report on game theory 20131214

Mr. Dickinson Cook International Relation

ESG Management School / Class 14

KIM Woojin

THE GAME THEORY OF NUCLEAR

PROLIFERATION

Game Theory of Nuclear Proliferation 2

International Relation Mr. Dickinson Cook

The Game Theory of Nuclear Proliferation

Table of Contents

I. Introduction: Game Theory of International Relations ......................... 3

Definition of Game Theory ....................................................................................... 3

Application of Game Theory to International Relations ......................................... 3

II. Dominant Strategy & Nash Equilibrium in Proliferation ..................... 4

Prisoner’s Dilemma ................................................................................................... 4

Dominant Strategy During The Cold War ............................................................... 5

Nash Equilibrium During The Cold War ................................................................ 6

Nuclear Proliferation Today ...................................................................................... 7

III. Conclusion: Strategies for The Optimal Choice .................................. 8

Carrot-and-stick Policy ............................................................................................... 8

Change The Rules of The Game ............................................................................... 8




I. Introduction: Game Theory of International Relations

Definition of Game Theory

Game Theory was first invented by the great genius mathematician of 20th century, John von Neumann.

Theoretic basis of the theory was established in <Theory of Games and Economic Behavior> which von

Neumann and his colleague and economist, Oscar Morgenstern published together in 1944. Then another

legendary American mathematician, John Forbes Nash made huge contribution to develop the game theory by

Nash equilibrium that generalized Cournot competition.

Simply defining, game theory is a study of strategic decision making. More formally, it is ‘‘the study of

mathematical models of conflict and cooperation between intelligent rational decision-makers’’.1 Since 1950’s,

game theory is being applied widely in not only microeconomics but also in most of economic field such as

macroeconomics, industry organization theory, international trade theory and theory of finance. Today its

importance has grown that it’s even used in political science and psychology as well as logic and biology.

Therefore it is no exaggeration to say that game theory is a basic theory to understand behaviors of economic

units and a number of economic phenomenon.

Application of Game Theory in International Relations

My first exposure to game theory was in my 16 by the book <게임이론(Game Thoery)> written by Globis

Management Institute, since then I studied it on my own. When given this work to analyze international

relations by my professor, Mr. Dickinson Cook, I thought it would be an interesting work if I apply my

knowledge in game theory into international relations. In my opinion, game theory can be used to explain the

behavior of states in the international system, especially under realist philosophies. By assuming that all actors

1 Wikipedia, http://en.wikipedia.org/wiki/Game_theory




in the system are rational and that the system is anarchic, game theory can be used to model a scenario and to

determine the most probable outcome.2 In consideration of limited pages, I will analyze specifically nuclear

proliferation during the Cold War(games with only 2 players) and today(games with multiple players) mostly

based on prisoner’s dilemma at the most basic level. The reason why I chose nuclear proliferation as a main

topic is 1) it’s what we are learning recently in the class and 2) it can be an intuitive example to explain the

prisoner’s dilemma.

II. Dominant Strategy & Nash Equilibrium in Proliferation

Prisoner’s Dilemma

Firstly understanding the concept of prisoner’s dilemma is essential before look into the real case of

proliferation during the Cold War.

Two members of a criminal gang are arrested and are in solitary confinement. The police doesn’t have

enough evidence so they offer a Faustian bargain(Deal with the Devil). Each prisoner has opportunities to

betray the other by confessing, or to cooperate with the other by remaining silent. Here’s how it goes:

If the prisoner A and B both betray the other, both of them serve 10 years in prison.

If A betrays but B remains silent, A will be released and B will serve 20 years in prison. (and vice

versa)

If A and B both remain silent, both of them will serve only 1 year in prison.

In this case, what is the best strategy that is most likely to be taken by two prisoners? Should they betray by

confessing or cooperate by keeping silent? In order to know the best strategy, we need to analyze each strategy

under two different conditions:

2 Answers, http://wiki.answers.com/Q/Game_theory_of_international_relations#slide2




When B betrays, A should betray as well to avoid from being sentenced to 20 years in prison. (vice

versa)

When B remains silent, A should betray in order to be set free. (vice versa)

In this case, to confess is always the best strategy regardless of the other’s decision. Thus the strategy, which

is better than another strategy for one player, no matter how that player’s opponents may play, is called

dominant strategy in game theory.3 However we cannot say that this is an optimal choice for the players

because they could have been sentenced only 1 year if both of them chose to remain silent. In other words,

opportunity cost can get bigger when a decision is taken by dominant strategy ironically. I’ll provide the

solution for reaching the optimal choice in the chapter 3.

Dominant Strategy During The Cold War

Then how can we explain the proliferation during the Cold War between the United States and Soviet Union

with prisoner’s dilemma? The game theory explains well the nuclear arms race between two superpowers

especially by its peak year 1962 when there was the Cuban missile crisis. The total number of nuclear weapons

in the U.S. Stockpile in October 1962 was approximately 26,400 and the Soviet Union approximately 3300.4 It

could already blow up the entire planet over many times.5 This cut-throat competition in nuclear weapons made

the whole world haunted by the fear of war. How was this kind of reckless and rash completion possible?

Actually this case is regarded basically the same as the previous prisoner’s dilemma if we substitute ‘confess’

with ‘acquire more nuclear weapons’ and ‘take the Fifth’ with ‘decrease or maintain nuclear weapons’. It was a

rational decision for the U.S. and Soviet Union to increase the quantity of nuclear weapons according to their

dominant strategy-‘acquire more nuclear weapons’.

3 Wikipedia, http://en.wikipedia.org/wiki/Dominant_strategy 4 S. Norris, Robert, The Cuban Missile Crisis: A Nuclear Order of Battle October/November 1962, 2012, p.8 5 Biddle, Sam, How Many Nukes Would It Take to Blow Up the Entire Planet?, 2012, http://gizmodo.com/5899569/how-many-nukes-would-it-take-to-blow-up-the-entire-planet




Nash Equilibrium During The Cold War

However the nuclear weapons stockpile has decreased in the U.S. and Soviet Union since its peak year 1969. At last count, in 2010, the Pentagon revealed it was the proud owner of 5,113 all-American nuclear warheads. That's down from a high of more than 31,000 in the late 1960s.6 How could it happen if every country sticks to its dominant strategy?

In reality, the nuclear arms race is close to a sequential game in the long-run, not a simultaneous game that we’ve seen until now. A sequential game is a game where one player chooses his action before the others choose theirs.7 In sequential games, there is one characteristic that simultaneous games do not have. And that characteristic enables players to cooperate which is impossible to occur in simultaneous games.

For instance, if the U.S. decides to increase nuclear weapons, we can expect Soviet Union to do so too based on its dominant strategy(tit-for-tat strategy). But what if the U.S. stops raising the number of nuclear weapons, would it be beneficial for Soviet Unions to keep increasing it?

The answer is ‘no’. In sequential games, fears of possible reprisal transcend the instant profit. Moreover the

both countries have already enough arms to blow up the earth even if oneside ceases its increase. Simply

speaking, dominant strategy doesn’t work anymore. Instead a group of players are in this situation if each one is

making the best decision that he or she can, taking into account the decisions of the others. When one player

takes a decision without considering opponent’s choice, he will lose the game, but as nobody wants to lose, the

game becomes in equilibrium. Nash equilibrium means this stable state of a system that involves several

interacting participants in which no participant can gain by a change of strategy as long as all the other

participants remain unchanged.8

6 Fung, Brian, The Number of Times We could Blow Up the Earth Is Once Again a Secret, National Journal, 2013, http://www.nationaljournal.com/nationalsecurity/the-number-of-times-we-could-blow-up-the-earth-is-once-again-a-secret-20130701 7 Wikipedia, http://en.wikipedia.org/wiki/Sequential_game 8 Khan Academy, https://www.khanacademy.org/economics-finance-domain/microeconomics/nash-equilibrium-tutorial/nash-eq-tutorial/v/more-on-nash-equilibrium




Nuclear Proliferation Today

Today there are 9 states with nuclear weapons: the US, China, France, Russia, the UK, India, Pakistan,

North Korea and Israle(under clear state). Despite Nuclear Test Ban Treaty in 1963 and Nuclear Non

Proliferation Treaty in 1968, there are more nuclear powers in the world. Why are there more than before? And

what is the result when there are more nuclear power nations?

To answer the first question, we need again basic prisoner’s dilemma. Assuming that there was a Non-

Proliferation Treaty, there are 2 states: the state A is a nuclear weapon state and the state B is non-nuclear

weapon state. However, both countries tend to be led to Lose-Lose situation by betraying each other.(see the

graphic 1 in appendix in page 10)

Second question is deeply concerned with the characteristic of the game with more than 3 players. Just like

the games with 2 players, games with multiple players also have Nash equilibrium-e.g. ‘all players increase

weapons’ or ‘all players decrease weapons’. However the mechanism to reach this equilibrium is more

ambiguous in games with multiple players than in two-player-games. That is to say, to anticipate other players’

strategies becomes so complicated which causes to grow fears of betrayal. In this context, influence of so-called

‘rogue state’ is so huge that it can create distrust overall and prevent the treaty to be enforced. For example,

North Korea withdrew from the NPT on January 10, 2003 and it built tension on the Korean peninsula and

reduced security from all in the end.

III. Strategies for The Optimal Choice

Carrot-and-stick Policy

Going back to prisoner’s dilemma once again. How can both prisoners only serve 1 year in prison which is

the optimal choice for them? For example, both of them swore each other not to reveal each other’s crimes in




any situation before. This vow can be seen as ‘treaty’ when it comes to international relations. But now we

know that it doesn’t give enough motivation to cooperate. But if the prisoners decided to impose penalty, e.g-

seizing all one’s property, when they break the promise, that alters the case. They would cooperate rather than

betraying due to bigger opportunity cost. On the same principal, I think that the NPT should carry stronger legal

binding force than now. The states that withdraw from the NPT should be sanctioned legally on any account.

At the same time, Adopting some appropriate conciliatory mesures are necessary to control rogue states

South Korean government launched the Sunshine Policy regarding North Korea during 1998-2008, that was the

last time that Koreans saw the light of a new dawn. I would like to emphasize on carrots instead of diplomatic

sticks when handling the rouge states as they normally have fewer things to lose than other states.

Change The Rules of The Game

In sequential games, there is one more way to win. It is to change the rules of the game by changing the

order or converting a sequential game into a simultaneous game. The one who is able to do so is who has more

information. However, ‘not credible strategy’ such as a promise that will clearly be broken or a threat, which

contains intentions that are an open book, cannot influence others.

In international relations, meaning of information in game theory can be expanded into military power,

economic power and natural resources, etc. I think that powerful states in the world have the key now. They

have power to sanction and to adopt a conciliatory policy, also to draw international agreements.

Although the number of nuclear powers increased, the total amount of nuclear weapons stockpiles has

decreased since 1986.(see the graphic 2 in appendix in page 9) The world is becoming better, I believe so. I’d

like to end with what President Eisenhower said: "In a nuclear war, there can be no victory--only losers."9

9 The American Presidency Project, http://www.presidency.ucsb.edu/ws/?pid=26506




Appendix

Graphic 1

http://wagingpeacetoday.blogspot.fr/2012/08/prisoners-dilemma-applied-to_10.html




Graphic 2

http://www.nrdc.org/nuclear/nudb/datab19.asp




Bibliography & Webography

A

Answers, http://wiki.answers.com/Q/Game_theory_of_international_relations#slide2

H

Biddle, Sam, How Many Nukes Would It Take to Blow Up the Entire Planet?, 2012, http://gizmodo.com/5899569/how-many-nukes-would-it-take-to-blow-up-the-entire-planet

K

Khan Academy, http://www.khanacademy.org/economics-finance-domain/microeconomics/nash-equilibrium-tutorial/nash-eq-tutorial/v/more-on-nash-equilibrium

T

S. Norris, Robert, The Cuban Missile Crisis: A Nuclear Order of Battle October/November 1962, 2012, p.8

Fung, Brian, The Number of Times We could Blow Up the Earth Is Once Again a Secret, National Journal, 2013, http://www.nationaljournal.com/nationalsecurity/the-number-of-times-we-could-blow-up-the-earth-is-once-again-a-secret-20130701

The American Presidency Project, http://www.presidency.ucsb.edu/ws/?pid=26506

W

Wikipedia, http://en.wikipedia.org/wiki/Game_theory

Wikipedia, http://en.wikipedia.org/wiki/Dominant_strategy

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