Game Theory and Applications PDF
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ECONOMIC 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 ® Academic Press, Inc. Harcourt Brace Jovanovich, Publishers San Diego New York Boston London Sydney Tokyo Toronto This book is printed on acid-free paper. @ Copyright © 1990 by Academic Press, Inc. All Rights Reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Academic Press, Inc. San Diego, California 92101 United Kingdom Edition published by Academic Press Limited 24-28 Oval Road, London NW1 7DX 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 CIP Printed in the United States of America 90 91 92 93 9 8 7 6 5 4 3 2 1 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.
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