A dynamic model of insurgency the case of the war in iraq (Larry Blank)

Peace Economics, Peace Science and

Public Policy

A Dynamic Model of Insurgency: The Case of

the War in Iraq

New Mexico State University, larryb@nmsu.edu

†New Mexico State University, cenomoto@nmsu.edu

‡New Mexico State University, dgegax@nmsu.edu

New Mexico State University, jmcgucki@nmsu.edu

††New Mexico State University, caisimmons@aol.com

Copyright c 2008 The Berkeley Electronic Press. All rights reserved.

Blank et al.: A Dynamic Model of Insurgency

1. Introduction

On March 20, 2003, the U.S.-led coalition moved into Iraq the day after President Bush declared war. According to Keegan (2005), the U.S.-led coalition faced an Iraqi army of 400,000 troops, twice its own size.1 Keegan further stated that the U.S.-led coalition was surprised when it met little resistance in the early days of the war:

The Ba'ath leaders and their party officials had disappeared, just as the army and the Republican Guard had disappeared. The disappearance of the soldiers was easily explained. They had taken off their uniforms and become civilians again (p. 3).

After the fall of Baghdad, the coalition soon began its reconstruction efforts building schools, hospitals, roads, the electrical power system and other parts of the Iraqi infrastructure. It became apparent that while many Iraqis welcomed the coalition and the prospect of democracy, there were others that despised what they felt was an occupational force. The kidnappings, attacks, and bombings, by the insurgents began. Keegan (2005) stated, "such attacks persisted during the winter of 2003-04 and by March 2004 had swelled into a full-scale insurgency."2 The coalition responded to insurgent attacks by sending more troops to Iraq. The presence of more U.S.-led troops, however, further inflamed the insurgents who stepped up their attacks. These attacks were not only against the coalition troops, but were also against Iraqi citizens who were thought to have any ties to the coalition; especially those in the newly created Iraqi police force and the new Iraqi army. The notion that the presence of the coalition increased the size of the insurgency is given by Hegghammer (2006) in which he stated,

There seems to be a broad consensus among terrorism experts that the U.S.-led invasion of Iraq in March 2003 has contributed negatively to the so-called "global war on terror." According to many analysts, the war and the subsequent occupation have increased the level of frustration in the Islamic world over American foreign policy and facilitated recruitment by militant Islamist groups. Moreover, Iraq seems to have replaced Afghanistan as a training ground where a new generation of Islamist militants can acquire military expertise and build personal relationships through the experience of combat and training camps (p. 11).

1 John Keegan, The Iraq War, (New York: Random House, 2005), p. 2. 2 Ibid, p. 220.

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There was also an escalation of sectarian violence. The Sunnis, who accounted for about 20% of Iraq's population, were now thrown into a position of being marginalized by the majority Shi'a and Kurdish leaders who dominated Iraq's transitional government after the overthrow of Saddam's regime. While Sunni extremists attacked the U.S.-led coalition who were seen as defenders and supporters of the new Iraqi government, other groups such as the Shi'a militiamen were also believed to be major contributors to the ongoing violence.3 The Shi'a militia were claimed to have been responsible for many attacks on Sunni civilians in retribution for the terrorist attacks of Sunni extremists that have killed many Shi'a. As Ghosh (2006) stated, "Civil wars are notoriously difficult to mediate without taking one side and it doesn't help that in Iraq, battling [Insurgent] Shi'ites and Sunnis seem to agree on only one thing: that the U.S. is ultimately to blame for the mess."4

The purpose of this paper is to develop a dynamic model of insurgency. The results will give insight into what has happened in the past and what may happen in the future, regarding interactions between the U.S.-led coalition and the insurgents within specifically defined definitions. The outline of the paper is as follows. In the next section, the literature on dynamic mathematical models of war will be reviewed. In section 3, a dynamic model of insurgency will be developed and in section 4, policy implications from the model will be discussed. In section 5, data from Iraq will be used to see if the model explains what has actually occurred and a summary and conclusions will be presented in the final section.

2. Dynamic Mathematical Models of War

Perhaps the most referred to book on the mathematics of war is that by Lanchester (1995). His original model of war was further elaborated on by Braun (1986), Onoda (1999), and Epstien (1985). Lanchester's original system of differential equations was given as:

dbdt = − rc

drdt = − bk

3 Jeffrey Gettleman, "Shiite Fighters Clash with G.I.'s and Iraqi Forces," The New York Times, Monday, March 27, 2006, p. A1, A10.

4 Aparisim Ghosh, "An Eye For an Eye," Time Magazine, March 6, 2006, p. 24.

Blank et al.: A Dynamic Model of Insurgency

where b represented the size of the "blue" force, r represented the size of the red force, and c and k are constants that represented the combat effectiveness of the red and blue forces respectively. The system of differential equations shows that the presence of the red force lowers the size of the blue force and the presence of the blue force lowers the size of the red force through combat losses.

Richardson (1978) specified a second type of conflict model or model of war. According to this model,

dxdt = ay −bx + g

dydt = cx −dy +m

where x represents the armaments of one nation, y represents the armaments of the other nation and a, b, c, d, g, and m are positive constants. This system of simultaneous differential equations suggests that one nation's armaments cause the other nation to increase its armaments. There is also a cost of armaments which decreases the nation's efforts to arm itself. Another type of mathematical model of conflict has been developed which focuses on guerrilla warfare. Sandler and Hartley (1995) stated, “Guerrilla warfare is another tactic used by an armed movement to overturn a government for the purpose of political change… Guerrillas rely on surprise and cover to harass and defeat government troops (p. 307).”

Intriligator and Brito (1988, 1990 and 1992), have modeled guerrilla warfare in different settings. In their 1988 paper, they specify their model as follows:

dxdt1 = α x1 x3 − β x1 x2

dxdt2 = γ x2 − δ x1 x2

dxdt3 = ε x1 − λ x2

Where x1(t) = number of guerrillas at time t, x2(t) = number of regular (government) soldiers at time t, x3(t) = size of the population controlled by the

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guerrillas at time t, and α, β, γ, δ, ε, and λ are positive constants (pp. 235-6).The first term in the first equation suggests that the number of guerrillas increases due to the interaction between the guerrillas and the size of the population under their control. Thus this first term shows the “recruitment” effect of increasing the size of the guerrilla force due to either more guerrillas present or having control over a larger share of the population. The second term in the first equation suggests that guerrilla losses are due to the interaction between the guerrillas and the government soldiers. Thus a large guerrilla force or a large government force will lead to more guerrilla losses.

In the second equation, the government’s recruitment effort increases with the size of the government force. Sandler and Hartley’s explanation for this is that the size of the government force is a proxy for the power or ability of the government to raise an armed force. The larger this ability or power, the greater will be the recruitment effort of the government. On the other hand, the size of the government force decreases with the interaction between the guerrillas and the government soldiers. The third equation is self explanatory with the size of the population controlled by the guerrillas being directly related to the number of guerrillas and inversely related to the number of government soldiers. The outcome of the model is dependent upon the coefficients of the x’s in the above three equations.

A recent mathematical model of terror support and recruitment has been offered by Faria and Arce (2005). They stated, “Terrorism is commonly defined as an act of violence against civilians in order to achieve political or religious goals. Among other objectives, terrorists seek publicity to make their cause known in order to increase its popular support (p. 263).” Faria and Arce develop a public support for terrorist activities function which depends on past terrorist activity. They also develop a terrorist recruitment function which depends on public support for terrorist activities. The authors state that there is an important difference between terrorist recruitment and guerrilla recruitment. Guerrilla recruitment depends upon the interaction between the guerrillas and the population they control. Terrorist recruitment can be a function of popular support for terrorist activity but it can also be independent of terrorist activity wherein hardliners join the movement simply because of their deep-seated beliefs. Faria and Arce go on to develop an equation describing terrorist attacks which depends on the number of terrorist recruits.

Sandler and Hartley (1995) define "insurrection" as another type of conflict. They stated, "An insurrection is a politically based uprising intended to overthrow the established system of governance and to bring about a redistribution of income... Dynamic and static considerations are germane and must be delineated when analyzing the interactive behavior of the government and insurgents... Information is either imperfect or one-sided when characterizing

Blank et al.: A Dynamic Model of Insurgency

insurrections; e.g., the government may not know the true strength of the insurgents (p. 307).”

In the next section, the above models will be combined and extended to form a model of insurgency.

3. A Dynamic Model of Insurgency

In the Richardson model, the armaments of one nation are directly related to the armaments of the opposing force. An arms race is one possible outcome. In the Lanchester model, the size of one army is inversely related to the size of the opposing force due to combat losses. A "reinforcement rate" is also introduced by Braun (1986) that only depends on t (time). In models of guerrilla warfare, recruitment by guerrillas depends on the interaction between the guerrillas and the population they control and guerrilla losses depend on the interaction between the guerrillas and the government soldiers. In models of terrorism, recruitment of terrorists depends on popular support for terrorist activity as well as the beliefs of hardliners who truly believe in the cause. In the case of the war in Iraq and the insurgency/terrorism that has developed, a model must be developed that includes features of all of the aforementioned models as well as features that are specific to the Iraq war. In the case of terrorism and insurgency, the terrorists and insurgents will not be wearing uniforms. They can engage in surprise attacks and disappear into the general population. Insurgents are defined as groups or factions who actively plan and/or carry out attacks against military and support personnel in the U.S.-led coalition. The U.S.-led coalition includes Iraqi police, military, United States armed forces and the 48 other member countries of the coalition.5 With this definition, we exclude sectarian and civil war violence because it is not against military personnel and may or may not end when the U.S.-led coalition leaves. The terrorists and insurgents may or may not be organized. They may consist of many small or large groups acting independently or they may consist of individuals acting alone. All insurgents fall under the definition whether they are acting alone or connected with Al Qaeda. The authors acknowledge that the different groups have different motivations, interests and methods but they are attacking U.S. led coalition forces and all have the same general goal, the death of coalition personnel. When the United States leaves Iraq, the Iraqi police and military will still be targets but not as U.S.-led personnel or coalition. Insurgents may not be interested in controlling territory but instead, may be politically motivated. Sandler and Hartley (1995), state that individuals can be involved in

5 List available at: http://www.whitehouse.gov/news/releases/2003/03/20030327-10.html

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terrorism and not insurrection (even though insurrections can host terrorist activity). That is, terrorists may commit bombings, kidnappings, and attacks, to promote their ideology and/or gain the release of prisoners, without the intention of overthrowing the government (p. 308). In the war in Iraq, the insurgents are interested in the removal from Iraq of what they see as the occupying force (the coalition) and they wish to have their place and stature in Iraq preserved, even if this results in the destruction of the present Iraqi government. According to the Iraq Study Group Report (2006),

Most attacks on Americans still come from the Sunni Arab insurgency. The insurgency comprises former elements of the Saddam Hussein regime, disaffected Sunni Arab Iraqis, and common criminals. It has significant support within the Sunni Arab community. The insurgency has no single leadership but is a network of networks. It benefits from participants’ detailed knowledge of Iraq’s infrastructure, and arms and financing are supplied primarily from within Iraq. The insurgents have different goals, although nearly all oppose the presence of U.S. forces in Iraq. Most wish to restore Sunni Arab rule in the country. Some aim at winning local power and control (p. 4).

To model this interaction between the insurgents in Iraq and the U.S.-led coalition, the presence of U.S.- led forces is characterized as having two opposing effects on the number of attacks by insurgents: 1) it leads to a decrease in the number of attacks by insurgents due to combat losses for the insurgents and 2) it leads to an increase in the number of insurgent attacks due to the added number of recruits who oppose U.S.- led forces in Iraq and due to the sheer number of U.S. and coalition troops who are now in harms way. These two effects are illustrated as follows:

where "I" represents the number of attacks by insurgents, "C" represents the number of U.S.-led coalition forces, γc is the positive combat effectiveness coefficient of the coalition , and ri represents the positive effect on insurgent attacks on the coalition forces due to the mere presence of the coalition. The first term suggests that insurgent attacks will decrease with the coalition presence. The symbol ri is used to represent the recruitment rate for the insurgents since a larger coalition presence leads to more insurgent recruits, directly or indirectly, who oppose what they view as an invasion of their land. The symbol ri, however,

Blank et al.: A Dynamic Model of Insurgency

is only loosely used as a recruitment rate since insurgent attacks (I) can increase as the size of the coalition (C) increases, simply due to the fact that more coalition troops are in harms way even though there are no new insurgent recruits. Thus equation (1) differs from the Lanchester and Intriligator/Brito models in that number of insurgent attacks rather than number of insurgents (or guerrillas) is used. In this framework, insurgent attacks (I) do not appear in the right-hand side of the equation. The reason for this is that it is assumed that a larger number of insurgent attacks does not directly lead to more insurgent recruits or more insurgent attacks over time. The recruitment of insurgents and insurgent attacks only depend on the presence of the U.S.-led coalition.6 However, the size of the coalition over time does depend on number of insurgent attacks (dC/dt = g(I)) as will be seen shortly. Thus the change in insurgent attacks, dI/dt, only indirectly depends on insurgent attacks. This differs from guerrilla warfare wherein guerrilla recruitment crucially and directly depends on the interaction of the guerrillas and the population they control. In fact, insurgency attacks and/or recruitment of insurgents are more in line with the terrorist model of Faria and Arce (2005) wherein terrorist activity can be dependent on past terrorist activity but it can also be independent of past terrorist activity due to the existence of hardliners that are recruited or attack more often, simply due to their deep-seated beliefs that their land is being occupied by foreign troops.

For similar reasons, insurgent attacks (I), do not lead to fewer insurgent attacks over time. (dI/dt is again, not directly a function of I). The mere presence of the U.S.-led coalition (on the right-hand side of equation (1)) can lead to fewer insurgent attacks due to the added cost imposed on the insurgents of initiating an attack. There need be no interaction between insurgents and the coalition to cause insurgent attacks to decrease. In models of guerrilla warfare, interaction between guerrillas and government soldiers is crucial in determining guerrilla losses.

An equation similar to equation (1) can be derived showing how insurgent attacks (I) affect the size of the coalition (C) over time. A given number of attacks by insurgents (I) leads to a decrease in U.S.-led forces (C) over time due to combat losses but it also leads to more U.S. troops sent to Iraq to decrease the number of attacks. These two effects are shown below.

where again "I" represents the number of attacks by insurgents, "C" represents the coalition size, γi represents the combat effectiveness of the insurgent attacks and is positive, and rc represents the recruitment rate of the coalition which is

6 See definition of Insurgent on page 9.

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positive. In this model, more coalition troops are recruited and sent into Iraq based on the number of insurgent attacks and not on the number of insurgents.7 It is these two opposing effects of "C" on dI/dt and "I" on dC/dt that differentiates this model from past models of conflict. Equations (1) and (2) give rise to the following system of simultaneous differential equations:

dIdt = (ri − γ c )C

(3)

dCdt = (rc − γ i )I

where ri and rc, the recruitment rate coefficients of the insurgents and the coalition, respectively, and γi and γc , the combat effectiveness coefficients of the insurgents and coalition, are all positive. Thus in the first equation of (3), insurgent attacks (I) increase over time given the coalition presence (C), if the recruitment rate of insurgents exceeds insurgent losses due to the combat effectiveness of the coalition. Similarly for the second equation in (3), the size of the coalition increases over time for a given number of insurgent attacks, if the recruitment rate of the coalition exceeds coalition losses. The system given in (3) can therefore be written as

dIdt = δ iC

(4)

dCdt = δ c I

where δi and δc are the "net" recruitment rates of the insurgents and coalition, respectively (recruitment rates minus losses due to combat).

There are several possible solutions to this system depending on the signs

7 Alternatively, the interaction between number of insurgents and size of the coalition could be used in place of insurgent attacks (I) in the right-hand side of equation (2). This model would then be similar to the guerrilla warfare models of Intriligator and Brito although the guerrilla warfare models still only allow for government forces to directly decrease the size of the guerrilla force and vice versa. This model of insurgency allows for the coalition presence to decrease and increase the number of insurgent attacks, and vice versa.