The Essential John Nash PDF
Preview The Essential John Nash
THE ESSENT A L J NASH 0 H N OD"'D .. HAROll) 'IV_ KUHN ANP SYLVIA N'5AR ....<"'0•••, ""<"0' . Copyright © 2002 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 3 Market Place, Woodstock, Oxfordshire OX20 ISY All Rights Reserved Seventh printing, and first paperback printing, 2007 Paperback ISBN-13: 978-0-691-09610-0 Paperback ISBN-I0: 0-691-09610-4 Library of Congress Control Number 2001095709 Cloth ISBN -13 978-0-691-09527-1 Cloth ISBN-I0. 0-691-09527-2 British Library Cataloging-in-Publication Data is available This book has been composed in Adobe Garamond and GIll Sans Printed on acid-free paper press.princeton.edu Printed in the United States of America 10 9 8 7 CONTENTS HAROLD W KUHN PRE FAC E vii SYLVIA NASAR INTRODUCTION xi Press Release--The Royal Swedish Academy of Sciences 1 2 Autobiography 5 PHOTO ESSAY 13 Editor's Introduction to Chapter 3 29 3 JOHN MILNOR The Game of Hex 31 Editor's Introduction to Chapter 4 35 4 The Bargaining Problem 37 v Editor's Introduction to Chapters 5, 6, and 7 47 5 Equilibrium Points in n-Person Games 49 6 Non-Cooperative Games Focsimi/e of PhD. Thesis 51 7 Non-Cooperative Games 85 vi 8 Two-Person Cooperative Games 99 Editor's Introduction to Chapter 9 115 9 Parallel Control 117 10 Real Algebraic Manifolds 127 II The Imbedding Problem for Riemannian Manifolds 151 Author's Note to 'The Imbedding Problem for Riemannian Manifolds" 209 12 Continuity of Solutions of Parabolic and Elliptic Equations 211 AFTERWORD 241 SOU RC ES 243 PREFACE HAROLD W. KUHN I have known John Nash for more than fifty years. We were graduate students together in the late 1940s. And, although he went off to MIT, we have never wholly lost touch. Even today, we have offices in the Mathematics Department of Princeton University on the same floor of Fine Hall. As his friend and colleague, it has been a great pleasure to co-edit this volume with his biographer, Sylvia Nasar. In Nasar's splendid introduction to this volume, we are taken on a guided tour of Nash's scientific life, starting with his brilliant early career, the decades of mental illness that followed, and the subsequent transformation of his life by the award of the Nobel prize in economics in 1994. For me, the defining moment that divided the period when he was in the depths of his despair from his reentry into a world that he had always deserved took place on a bench in front of a minimalist Japanese fountain at the Institute for Advanced Studies in Princeton in October 1994. Nash and I had just attended a seminar given by Herbert S. Wilf on the generation of identities for hypergeometric functions. I had called Nash earlier to ask him to lunch after the seminar but had not revealed my real purpose. Several months before, I had learned that it was almost vii certain that Nash would share the Nobel in an award that would recog nize the central importance of non-cooperative games in modern eco nomic theory. Using a variety of subterfuges, photos had been taken, a curriculum vitae had been assembled, an appointment to a nominal research position at Princeton had been arranged, and various support ing materials had been sent to Stockholm. In addition, with the active support ofJackie Savani, the press officer of Princeton University, prepa rations had been made for a press conference on the following day. Three viii days earlier, I had been informed that the Social Science Class of the Swedish Academy had approved the award unanimously, and I had been given permission to tell Nash the great news. I had already told Alicia Nash to take the day off from work and sworn her to secrecy. So, as we sat on the bench, enjoying the mild fall weather and the splendor of the Institute woods, I told John that he should be up at 6:30 A.M. the following morning to receive a phone call from Carl-Olof Jacobsen, secretary general of the Nobel Foundation, who would tell him that he was sharing the Prize in Economic Sciences in Memory of Alfred Nobel. John took the news very calmly; it appeared that his son, John David Stier, had sent him an article from the Boston Globe saying that he was in contention for the prize and that the only impediment was the fear that his mental illness might lead to behavior that would embarrass the King of Sweden. He seemed more interested in the fact that the prize was split three ways and, after taxes, the net amount would not be that much. We then went home to lunch where, after many objections, Nash met Jackie Savani. Nash refused to discuss any possible questions that might be asked at the press conference the next day. The press con ference went very well, largely due to John's highly developed sense of humor, which turned aside questions that probed his private life with quiet, always logical, answers. On the morning of the announcement, he avoided reporters by coming, at my invitation, to my undergradu ate course, where we were starting a section on game theory. It was a morning that those students will long remember. Recognition is a cure for many ills; although Nash's mental illness had faded into remission in the years preceding 1994, the announce ment of the Nobel prize signaled a new period in his life. The monetary amount of the prize was nontrivial (but subject to taxes in the United States, which is a surprise to most American Nobelists). Much more important was the recognition that was so long in coming. Although he is one of the most original mathematical minds of the twentieth century, the reasons for this delay are easy to understand. His formally published work consists of about fifteen papers, five in game theory and ten in pure mathematics, produced in the main during the ten-year period from 1949 to 1959. In the past forty years, he has published very little and many people who valued his early work thought ix he was dead. The truth, as we well know, is that he was suffering from a debilitating mental illness diagnosed as schizophrenia and was living a quiet and secluded life near the academic community of Princeton, sheltered physically and emotionally by his wife, Alicia Nash. After more than twenty-five years of life isolated in the main from academic activities, Nash began to emerge from the shadows of his mental illness and a number of friends and colleagues began to provide for him the rewards that he deserved. In 1993, at the instigation of Peter Sarnak and Louis Nirenberg, I joined them to collect and edit a sort of "Festschrift," consisting of papers in the fields in which Nash had made early and important contributions. Nash's name had been put forward for many years by a number of economists who are asked to make nominations for the Nobel Memorial Prize in Economics. In 1994, their efforts bore fruit. After the Nobel prize, honors appeared from all sides. He was elected to the National Academy of Sciences; the citation reads: "Nash is best known for his work in game theory. He has also made basic and extremely important contributions which have profoundly influenced differential geometry, real algebraic geometry, and partial differential equations." He was awarded the Steele Prize for a Seminal Contribution of the American Mathematical Society; the citation reads, in part: "This is one of the great achievements in mathematical analysis in this century." He has received honorary degrees from Carnegie Mellon University and the University of Athens, and many awards from mental-health organizations that wish to recognize his exemplary recovery. In preparing this volume, Sylvia Nasar and I wish to enlarge the ex tent of this recognition by making the most important contributions of HAROLD W KUHN Preface John Nash-both in game theory and in pure mathematics-available to a wider audience. We believe that, in this form, we may also bridge the gap berween the economists, on the one hand, and the pure math ematicians, on the other, each of whom has appreciated only part of Nash's scientific contributions. This book is not a "Complete Works"; we both wish John Nash well in his current research and hope to see many more significant works from him in the future. x Acknowledgments Sylvia Nasar and I thank the wonderful and wonderfully congenial team at Princeton University Press for their patience, skill, and, above all, infectious enthusiasm. In particular, we thank Linny Schenck, for her careful shepherding of the production process; Tracy Baldwin, for giving the book a very special appearance; Grady Klein, for giving us an inspired book cover; and Vickie Kearn and Peter Dougherty, for their calm stewardship throughout. All editors should be so lucky! We are also grateful to John and Alicia Nash, Martha Legg, and John Stier for their contributions, and to Avinash Dixit for his many useful suggestions. Working with old friends and making new ones was, for us, the best part of what has been a most rewarding experience. INTRODUCTION SYLVIA NASAR W hen Freeman Dyson, the physicist, greeted John Forbes Nash, Jr. at the Institute for Advanced Study one day in the early 1990s, he hardly expected a response. A mathematics legend in his twen ties, Nash had suffered for decades from a devastating mental illness. A mute, ghost-like figure who scrawled mysterious messages on black boards and occupied himself with numerological calculations, he was known around Princeton only as "the Phantom." To Dyson's astonishment, Nash replied. He'd seen Dyson's daughter, an authority on computers, on the news, he said. "It was beautiful," recalled Dyson. "Slowly, he just somehow woke up." Nash's miraculous emergence from an illness long considered a life sentence was neither the first, nor last, surprise twist in an extraordi nary life. The eccentric West Virginian with the movie star looks and Olym pian manner burst onto the mathematical scene in 1948. A one-line In describing John Nash's contributions to economics and mathematics. I drew from essays by Avinash Dixit, John Milnor, Roger Myerson. and Ariel Rubinstein as well as from my biography, A Beautifol Mind. Avinash Dixit and Harold Kuhn kindly commented on my draft. Any errors are, of course, mine alone. xi
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