l,o-Person 6ale lheOry The Essential Ideas by Anatol Rapoport ANN ARBOR THE UNIVERSITY OF MICHIGAN PRESS Copyright © by The University of Michigan 1966 All rights reserved Library of Congress Catalog Card No. 66-11082 Published in the United States of America by The University of Michigan Press and Simultaneously in Toronto, Canada, by Ambassador Books Limited Manufactured in the United States of America Preface Like most branches of mathematics, game theory has its roots in certain problems abstracted from life situa- tions. The situations are those which involve the necessity of making decisions when the outcomes will be affected by two or more decision-makers. Typically the decision- makers' preferences are not in agreement with each other. In short, game theory deals with decisions in conHict situations. A key word in what has just been said is abstracted. It implies that only the essential aspects of a situation are discussed in game theory rather than the entire situation with its peculiarities, ambiguities, and subtleties. If, how- ever, the game theoretician is asked "What are the essen- tial aspects of decisions in conHict situations?" his only honest answer can be "Those which I have abstracted." To claim more would be similar to maintaining that the essential aspect of all circular objects is their Circularity. This may be so for the geometer but not for someone who distinguishes coins from buttons and phonograph records from camera apertures. To be sure, the geometer deals not with "circular ob- jects" but with circles. That is to say, the conceptual act 6 Two-Person Game Theory of abstracting circularity from all circular objects was performed long enough ago to have been institutionalized in our language and in our science. Hence the geometer can assume that people who wish to study the geometric properties of circles will easily forget all the other prop- erties of circular objects, such as their color, the material from which they are made, or the uses to which they are put. The game theoretician is in a more difficult position. The aspects of decisions in conHict situations which he considers to be essential are not immediately evident to the mind as is the circularity of circular objects. And even after those aspects are brought into focus, they do not easily stay in focus. We are more emotionally in- volved with conHicts than with shapes of objects; and so the aspects which happen to be important to us keep in- truding into our conceptions of conHict. For this reason and, of course, also because the subject is new, there is still little understanding of what game theory is and what it is not, of what it could become and of what it cannot become because of certain inherent (not merely circumstance-imposed) limitations. On the other hand interest in game theory as a "science of ra- tional conHict" is extremely widespread in our age of competition, strategy, and gamesmanship. This interest is shared also by the "hard" behavioral scientists, ready to welcome a rigorous mathematically oriented concep- tual framework. This combination of lively interest and lack of sufficient acquaintance with the essential ideas of game theory has frequently led to regrettable misunder- standings and confusion; for example, about the uses and misuses of game theory in policy making, and about the relevance of game theory to the social sciences. This book is an attempt to introduce the theory of games to those interested in it in a way which would bring the essentials of the theory into the focus of atten- Preface 7 tion and keep them there. I suppose a disposition to read this book is sufficient evidence of interest; so I have no misgivings on that score. On the other hand, the question of what to presuppose about the reader's mathematical background is a difficult one. On formal grounds it is possible to maintain that hardly any knowledge of mathe- matics is required for understanding the essential ideas of game theory. If by knowledge of mathematics, one means an acquaintance with geometry, the calculus, and so forth, then it indeed appears to be true that very little of such knowledge is required beyond the ability to fol- low the process of solving some quite simple algebraic equations. However, the issue is not technical mathemat- ical knowledge but rather mathematical background, that is, certain habits of thought usually acquired only through the study of mathematics. Similarly, it is not the ability to play an instrument that is required in order to follow the development of a musical thought, say in a sym- phony, but rather "musicality," certain habits of listening. The ability to think mathematically is like the ability to listen musically. Some of this ability may be inborn; some may be acquired without technical training; and much of it comes with technical training. Therefore it is not to be denied that readers with some mathematical background will follow the ideas of game theory more easily than others. Clearly, however, this book is not addressed to mathematicians, who, if they wish to acquaint themselves with game theory, will turn to the standard treatises. The book is meant as a popular exposition of the subject, which (hopefully) penetrates it in depth. The difficulty which the nonmathematician experiences in reading mathematical works stems from two sources: a lack of experience with mathematical ideas, and a lack of experience with mathematical notation. In the case of game theory, the mathematical ideas are rather simple, their major source being set theory, which is almost en- 8 Two-Person Game Theory tirely self-contained and requires no "prerequisites."" However the notation of set theory is special and thus un- familiar to readers whose contact with mathematics has been through conventional elementary courses. I believe it is this special notation which makes difficulties for nonmathematicians who wish to familiarize themselves with game theory. Thus, I am told by my nonmathemat- ical colleagues that they find Games and Decisions by R. D. Luce and H. RaifIa difficult to read. In my opinion, that book has achieved a triumph in its lucid exposition of the essential ideas of game theory. I can conclude only that the difficulties stem from the fact that nonmathemat- ical readers shy away from unfamiliar mathematical no- tation which is still typographically prominent in Games and Decisions, in spite of the vast simplification over the original entirely uninhibited notation of the fundamental treatise (Theory of Games and Economic Behavior by J. von Neumann and O. Morgenstern). On the other hand, J. D. Williams' popular exposition, The Compleat Strategyst, from which difficult notation was eliminated, confines itself almost exclusively to the two-person zero- sum game. The idea embodied in this type of game is, perhaps, the foundation of game theory; but one cannot get an idea of a building by examining just the founda- tion. My original aim was to present the essential ideas of game theory (including its most interesting and challeng- ing departures from the two-person zero-sum game) for the general reader and for the social scientist, using the barest minimum of mathematical notation. I found this possible in the context of the two-person game; but I still have not succeeded in making an acceptable "translation" of the N-person game theory. Therefore, I have ventured o We may pOint out that the so-called "new math" now intro- duced to American children in grade schools places major em- phasis on set theory. The approach has been, as far as I know, a definite pedagogical success. Preface 9 to offer the present book first, hoping eventually to con- tinue with Essential Ideas of the N-person Game, when and if the notation problem is solved. In the present volume the reader will find only the mathematics of high school algebra and of very elemen- tary analytic geometry, except for an occasional deriva- tive. The only game-theoretical notation retained was that of the game matrix, which is quite easy to grasp and which is, at any rate, indispensable. In addition to the standard topics in the two-person game, a discussion of gaming theory is included, which, in my opinion, is an important link between abstract game theory and an ex- perimentally oriented behavioral science. Specific appli- cations to social science have not been stressed (these being discussed at length elsewhere); but the method- ological relations between game theory, decision theory, and social science are emphasized throughout. In contrast to the purely logical and mathematical ideas of game theory, the methodological ideas are controversial. I hope that the dividing line between facts and opinions has been made sufficiently clear. I am indebted to Prof. Robert M. Thrall, of The Uni- versity of Michigan and to Mrs. Claire Adler for helpful editorial suggestions; to my colleagues at the Mental Health Research Institute, and to The University of Michigan Press, for encouraging me to write this book. Whatever errors may have crept in despite the help I have received are, of course, my own. Anatol Rapoport