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Reprinted from “Bargaining and Markets”, ISBN 0-12-528632-5, Copyright 1990, with permission from Elsevier. References updated and errors corrected. Version: 2005-3-2. Bargaining and Markets This is a volume in ECONOMIC THEORY, ECONOMETRICS, AND MATHEMATICAL ECONOMICS A series of Monographs and Textbooks Consulting Editor: Karl Shell, Cornell University A list of recent titles in this series appears at the end of this volume. Bargaining and Markets Martin J. Osborne Department of Economics McMaster University Hamilton, Ontario Canada http://www.economics.utoronto.ca/osborne Ariel Rubinstein Department of Economics Tel Aviv University Tel Aviv, Israel http://arielrubinstein.tau.ac.il 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 c  1990 by Academic Press, Inc. All rights reserved. No part of this publicatio n may be reproduced or transmit ted in any form or by any means, electronic or mechanical, including photocopying, 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 Osborne, Martin J. Bargaining and Markets / Martin J. Osborne and Ariel Rubinstein p. cm. Includes bibliographical references. ISBN 0-12-528631-7 (alk. paper). – ISBN 0-12-528632-5 (pbk.: alk. paper) 1. Game Theory. 2. Negotiation. 3. Capitalism. I. Rubinstein, Ariel. II. Title. HB144.073 1990 380.1–dc20 90-30644 CIP Printed in the United States of America 90 91 92 93 9 8 7 6 5 4 3 2 1 Contents Preface ix 1. Introduction 1 1.1 Some Basic Terms 1 1.2 Outline of the Book 3 Notes 6 Part 1. Bargaining Theory 7 2. The Axiomatic Approach: Nash’s Solution 9 2.1 Bargaining Problems 9 2.2 Nash’s Axioms 11 2.3 Nash’s Theorem 13 2.4 Applications 17 2.5 Is Any Axiom Superfluous? 20 2.6 Extensions of the Theory 23 Notes 26 v vi Contents 3. The Strategic Approach: A Model of Alternating Offers 29 3.1 The Strategic Approach 29 3.2 The Structure of Bargaining 30 3.3 Preferences 32 3.4 Strategies 37 3.5 Strategies as Automata 39 3.6 Nash Equilibrium 41 3.7 Subgame Perfect Equilibrium 43 3.8 The Main Result 44 3.9 Examples 49 3.10 Properties of the Subgame Perfect Equilibrium 50 3.11 Finite versus Infinite Horizons 54 3.12 Models in Which Players Have Outside Options 54 3.13 A Game of Alternating Offers with Three Bargainers 63 Notes 65 4. The Relation between the Axiomatic and Strategic Approaches 69 4.1 Introduction 69 4.2 A Model of Alternating Offers with a Risk of Breakdown 71 4.3 A Model of Simultaneous Offers: Nash’s “Demand Game” 76 4.4 Time Preference 81 4.5 A Model with Both Time Preference and Risk of Breakdown 86 4.6 A Guide to Applications 88 Notes 89 5. A Strategic Model of Bargaining between Incompletely Informed Players 91 5.1 Introduction 91 5.2 A Bargaining Game of Alternating Offers 92 5.3 Sequential Equilibrium 95 5.4 Delay in Reaching Agreement 104 5.5 A Refinement of Sequential Equilibrium 107 5.6 Mechanism Design 113 Notes 118 Contents vii Part 2. Models of Decentralized Trade 121 6. A First Approach Using the Nash Solution 123 6.1 Introduction 123 6.2 Two Basic Models 124 6.3 Analysis of Model A (A Market in Steady State) 126 6.4 Analysis of Model B (Simultaneous Entry of All Sellers and Buyers) 128 6.5 A Limitation of Modeling Markets Using the Nash Solution 130 6.6 Market Entry 131 6.7 A Comparison of the Competitive Equilibrium with the Market Equilibria in Models A and B 134 Notes 136 7. Strategic Bargaining in a Steady State Market 137 7.1 Introduction 137 7.2 The Model 138 7.3 Market Equilibrium 141 7.4 Analysis of Market Equilibrium 143 7.5 Market Equilibrium and Compe titive Equilibrium 146 Notes 147 8. Strategic Bargaining in a Market with One-Time Entry 151 8.1 Introduction 151 8.2 A Market in Which There Is a Single Indivisible Good 152 8.3 Market Equilibrium 153 8.4 A Market in Which There Are Many Divisible Goods 156 8.5 Market Equilibrium 159 8.6 Characterization of Market Equilibrium 162 8.7 Existence of a Market Equilibrium 168 8.8 Market Equilibrium and Compe titive Equilibrium 170 Notes 170 9. The Role of the Trading Procedure 173 9.1 Introduction 173 9.2 Random Matching 175 9.3 A Model of Public Price Announcements 180 9.4 Models with Choice of Partner 182 9.5 A Model with More General Contracts and Resale 185 Notes 187 viii Contents 10. The Role of Anonymity 189 10.1 Introduction 189 10.2 The Model 190 10.3 Market Equilibrium 191 10.4 The No-Discount Assumption 195 10.5 Market Equilibrium and Competitive Equilibrium 197 Notes 197 References 199 Index 211 Preface The formal theory of bargaining originated with John Nash’s work in the early 1950s. In this book we discuss two recent developments in this theory. The first uses the tool of extensive games to construct theories of bargain- ing in which time is modeled explicitly. The second applies the theory of bargaining to the study of decentralized markets. We do not attem pt to survey the field. Rather, we select a small number of models, each of which illustrates a key point. We take the approach that a thorough analysis of a few models is more rewarding than short discussions of many models. Some of our selections are arbitrary and could be replaced by other models that illustrate similar points. The last section of each chapter is entitled “Notes”. It usually begins by acknowledging the work on which the chapter is based. (In general we do not make acknowledgments in the text itself.) It goes on to give a brief guide to some of the related work. We should stress that this guide is not complete. We include mainly references to papers that use the model of bargaining on which most of the book is based (the bargaining game of alternating offers). Almost always we give detailed proofs. Although this makes some of the chapters look “technical” we believe that only on understanding the proofs ix [...]... Section 2.6.3) Definition 2.1 A bargaining problem is a pair S, d , where S ⊂ R2 is compact (i.e closed and bounded) and convex, d ∈ S, and there exists s ∈ S such that si > di for i = 1, 2 The set of all bargaining problems is denoted B A bargaining solution is a function f : B → R2 that assigns to each bargaining problem S, d ∈ B a unique element of S This definition restricts a bargaining problem in a number... the bargaining solution must assign the same utility to each player Formally, the bargaining problem S, d is symmetric if d1 = d2 and (s1 , s2 ) ∈ S if and only if (s2 , s1 ) ∈ S SYM (Symmetry) If the bargaining problem S, d is symmetric, then f1 (S, d) = f2 (S, d) The next axiom is more problematic IIA (Independence of Irrelevant Alternatives) If S, d and T, d are bargaining problems with S ⊂ T and. .. of S , d If u1 , u2 , and h are differentiable, and 0 < zu < 1, then zu is the solution of u (1 − z) u1 (z) = 2 u1 (z) u2 (1 − z) (2.2) Similarly, zv is the solution of u1 (z) h (u2 (1 − z)) u2 (1 − z) = u1 (z) h (u2 (1 − z)) (2.3) The left-hand sides of equations (2.2) and (2.3) are decreasing in z, and the right-hand sides are increasing in z Further, since h is concave and h(0) = 0, we have h... pairs could result from many different combinations of agreement sets and preferences.) The objects of our subsequent inquiry are bargaining solutions A bargaining solution associates with every bargaining situation in some class an agreement or the disagreement event Thus, a bargaining solution does not specify an outcome for a single bargaining situation; rather, it is a function Formally, Nash’s central... “he” and sometimes “she” For example, in two-player games we treat Player 1 as female and Player 2 as male; in markets games we treat all sellers as female and all buyers as male We use “he” for generic individuals Preface xi Acknowledgments The detailed comments of Ken Binmore, Jeroen Swinkels, and Eric van Damme on a draft of the book guided us to significantly improve the accuracy of the arguments and. .. to understand better the circumstances under which a market is “competitive” In the theory of competitive equilibrium, the process by which the equilibrium price is reached is not modeled One story is that there is an agency in the market that guides the price The agency announces a price, and the individuals report the amounts they wish to demand and supply at this fixed price If demand and supply... (possibly incompatible) demands of the players; such a procedure may not satisfy I IA Without specifying the details of the bargaining process, it is hard to assess how reasonable the axiom is 2.3 Nash’s Theorem 13 The final axiom is also problematic and, like I IA, relates to the bargaining process PAR (Pareto Efficiency) Suppose S, d is a bargaining problem, s ∈ S, t ∈ S, and ti > si for i = 1, 2 Then... Note that the axioms SYM and PAR restrict the behavior of the solution on single bargaining problems, while INV and I IA require the solution to exhibit some consistency across bargaining problems 2.3 Nash’s Theorem Nash’s plan of deriving a solution from some simple axioms works perfectly He shows that there is precisely one bargaining solution satisfying the four axioms above, and this solution has... disagreement point to the origin and the solution f N (S, d) to the point (1/2, 1/2) (That is, αi = 1/(2(zi − di )) and βi = −di /(2(zi − di )), di = αi di + βi = 0, and αi fiN (S, d) + βi = αi zi + βi = 1/2 for i = 1, 2.) Since both f and f N satisfy INV we have fi (S , 0) = αi fi (S, d) + βi and fiN (S , 0) = αi fiN (S, d) + βi (= 1/2) for i = 1, 2 Hence f (S, d) = f N (S, d) if and only if f (S , 0) = f... this book, see Wilson (1987), Bester (1989b), and Binmore, Osborne, and Rubinstein (1992) For some basic topics in bargaining theory that we do not discuss, see the following: Schelling (1960), who provides an informal discussion of the strategic elements in bargaining; Harsanyi (1977), who presents an early overview of game-theoretic models of bargaining; and Roth (1988), who discusses the large body . Reprinted from Bargaining and Markets , ISBN 0-12-528632-5, Copyright 1990, with permission from Elsevier. References updated and errors corrected. Version: 2005-3-2. Bargaining and Markets This. each other and do not depend on Chapters 4 and 5. Thus, if you are interested mainly in the application of bargaining theory to the study of markets, you can read Chapters 2 and 3 and then some. that guides the price. The agency announces a price, and the individuals report the amounts they wish to demand and supply at this fixed price. If de- mand and supply are not equal, the agency adjusts