IALEIA Journal Vol 17 No 1

The Game is Afoot:

Applying Nash Equilibrium to the Iraqi Conflict

Hank J. Brightman, Ed.D.

[Editors Note: While on its face, the theme of this article may appear better suited for publication in a military journal, the key concepts of game theory presented herein can be applied to nearly any law enforcement-targeted entity or activity (e.g., organized crime, terrorism, etc.). The views expressed in this article do not necessarily represent the opinions of the U.S. Naval War College or IALEIA Journal. This research was completed in June 2006; well before the Iraq Study Group report was released or current levels of instability in Iraq manifested.]

The current conflict in Iraq may be examined by researchers as democratic nation building in an area of the world where such forms of government have historically not been the norm. It may also be studied as a model in humanitarian relief, given the large investment of the U.S. and the "coalition of the willing" towards improving public health infrastructure, education, and agriculture. However, little effort has been made to examine the present Iraqi Conflict from a mathematical perspective. Specifically, use of the widely-employed game theory concepts of Pareto improved and Pareto optimal strategies, Nash and preferred equilibriums, frequently analyzed in the fields of economics, law, finance, foreign policy development and trade have not been applied to Operation Iraqi Freedom.

Accordingly, the purpose of this article is to apply game theory to the Iraqi Conflict. Specifically, it will examine how, through application of game theory to this model, U.S. and coalition forces will ultimately suffer casualties at an increasing rate the longer they remain in Iraq. This will occur because both domestic insurgents and indigenous security forces 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 domestic insurgent (DI) attacks, with intelligence frequently provided by indigenous security forces (ISF).

For the purposes of this article, the author refers to ISF as the Iraqi military, state and local police, and DI as Al 91Awdah, The Popular Resistance for the Liberation of Iraq, Higher Command of the Mujahideen in Iraq and Jihad Brigades.

In order to fully understand how two seemingly disparate entities, indigenous security forces and domestic insurgents will ultimately work together in an effort to improve both of their respective positions, a basic discussion of game theory and the associated concepts of bargaining and equilibrium follows. Where applicable, mathematical terms (e.g., player, improved, optimal, corruption, preferred) have been italicized in order to distinguish them from their plain English counterparts.

I. Flashback to Logic 101: The Prisoners Dilemma

Many Journal readers may recall an activity played out in a college logic, mathematics or economics class known as the prisoners dilemma. Developed by Flood and Dresher at the RAND Corporation in 1950, the purpose of this scenario is to demonstrate that if two players, Suspect A and Suspect B, act only in their own self-interest both will suffer dire consequences. For example, if each suspect is held in a separate interrogation room and told that by either confessing to the crime or "ratting out" their accomplice each can reduce his or her own 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 is another's loss. If each of the two suspects condemns the other, the maximum penalty will be incurred by both. However, if both suspects 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 less individually (remaining silent), the benefit to both suspects will actually increase. This is because the State, lacking a confession or statement of the others guilt, will likely charge each with a lesser offense. The lesson learned from prisoners dilemma and similar scenarios is that players in competition with each other sometimes gain more by conspiring with one another than attempting to combat each other to the last.

II. Game Theory 101: A Primer

Mathematicians refer to scenarios such as the prisoners dilemma as simple form games (SFGs). A SFG, also referred to as a normal form game, commonly has two players, each of whom strives to receive the highest payoff at the end of a simultaneous move (i.e., seek what is referred to in economics as a Pareto optimal position). Payoffs are outcomes with real value to each player, and are determined through a process called quantification. Payoffs are quantified by those 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 indigenous security forces (ISF) and domestic insurgents (DI). An explanation as to why the United States and coalition forces are not considered to be players in this game will be discussed later in this article.

There are also extensive form games (EFGs) which feature two or more players engaged in multiple move-for-move exchanges. In EFGs, 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 simple form game, because both short-term and long-term payoff values must be considered. Moreover, as mathematicians Von Neumann and Morgenstern discovered, extensive form games are frequently not zero-sum games (i.e., one players loss does not always perfectly correlate with another players gain depending on the complexity of the rules); therefore, predicting the outcome based solely on the payoffs proves difficult at best. Because EFGs are distinguished by multiple moves, each player must possess both an overall broad strategy as they would in the simple form game as well as smaller sub-strategies to counter the other players moves throughout the game.

In the extensive form game, as time progresses, the model becomes susceptible to influence from outside forces, termed strange attractors. Because the 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 decrease the overall stability of the game.

III. Achieving Equilibrium: "Can't we all just get along?"

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 simple form game (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 even though she has not attained the jackpot. This is especially likely 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 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 extensive form game because it involves multiple turns, players, payoffs, strategies and sub-strategies). He may be willing to accept a smaller pot than if he had played through to the end if a new dealer is brought in later in the game (a strange attractor) who clearly knows the fine art of dealing.

As the players expectations for the ultimate payoff start to fade with the passing of time and--in the case of the EFG the destabilizing influence of strange attractors--each player begins to think about how, by negotiating with their opponent, he or she may be able to end the game without suffering additional losses. The point at which players start to work cooperatively towards agreement is referred to as bargaining towards equilibrium (or in economics, Pareto improvement). When both players have reached a point at which the highest aggregate payoff can be achieved, the game ends in preferred equilibrium.

However, the influence of strange attractors in a model that will become increasingly unstable (bifurcated) over time often yields the players to hasten their desire for a Pareto improved position vice a superior (Pareto optimal) position--even though their ultimate payoff may be lessened because they did not play through to the end of the game. The point at which both players reach Pareto improvement, despite the fact that they may have received a greater payoff had they waited is referred to as Nash equilibrium. First theorized by Princeton University professor John Nash, his equilibrium is sometimes referred to as an inchoate or interrupted equilibrium, because the players reach a point of compromise prior to the conclusion of the games ultimate payoff. There may be several Nash equilibriums at various points prior to achieving preferred equilibrium. Most simple form and extensive form games do not start out with players seeking to work cooperatively (i.e., striving for Pareto improvement). However, as each players "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 will be should they emerge victorious (an economic concept known as Rubenstein Bargaining). Ultimately, players strive to reach consensus if for no other reason than they wish to lessen their losses.

In applying Nash equilibrium to the prisoners dilemma, it becomes evident that this equilibrium point (both players confessing to the crime) will pre-empt 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, if played as an EFG, strange attractors are introduced into the model (e.g., so-called eyewitnesses, purported new evidence, etc.). Thus, the passing time and influence of strange attractors pre-empt achieving the preferred equilibrium and instead yields the inchoate or Nash equilibrium. As this article will examine in the ensuing section, the presence of U.S. and coalition forces in Iraq, especially over time, may actually be hastening a Nash response between indigenous security forces and domestic insurgents.

IV. Corruption Between Players: The Simple Form Game and the Iraqi Conflict

Equipped with a working knowledge of simple and extensive form games, Pareto improvement, Pareto optimal, and Nash and preferred equilibriums, it is possible not only to examine each players prospective payoffs, but also to 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 indigenous security forces (ISF) and domestic insurgents (DI). Admittedly, attempting to contain the myriad of security entities under the ISF umbrella is likely 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 which comprise the ISF category, along with the numerous hegemonic entities that make up the DI set, represent a variety of heterogeneous cultures, values, beliefs, and often competing interests.

Figure 1.1 provides the reader with a summary of payoffs that have been quantified for both players in the simple form, zero-sum game for the Iraqi Conflict. As shown in this figure, each players Pareto optimal strategy (point value = 4) is provided. The chart also identifies the respective quadrants in which the Nash and preferred equilibriums will occur.

In the simple form game shown in figure 1.1, 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., remain passive), thus achieving a Pareto optimal position for themselves. However, if this one-move simple form game 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 a Pareto optimal).

As illustrated, the preferred equilibrium in this SFG would be attained at the "3,3" quadrant, because the highest aggregate payoff is achieved at this point in the game. Remember, preferred equilibrium is not connected to the players Pareto optimal strategy, but rather is simply a mathematical expression for the point at which the greatest quantified payoff value can be derived.

As both players continue bargaining, the game moves from being competitive in nature to cooperative. This cooperation leads to increased communication, which yields further bargaining between players. Inflexible rules and intransigent positions become more elastic, and side payments are proffered to hasten agreement. At this point, the game is said to have become mathematically corrupted because the players are no longer following the rules that were established prior to initial play. The players have also moved from focusing on Pareto optimal positions to Pareto improved positions. Therefore, the inchoate or Nash equilibrium will inevitably occur, and equilibrium will be reached at the "2,2" quadrant.

When these concepts are applied to the SFG for the Iraqi Conflict, the challenges faced by U.S. and coalition forces in Iraq become readily apparent. Ultimately, the model will become mathematically corrupted. Both players will move from seeking Pareto optimal positions to Pareto improved positions, (i.e., indigenous security forces (ISF) and domestic insurgents (DI) will lessen their expectations, hastening equilibrium). Moreover, for reasons already discussed, Nash Equilibrium will pre-empt the two players from attaining the preferred equilibrium (the quadrant in which equilibrium the highest aggregate payoff value in the model will be achieved) wherein DI continue to carry out IED attacks throughout Iraq and ISF continues to arrest or kill terrorists.

It is important to understand that all equilibriums (Nash and preferred) can be thought of as solutions. Software such as the publicly-available Gambit application (originally developed by Turocy and McLennan in 1994 and now in its eleventh release) can be used to test the probability and frequency of these solutions occurring within the parameters of the model. Repeated test runs of the zero-sum Iraqi Conflict SFG yield the same result: a Nash response in which ISF and DI are willing to "sacrifice" U.S. and allied forces to achieve Pareto improvement is inevitable. There is already evidence to suggest that bargaining between players has begun, such as Prime Minister Al-Malikis recently-proposed "National Reconciliation Plan," which would afford partial amnesty to some domestic insurgents.

V. U.S. Interests in Iraq: The Extensive Form Game

One of the most frustrating aspects for the readers of this article (and for the soldiers, sailors, airmen, and marines who have dutifully served or given their lives in Iraq) is that mathematically speaking, neither the United States nor its coalition forces can be considered as players in the Iraqi Conflict simple form game. Such is the case because the United States cannot quantify payoffs. This also holds true in the extensive form game because the Americas citizenry does not have a direct, primary stakeholder interest in the Iraqi Conflict (i.e., they are not part of the quantification process). Only the Iraqi people--represented in this game by the two primary players, indigenous security forces and domestic insurgents--are fundamentally and intimately impacted 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, there have been very few conflicts in American history wherein the United States has had the ability to participate in the quantification process as primary player, save for the colonists in the American Revolutionary War, Union and Confederate forces in the Civil War, and U.S. intervention in World War II post the Japanese attack at Pearl Harbor. While no one should ever dismiss the brave and noble actions of U.S. forces in other conflicts, from an EFG perspective, the United States can only be mathematically considered to be a player when America is directly involved in the quantification of payoffs. In order to assume this role, the parties stakeholder interest must be of equal value to that of the other players. This is not to suggest that U.S. and coalition forces are not impacting the model or its two players (ISF and DI) in the Iraqi Conflict extensive form game. America and the coalition forces do play a role in the EFG and that is in the role of strange attractor.

For the purposes of the current situation in Iraq, U.S. and coalition forces, multinational business interests, third-party terrorist organizations, and other interested parties would all be considered to be strange attractors. The predominant role played by strange attractors in the Iraqi Conflict EFG is to hasten 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 simple form game presented earlier, the result is corruption of the EFG. Players begin working in cooperation (bargaining towards equilibrium) vice competing for a Pareto optimal position.

In the Iraqi Conflict, bargaining towards equilibrium means emergent conspiracies between the two players, ISF and DI, 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 household weapons caches. The recent revelation that the late terrorist leader Abu Musab al-Zarqawi's cell phone contained telephone numbers for some of Iraqs senior Interior ministry officials and lawmakers provides further evidence that Pareto improvement may have already commenced between indigenous security forces and domestic insurgents. Moreover, the insurgents ability to capture and kill three U.S. service members in retaliation for the alleged rape and murder of a 15-year old Iraqi girl suggests a level of access to operational security plans for U.S. forces previously unavailable to terrorist entities. Iraqi Interior Ministry were conspicuously absent on condemning the "honor" killing of the three soldiers, all of whom served in the same unit as the former solder who allegedly carried out the girls rape and murder.

Using the Gambit software application, the EFG for the Iraqi Conflict can be modeled from the perspective of domestic insurgents: player one in the dominant strategy position (i.e., DI makes the first move). The results, which are shown in figure 1.2, appear similar to those for the simple form game shown in figure 1.1.

VI Conclusions: Where do we go from here?

As both the SFG and EFG models show when applied to the Iraqi Conflict, both players (indigenous security forces and domestic insurgents) 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 pre-empt the preferred equilibrium. In the extensive form game, the presence of strange attractors such as U.S. and coalition forces, foreign terrorist entities, and other third-party interests, may only serve to hasten this process in an increasingly bifurcating model. The recent conflict between Israeli Forces and the Lebanese Hezbollah Party may introduce additional strange attractors into the model, further destabilizing conditions in Iraq thereby promoting the "2,2" Nash payoff even more quickly than initially predicted using the Gambit software application.

It is possible for the United States to assume a player role in Iraq vice serve as a strange attractor. However, to do so, the stakes for Americans would need to be equal to that of the Iraqi people in order for the quantification process to occur. The United States would need to commit hundreds of thousands--if not millions--of military and civilian personnel to Iraq for decades. U.S. and Iraqi culture and values would need to become inextricably linked. Each American would need to feel a stakeholder interest in Iraq evidenced through personal sacrifice in the form of military or civilian service in support of OIF or the rationing of U.S. 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 important to note that U.S. policy decisions take into account elements beyond the theoretical constructs of the simple or extensive form game. Even if it is not possible for America to obtain player status, there may be excellent reasons for the United States and coalition forces to remain in Iraq, such as the nation-building and humanitarian purposes referenced at the beginning of this article.

However, American policymakers and the public must be prepared to accept that if U.S. 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, it is imminent that domestic insurgents and Iraqi security forces will eventually arrive at Nash equilibrium.

SUGGESTED READINGS

Baird, D.G., Gertner, R.H. & Picker, R.C. (1994). Game theory and the law. Cambridge, MA: Harvard University Press

Davis, M.D. (1983). Game theory: A non-technical introduction. Mineola, NY: Dover Publications, Inc.

Turocy, T. (n.d./2006). Gambit: developers and maintainers [www page current at 07 July 2006]. URL: http://econweb.tamu.edu/gambit/support.html

Zlotkin, G. A (1993). Domain Theory for Task Oriented Negotiation. Proceedings of the Thirteenth International Joint Conference on Artificial Intelligence. San Mateo, CA: Morgan Kaufmann.

ABOUT THE AUTHOR

For the past seven years, Dr. Heath "Hank" Brightman has served as an Associate Professor and Chairperson of the Criminal Justice Department at Saint Peters College in Jersey City, New Jersey where he frequently applies game theory to organized crime investigations. He previously spent ten years in a variety of law enforcement, investigative, and intelligence analysis positions with the U.S. Department of the Interior and United States Secret Service. In addition to his civilian and academic experience, Lieutenant (Junior Grade) Brightman is an Information Warfare Officer (formerly known as Cryptology) with the United States Navy (Reserve Component) currently assigned to the War Gaming Department of the U.S. Naval War College. In this capacity, he has developed several war games and lectured on the application of non-linear inductive thinking in the Global War on Terror, and applying game theory to the Iraqi Conflict. His primary reserve billet assignment is as an Officer Instructor and Assistant Officer In Charge for the Center for Naval Leadership, Northeast Region.