In the 1983 cult classic movie Wargames, an experimental artificial intelligence named Joshua is given total control of the United States' nuclear arsenal. In the climactic scene, Joshua is asked to evaluate US nuclear strategy in order to determine the optimal path to a victory over the Soviet Union. His conclusion, that the only way to win is to not play the game, is a cinematic articulation of the conclusions of game theory, a branch of mathematics dealing with rational decision making. In game theory, a series of decisions are postulated, typically involving two or more players, and the expected outcome of the situation is arrived at by assessing the choices players will make in an attempt to maximize outcomes. Game theory has proved itself ideal for explaining the seemingly irrational choices that nuclear power make, such as the decision to engage in arms races, along side the supremely logical choices, such as the constant decision not to use those weapons.

The Normal Form

Game theory is a hugely advanced branch of mathematics which delves very deeply into the decision making process. Fortunately for the strategic thinker, military strategy between nations, particularly involving weapons of mass destruction, is easily represented in the table-style layout often referred to as the "Normal Form." The grid labeled "Figure 1" at right represents an example Normal Form game. The choices for Player 1 are represented as a row-choice (and are shown in blue) whereas the choices for Player 2 are represented as a column-choice (and are shown in red). In the intersecting cells, the expected payoffs for both Player 1 (blue) and Player 2 (red) are shown. Thus, we can see that if Player 1 chooses "cooperate" and Player 2 chooses "defect," Player 1 has an expected payout of 0 and Player 2 an expected payout of 3. Normal Form games like this one make the implicit assumption that all players have access to perfect information (meaning that they know the expected payouts in advance), but that no meaningful communication is possible between players outside of the decisions in the game – that is to say that they can not discuss their strategy.

The Prisoner's Dilemma

Figure 1 illustrates, not just how the Normal Form works, but also a situation called the Prisoners Dilemma. In this case, Player 1 and Player 2 are accomplices in a crime. They have each been arrested and each taken to separate interrogation rooms and may either cooperate (stick to their story and face the music) or defect (turn state's evidence in exchange for a lighter sentence). As the prisoners are unable to communicate with each other, each must decide if he trusts the other to stick to the story. Raw numbers make the Prisoners Dilemma a bit hard to grasp, so we'll assign some harder sounding language to them. Assume the following:

• 0: Life In Prison
• 1: 40 Years In Prison
• 2: 5 Years in Prison
• 3: Immediate Release

Looking at the choices available to the prisoners, we see that, from the point of view of Player 1 (blue) the most rational choice is to defect irrespective of what Player 2's choice is. This is a very powerful mathematical truth. It demonstrates that, by virtue of rational choice and a selfish desire to maximize expected outcome, neither Player 1 nor Player 2 will co-operate, which means that the optimal state of the game (Cooperate-Cooperate) and its optimal payout (2,2) will never be chosen. Rather, both players will choose to defect and will arrive at the (1,1) payout.

The Arms Race

This Prisoner's Dilemma scales perfectly to encompass the actions, not of prisoners but of nations. Rather than two prisoners in separate interrogation rooms we substitute two nations on either side of the world. Rather than the decision to stick to a story or turn State's evidence we substitute the choice between building schools or building nuclear weapons. Defecting (building bombs) becomes the rational choice for either power. Our nomenclature for our payouts has, of course, changed:

• 0: Complete loss of security.
• 1: Poor schools but safe (for now)
• 2: Temporary security and great schools
• 3: Perfect national security

Again, however, rational behavior seeking to selfishly maximize the expected outcome leads to the sub-optimal solution – the arms race.

The Nash Equilibrium

The set of choices upon which these games tend to settle is called the Nash Equilibrium. Proposed by John Forbes Nash, the equilibrium is the point "where no player has anything to gain by changing only his or her own strategy unilaterally." This is exactly the case with the Prisoner's Dilemma/Arms Race game, as neither side has reason to shift her action from "defect" to "cooperate" without a corresponding (and seemingly irrational) choice from the opponent. The Nash Equilibrium, however, elegantly explains the other side of super-power brinksmanship: peace through Mutually Assured Destruction. Or, to the more whimsical, the classic staple of 1950s greaser mythology: Chicken.

Chicken and Mutually Assured Destruction

The game of Chicken, first depicted in Rebel Without a Cause, pits two drivers in a battle of high-stakes wills. Drivers race towards certain death (a cliff face, or in later depictions, a head on collision with each other), achieving victory by being the last to flinch from death (by swerving or jumping from the car). Victory and defeat in the game are thus dwarfed by the larger implications of a negative stalemate: the certain death of both competitors. Figure 2 illustrates a game of Chicken. Unlike the Prisoner's Dilemma, the actual values involved have some significance in this game, so rather than use 0-3, the expected payouts have been weighted. Even so, to give some context, consider the following:

• 0: No loss of face.
• -1: Loss of status/argument
• +1: Win of status/argument
• -100: Death in a car crash

Chicken also serves as a theoretical model for Mutually Assured Destruction. Players, in this case nations, seek victory in the game, but not at the expense of the threatened catastrophic outcome. Communication between players takes place through the Three Cs of deterrence (more on this in Part 1 of this series), and the horrific possibility of nuclear annihilation leads one or both players to flinch in the face of total war.  

• 0: Coordinated relaxation of tensions.
• -1: Back down, possibly emboldening the enemy
• +1: Force the enemy to back down, possibly weakening them
• -100: Nuclear War

Key to understanding Chicken in the context of Mutually Assured Destruction is the recognition that this is not a choice between "launch" or "don't launch," but rather a decision between "continue towards war" or "back down." In the case of the 1962 Cuban Missile Crisis the choice for the US was between "blockade Cuba" and "let the Soviets keep their missiles," while the choice for Khrushchev was between "continue missile construction" and "withdraw from Cuba." The Chicken dynamic defined the crisis with the preferred solution for both parties representing the path to war. Chicken, and thus Mutually Assured Destruction, has two Nash Equilibriums, both involving one party "swerving." The closer the threat of Nuclear War, the more likely such a "swerve" is.

Conclusions

Game theory thus presents a logical and predictive explanation for the actions of nuclear nations. In the cold light of rationality, even the headlong rush towards war exhibited in the Arms Race and the razor thin edge of peace through Mutually Assured Destruction appear both logical and elegant solutions to the problem of the unimaginable power of the atom and the frightening rapidity of modern war. Despite the sense of security that comes with a mathematical proof of safety in a nuclear world, breakdowns do exist in the algebra of deterrence and dynamics can and will occur where the only rational choice is not peace, but war. These breakdowns in global stalemate of M.A.D. represent a real threat to global security, far more significant than the imagined Iraqi Weapons of Mass Destruction or even the phantoms of terrorism that lurk in the shadows of the global community and they are the subject of the third in this series, Destabilizing Dynamics.

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