Table Of Content

Universitext Editorial Board (North America) S. Axler F.W. Gehring K.A. Ribet Springer New York Berlin Heidelberg Barcelona Hong Kong London Milan Paris Singapore Tokyo Universitext Editors (North America): S. Axler, F.W. Gehring, and K.A. Ribet AksoyfKhamsi: Nonstandard Methods in Fixed Point Theory Andersson: Topics in Complex Analysis Aupetit: A Primer on Spectral Theory BachmanfNaricifBeckenstein: Fourier and Wavelet Analysis BalakrishnaniRanganathan: A Textbook of Graph Theory Balser: Formal Power Series and Linear Systems of Meromo rphic Ordinary Differential Equations Bapat: Linear Algebra and Linear Models (2nd ed.) Berberian: Fundamentals of Real Analysis BoossfBleecker: Topology and Analysis Borkar: Probability Theory: An Advanced Course BottcherfSilbermann: Introduction to Large Truncated Toeplitz Matrices CarlesonfGamelin: Complex Dynamics Cecil: Lie Sphere Geometry: With Applications to Submanifolds Chae: Lebesgue Integration (2nd ed.) Charlap: Bieberbach Groups and Flat Manifolds Chern: Complex Manifolds Without Potential Theory Cohn: A Classical Invitation to Algebraic Numbers and Class Fields Curtis: Abstract Linear Algebra Curtis: Matrix Groups DiBenedetto: Degenerate Parabolic Equations Dimca: Singularities and Topology of Hypers urfaces Edwards: A Formal Background to Mathematics I afb Edwards: A Formal Background to Mathematics II afb Foulds: Graph Theory Applications Friedman: Algebraic Surfaces and Holomorphic Vector Bundles Fuhrmann: A Polynomial Approach to Linear Algebra Gardiner: A First Course in Group Theory GardingfTambour: Algebra for Computer Science Goldblatt: Orthogonality and Spacetime Geometry GustafsonfRao: Numerical Range: The Field of Values of Linear Operators and Matrices Hahn: Quadratic Algebras, Clifford Algebras, and Arithmetic Witt Groups Holmgren: A First Course in Discrete Dynamical Systems HowefTan: Non-Abelian Harmonic Analysis: Applications of SL(2, R) Howes: Modem Analysis and Topology HsiehfSibuya: Basic Theory of Ordinary Differential Equations HumifMiller: Second Course in Ordinary Differential Equations HurwitzfKritikos: Lectures on Number Theory Jennings: Modem Geometry with Applications JonesfMorrisfPearson: Abstract Algebra and Famous Impossibilities KannanfKrueger: Advanced Analysis KellyfMatthews: The Non-Euclidean Hyperbolic Plane Kostrikin: Introduction to Algebra LueckingfRubel: Complex Analysis: A Functional Analysis Approach MacLanefMoerdijk: Sheaves in Geometry and Logic Marcus: Number Fields McCarthy: Introduction to Arithmetical Functions (continued after index) Peter Morris I ntroduction to Game Theory With 44 Illustrations Springer Peter Morris Mathematics Department Penn State University University Park, PA 16802 USA Editorial Board (North America): S. Axler F.w. Gehring Mathematics Department Mathematics Department San Francisco State University East Hall San Francisco, CA 94132 University of Michigan USA Ann Arbor, MI 48109 USA K.A. Ribet Mathematics Department University of California at Berkeley Berkeley, CA 94720-3840 USA Mathematics Subject Classifications (1991): 90Dxx, 90C05 Library of Congress Cataloging-in-Publication Data Morris, Peter, 1940- Introduction to game theory 1 Peter Morris. p. cm. - (Universitext) Includes bibliographical references and index. ISBN-13: 978-0-387-94284-1 e-ISBN-13: 978-1-4612-4316-8 DOl: 10.1007/978-1-4612-4316-8 1. Game theory. I. Title. QA269.M66 1994 94-6515 519.3-dc20 Printed on acid-free paper. © 1994 Springer-Verlag New York, Inc. All rights reserved. This work may not be translated or copied in whole or in part without the writ ten permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in con nection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. Production managed by Laura Carlson; manufacturing supervised by Gail Simon. Camera-ready copy prepared by the author using AMS-LATEX, Printed and bound by R. R. Donnelley & Sons, Harrisonburg, VA. 9876543 To my mother, Opal Morris, and to the memory of my father, W. D. Morris Preface The mathematical theory of games has as its purpose the analysis of a wide range of competitive situations. These include most of the recreations which people usually call "games" such as chess, poker, bridge, backgam mon, baseball, and so forth, but also contests between companies, military forces, and nations. For the purposes of developing the theory, all these competitive situations are called games. The analysis of games has two goals. First, there is the descriptive goal of understanding why the parties ("players") in competitive situations behave as they do. The second is the more practical goal of being able to advise the players of the game as to the best way to play. The first goal is especially relevant when the game is on a large scale, has many players, and has complicated rules. The economy and international politics are good examples. In the ideal, the pursuit of the second goal would allow us to describe to each player a strategy which guarantees that he or she does as well as possible. As we shall see, this goal is too ambitious. In many games, the phrase "as well as possible" is hard to define. In other games, it can be defined and there is a clear-cut "solution" (that is, best way of playing). Often, however, the computation involved in solving the game is impossible to carry out. (This is true of chess, for example.) Even when the game cannot be solved, however, game theory can often help players by yielding hints about how to play better. For example, poker is too difficult to solve, viii Preface but analysis of various forms of simplified poker has cast light on how to play the real thing. Computer programs to play chess and other games can be written by €Onsidering a restricted version of the game in which a player can only see ahead a small number of moves. This book is intended as a text in a course in game theory at either the advanced undergraduate or graduate level. It is assumed that the students using it already know a little linear algebra and a little about finite probability theory. There are a few places where more advanced mathematics is needed in a proof. At these places, we have tried to make it clear that there is a gap, and point the reader who wishes to know more toward appropriate sources. The development of the subject is introductory in nature and there is no attempt whatsoever to be encyclopedic. Many interesting topics have been omitted, but it is hoped that what is here provides a foundation for further study. It is intended also that the subject be developed rigorously. The student is asked to understand some serious mathematical reasoning. There are many exercises which ask for proofs. It is also recognized, however, that this is an applied subject and so its computational aspects have not at all been ignored. There were a few foreshadowings of game theory in the 1920's and 1930's in the research of von Neumann and Borel. Nevertheless, it is fair to say that the subject was born in 1944 with the publication of [vNM44]. The authors, John von Neumann and Oskar Morgenstern, were a mathematician and an economist, respectively. Their reason for writing that book was to analyze problems about how people behave in economic situations. In their words, these problems "have their origin in the attempts to find an exact description of the endeavor ofthe individual to obtain a maximum of utility, or, in the case of the entrepreneur, a maximum of profit." Since 1944, many other very talented researchers have contributed a great deal to game theory. Some of their work is mentioned at appropriate places in this book. In the years after its invention, game theory acquired a strange rep utation among the general public. Many of the early researchers in the subject were supported in part or entirely by the U.S. Department of De fense. They worked on problems involving nuclear confrontation with the Soviet Union and wrote about these problems as ifthey were merely inter esting complex games. The bloody realities of war were hardly mentioned. Thus, game theory was popularly identified with "war gaming" and was thought of as cold and inhuman. At the same time, the power of the theory was exaggerated. It was believed that it could solve problems which were, in fact, far too difficult for it (and for the computers of that time). Later, in reaction to this, there was a tendency to underestimate game theory. In truth, it is neither all-powerful nor a mere mathematician's toy without relevance to the real world. We hope that the usefulness of the theory will Preface ix become apparent to the reader of this book. It is a pleasure to thank some people who helped in moving this book from a vague idea to a reality. Mary Cahill, who is a writer of books of a different sort, was unfailing in her interest and encouragement. Linda Letawa Schobert read part of the first draft and offered many suggestions which greatly improved the clarity of Chapter I. Ethel Wheland read the entire manuscript and corrected an amazing number of mistakes, inconsis tencies, and infelicities of style. She is a great editor but, of course, all the errors that remain are the author's. Finally, thanks are due to several classes of students in Math 486 at Penn State who used early versions of this book as texts. Their reactions to the material had much influence on the final version. Contents Preface vii List of Figures xv 1. Games in Extensive Form 1 1.1. Trees 3 1.2. Game Trees 7 1.2.1. Information Sets 11 1.3. Choice Functions and Strategies 12 1.3.1. Choice Subtrees 13 1.4. Games with Chance Moves 20 1.4.1. A Theorem on Payoffs 22 1.5. Equilibrium N-tuples of Strategies 24 1.6. Normal Forms 29 2. Two-Person Zero-Sum Games 35 2.1. Saddle Points 36 2.2. Mixed Strategies 40 2.2.1. Row Values and Column Values 43 2.2.2. Dominated Rows and Columns 48 2.3. Small Games 52 2.3.1. 2 x nand m x 2 Games 55 2.4. Symmetric Games 59 2.4.1. Solving Symmetric Games 60