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Game Theory and Strategy AMS / MAA ANNELI LAX NEW MATHEMATICAL LIBRARY VOL 36 Game Theory and Strategy Philip D. Straffi n 10.1090/nml/036 GAME THEORY AND STRATEGY Philip D. Straffin Beloit College THE MATHEMATICAL ASSOCIATION OF AMERICA Fifth printing: December 2004 ©1993 by The Mathematical Association of America All rights reserved under International and Pan American Copyright Conventions Published in Washington, DC, by The Mathematical Association of America Library of Congress Catalog Card Number 92-064176 Print ISBN 978-0-88385-637-6 Electronic ISBN 978-0-88385-950-6 Manufactured in the United States of America NEW MATHEMATICAL LIBRARY published by The Mathematical Association of America Editorial Committee Paul Zorn, Chairman Anneli Lax St Olaf College New York University Basil Gordon University of California, Los Angeles Herbert J. Greenberg University of Denver Edward M. Harris Anne Hudson Armstrong State College Michael J. McAsey Bradley University Peter Ungar The New Mathematical Library (NML) was begun in 1961 by the School Mathematics Study Group to make available to high school students short expository books on various topics not usually covered in the high school syllabus. In three decades the NML has matured into a steadily growing series of some thirty titles of interest not only to the originally intended audience, but to college students and teachers at all levels. Previously pub- lished by Random House and L. W. Singer, the NML became a publication series of the Mathematical Association of America (MAA) in 1975. Under the auspices of the MAA the NML will continue to grow and will remain dedicated to its original and expanded purposes. Note to the Reader This book is one of a series written by professional mathematicians in order to make some important mathematical ideas interesting and understandable to a large audience of high school students and laymen. Most of the volumes in the New Mathematical Library cover topics not usually included in the high school curriculum; they vary in difficulty, and, even within a single book, some parts require a greater degree of concentration than others. Thus, while you need little technical knowledge to understand most of these books, you will have to make an intellectual effort. If you have so far encountered mathematics only in classroom work, you should keep in mind that a book on mathematics cannot be read quickly. Nor must you expect to understand all parts of the book on first reading. You should feel free to skip complicated parts and return to them later; often an argument will be clarified by a subsequent remark. On the other hand, sections containing thoroughly familiar material may be read very quickly. The best way to learn mathematics is to do mathematics, and each book includes problems some of which may require considerable thought. You are urged to acquire the habit of reading with paper and pencil in hand; in this way, mathematics will become increasingly meaningful to you. The authors and editorial committee are interested in reactions to the books in this series and hope that you will write to: Anneli Lax, Editor, New Mathematical Library, New York University, The Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, N.Y 10012. The Editors NEW MATHEMATICAL LIBRARY 1. Numbers: Rational and Irrational by Ivan Niven 2. What is Calculus About? by W. W. Sawyer 3. An Introduction to Inequalities by E. F. Beckenbach and R. Bellman 4. Geometric Inequalities by N. D. Kazarinoff 5. The Contest Problem Book I Annual High School Mathematics Examinations 1950-1960. Compiled and with solutions by Charles T. Salkind 6. The Lore of Large Numbers by P. J. Davis 7. Uses of Infinity by Leo Zippin 8. Geometric Transformations I by I. M. Yaglom, translated by A. Shields 9. Continued Fractions by Carl D. Olds 10. Replaced by NML-34 11. 1 Hungarian Problem Books I and II, Based on the Eö tvö s Competitions 12. J 1894-1905 and 1906-1928, translated by E. Rapaport 13. Episodes from the Early History of Mathematics by A. Aaboe 14. Groups and Their Graphs by E. Grossman and W. Magnus 15. The Mathematics of Choice by Ivan Niven 16. From Pythagoras to Einstein by K. O. Friedrichs 17. The Contest Problem Book II Annual High School Mathematics Examinations 1961-1965. Compiled and with solutions by Charles T. Salkind 18. First Concepts of Topology by W. G. Chinn and Ν . E. Steenrod 19. Geometry Revisited by H. S. M. Coxeter and S. L. Greitzer 20. Invitation to Number Theory by Oystein Ore 21. Geometric Transformations II by I. M. Yaglom, translated by A. Shields 22. Elementary Cryptanalysis— A Mathematical Approach by A. Sinkov 23. Ingenuity in Mathematics by Ross Honsberger 24. Geometric Transformations III by I. M. Yaglom, translated by A. Shenitzer 25. The Contest Problem Book III Annual High School Mathematics Examinations 1966-1972. Compiled and with solutions by C. T. Salkind and J. M. Earl 26. Mathematical Methods in Science by George Polya 21. International Mathematical Olympiads— 1959-1977. Compiled and with solutions by S. L. Greitzer 28. The Mathematics of Games and Gambling by Edward W. Packet 29. The Contest Problem Book IV Annual High School Mathematics Examinations 1973-1982. Compiled and with solutions by R. A. Artino, A. M. Gaglione, and N. Shell 30. The Role of Mathematics in Science by Μ . M. Schiffer and L. Bowden 31. International Mathematical Olympiads 1978-1985 and forty supplementary problems. Compiled and with solutions by Murray S. Klamkin 32. Riddles of the Sphinx by Martin Gardner 33. U.S.A. Mathematical Olympiads 1972— 1986. Compiled and with solutions by Murray S. Klamkin 34. Graphs and Their Uses by Oystein Ore. Revised and updated by Robin J. Wilson 35. Exploring Mathematics with Your Computer by Arthur Engel 36. Game Theory and Strategy by Philip D. Straffin, Jr. 37. Episodes in Nineteenth and Twentieth Century Euclidean Geometry by Ross Honsberger 38. The Contest Problem Book V American High School Mathematics Examinations and American Invitational Mathematics Examinations 1983-1988. Compiled and augmented by George Berzsenyi and Stephen B. Maurer 39. Over and Over Again by Gengzhe Chang and Thomas W. Sederberg 40. The Contest Problem Book VI American High School Mathematics Examinations 1989-1994. Compiled and augmented by Leo J. Schneider 41. The Geometry of Numbers by CD. Olds, Anneli Lax, and Giuliana Davidoff 42. Hungarian Problem Book III Based on the Eö tvö s Competitions 1929-1943 translated by Andy Liu 43. Mathematical Miniatures by Svetoslav Savchev and Titu Andreescu Other titles in preparation. Books may be ordered from: MAA Service Center P. O. Box 91112 Washington, DC 20090-1112 1-800-331-1622 fax: 301-206-9789 Contents Preface ix Part I Two-Person Zero-Sum Games 1 1 The Nature of Games 3 2 Matrix Games: Dominance and Saddle Points 7 3 Matrix Games: Mixed Strategies 13 4 Application to Anthropology: Jamaican Fishing 23 5 Application to Warfare: Guerrillas, Police, and Missiles 27 6 Application to Philosophy: Newcomb's Problem and Free Will 32 7 Game Trees 37 8 Application to Business: Competitive Decision Making 44 9 Utility Theory 49 10 Games Against Nature 56 Part Π Two-Person Non-Zero-Sum Games 63 11 Nash Equilibria and Non-Cooperative Solutions 65 12 The Prisoner's Dilemma 73 13 Application to Social Psychology: Trust, Suspicion, and the F-Scale 81 14 Strategic Moves 85 15 Application to Biology: Evolutionarily Stable Strategies 93 16 The Nash Arbitration Scheme and Cooperative Solutions 102 17 Application to Business: Management-Labor Arbitration 112 18 Application to Economics: The Duopoly Problem 118 Part III iV-Person Games 125 19 An Introduction to N-Person Games 127 20 Application to Politics: Strategic Voting 134 21 N-Person Prisoner's Dilemma 139 22 Application to Athletics: Prisoner's Dilemma and the Football Draft 145 23 Imputations, Domination, and Stable Sets 150 24 Application to Anthropology: Pathan Organization 161 25 The Core 165 26 The Shapley Value 171 27 Application to Politics: The Shapley-Shubik Power Index 177 28 Application to Politics: The Banzhaf Index and the Canadian Constitution 185 viii CONTENTS 29 Bargaining Sets 190 30 Application to Politics: Parliamentary Coalitions 196 31 The Nucleolus and the Gately Point 202 32 Application to Economics: Cost Allocation in India 209 33 The Value of Game Theory 213 Bibliography 217 Answers to Exercises 225 Index 241 Preface Given the long history of mathematics, and the longer history of human con- flict, the mathematical theory of conflict known as game theory is a surprisingly recent creation. John von Neumann published the fundamental theorem of two- person zero-sum games in 1928, but at that time it was simply a theorem in pure mathematics. The theory which provided a broader intellectual context, game theory, was begun in an intense period of collaboration between von Neumann and the economist Oskar Morgenstern in the early 1940's. Their work appeared as the monumental Theory of Games and Economic Behavior [1944]. The importance of the new subject and its potential for social science were quickly recognized. A. H. Copeland [1945] wrote: Posterity may regard this book as one of the major scientific achievements of the first half of the twentieth century. This will undoubtedly be the case if the authors have succeeded in establishing a new exact science— the science of economics. The foundation which they have laid is extremely promising. In the years since 1944 game theory has developed rapidly. It now has a central place in economic theory, and it has contributed important insights to all areas of social science. For a number of years I have taught a course titled "Game Theory and Strat- egy" in Beloit College's interdisciplinary division.
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