terror
cell. The expected utilities are: EUl = EUm = −(1− β)ρ/(1− δ) and EUp = −2ρ/(1− δ).To summarize the selected actions:
1. The principal offers no contracts.
2. Terror cell leader sets a = d.9Two papers that explore the importance of the outside option are Bueno de Mesquita (2005) and Berman and Laitin (2008).
23
3. N/A
4. Agents set xl,t = xm,t = 0.
5. N/A
6. The principal does not audit.
7. N/A
8. Principal does not reject, setting b ∈ nr.
9. The game repeats.
11.3 Transfer Contracting
Consider the setting where the principal uses transfer payments to get agents to match
their action to the state of the world. For simplicity, Assumption T is in place:
Assumption T: It is possible for the principal to make take-it-or-leave-it (TIOLI) con-
tracts to the agents in the terror cell.10
With Assumption T in place, the principal can commit to pay a transfer to agents af-
ter observing the action profiles. Consider what a transfer schedule of gl(xl,t, xm,t) =
(3β2−1)xl,t+
β2xm,t and gm(xm,t, xl,t) = (3β
2−1)xm,t+
β2xl,t would accomplish for two domestic
type agents. For the terror leader, the new utility function can be written as Ul,t = βρt− ρt.This transfer schedule makes the terror cell leader indifferent over his own actions and the ac-
tions of his partner.11 With this transfer schedule, it is equilibrium behavior for the terror cell
leader to partner with a domestic type agent and to match action to the state of the world.12
The expected utilities are EUl = EUm = −(ρ−βρ)/(1− δ) and EUp = −θ(4β− 2)ρ/(1− δ).
To summarize the selected actions:10While generally pushing aside the issue of credible commitment should be viewed with caution, there are two reasons why,
in this setting, assuming the possibility of binding contracts is actually fairly benign. First, while this game is only concernedwith a single terror cell, in reality AQI or the Haqqani Network uses many cells for conducting operations. By reneging onagreements for a single cell, these groups would damage their reputation which could lead to systemic repercussions throughoutthe organization. Second, in the case when the contract defines a transfer payment conditional on an action (as it is here), ahigh enough discount rate can make the principal’s contract self-enforcing. Therefore, assuming credible commitment is possibleis a technical simplification.
11
If the β2xm,t term was not included in the transfer schedule, agents would be indifferent over their action profiles, but would
have their utilities decreasing in the action profiles of their partner; without the β2xm,t term, the efficiency refinement would
imply both agents setting xl,t = xm,t = 0 would be the equilibrium.
12A different transfer contracting strategy exists where the agents form a heterogeneous terror cell and the principal offerscontracts to make them indifferent. While this different outcome is technically very similar, it does highlight the fact thatforeign fighters can be used in the transfer contracting setting.
24
1. The principal offers gl(xl,t, xm,t) = (3β2−1)xl,t+
β2xm,t and gm(xm,t, xl,t) = (3β
2−1)xm,t+
β2xl,t
2. Terror cell leader sets a = d.
3. N/A
4. Agents set xl,t = xm,t = ρt.
5. N/A
6. The principal does not audit.
7. N/A
8. Principal does not reject, setting b ∈ nr.
9. The game repeats.
11.4 Organizational Contracting with Auditing
The principal can require a heterogeneous terror cell to form. In a heterogeneous terror
cell in the non-repeated setting, it is a stage-game Nash equilibrium for the foreign and
domestic agents to match action to their type. However, for these agents, through repeated
interactions it constitutes a Pareto improvement to select some action profile that is closer
to matching action to the state of the world. Thus, in an infinitely repeated setting with a
heterogeneous cell and a high enough discount rate, a Pareto improving state can be reached
that is both better for the agents and better for the principal.13 To explain what occurs, the
folk theorems provide a great deal of insight.
Introducing the folk theorem in use here requires some notation. Let SNE denotes the
iterated stage game equilibrium for a heterogeneous cell. This is simply both agents se-
lecting their most preferred action profile (xl,t = 0 and xm,t = 1 for all t). Let EUl[SNE]
and EUm[SNE] denote the expected per-period payoffs for the terror cell leader and mem-
ber when they implement the stage game equilibrium. The expected-payoff convex hull as:
V = convex hull{(vl, vm) | ∃ (xl,t, xm,t) with EUl(xl,t, xm,t) = vl & EUm(xl,t, xm,t) = vm}.13Consider the strategies if period t = 1 was a one-shot game, ρ1 = 1
2, and a heterogeneous terror cell formed. In this game,
the Nash equilibrium has both agents matching action to their type (xl,t = 0 and xm,t = 1). This is inefficient because if bothplayers could commit to setting xl,t = xm,t = 1/2, then the cell would be committing the same violence profile without paying
the inefficiencies. In a repeated setting where ρt always equals 12
, a repeated prisoners’ dilemma type game arises, and theFolk Theorems imply that for a certain strategy profile and sufficiently high discount rate, always setting xl,t = xm,t = 1/2 is
a subgame perfect equilibrium. While ρt does not always equal 12
in the game here, through repeated interactions the sameintuition can be applied.
25
The V term can be thought of as the set of feasible per-period payoffs, in expectation. Fi-
nally, δ denotes the discount rate for the principal and agents.
Folk Theorem: For any (vl, vm) ∈ V with vl > EUl[SNE] and vm > EUm[SNE], there
exists a δ such that for all δ > δ there is a subgame-perfect equilibria with payoffs (vl, vm).
The Folk Theorem implies for a high enough discount rate, a range of feasible equilibria
are possible. For ease, this model will only consider a single efficient equilibrium, where a
heterogeneous terror cell is formed and agents match action to the state of the world. This
is Assumption C:
Assumption C : When agents form a heterogeneous cell, they will match action to the
state of the world.
There are reasons to suspect that the equilibria where both agents match action to the
state of the world is particularly focal. For a sufficiently high discount rate, not only is it the
simplest efficient equilibria to implement, but it is also what the principal would encourage
coordination upon. If this rationale is not compelling, it is also important to note that this
is a stronger assumption than necessary; the model in the main paper demonstrates that
even the in principal’s “worst case,” organizational contracting can still be better for the
principal. Here Assumption C provides simplicity, as the auditing procedure is already
notation-heavy. For these reasons, this assumption is in place, and the full set of conditions
on δ and ρ that allow for this assumption will be discussed in the following section.
The equilibrium is described below.
Recall that the audit procedure is in place to identify if operations were conducted by a
homogenous or heterogeneous terror cell. At the outset of the game, the principal defines
an auditing function as part of a contract. The auditing function assigns a likelihood of
auditing given the realization of actions (xl,t and xm,t) and the action utility for each period
t. In defining their supergame strategies, the agents choose between two types of profiles.
First is the “cooperative profile,” which has both agents set xl,t = xm,t = ρt. Next is the
“punishment profile,” which has the terror cell leader set y = f and xl,t = 0 and the terror
cell member set xm,t = 1.
The strategies are:
26
• Principal: For all periods, if auditing occurs and the terror cell is homogenous, disband
the cell. An audit occurs under two conditions:
– 1) Audit with probability defined by function γ(UaP,t).
– 2) Audit in period t+ 1 if xm,t 6= xl,t.
• Terror Cell Leader and Terror Cell Member: The terror cell leader sets y = f
and both agents start playing the strategies in the cooperative profile. If there is no
deviation, the terror cell leader continues to set y = f and agents continue playing the
strategies defined by the cooperative profile. If either (or both) agents deviate from the
actions defined in the cooperative profile, enter the punishment profile for the remainder
of the game.
One final term must be introduced. Letting K = max{ε, 1 − β}, the full auditing function
defining the probability of audit, γ(UaP,t
), is:14
γ(UaP,t
)=
(−UaP,t−K
2
)((2β−1)(1−δ)−U0
Aδ−β(ρ)δ
)if Ua
P,t≤ K
1 otherwise
In defining the expected utilities, it is useful to let γ =(K
2
) ( (2β−1)(1−δ)−U0
Aδ−β(ρ)δ
), which is the ex-
pected probability that the principal will audit. With this in place, the expected utilities are
EUl = −βρ/(1− δ), EUm = −(1− β)ρ/(1− δ), and EUP = −γΨ/(1− δ).
To summarize the selected actions:
1. The principal will audit under two conditions: 1) Audit with probability defined by
function γ(UaP,t), and 2) audit in period t + 1 if xm,t 6= xl,t. If auditing occurs and the
terror cell is homogenous, disband the cell.
2. Terror cell leader sets a = f .
3. N/A
4. Agents set xl,t = xm,t = ρt.
5. N/A
6. The principal audits based on the contract defined in stage 1.
14It should be noted that this is not the only auditing function that would work.
27
7. N/A
8. The terror cell will never be homogeneous; the principal will always set b ∈ nr.
9. The game repeats.
The mechanics of how this auditing scheme works will be left to the final section. For audit-
ing to be a feasible strategy, a series of inequalities must hold. The first set of inequalities
pertains to δ exceeding some cutoff point. The second inequality is a technical condition that
allows the strategy of both agents matching action to the state of the world to be feasible.
It should be noted that for a sufficiently high discount rate, there will be Pareto-efficent (for
the agents) strategies that do not require this additional condition.
The section below is concerned with payoffs.
11.5 Principal’s Decision
The principal maximizes her expected utility by deciding to use transfer contracts, orga-
nizational contracts, or no contracts at all. Using the payoffs from the respective strategies
gives Proposition A1:
Proposition A1 : Suppose Condition 1, Condition T, Assumption C, and As-
sumption T hold, and δ → 1. If organizational contracts lead to the worst-case for the
principal, then an equilibrium exists where the principal selects:
• Transfer Contracting Strategy if γΨ/(1− δ) > θ(4β − 2)ρ and 2ρ > θ(4β − 2)ρ,
• Organizational Contracting with Auditing if γΨ/(1−δ) ≤ θ(4β−2)ρ and 2ρ > γΨ/(1−δ),
• No Contracting Strategy otherwise.
Proof: This follows from payoffs above.
12 Mechanics of Organizational Contracting with Auditing
For organizational contracting with auditing to be feasible, four deviations from the strat-
egy above must be considered. The first two deviations involve a heterogeneous cell. These
28
are when the foreign agent or domestic agent deviates and sets their action equal to some
point that is not the state of the world. The third deviation to consider has the principal
align with a domestic type actor and deviate from the strategy of matching action to the
state of the world for at least one round (and then returning to a heterogeneous cell and
matching action to the state of the world). The forth deviation is the terror cell leader not
matching action to the state of the world in the heterogeneous cell, then forming a homoge-
neous cell (thus avoiding the terror cell member’s punishment phase).
To summarize, to prevent these deviations, there must be three conditions on δ (that δ
must be greater than a fixed value), and one technical condition (which is discussed in detail
below. Letting K = max{ε, 1− β}, these are:
−1 + 3β2
2βρ− ρ≤ δ
−β2− 1 + 2β
−1 + ρ+ 2β − 2βρ≤ δ
(−Ua
P,t −K)( (2β − 1)
2(−U0A − β(ρ))
)≤ δ
1− δ
3β − 2
4β − 2≤ ρ
12.1 First Deviation: Cell member deviates from xm,t = ρt to xm,t = 1 for one
round
To keep the terror cell member (foreign type) from not matching action to the state of the
world, the threat of entering the “punishment phase” must be sufficiently high. This would
require:
−β[1− (1 + ρt)
2
]− (1− β)(1− ρt)−
δ(β(
12
)+ (1− β)(1− ρ))
1− δ≤ −β(1− ρt) +−δβ(1− ρ)
1− δ
The left side of the inequality is the terror cell member’s expected utility from deviating from
the “cooperative phase” in period t to xm,t = 1 and then entering the punishment phase. The
right hand side is the expected utility from cooperating in period t (and all periods after).
Note that the incentive to deviate will be greatest when ρt = 0. Thus, the cell member
29
does better cooperating when the inequality below holds:
−1 + 3β2
2βρ− ρ≤ δ
This constraint also demands what will be known as the “technical condition.” It must be
that the terror cell member’s min-max payoff is less than his cooperative payoff. This holds
when:
3β − 2
4β − 2≤ ρ
Intuitively, this condition implies that in expectation, ρ is not too far away from the foreign
member’s ideal point. When ρ is close to zero, then the foreign type does strictly worse by
matching action to type rather than simply playing his stage game Nash equilibrium. It
should be mentioned that this technical condition is necessary because this section is only
considering the case when agents match action to the state of the world. As was shown in
the proof for Proposition 0 (in the body of the main paper), for a sufficiently high discount
rate a range of efficient and feasible equilibria exist. This technical condition ensures the
xl,t = xm,t = ρt action profile falls within the efficient and feasible set for a sufficiently high
δ.
12.2 Second Deviation: Cell leader deviates from xl,t = ρt to xl,t = 0 for one
round
To keep the terror cell leader (domestic type) from not matching action to the state of the
world, the threat of entering the “punishment phase” must be sufficiently high. This would
require:
−β(ρt2
)− (1− β)(ρt)−δ(β
(12
)+ (1− β)(ρ))
1− δ≤ −βρt +− δβρ
1− δ
The left side of the inequality is the terror cell leader’s expected utility from deviating from
the “cooperative phase” in period t to xl,t = 0 and then entering the punishment phase.15
The right hand side is the expected utility from cooperating in period t (and all periods after).
Note that the incentive to deviate will be greatest when ρt = 1. Thus, the cell member
15A deviation to xl,t = 0 is the most valuable deviation for the domestic agent, so satisfying this constraint would also satisfyany other deviation.
30
does better cooperating when the inequality below holds:
−β2− 1 + 2β
−1 + ρ+ 2β − 2βρ≤ δ
12.3 Third Deviation: Cell leader forms homogeneous cell for one round
Consider the setting where the principal deviates by partnering with a domestic type
agent for at least one round. There are two possible outcomes:
12.3.1 Preventing single round deviation
Here the terror cell leader forms a homogeneous terror cell for one round. In this setting
the terror cell member (domestic type) would select the stage-game Nash equilibrium strategy
and set xm,t = 0. Letting xl,t = ρt − ql denote the cell leader’s selected action (with ql ∈(0, ρt]), and letting π denote the probability that an audit occurs, the principal must define
the audit function to make the terror cell leader weakly worse off choosing this strategy:
−βρt − ql2− (1− β)ql + (1− π)
[−β(ρ)δ
1− δ
]+ π
[U0Aδ
1− δ
]≤ −βρt +− δβρ
1− δ
The left side of the inequality is the terror cell leader’s expected utility from deviating
from the “cooperative phase” in period t to forming a homogenous cell and setting xl,t = 0.
This would then invoke an audit with probability π, or the game would continue with a
heterogeneous cell with probability 1 − π. Note that doing this for one round and then
reverting back to the foreign type agent sets the agents back to the cooperative profile,
unless the cell leader is caught and the cell is disbanded. For the threat of audit to be a
sufficient deterrent, the following must hold:
βρt(1− δ)2(−U0
Aδ − β(ρ)δ)+
(32β − 1)(1− δ)−U0
Aδ − β(ρ)δql ≤ π
This defines the auditing scheme in relation to q, the amount each agent deviates from the
state of the world, and ρt, the state of the world. Importantly, both of these variables are
unobserved. However, the principal can condition the probability of audit on her realized
utility from the stage game (as will be shown below). In the case of a single round deviation,
it is useful to note the principal’s expected action utility has the property:
E[UaP,t
]= E [−ρt − |xm,t − ρt|+ εt] = E [−ql − ρt + εt] = −ql − ρt
31
12.3.2 Preventing multiple round deviation
The terror cell leader can also create a homogeneous terror cell for more than a single
round. If this is the case, the terror cell member and leader must select the same action
profile in all remaining rounds (lest they would trigger an audit with probability 1).
There are several ways forming a homogeneous cell for multiple rounds could work. In one
setting, the terror cell leader could plan to form a homogeneous cell for all rounds greater
than or equal to t, only stopping when the principal terminates the game. Alternatively, the
terror cell leader could establish a randomization procedure, where with some probability
he will terminate forming a homogeneous cell and revert to a heterogeneous cell. For an
auditing scheme to prevent both cases from occurring, it would be sufficient to prevent a
single-round deviation where both agents set xl,t = xm,t = ρt − q, with q ∈ (0, ρt].16 For the
auditing scheme to be effective, it must be that:
−β(ρt − q)− (1− β)q + (1− π)
[−β(ρ)δ
1− δ
]+ π
[U0Aδ
1− δ
]≤ −βρt +− δβρ
1− δ
The left side of the inequality is the terror cell leader’s expected utility from deviating from
the “cooperative phase” in period t to forming a homogenous cell and setting xl,t = xm,t =
ρt− q (these are the first two terms). This would then invoke an audit with probability π, or
the game would continue with a heterogeneous cell with probability 1− π. Note that doing
this for one round and then reverting back to the foreign type agent sets the agents back
to the cooperative profile, unless the cell leader is caught and the cell is disbanded. For the
threat of audit to be a sufficient deterrent, the threat of audit must increase with q by the
following inequality:
(2β − 1)(1− δ)−U0
Aδ − β(ρ)δq ≤ π
This defines the auditing scheme in relation to q, the amount each agent deviates from the
state of the world. While q is unobserved the principal can condition the probability of audit
on her realized utility from the stage game (as will be shown below). In the case of a single
round deviation, it is useful to note the principal’s expected action utility has the property:
E[UaP,t
]= E [−|xl,t − ρt| − |xm,t − ρt|+ εt] = E [−2q + εt] = −2q
It should be mentioned that this value is different that the value the expected utility takes
in the setting with a unilateral deviation.
16Note that while this is sufficient to prevent this type of action from happening, it may not necessarily be the most efficientauditing scheme to prevent this kind of deviation.
32
12.3.3 Preventing single and multiple round deviation: constructing the auditingfunction
A successful auditing function would prevent both types of deviation. To achieve this,
three issues must be considered. First is the rate at which the probability of audit increases
with q. In the inequalities above, the rate at which the probability of audit conditional on
q increases faster in the multiple-round deviation. Second, the probability of audit must be
non-negative and less than or equal to one. Third, in the case of a single-round deviation,
the expected probability of audit must be at least βρt(1−δ)2(−U0
Aδ−β(ρ)δ)regardless of what action the
terror cell leader selects. Through algebra, it can be shown that for K = max{ε, 1− β}, the
following audit procedure accommodates all these considerations:
γ(UaP,t
)=
(−Ua
P,t −K) ( (2β−1)(1−δ)
2(−U0Aδ−β(ρ)δ)
)if Ua
P,t≤ K
1 otherwise
Note that in order for this audit schedule to be feasible, it must be that(−Ua
P,t −K) ( (2β−1)
2(−U0A−β(ρ))
)≤
δ1−δ
12.4 Forth Deviation: Defecting and avoiding punishment
The final deviation to consider has the terror cell leader not matching action to the state
of the world in the heterogeneous cell, then forming a homogeneous cell (thus avoiding the
terror cell member’s punishment phase). This is prevented by the principal’s commitment
to audit in period t+ 1 when in period t action profiles do not match.
Part V
Leadership Targeting Analysis: Data,Alternative Explanations and Robustness
A Summary of Targeting Data, May 2007-April 2008
Below are summaries of the numbers of targeting events that occurred.
33
Level
Functional role Cell City Regional Senior
Admin 0 2 8 5
Facilitator 0 1 67 3
Finance 0 0 8 0
Intel/Security 1 8 4 0
Media 0 0 26 1
Military 28 14 18 1
Emir 0 24 25 8
Sharia 0 0 5 2
VBIED 5 8 19 0
Level
Location Cell City Regional Senior
Anbar 2 4 13 0
Baghdad 4 10 40 2
Bayji 0 5 3 1
Diyala 0 2 9 2
Kirkuk 0 1 1 1
Mosul 11 15 21 9
Salah al Din 2 6 18 2
Sulimaniyah 0 0 1 0
Functional Role
Location Admin Facil Finance Intel/Sec Media Mil Emir Sharia VBIED
Anbar 0 9 1 1 1 2 3 1 0
Baghdad 2 14 2 1 1 7 7 1 21
Bayji 2 3 0 1 0 1 2 0 0
Diyala 1 5 0 0 2 1 3 0 0
Kirkuk 0 0 0 0 0 0 3 0 0
Mosul 5 8 0 6 5 17 9 2 3
Salah al Din 1 11 1 0 1 4 9 1 0
Sulimaniyah 0 1 0 0 0 0 0 0 0
B Alternative Explanations
Future versions of this memo will consider more alternative explanations and will attempt
to address these. For now, the most compelling alternative explanations for the results are
below.
34
B.1 Alternative Explanation: Opportunistic Violence, Infighting, and Retalia-
tion
One explanation for the results above is that other Sunni or Shia groups were opportunisti-
cally increasing their attacks against AQI when the group had recently suffered a loss to their
leadership. In other words, the increase in local violence and local violence ratio stemmed
from other groups taking advantage of a destabilized AQI.
A second explanation would be that the in increase in local violence stemmed from infighting
within AQI to fill the leadership vacuum. Because the IBC dataset is unable to distinguish
violence against civilians from intra-group violence, the increase in recorded local violence
might stem from violence among members of AQI. This kind of behavior is also discussed in
Mexican cartels in Calderon et al. (2015).
A third explanation is that after a leadership strike, members of AQI retaliate against civil-
ians. After the capture or killing of a member of leadership, members of AQI might suspect
that civilians gave Coalition forces the necessary information to effectively target leadership.
For all cases, these explanations would create the pattern of leadership targeting and an
increase in civilian casualties.
B.2 Ruling out Alternative Explanations
The shortcoming of the explanations above is that they do not explain why the increase
in local violence ratio only seemed to occur for cell-level military leadership and not for
Emirs or upper-level military leadership. For each of these three explanations, there is no
distinguishing feature of cell-level military leadership that would only create the effect in
these cases. Instead, based on outside discussions of AQI embracing the M-form hierarchy
and giving cell leadership the ability to select operations, it seems plausible that removing
cell leadership left lower level members unconstrained, and this gave them the freedom to
act against civilians in the pursuit of local power or in an attempt to settle grievances.
C Robustness Checks
C.1 Population Normalized
The analysis in Table 2 is re-tested using population normalized leadership strikes and vi-
35
olence measures. The population levels in use are from the 2007 World Food Program’spopulation estimates, provided by the Empirical Studies of Conflict Program. The Table be-low demonstrates that normalizing for population does not substantively change the results.
(1) (2) (3) (4) (5) (6)
Sect Ratio I Sect Ratio II Fired Upon I Fired Upon II Sect I Sect II
F.D. Military, Cell 0.0283∗∗ 0.0283∗∗ -39.47∗∗∗ -39.47∗∗∗ 10.95+ 10.95+
(0.0108) (0.0108) (3.103) (3.103) (6.514) (6.514)
F.D. Military, Upper -0.0334 -0.0334 -0.451 -0.451 -30.51 -30.51
(0.0248) (0.0248) (14.41) (14.41) (21.72) (21.72)
F.D. Emir -0.0278∗∗ -0.0278∗∗ 14.79∗ 14.79∗ -9.981 -9.981
(0.0112) (0.0112) (6.648) (6.648) (6.465) (6.465)
F.D. Facilitator/Finance -0.00172 -0.00172 0.861 0.861 0.0370 0.0370
(0.00855) (0.00855) (2.786) (2.786) (9.950) (9.950)
F.D. Media -0.0229∗∗∗ -0.0229∗∗∗ 12.55 12.55 -15.58 -15.58
(0.00469) (0.00469) (16.80) (16.80) (9.718) (9.718)
F.D. Sharia -0.00938 -0.00938 5.727 5.727 -6.451 -6.451
(0.0248) (0.0248) (26.99) (26.99) (24.15) (24.15)
Full Controls No Yes No Yes No Yes
R-Squared 0.0322 0.0322 0.116 0.116 0.0774 0.0774
N 77 77 77 77 77 77
Standard errors in parentheses
All regressions run at governorate-month level with standard errors clustered at governorate level.
+ p < .15, ∗ p < 0.1, ∗∗ p < 0.05, ∗∗∗ p < 0.01
C.2 Dropping Governorates and Months
Another common robustness check for outliers is to re-run the above, though iterating
through by dropping a governorate. The results are below, with columns labeled by the
removed governorate, and with the local killing ratio as the dependent variable. Note that
Sulimaniyah is not included, as no cell level leadership targeting was recorded in this gover-
norate.
36
Table 1: Impact of Leadership Decapitation on Profile of Violence, Limited Area, Pop Controls
(1) (2) (3) (4) (5)Anbar Baghdad Diyala Ninewa Salah al Din
F.D. Military, Cell 0.0254∗∗ 0.0171∗∗ 0.0256∗∗∗ 0.0269∗∗∗ 0.0243∗∗
(0.012) (0.039) (0.004) (0.002) (0.021)Full ControlsR-Squared 0.0462 0.0459 0.0359 0.0185 0.0403N 66 66 66 66 66
p-values in parentheses
All regressions run at governorate-month level with standard errors clustered at governorate level.+ p < .15, ∗ p < 0.1, ∗∗ p < 0.05, ∗∗∗ p < 0.01
This table reveals that no governorate appears to be driving the result; the correlationbetween leadership targeting and the local killing ratio remains positive and statisticallysignificant at the 5% level in all cases.
A similar pattern emerges dropping individual months, where all results remain positiveand statistically significant at the 15% level.
Table 2: Impact of Leadership Decapitation on Profile of Violence, Limited Area, Pop Controls
(1) (2) (3) (4) (5)Jun 07 Jul 07 Aug 07 Sep 07 Oct 07
F.D. Military, Cell 0.0226∗∗ 0.0246∗∗∗ 0.0258∗∗∗ 0.0178∗∗ 0.0247∗∗
(0.013) (0.009) (0.000) (0.036) (0.012)Full ControlsR-Squared 0.0427 0.0368 0.0337 0.0222 0.0471N 70 70 70 70 70
p-values in parentheses
All regressions run at governorate-month level with standard errors clustered at governorate level.+ p < .15, ∗ p < 0.1, ∗∗ p < 0.05, ∗∗∗ p < 0.01
Table 3: Impact of Leadership Decapitation on Profile of Violence, Limited Area, Pop Controls
(1) (2) (3) (4) (5) (6)Nov 07 Dec 07 Jan 08 Feb 08 Mar 08 Apr 08
F.D. Military, Cell 0.0226+ 0.0278∗∗∗ 0.0226∗∗∗ 0.0340∗∗∗ 0.0328∗∗∗ 0.0220∗∗∗
(0.119) (0.003) (0.006) (0.001) (0.001) (0.002)Full ControlsR-Squared 0.0345 0.0473 0.0548 0.0615 0.0442 0.0322N 70 70 70 70 70 70
p-values in parentheses
All regressions run at governorate-month level with standard errors clustered at governorate level.+ p < .15, ∗ p < 0.1, ∗∗ p < 0.05, ∗∗∗ p < 0.01
37
References
BBC (2010) US says 80% of al-Qaeda leaders in Iraq removed.
Berman, E. and Laitin, D. D. (2008) Religion, Terrorism and Public Goods: Testing the club model, Journalof Public Economics, 92, 1942–1967.
Biddle, S., Friedman, J. A. and Shapiro, J. N. (2012) Testing the Surge: Why Did Violence Decline in Iraqin 2007?, International Security, 37, 7–40.
Brown, V. and Rassler, D. (2013) Fountainhead of Jihad, Hurst & Company, London.
Bueno de Mesquita, E. (2005) The Quality of Terror, American Journal of Political Science, 49, 515–530.
Calderon, G., Robles, G., Dıaz-Cayeros, A. and Magaloni, B. (2015) The beheading of criminal organizationsand the dynamics of violence in mexico, Journal of Conflict Resolution, p. 0022002715587053.
CTC (2005) Zawahiri’s Letter to Zarqawi, Tech. rep., Combating Terrorism Center, West Point, NY.
CTC (2007) Analysis of the State of ISI, Tech. rep., Combating Terrorism Center, West Point, NY.
Dressler, J. (2012) The Haqqani Network: A Foreign Terrorist Organization, Tech. rep., Institute for theStudy of War, Washington D.C.
Dressler, J. A. (2010) The Haqqani Network: From Pakistan to Afghanistan, Institute for the Study of War.
Fishman, B. (2009) Dysfunction and Decline: Lessons Learned From Inside Al Qa’ida in Iraq, Tech. rep.,Combating Terrorism Center, West Point, NY.
Wilson, J. Q. (1989) Bureaucracy: What government agencies do and why they do it, Basic Books.
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