Table Of Contentl,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
