**John Forbes Nash, Jr.**
Hilary Greenberg
Math 89s: Game Theory and Democracy
Professor Bray
03 December 2013
**Summary**
On June 13^{th}, 1928, John Forbes Nash, Jr. was born in Bluefield, West Virginia, a town located directly on the line between the bustling city of Chicago and Norfolk, Virginia. His father, John Forbes Nash, Sr. worked as an electrical engineer for the Appalachian Electric Power Company, and thus, the young Nash, often called Johnny, developed a deep curiosity for various mechanisms and technological inventions at a young age. Throughout his childhood, Nash treated activities that his peers found enjoyable and fascinating, such as sports and dances, as tedious distractions from his books and experiments. In grade school, Johnny’s teachers took note of his immaturity and social awkwardness, but failed to recognize any special intellectual gifts that he possessed. Around the age of thirteen or fourteen, the young Nash read E.T. Bell’s *Men of Mathematics*, an extraordinary book containing detailed biographical sketches of mathematicians, and found himself to be particularly attracted to Bell’s assurance that there were still many challenging problems that even the smartest mathematicians had not yet solved, and that he, an amateur, could explore extensively.
He entered Carnegie Tech. in Pittsburgh with the intention of studying chemical engineering, but with enough math credits to skip most entry-level courses, his professors encouraged him that mathematics would be a promising field. Nash was offered fellowships to enter as a graduate student at both Harvard and Princeton, and eventually decided that Princeton would be where he would continue his studies. After being exposed to economic ideas and problems through an elective course in International Economics at Carnegie Tech. and his thorough studies of the works of John von Neumann and Oskar Morgenstern, he developed his deep interest in game theory studies and focused on this as a graduate student. Nash was specifically fascinated by games in which there were several players, who could all potentially be winners if they cooperated, but who were not aware of all facts related to their situation. In his 28-page dissertation on non-cooperative games, he includes the definition and properties of equilibrium for competitive games, which was later referred to as the Nash equilibrium. In game theory, the Nash equilibrium is the solution of a game in which opposing players know the strategies of the other players and no one can gain an advantage by changing his or her own strategy. Initially, the implications of the Nash equilibrium were largely overlooked, except by the military, and this lack of recognition for his conclusions about non-cooperative games led him to move on to other areas of research. Over the next decade, he would travel between many famous academic institutions, where he boldly attacked a wide range of problems, from pure number theory to cosmology.
At the age of 30, after being named one of the brightest mathematicians in the world by Fortune magazine, Nash experienced the beginning of his mental disturbances. He started to hear alien voices and declared an urgent need to perform bizarre tasks. After months of trying to cover up for his strange behavior, Nash’s wife, Alicia, had him involuntarily hospitalized at a private psychiatric institution outside of Boston, where he was diagnosed with paranoid schizophrenia. For the next 30 years, he would experience a series of temporary recoveries along with a series of lapses, which would force him back into various mental institutions.
While the exact cause of his schizophrenia still remains a mystery, equally puzzling is the remarkable recovery that he made in the 1980’s. When he returned to the sane world, his ideas regarding game theory had become widely accepted by economists and he received a series of honors, including the John von Neumann Theory Prize for his conclusions about non-cooperative equilibria and a Nobel Prize in Economics, the most prestigious of all accolades. In 2012, Nash became a fellow of the American Mathematical Society, an impressive association of professional mathematicians. He has suggested hypotheses on mental illness, having faced one himself, and has developed work on the role of money in society. Covering a wide range of topics, John Forbes Nash, Jr. has proved himself to be a strong contributor to many areas of study and it will be interesting to see if his academic achievements will have further applications in the future.
**John Forbes Nash, Jr.**
John Forbes Nash, Jr. was a celebrated American mathematician who provided numerous contributions to the fields of game theory, differential geometry, and partial differential equations. Nash’s theories have provided insight into the forces that guide many aspects of daily life, including, but not limited to, market economics, politics, military theory, and evolutionary biology. Diagnosed with paranoid schizophrenia at the age of 30, John Nash is the subject of A Beautiful Mind, a book by Sylvia Nasar, which was later turned into a Hollywood film under the same title. The biographical text and movie focus on both Nash’s mathematical genius and his mental disturbance. Among other great accomplishments, Nash has been recognized globally for his discovery of non-cooperative equilibria, now called Nash equilibria, and in 1994, he received the Nobel Prize in Economics for his work in game theory at Princeton University.
On June 13^{th}, 1928, John Forbes Nash, Jr. was born in Bluefield, West Virginia, a town located directly on the line between the bustling city of Chicago and Norfolk, Virginia. The coalfields surrounding Bluefield once provided nearly all of the work in this remote little town, but due to its prime location, the town gradually became an important rail hub and attracted a prosperous white-collar class of ministers, managers, lawyers, and businessmen. Although Bluefield did not have a true “community of scholars,” which was apparent in the “intellectual hothouses of Budapest and Cambridge,” which produced John von Neumann, Norbert Wiener, and many other mathematical geniuses, the town had a sizeable group of men with talents in engineering and other scientific fields, attracted by the railroad, utility, and mining companies of the area (Nasar). His father, John Forbes Nash, Sr. worked as an electrical engineer for the Appalachian Electric Power Company, and thus, the young Nash, often called Johnny, developed a deep curiosity for various mechanisms and technological inventions at a young age. Throughout his childhood, Nash treated activities that his peers found enjoyable and fascinating, such as sports, dances, and church, as tedious distractions from his books and experiments. While he was not often found hanging around with companions for recreational purposes, he enjoyed serving as an example for risky scientific experiments in front of his peers. On one occasion, he held onto a large magnet, wired with electricity, to show how much current he could endure without even flinching. One of his classmates recalled one instance in which Nash was found constructing an elaborate system of mousetraps, after disappearing for hours from his post at Bluefield Supply and Superior Sterling (Nasar). Johnny’s parents and sister, Martha, constantly found themselves seeking out ways to make him a more “well-rounded” individual, but his lack of interest in childish hobbies, along with their efforts to change his nature only resulted in his withdrawal into his own world of solitude.
Unlike many other famous mathematicians, there is no evidence of notable mathematical strength in the Nash household. A born teacher, his mother, Virginia, received a degree from Martha Washington College and later at West Virginia University, where she focused on her studies in English and Latin. His father, John Nash, Sr., was not particularly knowledgeable in the concepts of abstract mathematics, even though his focus was in contemporary developments in science and technology, fields that often overlap with advanced mathematics. In grade school, Johnny’s teachers took note of his immaturity and social awkwardness, but failed to recognize any special intellectual gifts that he possessed. Ironically, the first hint of Johnny’s mathematical strength was when he “received a B- in fourth-grade arithmetic” (Nasar). His teacher reported that he constantly struggled to complete his works as his peers did, but it was obvious to his mother that he had merely found his own ways to solve problems. Later in his life as a student, Nash succeeded in showing his high school mathematics teacher that a proof could be completed in just two or three elegant steps, after she had struggled to produce this same proof in a laborious manner.
Around the age of thirteen or fourteen, Nash read E.T. Bell’s *Men of Mathematics*, an extraordinary book containing detailed biographical sketches of mathematicians who provided numerous contributions to many different aspects of the academic field. Bell’s book allowed him to explore true mathematics, which was, to him, “a heady realm of symbols and mysteries entirely unconnected to the seemingly arbitrary and dull rules of arithmetic and geometry taught in school” (Nasar). Nash was particularly attracted to Bell’s assurance that there were still many challenging problems that even the smartest mathematicians had not yet solved, and that he, an amateur, could explore extensively. Recalling his time reading as a high school student, Nash claims, “I remember succeeding in proving the classic Fermat theorem about an integer multiplied by itself p times where p is a prime” (Nash, Nobel Prize Biographical). This discovery of the thrill of mathematics ultimately led him to choose this field for his course of study. He entered Carnegie Tech. in Pittsburgh with the intention of studying chemical engineering, and possibly following in his father’s footsteps, but with enough math credits to skip most entry-level courses, his professors encouraged him that he was included in the group for which mathematics was a realistic choice for a profession.
Nash was offered fellowships to enter as a graduate student at both Harvard and Princeton, but because of Princeton’s favorable geographic location, the generosity of the fellowship, and the encouragement from professors there, he decided that this would be where he would continue his studies. After being exposed to economic ideas and problems through an elective course in International Economics at Carnegie Tech., he developed an idea, which led to the paper “The Bargaining Problem” (Nash, Nobel Prize Biographical). Further, this idea, along with his studies of the works of John von Neumann and Oskar Morgenstern, led to his interest in game theory studies as a graduate student at Princeton. In Fine Hall, the home of Princeton’s mathematics department, some of the finest thinkers of the age would gather each day for afternoon tea at half past three (Singh). In 1948, upon arriving at Princeton, Nash began gathering with this group and took a particular liking to the board games played at teatime, including chess, backgammon, and go. Fascinated by von Neumann’s research of games and ways to devise winning strategies, Nash soon invented a two-person zero-sum game with perfect information. This type of game involves both a winner and a loser, as indicated by “zero-sum,” and full awareness of all states of play by each player, as indicated by “perfect information” (Singh). Nash was specifically interested in games in which there were several players who could all potentially be winners if they cooperated, but who were not aware of all facts related to their situation. For example, he thoroughly examined the Prisoner's Dilemma, which is as follows:
“The police arrest two suspects, who are in fact both guilty, and question them in separate cells. If they are loyal to each other and refuse to turn state's evidence, then they will both be sentenced to one year in jail. But if one betrays the other, then he is immediately released and his partner is sentenced to five years. Finally, if the prisoners betray each other, then they are both sentenced to three years. The dilemma is this: Imagine you are one of the prisoners – what would you do, remain loyal or betray your partner?” (Singh).
The Prisoner’s Dilemma brings up the idea that remaining loyal would be most beneficial for both partners, as this would result in only one year of prison for both. However, due to the lack of awareness of the other partner’s decisions, remaining loyal could also result in five years of prison, whereas if both parties betray each other, they each only face three years in prison. The question ultimately comes down to how parties can cooperate to achieve a mutually beneficial outcome when betrayal is such a tempting strategy.
In his 28-page dissertation on non-cooperative games, he includes the definition and properties of equilibrium for competitive games, which was later referred to as the Nash equilibrium. In game theory, the Nash equilibrium is the solution of the game in which opposing players know the strategies of the other players and no one can gain an advantage by changing his or her own strategy. A group of players are in Nash equilibrium if each one is making the best decision possible, taking into account the decisions of the others in the game. Determining a specific solution provides a way of predicting what will happen if several opposing groups are making decisions at the same time, and if their ultimate outcomes depend on the decisions of other players.
Initially, the implications of the Nash equilibrium were largely overlooked, except by the military, which used it to develop cold war strategies. In theory, if superpowers could find a way to cooperate, as demonstrated in the Prisoner’s Dilemma situation, the vast expenditures required for war would be greatly reduced on both sides. Lack of recognition from economists for his research and conclusions about non-cooperative games led him to move on to other areas of research. Over the next decade he would travel between Princeton and MIT, where he boldly attacked a wide range of problems, from pure number theory to cosmology. With a limited knowledge of physics, “he developed a theory describing how gravity fields could generate a form of friction on photons and tried to convince Einstein of his ideas” (Singh). While theory was flawed, the fact that he dared to challenge Einstein on his own area of study is verification of Nash’s astonishing audacity.
At the age of 30, after being named one of the brightest mathematicians in the world by Fortune magazine in July 1958, Nash experienced the beginning of his mental disturbances. Struck suddenly with schizophrenia, Nash accused a fellow mathematician at Princeton of breaking into his office and stealing his ideas. He began to hear alien voices and declared an urgent need to establish a single world government. Furthermore, he turned down a prestigious chair position at the University of Chicago because he said he was “planning to become the Emperor of Antarctica” (Singh). At Princeton, while Nash was widely recognized for his contributions to mathematics, he also became the “Phantom of Fine Hall,” for it became apparent that he would scribble mysterious equations on the blackboards of classrooms in the middle of the night. After months of trying to cover up for his bizarre behavior, Nash’s wife, Alicia, and his colleagues at MIT had him involuntarily hospitalized at McLean Hospital, a private psychiatric institution outside of Boston. After his release, Nash resigned abruptly from MIT, withdrew his pension, and fled to Europe, where he intended to renounce his U.S. citizenship, until his wife had him deported back to the United States (PBS People & Events). For the next 30 years, he would experience a series of temporary recoveries along with a series of lapses, which would force him back into various mental institutions. In 1961, he endured insulin-coma therapy for a month and a half, five days per week. During his sessions, he was injected with insulin, and the theory behind it was that starving the brain of sugar would kill marginally functioning brain cells. Nash described his treatment as pure “torture” (Singh). The alternative treatment was electroshock therapy, but given that there was a risk of numbing his brain and potentially losing his inner genius, Nash and his family members refused this method. While the exact cause of his schizophrenia still remains a mystery, equally puzzling is the remarkable recovery that he made in the 1980’s.
During his absence from the sane world, his ideas regarding game theory had become widely accepted by economists and were beginning to be implemented as well. Upon returning to his academic realm, he received a series of honors, including the John von Neumann Theory Prize for his conclusions about non-cooperative equilibria and a Nobel Prize in Economics, the most prestigious of all accolades. In 2012, Nash became a fellow of the American Mathematical Society, an impressive association of professional mathematicians. He has suggested hypotheses on mental illness, having faced one himself, and has developed work on the role of money in society. Covering a wide range of topics, John Forbes Nash, Jr. has proved himself to be a strong contributor to many areas of study and it will be interesting to see if his academic achievements will have further applications in the future.
**Works cited**
Singh, Simon. "Between Genius and Madness." The New York Times, 14 June 1998. .
"John F. Nash Jr. - Biographical." *The Nobel Prize.* The Nobel Foundation, 1994. Web. .
Nasar, Sylvia. *A Beautiful Mind: A Biography of John Forbes Nash, Jr.* New York, NY: Simon & Schuster, 1998. Print.
"John F. Nash Jr." *Library of Economics and Liberty*. N.p., 2008. Web. .
"People & Events: John Nash (1928 - )." PBS, 1999. Web. .
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