Game Theory and Customary International Law

By Kurtis D. Loux

Kurtis is a junior degree seeking student at John Cabot University majoring in International Affairs. He transferred to JCU from Grand Rapids Community College and is originally from Rockford, Michigan. 


The world of international law can be described as the sandy ocean floor. The sand is constantly changing; getting pushed and pulled around with the forces of the tides and currents. So it is with international law, but particularly with Customary International Law (CIL). CIL is generally defined as the, “general and consistent practice of states followed by them from a sense of legal obligation.” (Goldsmith & Posner, 1999, p. 1113), meaning more than one state over a long period of time. From this definition, two important factors play crucial roles in defining what CIL is: State practice and opinio juris. State practice commonly refers to actions taken by states, whereas opinio juris refers to, as defined by the International Court of Justice, “a general practice accepted as law” (as cited in Swaine, 2002, p. 567). State practice and opinio juris do not just remain stagnant, however. Referring back to the ocean metaphor, instead of tides and currents exerting their force on the sand, the mixing and evolution of state practice and opinio juris over time change the landscape of CIL. Because of this constant shifting, it can be quite challenging to grasp how and why state behavior and ideals change. How can CIL be somewhat more readily be understood? Starting in the 1990’s, scholars in the field of CIL began to use something novel to more readily explain the behaviors of states: Game Theory.

Introduction to Game Theory

During the Coase Lecture Series at the University of Chicago Law School, Randal C. Picker (1994) defines Game Theory as, “a set of tools and a language for describing and predicting strategic behavior.” (p. 2). The “language” of Game Theory is clearly defined. Each scenario is called a “Game”, and the “Players” are the participatory parties to said game. Each game is comprised of a series of “Payoffs”, which can represent a value or a positive or negative motivations for players. It is critical to point out that the “Players” in these games are assumed to be rational and intelligent”, where rational is defined as, “decision-making behaviour…consistent with the maximization of subjective expected [payoff],” and intelligent is defined as, “[understanding] everything about the structure of the situation…” (Hviid, 1997, p.707). To better understand the language, we can use one of the most well-known models in Game Theory, “The Prisoner’s Dilemma”, as a tool to apply the basic language.

In a hypothetical situation, two suspected criminals are arrested. The police believe they were attempting to rob a store, but only have enough evidence to charge them for trespassing. In order to charge one of them for robbery, the police need one of them to turn on the other. Once at the police station, the criminals are put in individual rooms and each offered the following deal: if they both remain silent, they will be charged with trespassing and each spend one (1) year in prison; if one confesses and the other remains silent, the one that confesses will be set free while the silent one receives 10 years in prison; and if both of them confess, they each will receive five (5) years in prison.

    Player B
    Silent Confess
Player A


Silent 1,1 10,0
Confess 0,10 5,5

Figure 1: Prisoner’s Dilemma

As you can see from figure one, there are two players, Player A and Player B. By convention, Player A is always a man and Player B is always a woman, so that specific pronouns can be used without confusion. Each of them can make one of two choices; either to confess (defect) or to remain silent (cooperate). Each choice is accompanied by a payoff depending on what the other player does.  Suppose Player A remains silent while Player B confesses, what would the payoffs be for each individual? Player 1 would get the payoff “-10”, signifying the 10 years in prison he would receive, and Player 2 would get the payoff “0”, also corresponding to the amount of years spent in prison, or lack thereof.

Conventional Wisdom of CIL

Before getting into the theories of CIL according to Game Theory, it is important to get into what is considered the “Conventional Wisdom” of CIL. First, CIL is considered binding upon all states, whether a particular state agrees to it or not, as opposed to Treaty Law, which is only considered binding to the signatories of a particular treaty. On the other hand, if said treaties are to be amended or changed, all signatories must be notified so they can accept or reject the changes. Second, modern CIL relies mostly upon opinio juris, which is crucially important because opinio juris based CIL, “begins with general statements of rules rather than particular instances of practice.” (Roberts, 2001, p.258). Another thing to keep in mind is that modern CIL has the opportunity to expand given its extensive backing from international institutions to help CIL roll and move with the will of states. However, CIL still needs state practice in order to work, as Edward T. Swaine (2002) put it, “[opinio juris] without implementing usage is nothing more than rhetoric,” and, “state practice, without opinio juris, is just habit.” (p. 567-8).

  1. Theory of Goldsmith and Posner[1]

Now that we have a better understanding of the language and means that Game Theory employs, as well as some conventional wisdom about CIL, we can move on to how scholars use Game Theory to descriptively simplify the complexity that is CIL.

One of the landmark uses of Game Theory in CIL came from Jack L. Goldsmith and Eric A. Posner (1999). In their work entitled, “A Theory of Customary International Law”, they, “[use] simple game theoretical concepts to explain how CIL arises, why nations ‘comply’ with CIL as commonly understood, and how CIL changes.” (p. 1114). The basic idea that Goldsmith and Posner have is that nations “comply” with CIL only when it is in accordance to their national self-interest. In their Game Theoretic models, they found four (4) different scenarios that lead to what could be seen as cooperation under CIL; Coincidence of interest, coercion, true cooperation, and coordination. Each of the four scenarios are based off of the landmark Paquete Habana case, in which it was decided the CIL doctrine forbade the seizure of civilian fishing vessels from an enemy state.

Coincidence of interest

Figure 2: Coincidence of interest

For this first case, let’s assume “State j” is Player 1, and “State i” is Player 2. Let’s also assume that they are currently at war with each other, but share a common waterway and have navies to patrol and defend it. At the same time, both nations have civilian fishing boats that also use the waterway, but don’t have much value to the other player. Each player, like in the case of “The Prisoner’s Dilemma”, can make one of two choices; either attack the other state’s fishing boats or ignore them. In cases of coincidence of interest, as is shown in figure two, both players see an incentive to “ignore” the other and receive a higher payout, “irrespective of the action of the other.” (Goldsmith & Posner, 1999, p.1122). This gives the illusion that both players are playing by the same rule, similar to the CIL custom at work in the Paquete Habana. This is just an illusion, however, because if the fishing boats were worth more to the other Player, the payoff would be higher, and the Players might be more in favor of attacking.


Keeping with the same background information, let’s assume this time that “State j” is a very large, powerful state, while “State i” is a relatively smaller, weaker state. Let’s also assume that “State i” is attacking “State j”’s fishing vessels. “State j”, because he is powerful and the loss he would take destroying “State i”’s navy is small, can threaten “State i” with destruction of her navy. “State i”’s best option is that of ceasing attack, as her payoff is much better when she does what “State j” asks instead of crossing him, and will continue to follow that trend because of the high possibility of retaliation if she deviates. An important note is that coercion only works if the cost of attacking by “State j” is low. If the cost of attack is high and “State j” attempts to coerce “State i”, “State i” will not take the threat seriously and continue its attacks.


Under the third scenario, we’ll use what is called a “Bilateral Repeat Prisoner’s Dilemma”. The easiest way to understand this is that we use the same idea we did in Figure 1. However, instead of it being just a one-time interaction, the players do not have knowledge of when the game will end. Within the “Bilateral Repeat Prisoner’s Dilemma”, the payoff matrix looks as follows.


Figure 3: Bilateral Repeat Prisoner’s Dilemma

Let’s quickly assume Figure 3 is the payoff for a single interaction Prisoner’s Dilemma like that of Figure 1. Since both “State j” and “State i” can gain something from attacking the other, they will be more inclined to “attack” instead of “ignore” the other’s fishing vessels. This can be described as both states attempting to “cheat” the other in order to gain an advantage. Now, going forward, let’s assume that both states have no knowledge of when the game will end. Over the long term, both “State j” and “State i” will find it in their best interest to ignore the other, as their payoff will be greater. However, according to Goldsmith and Posner, there are three (3) main conditions that must be met in order for this “cooperation” to take hold.

First, both states, “must care about the future relative to the present.” (Goldsmith & Posner, 1999, p.1126). If either state is impatient or impulsive, they will not be able to continue a cooperative relationship into the future. In international relations, such states are called “Rogue States”, and are controlled by impulsive or irrational leaders, or failed states in which citizens live in relatively unstable political climates.

Second, the game, “must continue indefinitely, in the sense that players either expect it never to end or to end only with a sufficiently low probability.” (Goldsmith & Posner, 1999, p.1126). If either side knows when the game will end, it will have an incentive to “cheat” and gain a slight upper hand over the other. However, in real life, the game might actually never end. In a scenario of “State j” attacking “State i”’s fishing vessels, for example, it knows that even after the “game” is over, it will have to continue its relations in some fashion with “State i” and fears retaliation of doing so.

Lastly, “the payoffs from [attacking] must not be too high relative to the payoffs from [ignoring].” (Goldsmith & Posner, 1999, p.1126). If the payoffs in Figure 3 for mutual “attacking” were to change from 2 to 4, or mutual “ignoring” were to change from 3 to 1, the cooperation would break down and both states would change their strategy in favor of attacking the other. If all three (3) conditions are met, then true cooperation can take place.


The last of the scenarios proposed by Goldsmith and Posner is that of Coordination. While being similar to that of “Coincidence of Interest”, it differs in that the optimal outcome of one state depends on the choice that the other state makes. Let’s suppose that the payoff matrix for this scenario looks like this:


Figure 4: Coordination

Here, we can see that the best outcome for both “State j” and “State i” depend on what the other decides. In the words of Goldsmith and Posner, “Each state prefers to engage in X if the other state engages in X, and each state prefers to engage in Y if the other state engages in Y.” (Goldsmith & Posner, 1999, p.1127). However, the problem with “Coordination” is that both states must communicate their intended action beforehand and follow through with said action. If there is no ability to communicate an action, or if the communication is disregarded, then the possibility of “Coordination” ceases to exist. That being said, the overall agreement between both states ahead of time creates a situation in which diversion is unlikely. In this scenario, there is no reason for “State j” to say it will take action X, but then cheat and actually do action Y. This is because if both states agree on action X beforehand, and “State j” deviates from the plan, neither state gets anything.

  1. Aftermath of Goldsmith and Posner

In the years after Goldsmith and Posner published their paper, some scholars were quick to denounce the claims made. Mark A. Chinen, George Norman and Joel P. Trachtman, and Detlev F. Vagts have raised good points in rebuttal of Goldsmith and Posner’s paper. In Chinen’s paper, he underlines the notion that there is evidence for cooperation among states insofar as states freely bind themselves with treaties. He aptly raises the question, “Why would a state be willing to bind itself to a treaty?” (Chinen, 2001, p.160). It is unlikely a state would do so simply because of a coincidence of interest, as a state’s interests might change over time. One of the main reasons, Chinen argues, why states would subject themselves voluntarily to such measures would be if they find themselves in a cooperative game. However, Goldsmith and Posner’s argument is that cooperative games are rare due to the three (3) conditions that must be met. Yet treaties still occur, and the United Nations has over 40,000 treaties in its repository since the Second World War (as cited in Chinen, 2001, p.160). There are three (3) ways Chinen believes these treaties have come about; Coercion (which he believes unlikely, but nonetheless admits there are a few examples), existing conditions under which cooperation is possible, or some other factor, such as communication, that is not taken into account in cooperation or coordination games.

Let’s assume the last two options of Chinen’s theory are to be believed. If there are existing conditions under which cooperation is possible are more ubiquitous than previously thought, it follows that some cooperation among states may not take the form of written treaty, but instead that of CIL. The other option, the existence of other factors, such as communication, could also contribute to the reality of CIL. This is because, “communication facilitates greater cooperation…” (Chinen, 2001, p.161), and because states have an interest in communicating about issues important to them, they would move toward the first option in either establishing a treaty or adding to the underlying foundation of CIL. In either scenario, the end result is that state behavior is more stable than Goldsmith and Posner would have us believe. This idea of stable state behavior precedes Goldsmith and Posner’s theory, however. A year before their theory was published, Michael Byers conveys this sentiment by saying that CIL is, “the result of coordinated or at least…common behaviour, and that rules of international law therefore reflect the long-term interests of most…States.” (Byers, 1999, p.17).

In Norman and Trachtman’s paper, they offer a different rebuttal, insofar as they focus on the premise of Goldsmith and Posner’s argument itself. They do this by stating that, under a rationalist viewpoint of CIL, which both Goldsmith and Posner subscribe, “law has no motivating force.” (Norman & Trachtman, 2005, p.542). This is countered by the fact that CIL does, in fact, motivate states through self-interest. In addition, in Goldsmith and Posner’s argument, there is no place for opinio juris. This is contrary to the most common definition of CIL, where CIL is based upon both state practice and opinio juris. Without opinio juris, “CIL does not exist.” (Norman & Trachtman, 2005, p.542). Norman and Trachtman then set forth their own analysis of Game Theory and CIL, in which it is possible for opinio juris to exist, as well as the notion of a “Multilateral Repeat Prisoner’s Dilemma”, meaning in essence there are more than two states (or players) in the Prisoner’s Dilemma game.

Norman and Trachtman have created an entirely new “game” based upon the premise of the Prisoner’s Dilemma, but with a few new rules added that Goldsmith and Posner left out. First, all states (players) have the ability to communicate with each other. Second, because the game continues on indefinitely, each state may react to a previous action of another state at any time. Third, the game is infused with certain ideals of CIL, particularly Diplomatic Immunity. Lastly, each state has equal access to the information regarding the compliance of another. In this model, there is significant evidence to show, “that individuals contribute to the resolution of these problems in substantially greater amounts than the standard prisoner’s dilemma model would suggest.” (Norman & Trachtman, 2005, p.551).

Lastly, there is Detlev F. Vagts, who offers a scathing review of Goldsmith and Posner’s theory. In his rebuttal, Vagts states very clearly that what Goldsmith and Posner put forth in their theory is “anecdotal evidence as to episodes in which custom did not carry the day,” as well as adding that, “they have loaded the dice against custom.” (Vagts, 2004, p.1032). Vagts makes a good point; while reading through the Goldsmith and Posner’s theory, it appears that they have either left out cases in which CIL did “carry the day”, or at the very least neglected to give them a fair place. Vagts’ statement, then, that, “while titled ‘A Theory of Customary International Law’, [Goldsmith and Posner’s theory] is basically a theory against customary law.” (Vagts, 2004, p.1031) holds even more true.

III. Using Game Theory in CIL

As we have seen, there are a few crucial problems that arise when using Game Theory to study something as complex as CIL. First and foremost, Game Theory is unable to truly mirror the real world, and the more we simplify the game, the more information we lose. The best way to phrase this sentiment is, “Game Theory can never capture all real-world detail, including its highly nuanced decision making.” (Norman & Trachtman, 2005, p.542). The second problem with using Game Theory is that it forces us into a, “true/false Boolean dichotomy.” (Arfi, 2006, p.28). This problem becomes amplified when using Game Theory and CIL, as most of the time in CIL, there is no true/false or right/wrong, but rather the perplexing “sorta/kinda”. The third problem with Game Theory is that it requires us to “assume[2]” quite a bit, but most importantly that the players are “rational and intelligent”.  Lastly, the outcome of the game is fully dependent upon the rules one implements. This can be seen by the vastly different outcomes of the same basic game between Goldsmith and Posner’s “Bilateral Repeat Prisoner’s Dilemma” and Norman and Trachtman’s version with a different set of rules. This ambiguity of the “games” makes it hard for Game Theory to be taken seriously in academic and legal fields going back to the early 1990’s (Ayres, 1990, p.1315-7).

However, there is some positive reasoning in favor of using Game Theory and game theoretical models to help condense and understand various aspects of the complex decision making. First, while the criticism of Game Theory being considered too ambiguous to be taken seriously is valid, it is easily overcome by simply increasing the number of rules players must follow. That being said, the more rules one sets in a particular game, the less realistic the game becomes, and the purpose is lost. Norman and Trachtman bring these positive reasons for using Game Theory to describe CIL together nicely, saying, “The purpose of game-theoretic models is not to predict or prescribe behavior, but to generate testable hypotheses that, once tested, are expected to tell us something useful about the world.” (Norman & Trachtman, 2005, p.542). Game Theory is not perfect, and likewise cannot give us one hundred percent (100%) clarity when it comes to decision making in CIL, but at least it gives us a place to start.



Arfi, B. (2006). Linguistic Fuzzy-Logic Game Theory. The Journal of Conflict Resolution,
50(1), 28-57. Retrieved from
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Byers, M. (1999). Custom, Power and the Power of rules: International Relations and
Customary International Law. Retrieved from
Chinen, M. A. (2001). Game Theory and Customary International Law: A Response to
Professors Goldsmith and Posner. Michigan Journal of International Law, 23(1),
143-189. Retrieved from
Goldsmith, J., & Posner, E. (1999). A Theory of Customary International Law. The
University of Chicago Law Review, 66(4), 1113-1177. doi:10.2307/1600364
Hviid, M. (1997). Games Lawyers Play? Oxford Journal of Legal Studies, 17(4), 705-725.

Retrieved from
Norman, G., & Trachtman, J. (2005). The Customary International Law Game. The
American Journal of International Law, 99(3), 541-580. doi:10.2307/1602291

Roberts, A. E. (2001). Traditional and Modern Approaches to Customary International Law:
A Reconciliation. The American Journal of International Law, 95(4), 757-791.
Swaine, E. (2002). Rational Custom. Duke Law Journal, 52(3), 559-627.

Picker, Randel C. (1994). An Introduction to Game Theory and the Law. University of
Chicago Law School, (22), 1-20. Retrieved from

Vagts, Detlev F. (2004). International Relations Looks at Customary International Law: A
Traditionalists Defence. European Journal of International Law, 15(5), 1031-1040.

[1] The payoff matrices shown in this section (Figures 2,3,4) come directly from Goldsmith and Posner’s paper. I in no way take credit for their creation, and have included them to ease the explanation of the theory.

[2] Case in point, I have used the word “assume” nine (9) times in ten (10) pages.


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