APPLICATION OF GAME THEORY IN SOLVING

HUMAN CONFLICT PROBLEMS

WONG LING MING

UNIVERSITI TEKNOLOGI MALAYSIA

APPLICATION OF GAME THEORY IN SOLVING

HUMAN CONFLICT PROBLEMS

WONG LING MING

This thesis is submitted in fulfillment of the requirement for the award of the

Bachelor’s Degree in Science Education (Mathematic).

Faculty of Education

Universti Teknologi Malaysia

MARCH 2006

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Especially for my ever supportive, encouraging and loving dad, Wong Yiik Lung

and mom, Yong Pik Choo. My brother and sisters, Wong Ling Huo, Wong Ling Yu,

Wong Ling Nung, Wong Ling Yieng and my Lord.

With gratitude for all the love and support that all of you had given to me. You

all are the best in my life. You are all always warm in my heart.

I love you all. Thank you.

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ACKNOWLEDGEMENT

With this opportunity, I would like to express a billion thanks to Dr. Sabariah

Baharun as my project supervisor whom keeping me to complete this project.

Besides of her weekly schedule to meet me, she has guided me especially in my

writing skill and connection of the subtopic. She taught me much in writing in

English language besides making sure that I had all the necessary information and

examples in doing well with my project. I really must admit that her sharing has

helped me in many ways.

Nevertheless, I would like acknowledge my thanks to many individuals as my

encouraging supporters such as my best friends, my coursemates who also in doing

theses. We gave our support to each other.

Moreover, I would like to express my deepest thanks to my family for their

kindness and encouragement in whatever way. They really love me and play the

main and important roles not only in my project, but in my life also.

Finally, I thank you to all those involved directly and indirectly in helping me

to complete the project which I could not state out every one of them. Thank you for

everyone for their generosity and tolerance in doing all the things.

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ABSTRACT

Game theory is a branch of applied mathematics that studies strategic

situations where players choose different actions in an attempt to maximize their

returns. First developed as a tool for understanding economic behavior, game theory

is now used in many diverse academic fields, ranging from biology to philosophy

and its most recent application is by computer scientists using artificial intelligence

and cybernetics. It is aim of this project to investigate the mathematical applications

of the game theory in solving human conflicts. The initial attempt made is to study

the concept of game theory in particular the usage of mathematics in solving the

problems. The solution method of a noncooperative two person zero sum game, the

focus of this study, will be highlighted using Minimax Theorem. Several cases

representing the conflict problems are also solved using the Minimax Theorem with

the aid of Winplot and Matlab Programming.

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ABSTRAK

Teori permainan merupakan salah satu cabang dalam bidang matematik yang

mengkaji situasi strategik di mana pemain-pemain memilih tindakan yang berlainan

bermatlamat untuk memaksimumkan ganjaran mereka. Teori permainan ini pertama

sekali dikembangkan sebagai satu alat untuk memahami keadaan ekonomi, dan kini

ia digunakan dalam pelbagai bidang akademik, dari lingkungan biologi ke falsafah

serta aplikasi terbaru ini dipelopori oleh sainstis komputer dengan menggunakan

kecerdasan buatan dan siber-netik. Projek ini bertujuan untuk mengkaji aplikasi

secara matematik tentang teori permainan dalam menyelesaikan konflik manusia.

Projek ini memfokuskan kepada kaedah penyelesaian untuk permainan hasil tambah

sifar – dua orang (tak koperatif) yang menekankan penggunaan Teorem Minimax.

Beberapa kes yang menggambarkan masalah konflik juga diselesaikan dengan

Teorem Minimax berbantukan pengaturcaraan Winplot dan Matlab.

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CONTENTS

CHAPTER SUBJECTS PAGE

PENGESAHAN STATUS TESIS

SUPERVISOR ENDORSEMENT

TITLE i

DECLARATION ii

DEDICATION iii

ACKNOWLEDGEMENT iv

ABSTRACT v

ABSTRAK vi

CONTENTS vii

LIST OF TABLES x

LIST OF FIGURES xii

LIST OF SYMBOLS xiii

LIST OF APPENDIXES xv

CHAPTER I INTRODUCTION

1.1 Introduction 1

1.2 Project Background 2

1.3 Objectives 3

1.4 Scope 4

1.5 Summary and Outline of Project 4

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CHAPTER II LITERATURE REVIEW

2.1 Introduction 6

2.2 History of Game Theory 6

2.3 Game Theory 8

2.3.1 Elements of Game Theory 10

2.3.2 Division of Game Theory 12

2.3.2.1 Comparison of Noncooperative

and Cooperative Game 13

2.3.2.2 Noncooperative Games 14

2.4 Concluding Remarks 19

CHAPTER III SOLUTION CONCEPT OF THE MATRIX GAME

3.1 Introduction 20

3.2 Matrix 20

3.3 Minimax Theorem

(Von Neumann’s Theorem) 22

3.3.1 Minimax and Maximin

Criteria Principle 23

3.3.2 Minimax Theorem (Proof) 24

3.3.3 Saddle Point 28

3.3.4 Domination 30

3.4 Mixed Strategy 33

3.5 Larger Matrix Solution (m × n) 40

3.6 Concluding Remarks 46

CHAPTER IV GAME THEORY (MINIMAX THEOREM)

IN HUMAN CONFLICT PROBLEMS

4.1 Introduction 47

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4.2 Application of Minimax Theorem 49

4.3 Concluding Remarks 59

CHAPTER V CONCLUSION AND RECOMMENDATIONS

5.1 Introduction 60

5.2 Summary of Findings 60

5.3 Suggested Recommendations 62

REFERENCES 63

APPENDIXES 65-69

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LIST OF TABLES

No. Title Page

2.1 Payoff of Example 2.1 12

2.2 Comparison of Noncooperative

and Cooperative Game 13

2.3 Example of a reduced form from

game tree (Figure 2.5) 17

2.4 Bi-matrix Game

(A simple finite noncooperative game) 18

2.5 Simple Bimatrix Game (Nash Equilibrium). 18

2.6 Simple Bi-matrix Game 18

3.1 Game with Payoff A: m × n Game 21

3.2 Application of Minimax theorem 29

3.3 Application of domination 31

3.4 Elimination process in domination 32

3.5 Elimination process in domination 32

3.6 Elimination process in domination 32

3.7 (2 × 4 Game) 34

3.8 The Solution of Example 3.3 38

3.9 (Matrix 4 × 2 Game) 39

3.10 The solution of Example 3.4 40

4.1 Payoff Matrix for the Battle of the Bismark Sea 50

4.2 Application of Minimax Theorem in the Battle 50

4.3 Rule for the Game of Paper, Rock and Scissors 52

4.4 Payoff Matrix of the game of

Paper, Rock and Scissors 52

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4.5 Matrix of Fighters and Bombers 55

4.6 Application of Minimax Theorem in the matrix 55

4.7 Solution of Fighters and Bombers 58

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LIST OF FIGURES

No. Title Page

1.1 Project Flow 5

2.1 Summary of History 7

2.2 Elements of Game Theory 10

2.3 Hide-and-Seek Game 11

2.4 Division of the Game Theory 12

2.5 Head and Tail Game 15

3.1 Expectation against BI 35

3.2 Expectation against BII 35

3.3 Expectation against BIII 36

3.4 Expectation against BIV 36

3.5 Expectation of four graphs above 37

3.6 Combination of Four Equations 39

3.7 Probabilities of Player A and B 42

3.8 Probabilities of Player A and B 45

4.1 Probabilities of Player A and B 53

4.2 Fighters and Bombers 54

4.3 Graph 1 56

4.4 Graph 2 56

4.5 Combination of Graph 1 and 2 57

4.6 Probabilities of Player A and B 58

5.1 Flow Chart of Solution Method in Matrix 61

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LIST OF SYMBOLS

I - Set of player

i - Player

Si - Available actions

Γ - A system of game

iH - Payoff functions

E = v - Payoff

RP - Row player

CP - Column Player

m × n - Size of Matrix

A - Player A

B - Player B

N - Set of players

n - tuple - Represent the mathematical expectation

ii ∑∈σ - A set of strategies

π - Function of Game

Γi

Ni→∑∏=

∈π - Outcome set of the game

R→Γ:νi - Outcome

aij - expected payoff

X - Matrix

Ai - Strategy i for A

νb - Payoff (Infimum value)

Bj, - Strategy j for B

νa - Payoff (Supremum value)

f(x) - Real-valued Function

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xi - Probability of A of choosing ith row

xj - Probability of A of choosing jth column

Sm - Ordered m-tuple

Sn - Ordered n-tuple

( *ix , *

jy ) - Optimum solution as it is denoted

p - x

q - y

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LIST OF APPENDIXES

Appendix Title Page

A Some Types of Outdoor Game 65

B Example of Obstacles Game in Scout 66

C Winplot Programming 67

D Matlab Programming (Solving Overdetermined) 68

E Example 3.5: Solution in Matlab Programming 69

CHAPTER I

INTRODUCTION

1.1 Introduction

Game theory is a mathematical study of games. By a game we mean not only

recreational games like chess or poker. We also have in mind more serious “game”,

such as contract negotiation between a labor union and a corporation, war negotiation

or an election campaign (Elliott Mendelson, 2004). Thus, we can say that the word

“competition” might be more appropriate before the established term of “game”. In

this particular mathematical study, we are interested in finding optimal strategies to

get the best solution for this situation.

According to Martin J. Osborne and Ariel Rubinstein (1994), game theory is

a tool to help us to understand how a decision maker interacts in a situation. The

situation can be represented by a mathematical form. With the help of full

information about the situation, we can obtain the best and precise decision. In this

situation, decision maker will always choose the optimal strategies to get the best

payoff. Game can be played once or it can be repeated. We can use game theory to

interpret, explain and make prediction about the likely outcomes (payoff) of decision

problems.

Game theory is generally considered to have begun with the publication of

von Neumann and Mongerstern’s The Theory Of Games and Economic Behaviour in

1944. In this published book, it introduced the idea that the conflict could be

mathematically analyzed and provided the related terminology. In the recreational

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game such as the poker game (John McDonald, 1950), he identified that a player’s

winning or losing does not depend fully to his own moves but it also depends on the

moves of others.

There are times in a person’s life that he will face conflict in making his

decision. In this case, there is no involvement of other person and it is called ‘game

against nature’. As an example, Liza has to decide on how she would go on her own

to Skudai Parade whether it will be by bus or motorcycle. In such a condition, she

only considers her safety convenience. This is the simplest game where a single

player makes a decision for his/her own sake.

There are many complex problems (serious conflict) occurred in real world

phenomena. Here it is often natural to consider situations where the players must no

longer act as isolated individual. Serious conflict as stated before is bound to occur

in everyone life undoubtedly (two or more individuals). Every party will only aim

for what is the best for his/her own purpose and not considering about others’. Game

theory is useful in making decisions involving two or more decision makers solving

their conflicts. This often happens in the military world and business competition.

For instance a decision maker deciding on the best strategies will maximize his own

party’s benefits and in doing so will increase the losses of the other party.

1.2 Project Background

Mathematical models have been used to solve complex problems such as

those in social sciences, economics, psychology, and politics. Game theory is a

branch of applied mathematics that uses models to study interaction with formalized

incentive structure “game”. This makes it easier to analyses all the game in the

mathematical form or structure. Game theory helps in the making decision involving

conflicts. Generally, decisions should be made without letting other know his/her

decisions before the game is ended.

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Game theory is also applied in children’s game such as Paper, Rock and

Scissor game and Hide and Seek game. There are many often favorite games played

by children. Sometimes, they will make up the rules and change them at will to

make the game more interesting. Similarly, the same thing happens in business

transaction such as. These will happen the bargaining activities, negotiation,

combination or sharing the company and so on. Game theory helps to solve the

conflict between two or more individuals/companies when dealing with tender,

contract between two firms and anything related to business. So, the decision makers

need to make all decisions intensively to reduce the loss.

Game theory is also applied in military affairs to get the best strategies in

warfare. The war activity is solved by game theory rather than using the traditional

method. The war is a combinative nature of society appears to be resolving world

politics into two man game. Yet, not every player in the world is involved. A two or

more person/parties game may occur in military affairs and some games would end

with negotiation.

1.3 Objectives

This study has the following objectives:

i) To study the basic concept of game theory.

ii) To study the mathematical method in solving game theory problems,

particularly the Minimax Theorem.

iii) To solve three problems involving conflict (warfare and human conflict-

family affairs) using the solving method as stated in above.

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1.4 Scope

The scope of the study is to focus on the application of game theory in war

and human conflict. The concentration of study will be on:

i) Noncooperative zero sum game

ii) The discussion about Minimax and Maximin principle, saddle point,

domination

iii) Solution of matrix 2 × n, m × 2 and 3 × 3 by using Winplot and Matlab

Programming

1.5 Summary and Outline of Project

The result of the project is compiled in five chapters with their descriptions

given below:

Chapter 2 is devoted to the game theory itself in literature review. The reader

would be acknowledged in the period and inventor of the theory through the Section

2.2 in the history of game theory. By using some examples, the elements in the game

theory are introduced. In addition, Noncooperative and Cooperative games will be

discussed.

Chapter 3 illustrates basic concept of the theory. The Minimax Theorem,

served as the solving method of game theory is introduced. The strategy of the

matrix can be found in the form of pure or mixed strategy. Examples using Winplot

and Matlab Programming in solving the problem are also illustrated.

In Chapter 4, the reader will find some of the applications that make game

theory so interesting, for example in family affairs and two cases in solving the

problem in warfare. The discussed principle or method will be presented in this

chapter to complete the case study.

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Chapter 5 will conclude the throughout study and some recommendations for

the future research will also be suggested.

The outline of the project can be summarized as following Figure 1.1:

Figure 1.1: Project Flow

Literature Review (Chapter 2)

Basic concept of Game Theory (Chapter 3)

Solving Method-Minimax Theorem

Pure and Mixed Strategy

Application / Solution (Chapter 4)

Warfare (Two Cases) Human conflict problems (Family Affairs, One Case)

Conclusion and Recommendations (Chapter 5)

Introduction (Chapter 1)

Application of

Game Theory in Solving

Human Conflict Problems