Table Of ContentSOCIAL 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.
