Table Of ContentEric van Damme
Stability and Perfection
of Nash Equilibria
With 105 Figures
Springer-¥erlag Berlin Heidelberg New York
London Paris Tokyo
Prof. Dr. Eric van Damme
Department of Economics, University of Bonn,
Adenauerallee 24-42, D-5300 Bonn l/FRG
ISBN-13: 978-3-642-96980-5 e-ISBN-13: 978-3-642-96978-2
DOl: 10.1007/978-3-642-96978-2
Library of Congress Cataloging-in-Publication Data.
Damme, Eric van.
Stability and perfection of Nash equilibria / Eric van Damme.
p. cm. Bibliography: p. Includes index.
1. Game theory. 2. Equilibrium (Economics). 1. Title. II. Title: Nash equilibria.
HB144.D36 1987 339.5--dc19 87-27292
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2142/3020-543210
To Jeroen and Jessica
The rules of rational behavior must provide definitely for the possibility of
irrational conduct on the part of others. In other words: Imagine that we have
discovered a set of rules for all participants - to be termed as "optimal" or
"rational" - each of which is indeed optimal provided that the other participants
conform. Then the question remains as to what will happen if some of the
participants do not conform. If that should tum out to be advantageous for
them - and, quite particularly, disadvantageous to the conformists - then the
above "solution" would seem very questionable. We are in no position to give a
positive discussion of these things as yet - but we want to make it clear that
under such conditions the "solution," or at least its motivation, must be
considered as imperfect and incomplete. In whatever way we formulate the
guiding principles and the objective justification of "rational behavior," provisos
will have to be made for every possible conduct of "the others." Only in this way
can a satisfactory and exhaustive theory be developed. But if the superiority of
"rational behavior" over any other kind is to be established, then its description
must include rules of conduct for all conceivable situations - including those
where "the others" behaved irrationally, in the sense of the standards which the
theory will set for them.
John von Neumann and Oskar Morgenstern
Theory of Games and Economic Behavior,
Princeton University Press, Princeton, N.J.
(2nd ed. 1947, p. 32)
Preface
The last decade has seen a steady increase in the application of concepts from
noncooperative game theory to such diverse fields as economics, political science,
law, operations research, biology and social psychology. As a byproduct of this
increased activity, there has been a growing awareness of the fact that the basic
noncooperative solution concept, that of Nash equilibrium, suffers from severe
drawbacks. The two main shortcomings of this concept are the following:
(i) In extensive form games, a Nash strategy may prescribe off the equilibrium
path behavior that is manifestly irrational. (Specifically, Nash equilibria may
involve incredible threats),
(ii) Nash equilibria need not be robust with respect to small perturbations in the
data of the game.
Confronted with the growing evidence to the detriment of the Nash concept,
game theorists were prompted to search for more refined equilibrium notions with
better properties and they have come up with a wide array of alternative solution
concepts. This book surveys the most important refinements that have been
introduced. Its objectives are fourfold
(i) to illustrate desirable properties as well as drawbacks of the various
equilibrium notions by means of simple specific examples,
(ii) to study the relationships between the various refinements,
(iii) to derive simplifying characterizations, and
(iv) to discuss the plausibility of the assumptions underlying the concepts.
The book is addressed primarily to researchers who want to apply game
theory, but who do not know their way through the myriad of noncooperative
solution concepts. It can also be used as the basis for an advanced course in game
theory at the graduate level. It will be successful if it enables the reader to sift
the grain from the corn and if it can direct him to the concepts that are really
innovative.
Acknowledgements. Many colleagues and friends helped to shape my thinking on
these topics in the last several years. I am sure that, had I listened to their
invaluable advice more carefully I would have understood the subject better· and
I would have written a better book. Especially I want to express my thanks to
StefTijs, Jaap Wessels, Jan van der Wal, Reinhard Selten, Werner Giith, Roger
Myerson and Ehud Kalai for their help at crucial stages of the development. Of
course, the views expressed are not necessarily theirs, and I alone am responsible
for errors and mistakes.
x
Preface
Sincere appreciation is also extended to my wife Suzan, Ellen Jansen, Lieneke
Lekx, Netty Zuidervaart and Rolf-Peter Schneider who shared the effort in typing
the several versions of the manuscript. A special thanks also goes to the editorial
staff of Springer-Verlag, especially to Werner Muller for exerting just sufficient
pressure to get this book finished. Finally, I thank Jeroen and Suzan for their
patience, understanding and encouragement while the book was being written.
Bonn, July 1987 Eric E. C. van Damme
Organization
The Chaps. 1-6 of this book are virtually identical to the monograph
"Refinements of the Nash equilibrium concept" that I published with Springer
Verlag in 1983. Werner Miiller, the economics editor of Springer kindly asked me
to extend that monograph with some chapters illustrating the various equilibrium
concepts in specific examples. I was pleased to honor that request and I gladly
took this opportunity to clear up some inaccuracies and to include some recent
developments.
This book consists of 4 parts:
Part 1 (Chap. 1) provides a general introduction. It is argued that the solution of
a noncooperative game should be a Nash equilibrium but that not every Nash
equilibrium is eligible for the solution. Various refinements of the Nash concept
are introduced informally and simple examples are >used to illustrate these
concepts.
Part 2 (Chaps. 2-5) deals with normal form games. A great variety of refined
equilibrium concepts is introduced and relationships between these refinements
are derived, as well as characterizations of several of them. For a quick overview,
the reader should consult the Survey Diagrams 1 an(! 2 at the end of the book.
A main result, however, is that for normal form games there is actually little
need to refine Nash's concept since generically all Nash equilibria satisfy all
properties one could hope for.
Part 3 (Chap. 6) provides an introduction to extensive form games. Formal
definitions are given and elementary properties of several concepts (such as
(subgame) perfect equilibria and sequential equilibria) are derived. The main
result is that a proper equilibrium of the normal form induces a sequential
equilibrium in the extensive form. However, normal form properness does not
eliminate all "unreasonable" equilibria.
Part 4 (Chaps. 7-10) is devoted to specific applications, illustrating the strength
(resp. weakness) of the various concepts. These chapters are independent of each
other and familiarity with the basic notions from Chaps. 2 and 6 suffices to
follow the discussion.
The main theme in Chap. 7 is "how to implement concepts from cooperative
game theory by noncooperative methods?" The power ofthe subgame perfectness
concept is illustrated by means of a simple fair division problem, by means of
the Rubinstein bargaining model (which implements the Nash solution) and by
means of the Moulin model (that implements the KalaijSmorodinsky solution).
Furthermore, Nash's bargaining model is used to illustrate the essential equi
librium concept.
In Chap 8 we prove the Folk Theorem, which states that the set of cooperative
outcomes of the one-shot game coincides with the set of noncooperative out-
XII Organization
comes of the repeated game. This chapter shows that the subgame perfectness
concept has certain drawbacks and that it is not as restrictive as one might
initially think.
In Chap. 9 we turn to the biological branch of game theory. Here, the main
solution concept is that of evolutionarily stable strategies (ESS), i.e. of symmetric
Nash equilibrium strategies that satisfy a certain condition of neighborhood
stability. The relationships between (refinements of) the ESS concept and concepts
of strategic stability are studied. (An overview is given in Diagram 3.)
The ESS concept is very restrictive, however, (especially in extensive games) and
it is shown that to coarsen this concept one can use the same methods as
those that are used to refine the Nash concept.
In Chap. 10 we return to the question of whether a game is adequately
represented by its normal form, i.e. whether knowledge of the normal form is
sufficient to define rational behavior. In particular, it is investigated what kind
of restrictions the Kohlberg/Mertens concept of stability imposes on the beliefs
in a sequential equilibrium. Special attention is paid to signalling games, for which
several "intuitive criteria" for eliminating "unintuitive equilibria" are discussed
(see Diagram 4 for an overview). Many examples are given to illustrate the various
concepts.
Notational Conventions
We have tried to use standard notations as much as possible. The notation that is
specific to normal form games (resp. extensive form games) is introduced in
Sect. 2.1 (resp. 6.1). Chapter 9 uses the notation that is conventional in the
biological branch of game theory and this differs from our general notation for
normal form games. At this point, we confine ourselves to introducing some
terminology that is used throughout the book and that may not be completely
standard.
N denotes the set of the positive integers {1, 2, ... } (positive will always mean
strictly greater than 0). When dealing with an n-person game we will frequently
write Nfor {1, ... ,n}.
R denotes the set of real numbers and R m is the m dimensional Euclidean
space. For x,yeRm, we write x;;;;;y if Xj;;;;;Yj for all i. Furthermore, x<y means
Xj<Yj for all i. We writeR~ for the set of all xeRm which satisfy O;;;;;x andR~ +
for the set of all x e R mf or which 0 < x. Euclidean distance on R mi s denoted by (!
and A. denotes Lebesgue measure on Rm.
The set of all mappings from A to B is denoted by $' (A, B). Iffe $' (R~ +, Ri),
then "y is a limit point of a sequence {f(x)}X! 0" is used as an abbreviation for
"there exists a sequence {x(t)}teN such that x(t) converges to 0 andf(xt) converges
to y as t tends to infinity".
If A is a subset of some Euclidean space, than conv A denotes its convex hull
and 2A denotes the powerset of this set.
Let A and B be subsets of Euclidean spaces. A correspondence from A to B is
an element of $'(A, 2B). The correspondence F from A to B is said to be upper
semi-continuous if it has a closed graph, i.e. if {(x,F(x»; xeA} is closed.
The number of elements of a finite set A is denoted by IA I. If A is finite, and
L
fe$'(A,R) thenf(A):= f(a).
aeA
Indices can occur as subscripts or superscripts. Lower indices usually refer to
players. Upper indices usually stem from a certain numbering. For instance, when
dealing with an n-person normal form game, we write s} for the probability which
the mixed strategy Sj of player i assigns to the kth pure strategy of this player.
To avoid misunderstandings between exponents and indices, we will sometimes
write the basis of a power between brackets. Hence, (s})2 denotes the square of s~ .
Definitions are indicated by using italics. The symbol := is used to define
quantities. The symbol 0 denotes the end of a proof.
