Index and Stability in Bimatrix Games A Geometric-Combinatorial Appr
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Amdt von Schemde
Index and Stability in Bimatrix Games A Geometric-Combinatorial Approach
Springer
Author Amdt von Schemde Lilleborg gata 6 0480 Oslo Norway schemde @ gmail. com
Library of Congress Control Number: 2005929878
ISSN 0075-8442 ISBN-10 3-540-26366-7 Springer Berlin Heidelberg New York ISBN-13 978-3-540-26366-1 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9,1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springeronline.com © Springer-Verlag Berlin Heidelberg 2005 Printed in Germany The use of general descriptive names, 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: Camera ready by author Cover design: Erich Kirchner, Heidelberg Printed on acid-free paper
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To my parents
Preface
This work originates from my PhD thesis at the London School of Economics and Political Science. I am indebted to Bemhard von Stengel for his excellent supervision. He introduced me to the questions addressed in this work. I am thankful for the many hours of discussions, and also for his guidance and patience in times when results seemed far away. Also, I am grateful to Srihari Govindan and Robert Wilson for useful advice and encouragement. Furthermore, I would like to thank the members of the Mathematics Department for their general support. In particular, I thank Jackie Everid, David Scott and Mark Baltovic for their assistance, as well as Nic Georgiou and Luis Cereceda for their help on the final draft. There were many people who supported me personally. Foremost, I would like to thank my parents and Ane S. Flaatten for always being there for me. I am also beholden to Philipp Beckmann for his advice, and to Philip Hochstrate for his inspiration. Finally, I would like to thank the London School of Economics and Political Science (LSE), the Department of Mathematics at LSE and the UK Engineering and Physical Sciences Research Council (EPSRC) for financial support.
Oslo, July 2005
Amdt von Schemde
Contents
Introduction
1
1
Equilibrium Components with Arbitrary Index 7 1.1 Preliminaries 8 1.2 The Lemke-Howson Algorithm 15 1.3 Index Theory 18 1.4 Construction of Equilibrium Components with Arbitrary Index 24
2
A Reformulation of the Index for Equilibria in Bimatrix Games 2.1 The Dual Construction 2.2 Labelling and Ch
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