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Document created: 1 September 2007
Air & Space Power Journal- Fall 2007
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Dr. Hank J. Brightman*
You, the Iraqi army and police forces, don’t walk alongside the occupiers, because they are your archenemy.”1 This call for solidarity amongst indigenous security forces (ISF) and domestic insurgents (DI) by Shiite cleric and leader Muqtada al-Sadr in April 2007 is simply the latest evidence in support of this researcher’s three years of applying game theory to the Iraqi conflict. Although other studies have examined Operation Iraqi Freedom from perspectives such as democratic nation building in an area of the world where such forms of government historically have not been the norm, this article represents the first known effort to apply the game-theory concepts of “Pareto improved” and “Pareto optimal” strategies (named after Italian economist Vilfredo Pareto) as well as “Nash” and “preferred” equilibriums (the former named after American mathematician John Nash) to the Iraqi conflict.
Specifically, this article examines how, through application of game theory to this model, US and coalition forces will ultimately suffer casualties at an increasing rate the longer they remain in Iraq. This will occur because both DIs and ISFs will turn away from attacking each other towards a point of mathematical corruption. At this theoretical point, American and coalition troops will become the target of broad-based DI attacks, with intelligence frequently provided by ISFs. For the purposes of this article, ISF refers to the Iraqi military as well as state and local police, and DI refers to the various domestic insurgent groups within Iraq.
In order to fully understand how two seemingly disparate entities—ISFs and DIs—will ultimately work together in an effort to improve both of their respective positions, one must examine the basics of game theory and the associated concepts of bargaining and equilibrium. In the following discussion, such terms as player, improved, optimal, corruption, preferred, and so forth, are used in their mathematical rather than their usual sense.
Developed by Merrill Flood and Melvin Dresher at the RAND Corporation in 1950, the “prisoner’s dilemma”—an activity often played out in college logic, mathematics, and economics classes—demonstrates that if two players, suspect A and suspect B, act only in their own self-interest, both will suffer dire consequences.2 For example, if each suspect is held in a separate interrogation room and told that by either confessing to the crime or “ratting out” his or her accomplice, each can receive a reduced sentence, then both suspects will either implicate the other or confess to the crime. This is commonly referred to as a zero-sum game because one prisoner’s gain becomes the other’s loss. If each condemns the other, then both will incur the maximum penalty. However, if both confess independently, each will incur some penalty—albeit likely a lesser one because they have shown they are willing to “cooperate” with the authorities. Lastly, if the two suspects work together and adopt the common strategy that would appear at first blush to benefit each one less (remaining silent), the benefit to both will actually increase—because the State, lacking a confession or statement of the other’s guilt, will likely charge each with a lesser offense. The lesson learned from the prisoner’s dilemma and similar scenarios is that players in competition with each other sometimes gain more by conspiring than by attempting to combat each other to the last.
Mathematicians refer to scenarios such as the prisoner’s dilemma as simple form games (SFG)—also referred to as normal form games—which commonly have two players, each of whom strives to receive the highest payoff at the end of a simultaneous move (i.e., by seeking what is referred to in economics as a Pareto optimal position). One determines payoffs—outcomes with real value to each player—through a process called quantification, conducted by primary stakeholders who have a direct, vested interest in the outcome of the game. In the Iraqi conflict, the two players within the SFG are the ISFs and DIs. The United States and coalition forces are not considered players in this game (explained later in this article).
Additionally, in extensive form games (EFG)—which feature two or more players engaged in multiple move-for-move exchanges—players generally worry less about intermediate payoffs than the ultimate payoff at the conclusion of the game. Obviously, quantification of the EFG is far more complex than in the SFG because one must consider both short-term and long-term payoff values. Moreover, as mathematicians John von Neumann and Oskar Morgenstern discovered, EFGs are frequently not zero-sum games (i.e., one player’s loss does not always perfectly correlate with another player’s gain, depending on the complexity of the rules); therefore, predicting the outcome based solely on the payoffs proves difficult at best.3 Because EFGs are distinguished by multiple moves, players must possess an overall broad strategy (as they would in the SFG) as well as smaller substrategies to counter the other players’ moves throughout the game.
In the EFG, as time progresses, the model becomes susceptible to influence from outside forces, termed “strange attractors.” Because payoffs in the EFG are not as readily apparent and the rules are generally more complex than in the SFG, these strange attractors affect the players’ willingness to adhere to previously stated rules and therefore decrease the overall stability of the game.
As time elapses, both SFGs and EFGs become less stable due to player frustration (and, in some cases, physical fatigue). Accordingly, each player will begin to reduce his or her expectations for the ultimate payoff. Consider the gambler who feeds quarters into a slot machine for an hour. This is essentially a two-player SFG (the gambler and the house), consisting of a single turn, with the focus on an immediate payoff. Ultimately, the gambler will likely walk away from the “one-armed bandit” down $25 after 45 minutes without winning the jackpot—an especially likely outcome if the player is down to her last dollar (limited resources), has agreed to meet her sister-in-law in an hour to catch a Las Vegas show (time constraints), and is feeling pangs of hunger because she has not yet eaten lunch (player fatigue). Similarly, the professional poker player may be willing to cut his losses at five-card stud (an EFG because it involves multiple turns, players, payoffs, strategies, and substrategies) and accept a smaller pot rather than play through to the end and face a new dealer later in the game (a strange attractor) who clearly knows the fine art of dealing.
As players’ expectations for the ultimate payoff start to fade with the passing of time (in the case of the EFG, with the destabilizing influence of strange attractors), each player begins to think about how, by negotiating with the opponent, he or she might end the game without suffering additional losses. One refers to the point at which players start to work cooperatively towards agreement as bargaining towards equilibrium (or, in economics, Pareto improvement). When both players have reached a point at which they can achieve the highest aggregate payoff, the game ends in preferred equilibrium.
However, the influence of strange attractors in a model that will become increasingly unstable (bifurcated) over time often induces players to hasten their desire for a Pareto improved position instead of a superior (Pareto optimal) position—even though doing so may lessen their ultimate payoff because they did not play through to the end of the game. One refers to the point at which both players reach Pareto improvement, despite the fact that they may have received a greater payoff had they waited, as Nash equilibrium. First theorized by Princeton University professor John Nash, this equilibrium is sometimes described as an inchoate or interrupted equilibrium because the players reach a point of compromise prior to the conclusion of the game’s ultimate payoff.4 Several Nash equilibriums may exist at various points prior to achieving preferred equilibrium. Most SFGs and EFGs do not start out with players seeking to work cooperatively (i.e., striving for Pareto improvement). However, as each player’s “winner take all” strategy clearly becomes less viable with the passing of time, both players realize that the longer it takes to come to consensus and the more resources they expend in their individual quest for dominance, the smaller the ultimate payoff should they emerge victorious (an economic concept known as Rubinstein Bargaining).5 Ultimately, players strive to reach consensus if for no other reason than they wish to lessen their losses.
In applying Nash equilibrium to the prisoner’s dilemma, one sees that this equilibrium point (both players confessing to the crime) will preempt the preferred equilibrium (both players remaining silent). This is especially true with the passing of time (prisoners do not like being left alone in interrogation rooms) and, in the case of an EFG, if strange attractors are introduced into the model (e.g., so-called eyewitnesses, purported new evidence, etc.). Thus, the passing of time and the influence of strange attractors preempt achieving the preferred equilibrium and instead yield the inchoate or Nash equilibrium. The presence of US and coalition forces in Iraq, especially over time, may actually hasten a Nash response between ISFs and DIs.
Equipped with a working knowledge of SFGs, EFGs, Pareto improvement, Pareto optimal, and Nash and preferred equilibriums, one can not only examine each player’s prospective payoffs but also predict the point at which both the inchoate (Nash) and preferred equilibriums will occur in the Iraqi conflict. In order to identify these points, the remainder of this article assumes a two-player game, namely with ISFs and DIs. Admittedly, attempting to contain the myriad of security entities under the ISF umbrella will likely prove as much of a generalization as placing the many native terrorist organizations that exist in Iraq within the DI grouping. The many law-enforcement and military organizations that comprise the ISF category, along with numerous hegemonic entities that make up the DI set, represent a variety of heterogeneous cultures, values, beliefs, and often competing interests.
Figure 1 provides a summary of payoffs quantified for both players in the simple-form, zero-sum game for the Iraqi conflict as well as each player’s Pareto optimal strategy (point value equals four). It also identifies the respective quadrants in which the Nash and preferred equilibriums will occur.
Figure 1. Iraqi conflict as a simple form game
In figure 1’s SFG, the payoffs for both players are based on varying degrees of remaining active or passive. Each player hopes that the other will not move (i.e., will remain passive), thus achieving a Pareto optimal position for himself or herself. However, if this one-move SFG is repeated over and over again, it becomes clear to both players that neither is willing to remain passive. Over time, as player frustration increases, resources begin to dwindle, and fatigue sets in, the players will begin bargaining towards equilibrium (i.e., seeking Pareto improvement as opposed to Pareto optimal).
As illustrated, one would attain the preferred equilibrium in this SFG at the “3, 3” quadrant because the highest aggregate payoff occurs at this point in the game. One must remember that preferred equilibrium has no connection to the player’s Pareto optimal strategy; rather, it is simply a mathematical expression for the point at which one can derive the greatest quantified payoff value.
As both players continue bargaining, the game moves from a competitive to a cooperative mode, leading to increased communication, which in turn yields further bargaining between players. Inflexible rules and intransigent positions become more elastic, and the players proffer side payments to hasten agreement. At this point, the game is said to have become mathematically corrupted because the players are no longer following the rules established prior to initial play. They have also moved from focusing on Pareto optimal positions to Pareto improved positions. Therefore, the inchoate or Nash equilibrium will inevitably occur at the “2, 2” quadrant.
When one applies these concepts to the SFG for the Iraqi conflict, the challenges faced by US and coalition forces in Iraq become readily apparent. Ultimately, the model will become mathematically corrupted. Both players will move from seeking Pareto optimal to Pareto improved positions (i.e., ISFs and DIs will lessen their expectations, hastening equilibrium). Moreover, for reasons already discussed, Nash equilibrium will preempt the two players from attaining the preferred equilibrium (the quadrant in which equilibrium at the highest aggregate payoff value in the model will occur) wherein DIs continue to carry out attacks with improvised explosive devices throughout Iraq, and ISFs continue to arrest or kill terrorists.
It is important to understand that one can think of all equilibriums (Nash and preferred) as solutions. One can use software such as the publicly available Gambit application (originally developed by Theodore Turocy and Andrew McLennan in 1994 and now in its 11th release) to test the probability and frequency of these solutions occurring within the parameters of the model.6 Repeated test runs of the zero-sum Iraqi conflict SFG yield the same result: a Nash response in which ISFs and DIs are willing to “sacrifice” US and allied forces to achieve Pareto improvement is inevitable. Evidence already exists to suggest that bargaining between players has begun, such as Prime Minister Nouri al-Maliki’s proposed National Reconciliation Plan, which would afford partial amnesty to some DIs.7
Mathematically speaking, neither the United States nor its coalition forces can be considered players in the Iraqi conflict SFG because the United States cannot quantify payoffs. This also holds true in the EFG because America’s citizenry does not have a direct, primary-stakeholder interest in the conflict (i.e., they are not part of the quantification process). Only the Iraqi people—represented in this game by the two primary players (ISFs and DIs)—are fundamentally and intimately affected by the payoffs at each turn within the EFG, as well as by the ultimate payoff at the conclusion of the game.
Indeed, from a game-theory perspective, one finds very few conflicts in American history wherein US forces have had the ability to participate in the quantification process as a primary player, save for the colonists in the American Revolutionary War, Union and Confederate forces in the Civil War, and servicemen in the US intervention during World War II after the Japanese attack at Pearl Harbor. No one should ever dismiss the brave and noble actions of US forces in other conflicts, but from an EFG perspective, one can mathematically consider the United States a player only when America directly involves itself in the quantification of payoffs. For a party to assume this role, its stakeholder interest must have value equal to that of the other players. This is not to suggest that US and coalition forces do not affect the model or its two players (ISFs and DIs) in the Iraqi conflict EFG. Indeed, those forces function as strange attractors.
For the purposes of the current situation in Iraq, US and coalition forces, multinational business interests, third-party foreign-terrorist organizations, and other interested parties would all be considered strange attractors whose predominant role involves hastening the model towards equilibrium. As time progresses and the model continues to bifurcate, the EFG becomes inherently less stable; thus, strange attractors play a greater role in moving the players towards cooperative bargaining (Pareto improvement). As was the case in the SFG presented earlier, the EFG becomes corrupt. Players begin working in cooperation (bargaining towards equilibrium) rather than competing for a Pareto optimal position.
In the Iraqi conflict, bargaining towards equilibrium entails emergent conspiracies between the two players—ISFs and DIs—as the game becomes less stable. Police officers begin tipping off insurgents as to where raids will take place in exchange for protection from future attacks, and terrorists provide bribes to Iraqi soldiers in exchange for overlooking caches of household weapons. The revelation that the late terrorist leader Abu Musab al-Zarqawi’s cell phone contained telephone numbers for some of Iraq’s senior Interior Ministry officials and lawmakers provides further evidence that Pareto improvement may have already commenced between ISFs and DIs.8 In March 2006, Sgt Paul E. Cortez, Pfc Jesse Spielman, SPC James Barker, and Pvt Stephen D. Green raped and murdered 14-year-old Abeer Qassim al-Janabi and then killed her family.9 Subsequently, in September 2006, insurgents killed three US soldiers simply because they served in the same unit as the four former solders who carried out this heinous crime. Iraqi Interior Ministry officials refused to condemn the killing of the US soldiers, which Iraqis widely regarded as an “honor killing.”10 The insurgents’ ability to capture and kill US service members suggests a level of access to operational-security plans for US forces previously unavailable to terrorist entities.11
Using the Gambit software application, we can model the EFG for the Iraqi conflict from the perspective of DIs: player one in the dominant strategy position (i.e., DIs make the first move). The results (fig. 2) appear similar to those for the SFG (fig. 1).
As both the SFG and EFG models show when applied to the Iraqi conflict, both players (ISFs and DIs) will ultimately abandon their Pareto optimal strategies and instead begin bargaining towards equilibrium. When this happens, the model will become corrupted, and a Nash solution will preempt the preferred equilibrium. In the EFG, the presence of strange attractors such as US and coalition forces, foreign-terrorist entities, and other third-party interests may serve only to hasten this process in an increasingly bifurcating model. Release of The Iraq Study Group Report of December 2006, which specifically cites that “violence is increasing in scope and lethality,” coupled with increasingly nonlinear attacks against US and coalition forces (e.g., improvised chlorine chemical attacks, use of women as suicide bombers, etc.) suggests that the model explored in this article continues to destabilize.12 Moreover, additional conflicts between Israeli forces and the Lebanese Hezbollah Party may introduce additional strange attractors into the model, further hastening the “2, 2” Nash payoff even more quickly than initially predicted using the Gambit software application.
Figure 2. Iraqi conflict as an extensive form game
It is possible for the United States to assume a player role in Iraq rather than serve as a strange attractor. However, to do so, stakes for Americans would need to equal those of the Iraqi people in order for the quantification process to occur. The United States would have to commit hundreds of thousands—if not millions—of military and civilian personnel to Iraq for decades, which it could accomplish in the short term only by fully mobilizing all reserve-component forces and initiating a military draft to meet future needs. US and Iraqi culture and values would need to become inextricably linked. Each American would have to feel a stakeholder interest in Iraq, evidenced through personal sacrifice in the form of military or civilian service in support of Iraqi Freedom or the rationing of US goods to support the Iraqi people (comparable to rationing during World War II). Only then could America effectively participate in the quantification process. It is highly unlikely that present-day Americans or their elected representatives would be willing to commit to personal sacrifices, such as a military draft, war taxes, or the rationing of food and supplies. Accordingly, it is not mathematically possible for America to achieve player status in Iraq.
One must note that US policy decisions take into account elements beyond the theoretical constructs of the SFG or EFG. Even if America cannot obtain player status, excellent reasons may exist for the United States and coalition forces to remain in Iraq, such as nation-building and humanitarian purposes.
However, American policy makers and the public must be prepared to accept the fact that if US forces remain in Iraq, the soldiers, sailors, airmen, and marines bravely serving there will remain a strange attractor in a mathematical model that is destabilizing over time. Within this game, DIs and ISFs will eventually arrive at Nash equilibrium.
*Dr. Brightman is an associate professor and chairperson of the Criminal Justice Department at Saint Peter’s College in Jersey City, New Jersey, and an information warfare officer with the US Navy (Reserve Component), currently serving on active duty for special work and assigned to the War Gaming Department of the US Naval War College as a visiting professor.
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1. Associated Press, “Radical Cleric Calls on Iraqis to Halt Cooperation with U.S. Military,” FOXNews.com, 8 April 2007, http://www.foxnews.com/story/0,2933,264814,00 .html (accessed 10 April 2007).
2. Following World War II, private and quasi-governmental think tanks such as the RAND Corporation were established as a means to further research in areas such as war gaming and hostile-action contingency planning. Flood and Dresher were among the first of this new breed of mathematicians focusing on multiplayer simulations. Douglas Hofstadter expanded upon and clarified much of their original research in this area. See Douglas R. Hofstadter, Metamagical Themas: Questing for the Essence of Mind and Pattern (New York: Basic Books, 1985).
3. In 1944 Princeton University professors John von Neumann and Oskar Morgenstern authored their landmark economic paper “Expected Utility Theory,” which focused exclusively on the application of strategic game theory to social problems. Their technique was subsequently used to examine virtually every social-organizational problem imaginable, from settling antitrust disputes to the US-USSR arms race during the Cold War. “The von Neumann–Morgenstern Expected Utility Theory,” The History of Economic Thought, Bernard Schwartz Center for Economic Policy Analysis, http://cepa.newschool.edu/het/essays/uncert/vnmaxioms.htm (accessed 10 April 2007).
4. See Carlo C. Jaeger et al., “Decision Analysis and Rational Action,” working papers, chap. 3, http://www
.pdf (accessed 10 April 2007).
5. Ariel Rubinstein’s theory of 1982 states that in an alternating bargaining game in which one player makes an offer followed by the other, the ultimate value of the payoff will decrease the longer the game is played. Ultimately, the players will be willing to settle for a lesser payoff simply to end the game. See Lucy White, “Prudence in Bargaining: The Effect of Uncertainty on Bargaining Outcomes,” Harvard Business School, 9 December 2003, http://www.people.hbs.edu/lwhite/pdf/newfiles/ prudence%20in%20bargaining.pdf (accessed 10 April 2007).
6. The Gambit Software Application for Game Theory has been used extensively in mathematical modeling and simulation since its inception in 1994. The current version of the software, developed by Richard McKelvey, Andrew McLennan, and Theodore Turocy, allows the user to explore player strategies, contingencies, and outcomes for Nash equilibrium as well as for several other game-theory models. The current version of the Gambit software application is available free of charge at “Software Tools for Game Theory,” Gambit, http://econweb.tamu .edu/gambit/support.html (accessed 10 April 2007).
7. Prime Minister Nouri al-Maliki unveiled the National Reconciliation Plan on 6 June 2006 for the purpose of relieving tension between various religious and ethnic elements vying for power in Parliament. Under the terms of this plan, approximately 2,500 Iraqi detainees held in US and allied custody would be released without prejudice or further governmental action. Jaime Jansen, “Iraq Government to Release 2,500 Detainees in Reconciliation Bid,” Jurist: Legal News and Research, http://jurist.law.pitt .edu/paperchase/2006/06/iraq-government-to-release
-2500.php (accessed 10 April 2007).
8. Killed in a targeted bombing north of Baghdad by US and coalition forces on 8 June 2006, Abu Musab al-Zarqawi was considered the leading figure behind many of the bombings and kidnappings in Iraq, from the 2003 invasion up until his death. Al-Zarqawi was credited with implementing a complex terror-cell structure comparable to al-Qaeda’s other operations around the world. Ellen Knickmeyer and Jonathan Finer, “Insurgent Leader Al-Zarqawi Killed in Iraq,” washingtonpost.com, 8 June 2006, http://www.washingtonpost.com/wp-dyn/content/article/ 2006/06/08/AR2006060800114.html (accessed 10 April 2007).
9. In February 2007, Sergeant Cortez accepted a plea bargain and will be eligible for parole in 10 years. Specialist Barker was sentenced to 90 years in a military prison, and Private Green was dishonorably discharged. “U.S. Soldier Sentenced to 100 Years for Iraq Rape, Killing,” Aljazeera.com, 23 February 2007, http://www.aljazeera.com/me.asp?service_ID=13000 (accessed 10 April 2007).
10. See Associated Press, “Group: Soldiers Killed over Rape-Slaying,” USA Today, 11 July 2006, http://www.usatoday .com/news/world/iraq/2006-07-10-group-claim_x.htm (accessed 10 April 2007).
11. DI access to the operations data of US and coalition forces appears to be growing in Iraq. In addition to knowing the details with respect to where and when US forces would be located on the day this ambush took place, The Iraq Study Group Report notes that the alliance between insurgent groups and government officials in some cases has compromised operational security. James A. Baker III et al., The Iraq Study Group Report (New York: Vintage Books, 2006), 5, http://permanent.access.gpo
.gov/lps76748/iraq_study_group_report.pdf (accessed 10 April 2007). Lastly an attack in Karbala on 22 January 2007, which left 27 people dead, including two marines, went undetected because the terrorists had US and Iraqi military uniforms and passes. Damien Cave, “Troops Killed by ‘Insurgents’ Wearing US Army Uniforms,” Infowars.com, 22 January 2007, http://www.infowars.com/articles/iraq/troops_killed_by_insurgents_wearing_us
_army_uniforms.htm (accessed 10 April 2007).
12. Baker et al., Iraq Study Group Report, xiii.
The conclusions and opinions expressed in this document are those of the author cultivated in the freedom of expression, academic environment of Air University. They do not reflect the official position of the U.S. Government, Department of Defense, the United States Air Force or the Air University
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