A Beautiful Math Read online

  A BEAUTIFUL MATH

  JOHN NASH, GAME THEORY, AND THE MODERN QUEST FOR A CODE OF NATURE

  TOM SIEGFRIED

  JOSEPH HENRY PRESS

  Washington, D.C.

  * * *

  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 American 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

  519.3—dc22

  2006012394

  Copyright 2006 by Tom Siegfried. All rights reserved.

  Printed in the United States of America.

  Copyright © 2008/2009 Mobipocket.com. All rights reserved.

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  Preface

  Shortly after 9/11, a Russian scientist named Dmitri Gusev proposed an explanation for the origin of the name Al Qaeda. He suggested that the terrorist organization took its name from Isaac Asimov's famous 1950s science fiction novels known as the Foundation Trilogy. After all, he reasoned, the Arabic word "qaeda" means something like "base" or "foundation." And the first novel in Asimov's trilogy, Foundation, apparently was titled "al-Qaida" in an Arabic translation.

  In Asimov's 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 empire's 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 civilization's eventual rebirth.

  At least that's what he told the empire's authorities.

  In fact, Asimov's hero, a mathematician named Hari Seldon, created a community of scientists devoted to manipulating the future. Seldon actually formed two foundations—one 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 surreptitiously, intervening at key points in history to nudge events along Seldon's chosen path.

  Seldon's plan for controlling human affairs was based on a mathematical system that he invented called psychohistory. It enabled Seldon to predict political, economic, and social trends; foresee the rise and fall of governments; and anticipate the onset of wars and periods of peace.

  I don't think Osama bin Laden is Hari Seldon. But it's 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 society's 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 Seldon's strategies. (Certainly Osama bin Laden's occasional taped messages are eerily similar to Seldon's 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 Asimov's assertion that "violence is the last refuge of the incompetent."

  But in fact, Asimov's series did inspire some real-world imitators: not terrorists, but scientists—scientists seeking the secrets of Hari Seldon's psychohistory. If there is a real-life Hari Seldon, it is not Osama bin Laden, but John Forbes Nash.

  Nash's life, chronicled so engagingly by Sylvia Nasar in A Beautiful Mind, is a story of the struggles of a brilliant but troubled man. Nash's math, for which he won a Nobel Prize, is an entirely different tale, still unfolding, about science's 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 infiltrated nearly every field of modern science, especially those concerned with human nature and behavior. It has begun to establish links with the physical sciences as well, and ultimately, I suspect, it will forge a merger of all the sciences in the spirit of Asimov's 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 applications. My focus is rather on how various manifestations of game theory built on Nash's 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 Asimov's 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 pioneers 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 objections 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 theory's prominence 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 prominent 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 science of human behavior. Game theory is showing signs of playing an increasingly important role in that endeavor. It's a story of exploration along the shoreline separati
ng the continent of knowledge from an ocean of ignorance, and I think it's 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 I've 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

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  Contents

  Introduction 1

  1 Smith's Hand Searching for the Code of Nature 11

  2 Von Neumann's Games Game theory's origins 27

  3 Nash's Equilibrium Game theory's foundation 51

  4 Smith's Strategies Evolution, altruism, and cooperation 73

  5 Freud's Dream Games and the brain 93

  6 Seldon's Solution Game theory, culture, and human nature 110

  7 Quetelet's Statistics and Maxwell's Molecules Statistics and society, statistics and physics 126

  8 Bacon's Links Networks, society, and games 144

  9 Asimov's Vision Psychohistory, or sociophysics? 164

  10 Meyer's Penny Quantum fun and games 182

  11 Pascal's Wager Games, probability, information, and ignorance 197

  Epilogue 217

  Appendix: Calculating a Nash Equilibrium 225

  Further Reading 230

  Notes 233

  Index 249

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  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 exist—not now, not really, and not for a long time. But there are many research enterprises under way in the world today that share the goal of better understanding human behavior in order to foresee the future. At the foundation of these enterprises are mathematical methods closely resembling Asimov's psychohistory. And in the midst of it all is the work of a mathematician named John Forbes Nash.

  Brilliant but odd, intellectually sophisticated but socially awkward, Nash dazzled the world of mathematics in the 1950s with astounding and original results in several arenas. He rattled the routines at Princeton University and the Rand Corporation in California with both his mental magnificence and his disruptive behavior. By now, the subsequent tragic aspects of Nash's life story are familiar to millions of people, thanks to the Oscar-winning movie starring Russell Crowe, and Sylvia Nasar's A Beautiful Mind, the acclaimed book on which the movie was based. Yet while book and movie probed the conflicting complexities in Nash the man, neither delved deeply into Nash's math. So for most people today, his accomplishments remain obscure. Within the world of science, though, Nash's math now touches more disciplines than Newton's or Einstein's. What Newton's and Einstein's math did for the physical universe, Nash's math may now be accomplishing for the biological and social universe.

  Indeed, had mental illness not intervened, Nash's name might today be commonly uttered in the same breath with those scientific giants of the past. As it is, he made important contributions to a few mathematical specialties. But he achieved his greatest fame in economics, the field in which he shared the 1994 Nobel Prize with John Harsanyi and Reinhard Selten for their seminal work on the theory of games—the math that analyzes how people make choices in contests of strategy.

  Game theory originated in efforts to understand parlor games like poker and chess, and was first fully formulated as a mathematical tool for describing economic behavior. But in principle, game theory encompasses any situation involving strategic interaction—from playing tennis to waging war. Game theory provides the mathematical means of computing the payoffs to be expected from various possible choices of strategies. So game theory's math specifies the formulas for making sound decisions in any competitive arena. As such, it is "a tool for investigating the world," as the economist Herbert Gintis points out. But it is much more than a mere tool. "Game theory is about how people cooperate as much as how they compete," Gintis writes. "Game theory is about the emergence, transformation, diffusion and stabilization of forms of behavior."3

  Nash did not invent game theory, but he expanded its scope and provided it with more powerful tools for tackling real-world problems. At first, though, the depth of his accomplishment was little appreciated. When his revolutionary papers appeared, in the early 1950s, game theory briefly became popular among Cold War analysts who saw similarities between international aggression and maximizing profits. But within economics, game theory remained mainly a curiosity. "It didn't take off," the economist Samuel Bowles told me. "Like a lot of good ideas in economics, it just fell by the wayside."4

  In the 1970s, though, evolutionary biologists adopted game theory to study the competition for survival among animals and plants. And in the 1980s, economists finally began to use game theory in various ways, finding it especially helpful in designing actual experiments to test economic theory. By the late 1980s game theory had re-emerged in economics in a big way, leading to Nash's 1994 Nobel.

  Even before then, game theory had already migrated into the curricula of many scientific disciplines. You could find it taught in departments not only of mathematics and economics and biology but also political science, psychology, and sociology. By the opening years of the 21st century, game theory's uses had spread even wider, to fields ranging from anthropology to neurobiology.

  Today, economists continue to use game theory to analyze how people make choices about money. Biologists apply it to scenarios explaining the survival of the fittest or the origin of altruism. Anthropologists play games with people from primitive cultures to reveal the diversity of human nature. And neuroscientists have joined the fun, peering inside the brains of game-playing people to discover how their strategies reflect different motives and emotions. In fact, a whole new field of study, called neuroeconomics, has taken shape, mixing game theory's methods with brain-scanning technology to detect and measure neural activity corresponding to human judgments and behavior. "We're quantifying human experience," says neuroscientist Read Montague, "in the same way we quantify airflow over the wings of a Boeing 777."5

  In short, Nash's math—with the rest of modern game theory built around it—is now the weapon of choice in the scientist's arsenal on a wide range of research frontiers related to human behavior. In fact, Herbert Gintis contends, game theory has become "a universal language for the unification of the behavioral sciences."6

  I think it might go even farther than that. Game theory may become the language not just of the behavioral sciences, but of all the sciences.

  As science stands today, that claim is rather bold. It might even be wrong. But game theory already has conquered the social sciences and invaded biology. And it is now, in the works of a few pioneering scientists, forming a powerful alliance with physics. Physicists, of course, have always sought a unity in the ultimate description of nature, and game theory may have the potential to be a great unifier.

  That realization hit me in early 2004, when I read a paper by physicist-mathematician David Wolpe
rt, who works at NASA's Ames Research Center in California. Wolpert's paper disclosed a deep connection between the math of game theory and statistical mechanics, one of the most powerful all-purpose tools used by physicists for describing the complexities of the world.

  Physicists have used statistical mechanics for more than a century to describe such things as gases, chemical reactions, and the properties of magnetic materials—essentially to quantify the behavior of matter in all sorts of circumstances. It's a way to describe the big picture when lacking data about the details. You can't track every one of the trillion trillion molecules of air zipping around in a room, for instance, but statistical mechanics can tell you how an air conditioner will affect the overall temperature.

  It's no coincidence that statistical mechanics (which encompasses the kinetic theory of gases) is the math that inspired Asimov's heroic mathematician, Hari Seldon, to invent psychohistory. As Janov Pelorat, a character in the later novels of the Foundation series, explained: