Social and Economic Networks in Cooperative Game Theory PDF
Preview Social and Economic Networks in Cooperative Game Theory
SOCIAL AND ECONOMIC NETWORKS IN COOPERATIVE GAME THEORY THEORY AND DECISION LIBRARY General Editors: W. Leinfellner (Vienna) and G. Eberlein (Munich) Series A: Philosophy and Methodology of the Social Sciences Series B: Mathematical and Statistical Methods Series C: Game Theory. Mathematical Programming and Operations Research Series D: System Theory, Knowledge Engineering and Problem Solving SERIES C: GAME THEORY, MATHEMATICAL PROGRAMMING AND OPERATIONS RESEARCH VOLUME 27 Editor-in-Chief' H. Peters (Maastricht University); Honorary Editor: S. H. Tijs (University of Tilburg); Editorial Board: E.E.C. van Damme (University of Tilburg), H. Keiding (Copenhagen), I.-F. Mertens (Louvain-la-Neuve), H. Moulin (Rice University, TX, USA), S. Muto (Tokyo University, Japan), T. Parthasarathy (Indian Statistical Institute, New Delhi), B. Peleg (Jerusalem), T. E. S. Raghavan (Chicago, IL, USA), 1. Rosenmiiller (Bielefeld), A. Roth (Pittsburgh, PA, USA), D. Schmeidler (Tel-Aviv), R. Selten (Bonn), W. Thomson (University of Rochester, NY, USA) Scope: Particular attention is paid in this series to game theory and operations research, their formal aspects and their applications to economic, political and social sciences as well as to socio-biology. It will encourage high standards in the application of game-theoretical methods to individual and social decision making. The titles published in this series are listed at the end of this volume. SOCIAL AND ECONOMIC NETWORKS IN COOPERATIVE GAMETHEORY by MARCO SLIKKER Technische Universiteit Eindhoven, The Netherlands and ANNE VAN DEN NOUWELAND University of Oregon, U.SA . .... " SPRINGER SCIENCE+BUSINESS MEDIA, LLC ISBN 978-1-4613-5619-6 ISBN 978-1-4615-1569-2 (eBook) DOI 10.1007/978-1-4615-1569-2 Library of Congress Cataloging-in-Publication Data A C.I.P. Catalogue record for this book is available from the Library of Congress. Copyright It> 2001 Springer Science+Business Media New York Originally published by Kluwer Academic Publishers in 2001 Softcover reprint of the hardcover 1s t edition 2001 AII rights reserved. No part of this publicat ion may be reproduced, stored in a retrieval system or transmitted in any form or by any means, mechanical, photo-copying, recording, or otherwise, without the prior written permis sion of the publisher, Springer Science+Business Media, LLC Printed on acid-free paper. Contents Preface IX Part I Social and Economic Networks in Cooperative Situations 1. GAMES AND NETWORKS 3 1.1 Coalitional games 3 1.2 Networks 13 2. RESTRICTED COOPERATION IN GAMES 21 2.1 Games with restricted communication 21 2.2 The Myerson value 29 2.3 Other allocation rules 43 3. INHERITANCE OF PROPERTIES IN COMMUNICATION SITUATIONS 53 3.1 Superadditivity 54 3.2 Balancedness and totally balancedness 55 3.3 Convexity 58 3.4 Average convexity 62 3.5 Core-inclusion of the Shapley value 73 3.6 Population monotonic allocation schemes 81 3.7 Review and remarks 86 4. VARIANTS ON THE BASIC MODEL 89 4.1 Games with coalition structures 90 4.2 Hypergraph communication situations 95 4.3 Probabilistic communication situations 100 4.4 NTU communication situations 107 4.5 Reward communication situations 115 4.6 Directed communication situations 124 VI NETWORKS IN COOPERATIVE GAME THEORY Part II Network Formation 5. NONCOOPERATIVE GAMES 135 5.1 Games in extensive form 135 5.2 Games in strategic form 140 6. A NETWORK-FORMATION MODEL IN EXTENSIVE FORM153 6.1 Description of the model 153 6.2 Some examples 155 6.3 Weighted majority games 160 6.4 Symmetric convex games 164 7. A NETWORK-FORMATION MODEL IN STRATEGIC FORM173 7 .1 Description of the model 174 7.2 Nash equilibrium and strong Nash equilibrium 179 7.3 Undominated Nash equilibrium and coalition-proof Nash equilibrium 183 7.4 Comparison of the network-formation models 189 7.5 Remarks 190 8. NETWORK FORMATION WITH COSTS FOR ESTABLISHING LINKS 193 8.1 The cost-extended Myerson value 193 8.2 Network-formation games in extensive form 198 8.3 Network-formation games in strategic form 202 8.3.1 Nash equilibrium and strong Nash equilibrium 203 8.3.2 Undominated Nash equilibrium 205 8.3.3 Coalition-proof Nash equilibrium 206 8.4 Extensions 209 8.5 Comparison of the network-formation models 210 9. A ONE-STAGE MODEL OF NETWORK FORMATION AND PAYOFF DIVISION 213 9.1 The model 213 9.2 Nash equilibrium 218 9.3 Strong Nash equilibrium 223 9.4 Coalition-proof Nash equilibrium 229 10.NETWORK FORMATION AND POTENTIAL GAMES 249 10.1 Potential games 249 10.2 A representation theorem 254 10.3 Network formation 257 10.4 Remarks 262 l1.NETWORK FORMATION AND REWARD FUNCTIONS 265 11.1 Pairwise stability 265 11.2 Weak and strong stability 271 Contents VB 11.3 Dynamic models of network formation 276 References 281 Notations 287 Index 289 Preface In many social or economic settings, communication between the par ticipants is important for the dissemination of information. Much of this communication takes place through networks, systems of decentral ized bilateral relationships between the participants. Networks are the primary vehicle for the diffusion of information on job openings, busi ness opportunities, and new products, to name a few. The importance of social and economic networks has been extensively documented in empirical work. In recent years, there has been increased attention for theoretical mod els that help us understand how networks affect economic outcomes and that analyze how networks emerge. In this book, we present a coherent overview of a branch of theoretical literature that studies the influence and formation of networks in social and economic situations in which the relations between participants who are not included in a particular participant's network are not of consequence to this participant. This means that we concentrate on a particular strain of the theoretical liter ature and exclude others. The strain that we report on is arguably not the most general one. However, it is one for which research has been very fruitful. We hope that the reader of this book will be inspired by it to study the influence and formation of networks in more complicated settings and that this will lead to progress in this important branch of research. We have organized the material in two parts. In part I we concen trate on the question how network structures affect economic outcomes. In chapter 1 we discuss the basic concepts and results in cooperative game theory and graph theory that are used throughout the book. In chapter 2 we show how networks are integrated into a coalitional game to form a so-called network-restricted game and we define several al- x NETWORKS IN COOPERATIVE GAME THEORY location rules for games with communication restrictions. One of the rules we discuss is the Myerson value, which plays an important role in most of the network-formation models in part II. We discuss this rule and its properties extensively. In chapter 3 we study how restrictions on communication influence various properties of coalitional games, such as sllperadditivity, balancedness, and convexity. We consider conditions on the underlying network that are necessary and sufficient to guarantee that such desirable properties of the underlying game are still satisfied by the corresponding network-restricted game. In chapter 4 we discuss several variants of the basic model in chapter 2. These variants involve alternative representations of restrictions on communication, of the eco nomic possibilities of the players, or a combination of both. In part II of the book we study the formation of networks by agents who engage in a network-formation process to be able to realize the pos sible gains from cooperation. The agents form bilateral relations with each other if doing so is to their advantage. We study which networks are formed under various assumptions about the process of network forma tion. We start part II with chapter 5, in which we introduce the concepts from noncooperative game theory that will be used in the chapters to come. We discuss games in extensive form and games in strategic form and several solution concepts for such garnes, all of which are based on Nash equilibrium, which is the standard solution concept in noncoop erative game theory. In chapter 6 we consider extensive-form games of network formation that describe network-formation processes in which links are formed one at a time and players observe which links are formed as the process progresses. The models in this chapter use an exogenously given allocation rule to describe the payoffs to the agents in any of the networks that they can form. In chapter 7 we study strategic-form games of network formation that describe network-formation processes in which agents have to decide on the formation of links in a setting where they are not aware of which other links may have been formed. The models in this chapter also use exogenously given allocation rules to determine agents' payoffs. In chapter 8 we incorporate costs for the formation of communication links into the network-formation models in chapters 6 and 7. Our main interest is to analyze the influence of costs for estab lishing links on the networks that result according to several equilibrium concepts. In chapter 9 we study a model of network formation in which players bargain over the formation of links and the division of the pay offs simultaneously. This makes the model very different from those in previous chapters, where bargaining over payoff division occurred only when the network-formation stage had been completed. In chapter 10 we revisit the network-formation model in strategic form of chapter 7. \Ve Preface Xl study the conditions under which these games are potential games. For such games, all the information that is necessary to determine Nash equi libria can be captured in a single function and we can apply a refinement of the Nash-equilibrium concept to study which network structures are formed according to this refinement. In chapter 11 we consider questions related to the formation of networks in cases where a reward function gives the value of each network. Reward functions allow us to model situations in which the value that can be obtained by a group of agents does not depend solely on whether they are connected or not, but also on exactly how they are connected to each other. We have worked hard to make the book self-contained and, in princi ple, it is possible to read it without any prior knowledge of its subject. However, the proofs rely heavily on mathematics and it might be hard to read those without some level of proficiency in several areas of math ematics. However, the reader should still be able to get a feel for the types of results that we report if he or she skips the proofs. This book should appeal to researchers who are interested in networks and it could form the basis for a graduate course on networks. We have opted to consistently use 'he' whenever we need a third person singular pronoun. If there were a gender-neutral third-person singular pronoun available, we would have used that, but one simply does not exist. In many languages, the convention is to use the male pronoun when the gender is unknown. In such situations, the male pronoun refers to both males and females. We have decided to follow this convention because we believe that the use of he/she will divert the readers' attention from the main issues. Acknow ledgements We would like to thank the people whose contributions to this book are greatly appreciated. This, of course, includes all our co-authors. It also includes Stef Tijs, who introduced us to game theory and who motivated us to write this book. We thank Van Kolpin for proof-reading parts of the book for us and we thank Larry Singell for answering our many questions about the English language. All remaining errors are, of course, ours.
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