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Game Theory: Mathematical Models of Conflict PDF

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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 WOODHEAD PUBLISHING Oxford Cambridge Philadelphia NewDelhi PublishedbyWoodheadPublishingLimited, 80HighStreet,Sawston,CambridgeCB22 3HJ,UK www.woodheadpublishing.com Woodhead Publishing, 1518WalnutStreet.Suite1100,Philadelphia. PA19102-3406, USA Woodhead PublishingIndiaPrivateLimited,0-2, VardaanHouse,7/28AnsariRoad, Daryaganj,NewDelhi- 110002,India www.woodheadpublishingindia.com Firstpublishedin2000byHorwood PublishingLimited; reprinted2004 Reprinted byWoodhead PublishingLimited,2011 ©A.J.Jones,2000 Theauthorhasassertedhermoralrights Thisbookcontainsinformationobtainedfromauthenticandhighlyregardedsources.Reprinted material isquotedwithpermission,andsourcesareindicated. Reasonableeffortshavebeen madetopublishreliabledataandinformation, buttheauthorandthepublishercannotassume responsibility forthevalidityofallmaterials.Neithertheauthornorthepublisher,noranyone elseassociatedwiththispublication,shallbeliableforanyloss,damageorliabilitydirectlyor indirectlycausedorallegedtobecausedbythisbook, Neitherthisbooknoranypartmaybereproducedortransmittedinanyformorbyany means,electronicormechanical, includingphotocopying,microfilmingandrecording,orby anyinformationstorageorretrievalsystem,withoutpermission inwritingfromWoodhead PublishingLimited. TheconsentofWoodhead PublishingLimiteddoesnotextendtocopyingforgeneral distribution,forpromotion, forcreatingnewworks,orforresale.Specificpermissionmustbe obtainedinwritingfromWoodhead PublishingLimitedforsuchcopying. Trademarknotice:Productorcorporatenamesmaybetrademarksorregisteredtrademarks,and areusedonlytoridentificationandexplanation,withoutintenttoinfringe. BritishLibraryCataloguinginPublicationData AcataloguerecordforthisbookisavailablefromtheBritishLibrary ISBN978-1-898563-14-3 PrintedbyLightningSource. 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).

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