Table Of ContentN-Person Game Theory
Concepts and Applications
by Anatol Rapoport
ANN ARBOR
THE UNIVERSITY OF MICHIGAN PRESS
Copyright © by The University of Michigan 1970
All rights reserved
SBN 472-00117-5
Library of Congress Catalog Card No. 79-83451
Published in the United States of America by
The University of Michigan Press and simultaneously
in Don Mills, Canada, by Longmans Canada Limited
Manufactured in the United States of America
Preface
This book is a sequel to Two-Person Game Theory: The
Essential Ideas (University of Michigan Press, 1966). It
is addressed to the same audience: to people with little
mathematical background but with an appetite for rig-
orous analysis of the purely logical structure of strategic
conflict situations.
In the preface to Two-Person Game Theory I explained
why I found it necessary to separate the expositions of
Two-person and N-person game theory. The former can
be presented with a minimum of (mostly familiar) math-
ematical notation; the latter cannot. I remain convinced
that unfamiliar mathematical notation scares at least as
many people away from mathematical treatments of im-
portant subjects as the difficulty of follOwing mathe-
matical reasoning. The situation in the study of Russian
is somewhat similar but with an important difference.
Many people think that Russian is difficult to learn be-
cause it is written in an unfamiliar alphabet. Russian is,
to be sure, comparatively difficult for non-Slavic speak-
ers, but certainly not because of its non-Latin alphabet.
The Cyrillic alphabet can be learned in an hour. The
difficulties stem largely from the fact that Russian is a
more inflected language than the modem Germanic and
Romance languages, so that there is more grammar to
learn. I wish I could say the same for mathematics: that
the difficulties of notation are trivial and that only the
difficulties of mathematical "grammar" (mode of reason-
ing) need to be overcome. Unfortunately, this is not the
case. Mathematical notation and mathematical reasoning
are much more intertwined than alphabet and grammar;
6 N-Person Game Theory
so that one cannot really learn to read mathematical nota-
tion without acquiring a certain degree of mathematical
maturity.
The interdependence between mathematical notation
and mathematical logic suggests the task of mathematical
pedagogy: one must constantly emphasize the essential
connections between the symbols and the concepts for
which they stand. This emphasis is particularly impor-
tant in set theory, one of the mathematical pillars on
which N-person game theory rests. When the reader has
learned to associate quickly the concepts with their rep-
resentations, he is well on the way toward understanding
set-theoretic reasoning and has mastered one half of the
conceptual repertoire that underlies game theory. The
other half is the notion of multi-dimensional space as the
set of all possible n-tuples of numbers. In Two-person
game theory, this notion presents no difficulty. There be-
ing only two players, all possible payoffs of a game are
pairs of numbers, representable on two-dimensional dia-
grams. If n = 3, we can still resort to projections of three-
dimensional figures. For n > 3, visual intuition fails. One
must learn to think in terms of visually unrepresentable
«spaces." Here again, once one has learned to "read"
properly, the conceptual difficulties begin to resolve
themselves quite rapidly.
The Introduction is a summary of mathematical con-
cepts that I believe to be sufficient for understanding the
essential ideas of N-person game theory. The ideas them-
selves (mostly in a purely logical context) are presented
in Part I. «Applications" are discussed in Part II. The In-
troduction to Part II will hopefully forestall misunder-
standing concerning the meaning of «applicatiOns" in the
context of game theory. More will be said on this matter
in the last two chapters.
The scope of the book covers the essential ideas de-
veloped in the original formulation of N-person game
theory by Von Neumann and Morgenstern and the sub-
Preface 7
sequent extensions by the present generation of game
theoreticians. In their book Games and Decisions, Luce
and Raiffa have already covered practically all of the
significant advances up to 1957. Since then, two more
volumes of Contributions to the Theory of Games (An-
nals of Mathematics Studies series, Princeton University
Press) have appeared, as well as many separate journal
articles, some proceedings of conferences on game theory,
and numerous memoranda and preprints. These were my
main sources.
The reader will note that the authors cited are pre-
dominantly American and Israeli. This reHects the con-
tinued interest in the United States and in Israel in the
application potential of game theoretic ideas to social
science. There is also a large Russian literature; but, to
the extent that I have examined it, it is of interest only
to the mathematical specialist, and so falls outside the
scope of this book.
I take pleasure in thanking the University of Michigan
Press for continued encouragement. I am indebted to
Professors R. M. Thrall and to William F. Lucas for
their critical reading of the manuscript and for many
helpful suggestions. My heartfelt gratitude goes once
more to Claire Adler, who has given invaluable editorial
assistance, and to Dorothy Williams Malan for help in
the preparation of the manuscript.
Contents
Introduction: Some Mathematical Tools 11
Part I. Basic Concepts
1. Levels of Game-theoretic Analysis 45
2. Three-level Analysis of Elementary Games 68
3. Individual and Group Rationality 87
4. The Von Neumann-Morgenstern Solution 93
5. The Shapley Value 106
6. The Bargaining Set 114
7. The Kernel 125
8. Restrictions on Realignments 137
9. Games in Partition Function Form 145
10. N-Person Theory and Two-Person Theory Com-
pared 158
11. Harsanyi's Bargaining Model 170
Part II. Applications
Introduction to Part II 183
12. A Small Market 186
13. Large Markets 196
14. Simple Games and Legislatures 207
15. Symmetric and Quota Games 222
16. Coalitions and Power 234
10 N-Person Game Theory
17. Experiments Suggested by N-Person Game
Theory 254
18. "So Long Sucker": A DO-it-yourself Experiment 271
19. The Behavioral Scientist's View 284
20. Concluding Remarks 301
Notes 311
References 317
Index 321
