Game Theory: Mathematical Models of Conflict

307 Pages·2000·20.877 MB·English

by  A.J. Jones

Game Theory:
Mathematical Models ofConflict
Dedication:
To Tina
Talking of education, people have now a-days" (said he) "got a strange opinion
thatevery thing should betaught by lectures. Now, Icannot seethat lectures can do
so much good as reading the books from which the lectures are taken. I know
nothing that can be best taught by lectures, except where experiments are to be
shewn. You may teach chymestry by lectures- You might teach making ofshoes
by lectures!"
James Boswell: LifeofSamuelJohnson. /766 (1709-1784)
ABOUT OUR AUTHOR
AntoniaJ.Jones graduated from theUniversityofReading withadouble first
in mathematics and physics, and then gained a Ph.D. from Cambridge in
number theory. During her distinguished career she has published numerous
papers across a multi-faceted range of topics in number theory, genetic
algorithms, neural networks, adaptive prediction, control andsynchronisation
of chaotic systems and other works on non-linear modelling and noise
estimation. She has worked in mathematics and computer science at many
renowned academic institutions in the UK and USA, including the Institute
forAdvanced Studies,Princeton N.J.,UniversityofColorado, Boulder,Royal
Holloway College, London, Imperial College, London, and the Computing
Research Laboratory, NMSU Las Cruces. She is a Life Fellow of the
Cambridge Philosphical Society.
She has helped to prepare artificial intelligence and neural network courses
for Control Data Corporation and Bellcore, acted as consultant in advanced
computing techniques for anumber of majorcompanies, has been an invited
visitortoGMD,Cologne,Germany, theOregonGraduateInstitute,Weizmann
Institute, Israel, Coppe Sistemas, UF Rio de Janeiro, and has a continuing
research relationship with Laborat6rio de Inteligencia Computacional, Sao
Carlos-USP, Brazil.
Currently Professor ofEvolutionary and NeuralComputing atthe University
of Wales, Cardiffshe has settled in afarm in the Brecon Beacons. The farm
provides apeaceful environmentforresearchandstudyandenables academic
visitors from around the world to enjoy Welsh culture and the beauty ofthe
mountains.
Game Theory:
Mathematical Models of Conflict
A.J. Jones
Professor ofComputerScience
University ofCardiff
WP
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Table of contents
Author's Preface ix
Glossary of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii
1The name of the game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1
1.1INTRODUCTION 1
1.2EXTENSIVE FORMS AND PURE STRATEGIES 2
1.3NORMAL FORMS AND SADDLE POINTS 8
1.4MIXED STRATEGIES AND THE MINIMAX THEOREM 17
1.5DOMINANCE OF STRATEGIES 21
1.62 x n AND SYMMETRIC GAMES . . . . . . . . . . . . . . . . . . . . . . . . . . .. 24
1.7OTHER KINDS OF TWO PERSON ZERO SUM GAMES 33
PROBLEMS FOR CHAPTER 1 40
CHAPTER REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 45
2 Non-cooperative Games 47
2.1 EXTENSIVE FORMS AND EQUILIBRIUM N-TUPLES 47
2.2 NORMAL FORMS AND MIXED STRATEGY EQULIBRIA . . . . . . . . .. 54
2.3 DISCUSSION OF EQUILIBRIA ... . . . . . . . . . . . . . . . . . . . . . . . . . .. 61
2.4 PRELIMINARY RESULTS FOR 2-PERSON ZERO SUM GAMES ..... 63
2.5 THE MINIMAX THEOREM FOR MATRIX GAMES . . . . . . . . . . . . . .. 67
2.6 PROPERTIES OF MATRIX GAMES 73
2.7 SIMPLIFIED 2-PERSON POKER 81
2.8 CONTINUOUS GAMES ON THE UNIT SQUARE 87
PROBLEMS FOR CHAPTER 2 94
CHAPTER REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 98
3 Linear Programming and Matrix Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 100
3.1 INTRODUCTION 100
3.2 PRELIMINARY RESULTS 106
3.3 DUALITY THEORY 109
3.4 THE GEOMETRIC SITUATION 117
3.5 EXTREME POINTS OF THE FEASIBLE REGION 123
3.6 THE SHAPLEY-SNOW PROCEDURE FOR GAMES . . . . . . . . . . . . .. 126
3.7 MIXED CONSTRAINTS, SLACK VARIABLES. AND THE
TABLEAU 136
3.8 THE PIVOT OPERAnON .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 140
3.9 ARTIFICIAL VARIABLES 147
PROBLEMS FOR CHAPTER 3 157
CHAPTER REFERENCES 160
vi Table of contents
4 Cooperative games 161
4.1 UTILmES AND SCALES OF MEASUREMENT 161
4.2 CHARACTERISTIC FUNCTIONS 164
4.3 IMPUTATIONS 174
4.4 STRATEGIC EQUIVALENCE 177
4.5 DOMINANCE OF IMPUTATIONS . . . . . . . . . . . . . . . . . . . . . . . . . .. 182
4.6 VON NEUMANN-MORGENSTERN SOLUTIONS 188
4.7 THE EDGEWORTH TRADING MODEL 195
4.8 THE SHAPLEY VALUE 202
PROBLEMS FOR CHAPTER 4 206
CHAPTER REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 208
5 Bargaining Models 210
5.1 INTRODUCTION 210
5.2 GRAPHICAL REPRESENTATION OF GAMES AND STATUS QUO
POINTS 211
5.3 THE NASH BARGAINING MODEL 215
5.4 THE THREAT GAME . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 221
5.5 SHAPLEY AND NASH SOLUTIONS FOR BIMATRIX GAMES 224
5.6 OTHER BARGAINING MODELS 231
PROBLEMS FOR CHAPTER 5 235
CHAPTER REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 236
Appendix 1Fixed Point Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 237
REFERENCES 238
Appendix II Some Poker Terminology 239
Solutions to problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 242
CHAPTER 1 242
CHAPTER 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 255
CHAPTER 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 267
CHAPTER 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 272
CHAPTER 5 275
INDEX 281
vii
LIST OF FIGURES
Figure 1.1Game tree for 2-2 Nim. . . 3
Figure 1.2This does not happen. . 4
Figure 1.3Game tree for Renee v Peter. . . 5
Figure 1.4Game tree for Red-Black . 6
Figure 1.5Extensive form of Russian roulette. . 15
Figure 1.6Graphical representation of Example 1.6. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Figure 1.7Graphical representation of Example 1.6. . . 26
Figure 1.8A's Gain floor in Example 1.7 . 27
Figure 1.9An area division game . 33
Figure 1.10Coordinates for the pursuit game . 37
Figure 2.1 Truncating r. . . 49
Figure 2.2 Set of mixed strategies ~(j) (mj=3) . 56
Figure 2.3 Convex and non-convex sets. . . 57
Figure 2.4 Convex and concave functions. . . 63
Figure 2.5 The set r (m =2, n =3). . . 68
Figure 2.6 Support planes to S(m =2). . 69
Figure 2.7 rand 0A (m =2). Because 0A (1 r 4= <p Ais r-admissible. . . 69
Figure 2.8 rand 0Ao(m =2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
Figure 2.9 Supportplane to 0AO' ..•..•.....•..•................•... . 70
Figure 2.10 VariationofPix, y). . . . . . . . . . . . . . . . . . . . . . . . . • . . • . . . . . . . . . . . . . . . 88
Figure 3.1 H separates C and b. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
Figure 3.2 Feasible region for the minimum problem (3.34) . 118
Figure 3.3 Feasible region for the maximum problem (3.35). . . 119
Figure 4.1 Utility for money need not be linear and may vary from one individual to another.
162
Figure 4.2 Imputations which satisfyx3 s 2/3, . 187
Figure 4.3 The core: those imputations which satisfy all three inequalities in Example4.3.
188
Figure 4.4 The three line segments which form W . 192
Figure 4.5 Stable sets for small a. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
Figure 4.6 Indifferencecurves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
Figure 4.7 -The Edgeworth Box. . . 196
Figure 4.8 The contractcurve CD. . 196
Figure 5.1 Payoffpolygon. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
Figure 5.2 Graph for Example 5.1. . 213
Figure 5.3 Graph for Example 5.2. . . 215
Figure5.4 Graph for Example 5.3. . . 221
Figure 5.5 Smooth Pareto optimal boundary. . . 222
Figure 5.6 Polygonal Pareto optimal boundary ABCD. . . . . . . . . . . . . . . 222
Figure 5.7 Diagram for Example 5.4 . 227
Figure 5.8 Diagram for Example 5.5. . . 228
Figure 5.9 Battle ofthe sexes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
viii List of Figures
Figure5.10BoSregarded asacooperative game. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 233
Figure5.11 Alternative negotiation scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 233
FigureA 1Illustrating theexistance of afixed point foracontinuous functionon [0, 1]. 237
FigureS 1Matching Penniesby choosing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 242
FigureS2 Matching Penniesby tossing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 242
FigureS3AttackonConcord. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 243
FigureS4 PeaceCorps vFlies. 243
FigureS5 AcesandQueens. ...............................•........... 244
FigureS6Graphical solution forQI-6. 245
FigureS7Hi-La. 246
FigureS8Payoff to Ain 1-2Nim. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 247
FigureS9Graph forBin QI-9(a). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 248
FigureS 10Graphfor AinQI-9(b). 249
FigureS 11Extensiveformfor Kingsand Aces. ...............................• 250
FigureS 12Graphical solution forQl-ll. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 251
FigureS 13Daughter-Mother-Fathergame. 252
FigureS 14Graph forQ2-1O. 260
FigureS 15AnnandBill'ssimplified Poker. 261
FigureS 16 Geometric interpretation for theequations ofQ3-2. . . . . . . . . . . . . . . . . . . . . . .. 268
FigureS 17DiagramsforQ4-3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 273
FigureS 18Diagram forQ5-1. 276
FigureS 19Diagram forQ5-2. 277
Author's Preface
This book originally evolved from a series of lectures in the theory of games which I gave
someyears agoat RoyalHollowayCollege.Ihopethatitwillserve tocomplimenttheexisting
literature and perhaps encourage other teachers to experiment by offering a course in game
theory. My experience has been that students are keen to learn the subject and, although it is
notalways whattheyexpect, theyenjoyit,providedonedoes notdwellon thetechnically more
difficultproofs.Oneattractive aspectofthesubjectisthat,unlikeotherquite puremathematical
disciplines, game theory offers room for interesting debates on, for example, what constitutes
rational behaviour (the classical example being the Prisoner's Dilemma), how one might
approach the general problem of social policy choice", and the nature of competitive markets
in economic theory.
Beforeone is able toappreciate the virtueof proving thingsit is first necessary to have afinn
intuitive grasp of the structure in question. With this objective in view I have written Chapter
I with virtually no proofs at all. After working through this chapter the reader should have a
finn grasp of the extensive form, the idea of a purestrategy, the normal form, the notion of a
mixedstrategy, and the minimax theorem, and beable to solve a simple two-person zero sum
gamestarting from the rules. These ideas, particularly that of a pure strategy, are quite subtle,
andthesubtletiescanonlybegraspedbyworkingthroughseveraloftheproblems providedand
making the usual mistakes.
BeyondChapter 1there arejust twoitems whichshould be included in any course: the Nash2
equilibrium point as a solution concept for a n-person non-cooperative game, which is
introduced early in Chapter 2, and the brief discussion of utility, which is postponed until the
beginning of Chapter 4. Granted these exceptions, Chapters 2-5 are logically independent and
can be taken in any order or omitted entirely! Thus a variety of different courses can be
1Some very nice examples of such mathematical analyses can be found in Game Theory
and Social Choice - Selected papers ofKenjiro Nakamura, Ed. Mitsuo Suzuki, Keiso Shobo
Publishing Co. 2-23-15 Koraku, Bunkyo-ku, Tokyo, Japan, 1981. Unfortunately Kenjiro
Nakamura,a graduate of theTokyoInstitute ofTechnology, died at the early age of 32.These
selected papers are his gift to the subject to which he had devoted his life.
2 An excellent account of the life of John Nash, who was awarded the Nobel Prize in
Economics in 1994,is recently available in:A Beautiful Mind, Sylvia Nasar, Faber and Faber
(Simon & Schuster in the USA), 1998,ISBN 0-571-17794-8).