Table Of ContentECONOMIC THEORY, ECONOMETRICS, AND
MATHEMATICAL ECONOMICS
Edited by Karl Shell, Cornell University
Previous Volumes in the Series
Erwin Klein, Mathematical Methods in Theoretical Economics: Topological and
Vector Space Foundations of Equilibrium Analysis
Paul Zarembka, editor, Frontiers in Econometrics
George Horwich and Paul A. Samuelson, editors, Trade, Stability, and Macro
economics: Essays in Honor of Lloyd A. Metz 1er
W. T. Ziemba and R. G. Vickson, editors, Stochastic Optimization Models in
Finance
Steven A. Y. Lin, editor, Theory and Measurement of Economic Externalities
Haim Levy and Marshall Sarnat, editors, Financial Decision Making under
Uncertainty
Yasuo Murata, Mathematics for Stability and Optimization of Economic Systems
Jerry S. Kelly, Arrow Impossibility Theorems
Fritz Machlup, Methodology of Economics and Other Social Sciences
Robert H. Frank and Richard T. Freeman, Distributional Consequences of Di
rect Foreign Investment
Elhanan Helpman and Assaf Razin, A Theory of International Trade under
Uncertainty
Edmund S. Phelps, Studies in Macroeconomic Theory, Volume I: Employment
and Inflation, Volume 2: Redistribution and Growth
Marc Nerlove, David M. Grether, and José L. Carvalho, Analysis of Economic
Time Series: A Synthesis
Michael J. Boskin, editor, Economics and Human Welfare: Essays in Honor of
Tibor Scitovsky
Carlos Daganzo, Multinomial Probit: The Theory and Its Application to Demand
Forecasting
L. R. Klein, M. Nerlove, and S. C. Tsiang, editors, Quantitative Economics
and Development: Essays in Memory ofTa-Chung Liu
Giorgio P. Szegö, Portfolio Theory: With Application to Bank Asset Man
agement
M. June Flanders and Assaf Razin, editors, Development in an Inflationary
World
List continues at the end of this volume.
Previous Volumes in the Series (Continued)
Thomas G. Cowing and Rodney E. Stevenson, editors, Productivity Measure
ment in Regulated Industries
Robert J. Barro, editor, Money, Expectations, and Business Cycles: Essays in
Macroeconomics
Ryuzo Sato, Theory of Technical Change and Economic Invariance: Application
of Lie Groups
Iosif A. Krass and Shawkat M. Hammoudeh, The Theory of Positional Games:
With Applications in Economics
Giorgio Szegö, editor, New Quantitative Techniques for Economic Analysis
John M. Letiche, editor, International Economic Policies and Their Theoretical
Foundations: A Source Book
Murray C. Kemp, editor, Production Sets
Andreu Mas-Colell, editor, Noncooperative Approaches to the Theory of Perfect
Competition
Jean-Pascal Benassy, The Economics of Market Disequilibrium
Tatsuro Ichiishi, Game Theory for Economic Analysis
David P. Baron, The Export-Import Bank: An Economic Analysis
Real P. Lavergne, The Political Economy of U.S. Tariffs: An Empirical Analysis
Halbert White, Asymptotic Theory for Econometricians
Thomas G. Cowing and Daniel L. McFadden, Microeconomic Modeling and
Policy Analysis: Studies in Residential Energy Demand
V. I. Arkin and I. V. Evstigneev, translated and edited by E. A. Medova-Demp-
ster and M. A. H. Dempster, Stochastic Models of Control and Economic
Dynamics
S vend Hylleberg, Seasonality in Regression
Jean-Pascal Benassy, Macroeconomics: An Introduction to the Non-Walrasian
Approach
C. W. J. Granger and Paul Newbold, Forecasting Economic Time Series, Second
Edition
Marc Nerlove, Assaf Razin, and Efraim Sadka, Household and Economy: Wel
fare Economics of Endogenous Fertility
Thomas Sargent, Macroeconomic Theory, Second Edition
^Yves Balasko, Foundations of the Theory of General Equilibrium
Jean-Michel Grandmont, Temporary Equilibrium: Selected Readings
J. Darrell Duffie, An Introductory Theory of Security Markets
Ross M. Starr, General Equilibrium Models of Monetary Economics
S. C. Tsiang, Finance Constraints and the Theory of Money: Selected Papers
Masanao Aoki, Optimization of Stochastic Systems: Topics in Discrete-Time
Dynamics, Second Edition
Peter Diamond and Michael Rothschild, Uncertainty in Economics: Readings
and Exercises, Revised Edition
Martin J. Osborne and Ariel Rubinstein, Bargaining and Markets
Game Theory
and Applications
Edited by
Tatsuro Ichiishi
Department of Economics
The Ohio State University
Columbus, Ohio
Abraham Neyman
Institute for Decision Sciences
State University of New York at Stony Brook
Stony Brook, New York
and The Hebrew University
Jerusalem, Israel
Yair Tauman
Department of Economics
The Ohio State University
Columbus, Ohio
and Tel Aviv University
Tel Aviv, Israel
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Library of Congress Cataloging-in-Publication Data
Game theory and applications / edited by Tatsuro Ichiishi, Abraham
Neyman, Yair Tauman.
p. cm.
Includes bibliographical references.
ISBN 0-12-370182-1 (alk. paper)
1. Game theory. I. Ichiishi, Tatsuro. II. Neyman, Abraham.
III. Tauman, Yari.
HB144.G37 1990
519.3-dc20 90-30648
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Contributors
Numbers in parentheses indicate the pages on which the authors' contributions begin.
Robert Aumann (158), Institute of Mathematics, The Hebrew University, Givat
Ram 91904, Jerusalem, Israel
M. L. Balinski (373), Laboratoire d'Econometrie de ΓEcole Polytechnique,
75005 Paris, France
Bernard Cornet (375), Department de Mathématiques, Université Paris I,
Pantheon-Sorbonne, 75634 Paris Cedex 13, France
Stephen Ellner (377), Biomathematics Program, Department of Statistics,
Box 8203, North Carolina State University, Raleigh, North Carolina
27695-8203
Ky Fan (358), Department of Mathematics, University of California, Santa Bar
bara, Santa Barbara, California 93106
Françoise Forges (64), CORE, Université Catholique de Louvain, Louvain-la-
Neuve, Belgium
David Gale (373), Department of Mathematics, University of California,
Berkeley
Sergiu Hart (166, 380), School of Mathematical Sciences, Tel Aviv University,
69978 Tel Aviv, Israel
Martin Hellwig (381), Institute fur Volkswirtschaft, Universität Basel, Postfach,
CH-4003 Basel, Switzerland
Ron Holzman (383, 384), Department of Applied Mathematics and Computer
Science, The Weizmann Institute of Science, Israel
Tatsuro Ichiishi (385), Department of Economics, The Ohio State University,
Columbus, Ohio 43210-1172
Ehud Kalai (131), Department of Managerial Economics and Decision Sciences,
J. L. Kellogg Graduate School of Management, Northwestern University,
Evanston, Illinois 60208
Morton I. Kamien (273), J. L. Kellogg Graduate School of Management, North
western University, Evanston, Illinois 60208
Elon Kohlberg (3), Graduate School of Business, Harvard University, Soldiers
Field, Boston, Massachusetts 02163
Vili
CONTRIBUTORS IX
Ehud Lehrer (384), Department of Managerial Economics and Decision Sci
ences, J. L. Kellogg Graduate School of Management, Northwestern Uni
versity, Evanston, Illinois 60208
Wolfgang Leininger (381, 388), Department of Economics, Universität Dort
mund, D-4600 Dortmund 50, West Germany
Anat Levy (390), Department of Economics, Tel Aviv University, 69978 Tel
Aviv, Israel
Nathan Linial (384), Department of Computer Science, The Hebrew University
of Jerusalem, 91904 Jerusalem, Israel
William F. Lucas (300), Claremont Graduate School, Department of Mathemat
ics, Claremont, California 91711
Michael Maschler (183), Hebrew University of Jerusalem, Institute of Mathe
matics and Computer Science, Givat Ram 91904, Jerusalem, Israel
Richard D. McKelvey (317), Division ofH&SS, California Institute of Tech
nology, Pasadena, California 91125
Richard P. McLean (390), Department of Economics, New Jersey Hall, Rutgers
University, New Brunswick, New Jersey 08903
Jean-François Mertens (77), CORE, Université Catholique de Louvain, Louvain,
Belgium
Shaul Mishal (336), Department of Political Science, Tel Aviv University, Ramat
Aviv, 69978 Tel Aviv, Israel
Dov Monderer (392), The Technion, Haifa, Israel
Hervé Moulin (282), Department of Economics, Duke University, Durham,
North Carolina 27706
Abraham Neyman (380), Institute for Decision Sciences, State University of New
York at Stony Brook, Stony Brook, New York 11794, and The Hebrew Uni
versity, Jerusalem, Israel
Andrzej S. Nowak (393), Institute of Mathematics, Wroclaw Technical Univer
sity, 50-370 Wroclaw, Poland
Akira Okada (398, 401), Graduate School of Policy Science, Saitama Univer
sity, Urawa, Saitama 338, Japan
Bezalel Peleg (176), Department of Mathematics, The Hebrew University, Givat
Ram, 91904 Jerusalem, Israel
Hans Peters (404), Department of General Science, University ofLimburg, The
Netherlands
Roy Radner (407), AT & T Bell Labs, Murray Hills, New Jersey
Joachim Rosenmüller (216), Department of Economics, Universität Bielefeld,
B-4800 Bielefeld 1, West Germany
Alvin E. Roth (232), Department of Economics, University of Pittsburgh, Pitts
burgh, Pennsylvania 15260
Mark Satterthwaite (408), Department of Managerial Economics and Decision
Sciences, J. L. Kellogg Graduate School of Management, Northwestern
University, Evanston, Illinois 60208
X CONTRIBUTORS
David Schmeidler (336), Department of Economics, The Ohio State University,
Columbus, Ohio, 43210-1172, and Faculty of Statistics, Tel Aviv Univer
sity, Ramat Aviv, 69978 Tel Aviv, Israel
Andrew Schotter (407), C. V. Starr Center, Economics Department, New York
University, New York, New York 10003
Itai Sened (336), Department of Political Science, University of Rochester,
Rochester, New York 14627
Avi Shmida (377), Institute of Life Sciences, Botany Department, The Hebrew
University of Jerusalem, Givat Ram, 91904 Jerusalem, Israel
Martin Shubik (252), Department of Economics, Yale University, New Haven,
Connecticut 06520-2125
Sylvain Sorin (46), Department de Mathématique, Université Louis Pasteur,
67084 Strasbourg, France
Yair Tauman (273), Department of Economics, The Ohio State University, Co
lumbus, Ohio 43210, and School of Management, Tel Aviv University, Tel
Aviv, Israel
Rastislav Telgârsky (409), Technical Solutions, Incorporated, Mestila Park,
New Mexico 88047
William Thomson (187), Department of Economics, University of Rochester,
Rochester, New York 14627
Stef Tijs (410), Department of Mathematics, University ofNijmegen, Toernooi-
veld, The Netherlands
Peter Wakker (404), Department of Mathematical Psychology, University ofNij
megen, The Netherlands
Steve Williams (408), Math Center-Leverone Hall, Northwestern University,
Evans ton, Illinois 60208
Myrna Holtz Wooders (413, 416), Department of Economics, University of To
ronto, Toronto, Ontario M5S 1A1, Canada
William R. Zame (416), Department of Economics, University of Califor
nia at Los Angeles, Los Angeles, California 90024, and Department of
Mathematics, State University of New York at Buffalo, Buffalo, New York,
14214-3093
Shmuel Zamir (273), Department of Statistics, The Hebrew University of Jeru
salem, Mountscopus, 91905 Jerusalem, Israel
Preface
The works collected in this volume are based on presentations at the Interna
tional Conference on Game Theory and Applications held at The Ohio State
University, June 18-24, 1987. The first part of the volume contains 19 papers
that are based on the invited lectures, most of which are surveys of the recent
development of specific areas in game theory, pure or applied. These papers are
classified here according to the subjects they address. The second part of the
volume contains 21 selected detailed abstracts based on the talks in the contrib
uted paper sessions, assembled here alphabetically according to the authors'
last names.
We would like to express our gratitude to the contributors to this volume and
all the participants of the conference. The financial support for the conference,
provided by the Sloan Foundation, the U.S. National Science Foundation, and
The Ohio State University, is gratefully acknowledged.
TATSURO ICHIISHI
ABRAHAM NEYMAN
YAIR TAUMAN
XI
Refinement of Nash Equilibrium: The Main Ideas
ELON KOHLBERG
Harvard University
1. INTRODUCTION
The purpose of this paper is to describe the main ideas in the literature on
refinements of Nash equilibrium. The number of refinements that have been pro
posed is staggering. Among others, there are subgame-perfect, perfect, proper,
sequential, persistent, justifiable, neologism-proof, stable, perfect-sequential,
intuitive, divine, undefeated, and explicable equilibria (see Selten, 1965, 1975;
Myerson, 1978; Kreps and Wilson, 1982; Kalai and Samet, 1984; McLennan,
1985; Farrell, 1985; Kohlberg and Mertens, 1986; Grossman and Perry, 1986;
Cho and Kreps, 1987; Banks and Sobel, 1987; Okuno-Fujiwara and Postlewaite,
1987; Reny, 1987, respectively). This proliferation does not stem from differ
ences in the underlying concepts; on the contrary, all the refinements represent
attempts to formalize the same two or three intuitive ideas.
Rather than discuss the intricate distinctions and relationships between the
various refinements,1 I will try to describe the basic ideas, to demonstrate their
power in applications, and to discuss some of the difficulties in their formaliza-
tion. Along the way I will prove several results which may be of interest in their
own right.
2. BACKWARD INDUCTION AND FORWARD INDUCTION
2.1 SELF-ENFORCING AGREEMENTS
In a noncooperative game, each player must choose his strategy in ignorance
of what the other players do. Communication is impossible, and there is no
mechanism whereby agreements can be enforced. Given such a game, can we
predict what rational players will actually do?
Except when the game is unusually simple (e.g., when all players have domi
nant strategies) our answer must be "No!" The reason, of course, is that in
'The interested reader is referred to the excellent book by van Damme (1987).
3
GAME THEORY AND APPLICATIONS Copyright © 1990 by Academic Press, Inc.
All rights of reproduction in any form reserved.
4 KOHLBERG
determining his own choice each player must take into account what the others
would do. This leads to an endless chain of deductions about the other players,
about the other players' deductions, etc.
The main idea of noncooperative game theory, due to Cournot (1838) and
Nash (1951), is to make a bold simplification and, instead of asking how the
process of deductions might unfold, ask where its rest points might be. Thus a
(Cournot-)Nash equilibrium is a list of strategies, one for each player, such that
each player's strategy maximizes his own payoff given the other players' strate
gies. The great advantage of this simplification is, of course, that it leads to a
well defined system of inequalities, which always has a solution (this is Nash's
basic existence theorem2).
But what is the relevance of the equilibria for predicting how the players will
play? Possibly none: there is absolutely no reason for the actual strategic choices
to constitute an equilibrium. Indeed, the only connection between equilibrium
and the outcome is this: if the actual choices were known to all the players
in advance of play, then they must have constituted an equilibrium. In other
words, the equilibria of the game do not delineate the possible outcomes; rather,
they delineate the set of self-enforcing agreements, or self-enforcing norms of
behavior.
This is rather an awkward connection.3 But, like it or not, it is the one we
must work with if we are to retain the great simplifying power of the equilibrium
concept. Luckily, in many applications it turns out that one can learn a great deal
by discovering which norms of behavior are robust to selfish profit maximization
or, alternatively, which agreements can hold up despite the absence of an en
forcement mechanism.
2.2 THE NEED FOR REFINEMENTS
For quite some time after Nash's definition of equilibrium, it was taken for
granted that all Nash equilibria were self-enforcing. However, Selten (1965)
pointed out that this was not the case.
Below is Selten's basic example. There are two equilibria,4 as indicated in
Figures 1 and la. However, only the former is self-enforcing. The idea is that
even if the players had (explicitly or implicitly) coordinated on (Y,R) nevertheless
2 Assume there are finitely many players, each with finitely many pure strategies. The equilibrium
may, of course, require randomizations ("mixed strategies").
3The connection is much less awkward when each "player" represents a whole population of
individuals facing the same alternatives and payoffs. Thus a (mixed) strategy is simply a distribu
tion over the choices made by individuals in the population, and the equilibrium condition is that
each individual maximize his payoff assuming that the distribution over the choices of other players
(i.e., other populations) will be unaffected by his choice.
4Figure la represents a Nash equilibrium because, given /'s choice Y, any choice for // is payoff-
maximizing, in particular R.
