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

Blank et al.: A Dynamic Model of Insurgency

of the net recruitment rate coefficients, δi and δc. The first case considered (Case

where A and B are constants.8 The equilibrium solution to (4) is I = 0 and C = 0. C=0 when the U.S.-led coalition is gone and the Iraqi police and military are no longer supported by the United States. This equilibrium is a result of the insurgency model in which dI/dt = dC/dt = 0 only when there is no coalition presence and no further insurgency attacks. It is assumed in the model of insurgency that the coalition is not concerned about permanent occupation of the territory and the insurgent attacks are only in response to the U.S.-led coalition presence. While the absence of the U.S.-led coalition may result in a continuing civil war between the Sunnis and the Shiites, this conflict would be part of a larger conflict model that goes beyond the dynamic model of insurgency specified in the system of equations given in (4).

The equilibrium, I = 0 and C = 0, is furthermore an unstable solution since the eigenvalues of the system are ± δ iδ c . 9 The orbits or phase trajectories of

the system are given by the equation dC/dI = (δcI)/(δiC).10 Integrating this separable equation results in δiC2 -δcI2 = V, where V is a constant, which yields the curves plotted in Figure 1.

Only the quadrant with positive values of "I" and "C" is relevant since the variables represent the number of insurgent attacks and the size of the U.S.-led coalition, respectively. The solution given in (5) and also shown in Figure 1, shows that there is no winner in this case. With positive net recruitment rates, δc

Peace Economics, Peace Science and Public Policy, Vol. 14 [2008], Iss. 2, Art. 1

and δi both positive, the number of attacks by insurgents increases with the size of the coalition and the size of the coalition increases with the number of insurgent attacks. There is no end to the conflict and the size of the coalition and insurgent attacks continually increase over time.

Figure 1. δi > 0 and δc > 0

[dC/dt = 0] C

- +

In case 2, the net recruitment rates for the insurgents and coalition in the equation system (4) are both negative (δi < 0 and δc < 0). The presence of U.S.- led forces leads to a decrease in the number of insurgent attacks and insurgent attacks lead to a decrease in the number of coalition troops since the combat effectiveness of each side outweighs recruiting efforts on both sides. The equilibrium solution to the system is again I = 0, C = 0, which also turns out to be an unstable solution with the same eigenvalues as given in case 1. The orbits or phase trajectories are given by the same equation as in case 1 and again the phase diagram is that of a saddle point. The directional arrows on the phase trajectories

Blank et al.: A Dynamic Model of Insurgency

are, however, different in this case. This is illustrated in the phase diagram in Figure 2.11

Figure 2. δi < 0 and δc < 0

[dC/dt = 0] C

+ -

V = 0

+ I

Again, the relevant quadrant contains positive "I" and "C" values and the phase trajectories are again given by, δiC2 - δcI2 = V, where δiC2 < 0 and - δcI2 > 0 since δi < 0 and δc < 0 in case 2. The coalition (C) will win if C > 0 and I = 0 [the coalition is still present when the insurgent attacks have been eliminated which occurs at a point such as R in figure 2] which occurs when V < 0. Thus, a winning result for the coalition (V < 0) requires increasing the combat effectiveness of the coalition (which makes δi more negative), lowering the recruitment rate of the insurgents (which makes δi more negative), reducing the combat effectiveness of the insurgents (which makes δc less negative), or

11 See Braun page 405.

Peace Economics, Peace Science and Public Policy, Vol. 14 [2008], Iss. 2, Art. 1

increasing the recruitment rate of the coalition (which makes δc less negative). If V > 0, the coalition will be eliminated before the insurgent attacks stop and the insurgents will win the conflict. The conflict ends at a point such as S in Figure 2. The final possibility is if V = 0. Here both the coalition and insurgent attacks are jointly eliminated, resulting in a tie.12

In case 3, the net recruitment rate of the insurgents is negative (δi < 0) but the net recruitment rate of the coalition is positive (δc > 0). The coalition presence decreases the number of insurgent attacks on net and insurgent attacks lead to an increase the size of the coalition on net. This leads to the following system of simultaneous differential equations:

dIdt = − ζ iC

(6)

dCdt = δ c I

where ζi ≡ - δi . In this example, ζi is the negative of the net recruitment rate of the insurgents and is positive as is δc , the net recruitment rate of the coalition. The general solution to (6) is thus given by:

The eigenvalues of this system are ± (ζ iδ c ) j , where j is the imaginary unit, and

the phase diagram for this system appears in Figure 3. As can be seen from the directional arrows on the phase trajectories (quadrant with positive “I” and “C” values), the coalition will win the conflict. The intuition of this case is clear. If the coalition’s presence leads to a decrease in the number of insurgent attacks over time on net (a negative net recruitment rate for the insurgents, δi < 0 and dI/dt = δiC) and the insurgent attacks lead to an increase in the number of U.S.-led troops over time on net (a positive net recruitment rate for the coalition, δc > 0 and

12 Ibid

Blank et al.: A Dynamic Model of Insurgency

dC/dt = δcI), the U.S.-led coalition will win the conflict. In Figure 3 (northeast quadrant), the insurgent attacks (I) will be eliminated while the coalition remains in Iraq. The coalition can help bring about this result by again lowering the recruitment rate of the insurgents (which makes δi more negative), raising the combat effectiveness of the coalition (which makes δi more negative), raising the recruitment rate of the coalition (which makes δc larger and positive), and reducing the combat effectiveness of the insurgents (which makes δc larger and positive).

Figure 3. δi < 0 and δc > 0

[dC/dt = 0] C

- +

-

[dI/dt = 0]

+ I

In case 4, the net recruitment rate of the insurgents is positive (δi > 0) and the net recruitment rate of the coalition is negative (δc < 0). The presence of U.S.-led forces (the coalition) leads to an increase in the number of insurgent attacks on net and insurgent attacks lead to a decrease in the number of U.S.-led troops on net. This system of differential equations is shown below.

Peace Economics, Peace Science and Public Policy, Vol. 14 [2008], Iss. 2, Art. 1

dIdt = δ iC

(8)

dCdt = − ζ c I

where ζc ≡ - δc . In this example, ζc is the negative of the net recruitment rate of the coalition and is positive as is δi , the net recruitment rate of the insurgents.

The eigenvalues of the system are ± (ζ cδ i ) j , and the phase diagram of the

system appears in Figure 4. What differentiates case 4 from case 3 is the direction of the arrows on the phase trajectories. In case 4, the insurgents win over the U.S.-led forces. The reason is clear. If the presence of the U.S.-led coalition leads to more insurgent attacks over time on net (due to a positive net recruitment rate for the insurgents, δi > 0 and dI/dt = δiC) and the insurgent attacks lead to a decrease in the number of U.S.-led troops over time on net (due to a negative net recruitment rate for the coalition, δc < 0 and dC/dt = δcI), the insurgents will win the conflict. In Figure 4, the conflict ends in the northeast quadrant with an eliminated U.S.-led coalition.

Figure 4. δi > 0 and δc < 0

[dC/dt = 0]

C

+ -

+

[dI/dt = 0]

- I

Blank et al.: A Dynamic Model of Insurgency

These four cases that have been illustrated are summarized in Table 1. In the next section, policy implications from the model will be presented.

Table 1. Summary of Four Cases

The dynamic model of insurgency is given by the following system of simultaneous equations:

dIdt = (ri − γ c )C = δ iC dCdt = (rc − γ i ) I = δ c I

where the variables are defined as follows:

I ≡ the number of insurgent attacks on the coalition C ≡ the size of the coalition

ri ≡ the recruitment rate coefficient of the insurgents rc ≡ the recruitment rate coefficient of the coalition

γc ≡ the combat effectiveness coefficient of the coalition γi ≡ the combat effectiveness coefficient of the insurgents

δi and δc are defined to be the net recruitment rate coefficients (recruitment rates minus combat losses due to the combat effectiveness of the opposing side) of the insurgents and coalition, respectively.

The model yields four cases:

Case (1): δi > 0 and δc > 0 and the net recruitment rates of the insurgents and coalition are both positive. For a given level of (C), insurgent attacks increase over time. For a given level of (I), the size of the coalition increases over time.

Outcome: Neither side wins and the conflict escalates with C and I increasing over time.

Case (2): δi < 0 and δc < 0 and the net recruitment rates of the insurgents and coalition are both negative. For a given level of (C), insurgent attacks decrease over time. For a given level of (I), the size of the coalition decreases over time.

Outcome: For the coalition to win the conflict (C > 0 and I = 0), certain conditions must be satisfied. Movement toward the conditions for coalition victory require a) increasing the combat effectiveness of the coalition, b) lowering the recruitment rate of the insurgents, c) reducing

the combat effectiveness of the insurgents, or d) increasing the recruitment rate of the coalition.

Case (3): δi < 0 and δc > 0 with a negative net recruitment rate of the insurgents and a positive net recruitment rate of the coalition. For a given level of (C), insurgent attacks decrease over time. For a given level of (I), the size of the coalition increases over time.

Outcome: The coalition wins the conflict.

Case (4): δi > 0 and δc < 0 with a positive net recruitment rate of the insurgents and a negative net recruitment rate of the coalition. For a given level of (C), insurgent attacks increase over time. For a given level of (I), the size of the coalition decreases over time.

Outcome:The insurgents win the conflict.

Peace Economics, Peace Science and Public Policy, Vol. 14 [2008], Iss. 2, Art. 1

4. Discussion and Policy Implications

The model as outlined in the previous section, contains four parameters that determine the outcome of the war: 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. Each of these parameters affects the outcome of the conflict in a very predictable way. If the recruitment rate coefficient of the insurgents sufficiently increases and/or the combat effectiveness of the insurgents sufficiently increases, a victory for the insurgents results. If the recruitment rate coefficient of the coalition sufficiently increases and/or the combat effectiveness of the coalition sufficiently increases, a victory for the coalition results. These four coefficients are not fixed throughout the conflict but vary continuously from week to week and day to day due to changing tactics, strategies, policies, and expectations. The combat effectiveness of the coalition can be increased for example, through the use of better weapons, superior tactics, and superior leadership. The recruitment rate of the insurgents can be reduced by providing better job and educational opportunities for those who would otherwise join the insurgency. This recruitment rate can also be reduced by guarding the borders and working with neighboring nations to reduce the inflow of foreign fighters into Iraq who would join the insurgency. (The Brookings Institution Report, “Iraq Index: Tracking Variables of Reconstruction & Security in Post-Saddam Iraq,” gives estimates of 800 to 2000 foreign fighters in the insurgency from April 2006 to September 2006). A "soft partition" as suggested by Senator Biden and others wherein Iraq is separated into sectarian regions, each with its own local government and police force, with some sort of loose federal government to oversee the entire nation and distribution of oil revenues, may also help lower the overall level of violence in Iraq and reduce the recruitment rate of the insurgency. The combat effectiveness of the insurgents can be reduced by equipping the coalition troops with heavily armored vehicles and better surveillance. Better communication with the Iraqi citizens may also lead to more information about the insurgency, its movements, and its plans of attack which would further reduce the combat effectiveness of the insurgents. Finally, a sufficient increase in the recruitment rate of coalition troops can lead to a coalition victory as predicted by the model.

At the end of 2006 and in January 2007, a “surge” in U.S. troops to Iraq was being discussed. Wright and Baker (2006, Dec. 19, Washingtonpost.com) stated, “The Bush administration is split over the idea of a surge in troops to Iraq, with White House officials aggressively promoting the concept over the unanimous disagreement of the Joint Chiefs of Staff, according to U.S. officials familiar with the intense debate… The Pentagon has cautioned that a modest surge could lead to more attacks by al-Qaeda, provide more targets for Sunni

Blank et al.: A Dynamic Model of Insurgency

insurgents and fuel the jihadist appeal for more foreign fighters to flock to Iraq to attack U.S. troops, the officials said.” In September of 2007, General David Petraeus reported that the surge had been successful in reducing violence in Baghdad. "Casualties among U.S. troops were down slightly in July and August but are surpassing last year's levels." (Time Magazine, September 17, 2007, pg. 32). In the context of the model of insurgency presented in section 3, the surge is represented as an increase in the recruitment rate of the coalition which should have contributed to an outcome favorable to the coalition.

With the surge, the coalition has also increased its ranks with former Sunni insurgents who are working with the coalition to fight al-Qaeda.

Al-Qaeda violently overplayed its hand and started randomly killing Sunnis who refused to ally themselves with the terrorist organization. And in some places, America won the Sunnis over the old-fashioned way: by paying them. The question is how widely the Anbar model can be applied elsewhere. It is easy to forget that Anbar is the one part of Iraq that is largely Sunni and thus doesn't suffer from the same kind of civil strife that upends order in other parts of the country (Time Magazine: September 17, 2007, pp. 34-35).

In the context of the model of insurgency, this coalition strategy has further increased the recruitment rate coefficient of the coalition and reduced the recruitment rate of the insurgents, both in the direction necessary to establish conditions for coalition victory.

If the surge continues to be successful in reducing violence in Iraq and if strategies such as a "soft partition" along with a withdrawal of U.S. forces to remote bases in Iraq occurs, then a situation such as depicted in figures (2--point R) and (3) could come about. In those cases the number of insurgent attacks is eliminated but a coalition presence still remains.

As stated in U.S. News & World Report:

There is agreement that the surge is, to an extent, working, in that sectarian violence is down in Iraq, according to military figures....

Others have floated the idea that perhaps a downturn in violence is the byproduct of the ethnic cleansing that has already cleared out many mixed neighborhoods in Baghdad and sent some 2 million Sunnis fleeing to Syria and Jordan (September 17, 2007, pg. 27).

Peace Economics, Peace Science and Public Policy, Vol. 14 [2008], Iss. 2, Art. 1

Thinkprogress.org stated:

Bush's surge has escalated ethnic cleansing. Shiites have cleared the western half of Baghdad of thousands of Sunnis, who once dominated the area.... Center for American Progress analyst Brian Katulis estimated that Baghdad, which once used to be a 65 percent Sunni majority city, is now 75 percent Shia (http://thinkprogress.org/2007/09/06/sunni-shia-baghdad/).

Thus a "soft partition of Iraq" is already occurring. To the extent that it is contributing to the overall reduction in violence in Iraq, it may be a strategy or end result in the coalition's favor serving to reduce the recruitment rate of the insurgents.

As U.S. News & World Report further stated:

The bulk of the U.S. debate, centers on how many U.S. troops are needed in Iraq and for how long. At the heart of this debate are very real constraints on the U.S. military. Extending the surge past the spring would require lengthening the tours of the troops from 15 to 18 months--something top military leaders have repeatedly said they are unwilling to do given the extreme stresses of combat on their soldiers... but while the tasks ahead remain considerable, a smaller force level in Iraq would be far more sustainable. There remain, of course some calls in Congress for an immediate withdrawal of all troops. Privately, some military official estimate that the military could comfortably keep 10 brigades (which each number some 3,500 troops) in Iraq and three brigades in Afghanistan "indefinitely," (September 17, 2007, pg. 30).

In the model of insurgency presented in this paper in section 3, the equilibrium solution, C = 0, I = 0, with no coalition presence and an end to insurgent attacks (Cases 1 and 2), was shown to be an "unstable equilibrium," and thus not a viable solution. A victory for the coalition could occur in Cases 2 and 3, wherein I = 0 and C > 0, with an end to insurgent attacks but a remaining coalition presence. This could possibly occur if the "soft partition" continued to keep violence down and if the remaining coalition forces were used to guard the borders, continue training the Iraqi military, and fight the continued presence and disruptions caused by al-Qaeda.