Simple Adaptive Strategies: From Regret-Matching to Uncoupled Dynamics

335 Pages·2013·2.29 MB·English

by  Sergiu Hart

Simple AdAptive
StrAtegieS
From Regret-Matching to
Uncoupled Dynamics
World Scientific Series in Economic Theory
(ISSN: 2251-2071)
Series Editor: Eric Maskin (Harvard University, USA)
Published
Vol. 1 Equality of Opportunity: The Economics of Responsibility
by Marc Fleurbaey and François Maniquet
Vol. 2 Robust Mechanism Design: The Role of Private Information and
Higher Order Beliefs
Dirk Bergemann and Stephen Morris
Vol. 3 Case-Based Predictions: An Axiomatic Approach to Prediction,
Classification and Statistical Learning
Itzhak Gilboa and David Schmeidler
Vol. 4 Simple Adaptive Strategies: From Regret-Matching to
Uncoupled Dynamics
Sergiu Hart and Andreu Mas-Colell
Forthcoming
Robustness and Macroeconomics
Lars Peter Hansen (University of Chicago) &
Thomas Sargent (New York University, USA)
Epistemic Game Theory
Adam Brandenburger (New York University, USA)
Decision Theory
Wolfgang Pesendorfer (Princeton University, USA) &
Faruk Gul (Princeton University, USA)
Leverage and Default
John Geanakoplos (Yale University, USA)
Leverage Cycle, Equilibrium and Default
Vol. 2: Collateral Equilibrium and Default
John Geanakoplos (Yale University, USA)
Learning and Dynamic Games
Dirk Bergemann (Yale University, USA) & Juuso Valimaki (Aalto University, Finland)
Alisha - Simple Adaptive Stragegies.pmd 1 11/5/2012, 3:59 PM
World Scientific Series in Economic Theory – Vol. 4
Simple AdAptive
StrAtegieS
From Regret-Matching to
Uncoupled Dynamics
Sergiu Hart
The Hebrew University of Jerusalem
Andreu Mas-Colell
Universitat Pompeu Fabra, Barcelona
with the collaboration of
Yakov Babichenko
Amotz Cahn
Yishay Mansour
David Schmeidler
World Scientific
NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI
Published by
World Scientific Publishing Co. Pte. Ltd.
5 Toh Tuck Link, Singapore 596224
USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
Library of Congress Cataloging-in-Publication Data
Hart, Sergiu.
Simple adaptive strategies : from regret-matching to uncoupled dynamics / by Sergiu Hart
(The Hebrew University of Jerusalem, Israel) & Andreu Mas-Colell (Universitat Pompeu Fabra,
Spain).
p. cm -- (World Scientific series in economic theory, ISSN 2251-2071 ; v. 4)
Includes bibliographical references and index.
ISBN 978-9814390699
1. Games of strategy (Mathematics) 2. Heuristic algorithms. I. Mas-Colell, Andreu. II. Title.
QA270.H37 2012
519.3--dc23
2012023203
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Copyright © 2013 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
electronic or mechanical, including photocopying, recording or any information storage and retrieval
system now known or to be invented, without written permission from the Publisher.
For photocopying of material in this volume, please pay a copying fee through the Copyright
Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to
photocopy is not required from the publisher.
In-house Editor: Alisha Nguyen
Typeset by Stallion Press
Email: [email protected]
Printed in Singapore.
Alisha - Simple Adaptive Stragegies.pmd 2 11/5/2012, 3:59 PM
November20, 2012 6:2 9in x 6in b1333-fm Simple AdaptiveStrategies
Simply, to Priya and Esther
v
November20, 2012 6:2 9in x 6in b1333-fm Simple AdaptiveStrategies
TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk
November20, 2012 6:2 9in x 6in b1333-fm Simple AdaptiveStrategies
FOREWORD
Nash equilibrium—a configuration of strategies from which no player has
theincentivetodeviateunilaterally—isthecentralsolutionconceptinnon-
cooperative game theory. That is, in both the theoretical and applied lit-
eratures, it constitutes the leading prediction of how players make their
strategic choices. Left unstated in its definition, however, is how a Nash
equilibrium might actually come about.
Onepossibilityisthatequilibriumarisesthroughintrospection:aplayer
expects others to play their Nash equilibrium strategies, and so he finds it
payoff-maximizing to do the same. But this still doesn’t explain why he
wouldanticipatethatotherplayerswilluseequilibriumstrategies,i.e.,why
he would have “equilibrium expectations.” As the literature on rational-
izability makes clear, even common knowledge of players’ rationality and
uniqueness of equilibrium do not guarantee such expectations.
However,itisconceivablethatthroughrepeatedexperienceofthesame
game, a player might learn to expect others to play equilibrium strategies.
ThisistheissuethatSergiuHartandAndreuMas-Colellhavebeenexplor-
ing in their important work on adaptive dynamics. Interestingly, they con-
clude that learning—at least, learning of a reasonably simple kind—will
not,ingeneral,leadto Nashequilibriumexpectations(unless playersknow
one another’s payoff functions or there is some other way in which players
are “coupled”). However, a form of adaptive learning, called regret match-
ing—whereina playerisapttouse astrategythatwouldhavedonewellin
previous play—will lead in the long run to correlated equilibrium: a Nash
equilibrium in which players’ randomizations need not be independent of
one another.
I amdelightedthatSergiuandAndreu haveagreedto collecttwelve of
theirmostsignificantpapersinthisarea—togetherwithanewanddetailed
introduction—in this volume.
Eric Maskin
Editor-in-Chief, World Scientific Series in Economic Theory
vii
November20, 2012 6:2 9in x 6in b1333-fm Simple AdaptiveStrategies
TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk
November20, 2012 6:2 9in x 6in b1333-fm Simple AdaptiveStrategies
CONTENTS
Foreword vii
Eric Maskin
Introduction xvii
Sergiu Hart and Andreu Mas-Colell
Part I Correlated Equilibria 1
Chapter 1. Existence of Correlated Equilibria 3
Sergiu Hart and David Schmeidler
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Finite Games . . . . . . . . . . . . . . . . . . . . . . . . . 4
3 Infinitely Many Players . . . . . . . . . . . . . . . . . . . . 7
4 Infinitely Many Players and Strategies . . . . . . . . . . . 11
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Part II Regret Matching 15
Chapter 2. A Simple Adaptive Procedure Leading
to Correlated Equilibrium 17
Sergiu Hart and Andreu Mas-Colell
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2 The Model and Main Result . . . . . . . . . . . . . . . . . 20
3 No Regret and Blackwell Approachability. . . . . . . . . . 24
4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
Appendix. Proof of the Main Theorem . . . . . . . . . . . . . . 36
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
ix