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  1. 1. JOSEPH HENRY PRESS Washington, D.C. A BEAUTIFUL MATHA BEAUTIFUL MATH JOHN NASH, GAME THEORY, AND THE MODERN QUEST FOR A CODE OF NATURE T O M S I E G F R I E D
  2. 2. Joseph Henry Press 500 Fifth Street, NW Washington, DC 20001 The Joseph Henry Press, an imprint of the National Academies Press, was created with the goal of making books on science, technology, and health more widely available to professionals and the public. Joseph Henry was one of the founders of the National Academy of Sciences and a leader in early Ameri- can science. Any opinions, findings, conclusions, or recommendations expressed in this volume are those of the author and do not necessarily reflect the views of the National Academy of Sciences or its affiliated institutions. Library of Congress Cataloging-in-Publication Data Siegfried, Tom, 1950- A beautiful math : John Nash, game theory, and the modern quest for a code of nature / Tom Siegfried. 1st ed. p. cm. Includes bibliographical references and index. ISBN 0-309-10192-1 (hardback) ISBN 0-309-65928-0 (pdfs) 1. Game theory. I. Title. QA269.S574 2006 519.3dc22 2006012394 Copyright 2006 by Tom Siegfried. All rights reserved. Printed in the United States of America.
  3. 3. Preface Shortly after 9/11, a Russian scientist named Dmitri Gusev pro- posed an explanation for the origin of the name Al Qaeda. He suggested that the terrorist organization took its name from Isaac Asimovs famous 1950s science fiction novels known as the Foun- dation Trilogy. After all, he reasoned, the Arabic word qaeda means something like base or foundation. And the first novel in Asimovs trilogy, Foundation, apparently was titled al-Qaida in an Arabic translation. In Asimovs books, Foundation referred to an organization dedicated to salvaging a decaying galactic empire. The empire was hopeless, destined to crumble into chaos, leaving civilization in ruins for 30,000 years. Foreseeing the inevitability of the empires demise, one man devised a plan to truncate the coming era of darkness to a mere millennium. His strategy was to establish a foundation of scholars who would preserve human knowledge for civilizations eventual rebirth. At least thats what he told the empires authorities. In fact, Asimovs hero, a mathematician named Hari Seldon, created a community of scientists devoted to manipulating the fu- ture. Seldon actually formed two foundationsone in a remote but known locale (sort of like Afghanistan), the other in a mystery location referred to only with riddles. Foundation I participated openly in the affairs of the galaxy. Foundation II operated surrep- titiously, intervening at key points in history to nudge events along Seldons chosen path. Seldons plan for controlling human affairs was based on a iii
  4. 4. iv PREFACE mathematical system that he invented called psychohistory. It en- abled Seldon to predict political, economic, and social trends; fore- see the rise and fall of governments; and anticipate the onset of wars and periods of peace. I dont think Osama bin Laden is Hari Seldon. But its not so far-fetched to believe that the organizers of the real Al Qaeda perceived Western civilization as an empire in decay. Or that they anointed themselves as societys saviors, hoping to manipulate events in a way that would lead to a new world order more to their liking. So perhaps they adopted some of Hari Seldons strategies. (Certainly Osama bin Ladens occasional taped messages are eerily similar to Seldons video appearances from time to time, prepared before his death for delivery decades or even centuries later.) Of course, any such link to Asimov changes nothing about terrorism. Al Qaeda gains no justification for atrocity from any connection to science fiction. And frankly, the similarities seem rather superficial. Had the terrorists really studied Foundation, they would have noticed Asimovs assertion that violence is the last refuge of the incompetent. But in fact, Asimovs series did inspire some real-world imita- tors: not terrorists, but scientistsscientists seeking the secrets of Hari Seldons psychohistory. If there is a real-life Hari Seldon, it is not Osama bin Laden, but John Forbes Nash. Nashs life, chronicled so engagingly by Sylvia Nasar in A Beau- tiful Mind, is a story of the struggles of a brilliant but troubled man. Nashs math, for which he won a Nobel Prize, is an entirely different tale, still unfolding, about sciences struggle to cope with the complexities of collective human behavior. At the same time Asimov was publishing his Foundation books, Nash was publishing papers establishing foundational principles for a science called game theory. Game theory is the science of strategy; its formulas tell you what choices to make to get the best deal you can get when interacting with other people. Originally formulated to be applied to economics, game theory has now infil- trated nearly every field of modern science, especially those con- cerned with human nature and behavior. It has begun to establish
  5. 5. PREFACE v links with the physical sciences as well, and ultimately, I suspect, it will forge a merger of all the sciences in the spirit of Asimovs psychohistory. At least that is the prospect that I explore in this book. Game theory is a rich, profound, and controversial field, and there is much more to it than you could find in any one book. What follows is in no way a textbook on game theory. Nor do I attempt to give any account of its widespread uses in economics, the realm for which it was invented, or the many variants and refinements that have been developed to expand its economic ap- plications. My focus is rather on how various manifestations of game theory built on Nashs foundation are now applied in a vast range of other scientific disciplines, with special attention to those arenas where game theory illuminates human nature and behavior (and where it connects with other fields seeking similar insights). I view these efforts in the context of the ancient quest for a Code of Nature describing the laws of human behavior, a historical precursor to Asimovs notion of psychohistory. As with all my books, I try to give any interested reader a flavor of what scientists are doing at the frontiers of knowledge, where there are no guarantees of ultimate success, but where pio- neers are probing intriguing possibilities. There are scientists who regard some of this pioneering work as at best misguided and at worst a fruitless waste of time. Consequently, there may be objec- tions from traditionalists who believe that the importance of game theory is overstated or that the prospects for a science of society are overhyped. Well, maybe so. Time will tell. For now, the fact is that game theory has already established itself as an essential tool in the behavioral sciences, where it is widely regarded as a unifying language for investigating human behavior. Game theorys promi- nence in evolutionary biology builds a natural bridge between the life sciences and the behavioral sciences. And connections have been established between game theory and two of the most promi- nent pillars of physics: statistical mechanics and quantum theory. Certainly many physicists, neuroscientists, and social scientists from various disciplines are indeed pursuing the dream of a quantitative
  6. 6. vi PREFACE science of human behavior. Game theory is showing signs of play- ing an increasingly important role in that endeavor. Its a story of exploration along the shoreline separating the continent of knowl- edge from an ocean of ignorance, and I think its a story worth telling. I owe much gratitude to those who helped make this book possible, particularly the many scientists who have discussed their research with me over the years. Their help is acknowledged by their presence in the pages that follow. Many other friends and colleagues have listened patiently while Ive shaped my thoughts on this book during conversations with them. They know who they are, and I appreciate them all. The one person I want to thank by name is my wife, Chris, who really made it possible for me to write this book, because she has a job. Tom Siegfried Los Angeles, California
  7. 7. Contents vii Introduction 1 1 Smiths Hand 11 Searching for the Code of Nature 2 Von Neumanns Games 27 Game theorys origins 3 Nashs Equilibrium 51 Game theorys foundation 4 Smiths Strategies 73 Evolution, altruism, and cooperation 5 Freuds Dream 93 Games and the brain 6 Seldons Solution 110 Game theory, culture, and human nature 7 Quetelets Statistics and Maxwells Molecules 126 Statistics and society, statistics and physics 8 Bacons Links 144 Networks, society, and games
  8. 8. 9 Asimovs Vision 164 Psychohistory, or sociophysics? 10 Meyers Penny 182 Quantum fun and games 11 Pascals Wager 197 Games, probability, information, and ignorance Epilogue 217 Appendix: Calculating a Nash Equilibrium 225 Further Reading 230 Notes 233 Index 249 viii CONTENTS
  9. 9. 1 Introduction Could not mind, as well as mindless motion, have an underlying order? Emperor Cleon to Hari Seldon, Prelude to Foundation Isaac Asimov excelled at predicting the future. In one of his early science fiction stories, he introduced pocket calculators decades before you could buy them at Radio Shack. In a later book, he described a digital camera transmitting photos directly to a computer via WiFi.1 He just forgot to mention that you could also use the same device to make phone calls. And in his most celebrated work, a series of 1950s science fiction novels known as the Foundation Trilogy, Asimov foresaw a new kind of science called psychohistory, capable itself of forecasting political, economic, and social events. Psychohistory, as Asimov envisioned it, was the science of human behavior reduced to mathematical equations.2 Real-life psychohistory does not yet existnot now, not re- ally, and not for a long time. But there