The Essential John Nash

by John F. Nash

Book Overview

When John Nash won the Nobel prize in economics in 1994, many people were surprised to learn that he was alive and well. Since then, Sylvia Nasar's celebrated biography A Beautiful Mind, the basis of... This description may be from another edition of this product.

• Format: Hardcover
• Language: English
• ISBN: 0691095272
• ISBN13: 9780691095271
• Release Date: December 2001
• Publisher: Princeton University Press
• Length: 272 Pages
• Weight: 1.15 lbs.
• Dimensions: 0.9" x 6.0" x 9.0"

Customer Reviews

The Nash equilibrium

This is a faithful publication of the original John Nash Econometrica articles on game theory, as well as contributions by Harold Kuhn, Sylvia Nasar, and others that review Nash's work and his struggle with schizophrenia. It has a new retrospective introduction by John Nash well worth reading for his insights.

excellent

Personally, I found this book to be very interestring. The proofs and ideas are presented in clear and non-rigomorphic fashion. One is able to read the works of Nash in the way he himself presented them, and hopefully appropriate some mental strategies used by this genius. There is much that goes on behind the scene of creation of proofs. I think mathematicians of today would greatly benefit from availability of larger number of books which would contain the mathematical works in the way they were originally presented. This is certainly a major step in that direction.

A Most Welcome Mathematical Banquet

I can't begin to express how deeply satisfying it was to peruse these papers by John Nash. You almost felt you were right there at his side, as he penned them.There is even something in the book for non-mathematical types: Sylvia Nasar's Introduction and the autobiographical essay (Chapter Two). But for me the greatest interest resided in the remaining chapters: 4-11.Of these, I particularly enjoyed reading the original presentation of Nash's Thesis on 'Non-Cooperative Games' (Chapter 6), and was fascinated not only with the air-tight logic of his proofs, but the use of hand written-in symbols.Of course, Chapter 7 is just the re-hashing of Ch. 6, but in proper type-set form, rather than Nash's original script. But - give me the former any day! Reading the original form and format almost made me feel like Nash's Thesis aupervisor, including the same excitement of a new discovery!Chapter 8 'Two person Cooperative Games' nicely extends the mathematical basis to cover this species of interaction.(And in many ways, people will find the cooperative game model easier to understand than the non-cooperative).Chapter 9 is important because it delves into the issue of parallel control, and logical functions such as used in high speed digital computers. This chapter was of much interest to me since particular aspects of parallel control figured in my own model of consciousness - recently presented in Chapter Five of my book, 'The Atheist's Handbook to Modern Materialism'. Astute readers who read both books will quickly see the analog between the Schematic of Logical Unit Function (p. 122) and my own Figure 5-13 ('Development of Neural Assemblies', p. 156). I enjoyed Chapter 10, 'Real Algebraic Manifolds' because of my ongoing interest in Algebraic Topology, and especially homology and homotopy theory. In his chapter, Nash presents a cornucopia of methods for representation, which I am still playing with for different manifolds. Chapter 11, 'The Imbedding Problem for Riemannian Manifolds', is a delight for anyone familiar with Einstein's General Relativity, or even differential geometry. When you read through this chapter, you also will understand why Nash is still very interested (and involved) in research to do with general relativity and cosmology. Particularly fun for me was his section on 'Smoothing of Tensors' (p. 163) and 'Derivative Size Concept for Tensors' (p. 164).Chapter 12, 'Continuity of Solutions of Parabolic and Elliptic Equations' is like 'dessert' for anyone who is intensely interested (as I am) in modular functions, which themselves are related intimately to elliptic equations.In short, I think this book has something for both mathematicians and non-math types alike. Obviously, the former are likely to get more out of it, so the question the latter group must ask is whether the purchase is worth satiating their curiosity about Nash.I know how I would answer, even if I couldn't tell a derivative from a differential. However, this book can be

An excellent compilation

Having written about the life of the mathematician John Nash in the excellent biography "A Beautiful Mind", Sylvia Nasar teams up with the mathematician Harold W. Kuhn to produce a book that introduces the mathematical contributions of Nash, something that was done only from a "popular" point of view in Nasar's biography. For those who have the background, this book is a fine overview of just what won Nash acclaim in the mathematical community, and won him a Nobel Prize in economics. It is always easy to dismiss ideas as trivial after they have been discovered and have been put into print. This is apparently what John von Neumann did after discussing with Nash his ideas on noncooperative games, dismissing his ideas as a mere "fixed point theorem". At the time of course, the only game-theoretic ideas that had any influence were those of von Neumann and his collaborator, the Princeton economist Oskar Morgenstern. The rejection of ideas by those whose who hold different ones is not uncommon in science and mathematics, and, from von Neumann's point of view at the time, he did not have the advantage that we do of examining the impact that Nash's ideas would have on economics and many other fields of endeavor. Therefore, von Neumann was somewhat justified, although not by a large measure, in dismissing what Nash was proposing. Nash's thesis was relatively short compared to the size on the average of Phd theses, but it has been applied to many areas, a lot of these listed in this book, and others that are not, such as QoS provisioning in telecommunication and packet networks. The thesis is very readable, and employs a few ideas from algebraic topology, such as the Brouwer fixed point theorem. The paper on real algebraic manifolds though is more formidable, and will require a solid background in differential geometry and algebraic geometry. However, from a modern point of view the paper is very readable, and is far from the sheaf and scheme-theoretic points of view that now dominate algebraic geometry. It is interesting that Nash was able to prove what he did with the concepts he used. The result could be characterized loosely as a representation theory employing algebraic analytic functions. These functions are defined on a closed analytic manifold and serve as well-behaved imbedding functions for the manifold, which is itself analytic and closed. These manifolds have been called 'Nash manifolds' in the literature, and have been studied extensively by a number of mathematicians. I first heard about John Nash by taking a course in algebraic topology and characteristic classes in graduate school. The instructor was discussing the imbedding problem for Riemannian manifolds, and mentioned that Nash was responsible for one of the major results in this area. His contribution is included in this book, and is the longest chapter therein. Here again, the language and flow of Nash's proof is very understandable. This is another example of the difference in the

Fascinating Reading

Even without the Nobel Prize for Economics, the outstanding movie by Ron Howard ("A Beautiful Mind"), or the exceptional biography by Sylvia Nasar (also "A Beautiful Mind"), Professor John Nash would a legend. While cursed with severe mental illness, Dr. Nash was and is an extraordinary man. His contributions to game theory were so ahead of their time it took over 30 years for economists and business leaders to apply them fully. When they were applied, they advanced everything from international trade talks and arms control treaties, to radio frequency auctions and the study of evolutionary biology. Dr. Nash's work has had a profound, highly practical impact on negotiation and decision making throughout business and government. He created a path toward win-win solutions to complex, multi-party agreements.This book is largely a collection of Dr. Nash's own writings, each a significant contribution to mathematics or economics. Nash's papers are thoughtfully introduced and explained - thankfully so given the complexity of Nash's writings. Also included is Nash's own touching and revealing autobiography. The result is a compelling glimpse inside the thought processes of a genius - a beautiful mind indeed. Thanks to Harold Kuhn and Sylvia Nasar for pulling this wonderful collection together.