Game Theory in the Social Sciences Concepts . :r, ■{ и ■ * Solutions f " 3 F-Ц-' h A/ . \: ■^nri 1/ 2 \3 C - 1/ 2 \3 f . O3 O4 O3 O^ v -^Ov)-*; 1 Ы Martin Shubik ...... Game Theory in the Social Sciences Concepts and Solutions Game Theory in the Social Sciences Concepts and Solutions Martin Shubik The MIT Press Cambridge, Massachusetts London, England Third printing, 1985 First MIT Press Paperback Edition, 1984 © 1982 by The Massachusetts Institute of Technology All rights reserved. No part of this book may be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording, or by any infor­ mation storage and retrieval system, without permission in writing from the publisher. This book was set in VIP Baskerville by DEKR Corporation and printed and bound in the United States of America. Library of Congress Cataloging in Publication Data Shubik, Martin. Game theory in the social sciences. Bibliography: p. Includes index. 1. Social sciences—Mathematical models. 2. Game theory. I. Title. H61.25.S49 300M'5193 82-63 ISBN 0-262-19195-4 (hardcover) AACR2 ISBN 0-262-69091-8 (paperback) To Claire and Julie. To Marian. And to Lloyd, who was the only person who understood what I was trying to say and was able to explain it to me. Contents Acknowledgments ix 1 Models and Their Uses 1 2 Decision Makers 16 3 The Rules of the Game: Extensive and Strategic 33 Forms 4 Individual Preferences and Utility 81 5 Group Preferences and Utility 109 6 Characteristic Function, Core, and Stable Sets 127 7 The Value Solution 179 8 Two-Person Zero-Sum Games 217 9 Noncooperative Solutions to Non-Constant-Sum 240 Games 10 Other Noncooperative Solutions and Their 306 Applications 11 Further Cooperative Solution Concepts 336 12 On the Applications of Game Theory 368 Appendix A Some Details on Preference and 417 Utility Appendix B Bibliography on Stable-Set Solutions 425 Appendix C Games against Nature 431 Bibliography 434 Index 493 Acknowledgments This book is the outgrowth of many years of joint work with Lloyd Shapley. Our collaboration has been deep and fruitful to me, and it seems strange to be publishing this work under my own name. I feel that it does merit publication, though, and Lloyd has graciously con­ sented to have it published in this form. I neither question his deci­ sion nor attempt to read motivations into it. Preliminary versions of chapters 1-6 have appeared as jointly au­ thored RAND Memoranda R-904/1, 904/2, 904/3, 904/4, and 904/6. Chapter 7 contains much material prepared jointly for a RAND Memorandum which was not finished. Much of the material in chap­ ters 8-11, as well as some of the writing, comes from joint notes or files. Chapter 12 is primarily the product of this author, although even here I have benefited from Lloyd’s wide knowledge and critical insights. I am indebted to many colleagues who have taught me much over the years and corrected my many mathematical errors. These include especially Pradeep Dubey, Robert Weber, Herbert Scarf, Gerard De- breu, Andreu Mas-Colell, Robert Wilson, Robert Aumann, Michael Maschler, John Harsanyi, Reinhard Selten, Edward Paxson, Matthew Sobel, Richard Levitan, Charles Wilson, and David Schmeidler. My thanks also go to Bonnie Sue Boyd, Glena Ames, Karen Marini, and several others for excellent mathematical typing and support in the all too long gestation period for this book. I am grateful to John Nash and John Milnor who have kindly allowed me to reproduce some of their work. And finally, I wish to acknowledge both the Office of Naval Research and the National Science Foundation, under whose generous support much of the research leading to the writing of this work was done. 1 Models and Their Uses 1.1 Introduction The objective of this work is to develop a fruitful application of the mathematical theory of games to the subject matter of general eco­ nomic theory in particular and to suggest other applications to the behavioral sciences in general. After the general orientation and mathematical preparations that fill a major portion of this volume, the work will proceed through a series of detailed examinations of particular game-theoretic models representing basic economic pro­ cesses. Throughout I shall endeavor to maintain contact with the view­ points, formulations, and conclusions of traditional economic analy­ sis, as well as with the growing body of published work in economic game theory. Nevertheless, much of the material is new—in its view­ point, formulation, or conclusions—and is offered as a contribution to a fundamental reshaping of the methodology of theoretical eco­ nomics and other social sciences. The economic models treated will cover a reasonably broad portion of static economics, and the game-oriented methodology employed will be reasonably systematic in its application, but I cannot pretend to completeness on either score. Indeed, I feel that a comprehensive, monolithic theory is probably not worth striving for, at least at pres­ ent, and that the pluralistic approach is the only reasonable way to make progress toward a fully effective mathematical theory of eco­ nomic activity. Two arguments in support of this view merit discussion. The first has to do with the overall purpose of mathematical models in the social sciences. The usefulness of mathematical methods—game-the­ oretic or not—depends upon precision in modeling, and in economics as elsewhere, precise modeling implies a careful and critical selectiv­ ity. The nature of the particular question under investigation will determine both the proper scope of the mathematical model and the proper level of detail to be included. Compatibility between different models in the same subject area is desirable, perhaps, but it is a Models and Their Uses luxury not a necessity. A rigorously consistent superstructure, into which the separate models all nicely fit, is too much to expect perhaps even in principle. A patchwork theory is to be expected, even wel­ comed, when one is committed as in the social sciences to working with mathematical approximations to a nonmathematical reality. The second and more remarkable argument for pluralism stems from the nature of multiperson interaction. The general n-person game postulates a separate “free will” for each of the contending parties and is therefore fundamentally indeterminate. To be sure, there are limiting cases, which game-theorists call “inessential games,” in which the indeterminacy can be resolved satisfactorily by applying the familiar principle of self-seeking utility maximization or individ­ ual rationality. But there is no principle of societal rationality, of comparable resolving power, that can cope with the “essential” game, and none is in sight. Instead, deep-seated paradoxes, challenging our intuitive ideas of what kind of behavior should be called “rational,” crop up on all sides as soon as we cross the line from “inessential” to “essential.” Three or four decades ago the “n-person problem”—as this con­ ceptual puzzle will be called—received its first clear mathematical formulation at the hands of John von Neumann and Oskar Morgen­ stern (1944, esp. pp. 8-15). Beginning with their work, a surprisingly large number of ingenious and insightful solution concepts for n- person games have been proposed by many different authors. Each solution probes some particular aspect of societal rationality, that is, the possible, proposed, or predicted behavior of rational individuals in mutual interaction. But all of them have had to make serious compromises. Inevitably, it seems, sharp predictions or prescriptions can be had only at the expense of severely specialized assumptions about the customs or institutions of the society being modeled. The many intuitively desirable properties that a solution ought to have, taken together, prove to be logically incompatible. A completely sat­ isfying solution is rarely obtained by a single method of attack on any multiperson competitive situation that is “essential” in the above sense. Instead, we find that in any given application some of the available solution concepts may provide only halfway or ambiguous insights, while others may miss the point entirely. Since no one definition of solution is uniformly superior to all the rest, we shall repeatedly have Models and Their Uses to “solve” the same economic model from several different stand­ points. Out of the various answers, whether they agree or disagree, we may come to a deeper understanding of the real economic situ­ ation and its social context. This may seem an unsatisfactory state of affairs. But the class of phenomena with which game theory is concerned—the behavior of independent decision makers whose fortunes are linked in an inter­ play of collusion, conflict, and compromise—is one that has eluded all previous attempts at systematic mathematical treatment. Nor is game theory a wholly new departure for economics and the social sciences. On the contrary, we shall often find that one or another of the game solutions,/or a particular model, corresponds to a standard, classical solution. What is new is the systematic use of the resources of game theory, inconclusive as they may be; this provides both the technique and the raison d’être of our investigations. 1.2 Plan of the Work The present work is divided into several major parts, each essentially self-contained. This policy suits the subject matter as well as the specialized interests of some prospective readers. It also provides flexibility in exposition, such as the freedom to choose new mathe­ matical notations convenient to the matter at hand. It inevitably entails some repetition of basic definitions and explanatory material, but even the straight-through reader may find this repetition enlight­ ening since each reprise is fashioned to a new set of circumstances.