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Erie van Damme Stabilityand Perfection of Nash Equilibria Second, Revised and Enlarged Edition With 105 Figures Springer-Verlag Berlin Heidelberg GmbH Pfof. Of. Eric van Oamme CentER for Economic Research Tilburg University, PO Box 90153 5000 LE Tilburg, The NetherIands ISBN 978-3-540-53800-4 Library of Congress Cataloging-in·Publication Data Damme, Eric van. Stability and perfection of Nash equilibriajEric van Damme. - 2nd, rev. and enl. ed. Includes bibliographical references and index. ISBN 978-3-540-53800-4 ISBN 978-3-642-58242-4 (eBook) DOI 10.1007/978-3-642-58242-4 1. Game theory. 2. Equilibrium (Economics) 1. Title. HBI44.D36 1991 91-10656 339S-<lc20 CIP This work is subject to copyright. AII rights are reserved, whether the whole or part of the material is concemed, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its current version, and a copyright fee must always be paid. © Springer-Verlag Berlin Heidelberg 1987, 1991 2nd printing 1996 Origina1ly published by Springer-Verlag Berlin Heidelberg New York in 1991 The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: With a system of the Springer Produktions-Gesellschaft, Berlin. Dataconversion: Briihlsche Universitătsdruckerei, Giessen. 42/3020-543210 To Jeroen, Jessica and Jean-Paul 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 turn 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 to the Second Edition I have been pleased with the favourable reception of the first edition of this book and I am grateful to have the opportunity to prepare this second edition. In this revised and enlarged edition I corrected some misprints and errors that occurred in the first edition (fortunately I didn't find too many) and I added a large number of notes that give the reader an impression of what kind of results have been obtained since the first edition was printed and that give an indication of the direction the subject is taking. Many of the notes discuss (or refer to papers discussing) applications of the refinements that are considered. Of course, it is the quantity and the quality of the insights and the applications that lend the refinements their validity. Although the guide to the applications is far from complete, the notes certainly allow the reader to form a good judgement of which refinements have really yielded new insights. Hence, as in the first edition, I will refrain from speculating on which refinements of Nash equilibria will survive in the long run. To defend this position let me also cite Binmore [1990] who compares writing about refinements to the Herculean task of defeating the nine-headed Hydra which grew too heads for each that was struck off. It is a pleasure to have the opportunity to thank my secretary, Marjoleine de Wit, who skilfully and, as always, cheerfully typed the manuscript and did the proofreading. Tilburg, March 1991 Preface to the First Edition 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, off the equilibrium path a Nash strategy may prescribe 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 Seiten, 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. XII 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 Muller, 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 and 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 of the 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-