john nash game theory dissertation

John Nash and his contribution to Game Theory and Economics

Distinguished Research Professor of Economics, University of Technology Sydney

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A two-page paper published by John Nash in 1950 is a seminal contribution to the field of Game Theory and of our general understanding of strategic decision-making. That paper, “ Equilibrium points in N-person games ”, introduced a cornerstone concept which came to be known as Nash equilibrium .

Game theory is concerned with situations where decisions interact – where the “payoff” or reward for a decision maker depends not only on his or her own decision but also on the decisions of others.

Such situations are pervasive in real life. The payoff for a buyer in an auction, for example, depends not only on the amount he bids but also on the bids of the other buyers. If the buyer’s bid is not the highest, then he loses the auction. Likewise, the profit realised by a firm depends not only on the price it sets for its product but also on the prices set by its competitors. In a tennis match, the likelihood the server will win a point depends on whether she delivers the serve to the receiver’s left or right and whether the receiver correctly anticipates it.

Auctions, price setting and tennis are all examples of “non-cooperative” strategic interactions that mathematicians and economists refer to as “games”. They are non-cooperative because decision makers take their actions independently and are unable to enter into binding agreements with others regarding their actions, either because such agreements are illegal (when setting prices) or because they have no incentive to do so (as in tennis).

The notion of Nash equilibrium, developed in Nash’s 1950 paper, is the basis of how economists predict the outcome of strategic interactions.

Informally, a Nash equilibrium is a list of actions, one for each decision maker, such that each decision maker’s action is best for him, given the actions of the others. Such a list of actions is an equilibrium (or stable point), since no decision maker has an incentive to change his action.

Consider a driver approaching an intersection. She stops when she approaches a red light and she continues without concern when she approaches a green light. It is a Nash equilibrium when all drivers behave this way. When approaching a red light it is best to stop since the crossing traffic has a green light and will continue. When approaching a green light it is best to continue since the crossing traffic has a red light and will stop. Thus it is in each driver’s own interest to play her part in the equilibrium, given that everyone else does. No traffic cop is required.

Nash equilibrium also allows for the possibility that decision makers follow randomised strategies. Allowing for randomisation is important for the mathematics of game theory because it guarantees that every (finite) game has a Nash equilibrium.

Randomisation is also important in practice in commonly played games such as Two-up, Rock-Paper-Scissors, poker and tennis. We all know from our own experience how to play Rock-Paper-Scissors against a sophisticated opponent: play each action with equal probability, independently of the actions and outcomes in past plays. Indeed, this is exactly what Nash equilibrium predicts. Nash’s theory applies to any game with any number of decision makers, whereas John von Neumann’s 1928 Minimax Theorem applies only to “zero-sum” games with two players.

Interestingly, data collected from championship tennis matches has shown that the serve-and-return behaviour of professional players is consistent with both von Neumann’s Minimax Theory and Nash’s generalisation.

Nash’s work dealt with games in which each decision maker takes his or her action without knowing the actions taken by others, and in which no decision-maker has private information. John Harsanyi extended the notion of Nash equilibrium to deal with strategic interactions, such as in auctions, in which decision-makers have private information. (In an auction, buyers know the value they place on the item being sold, but they don’t know how other buyers value the item.)

Reinhardt Selten extended the notion of Nash equilibrium to deal with dynamic interactions, in which one or more decision-makers observe the action of another before taking their own action. In 1994, Nash shared the Nobel Prize in Economics with Harsanyi and Selten for these contributions.

While Nash is best known for his contribution to non-cooperative game theory, he also made a seminal contribution to cooperative game theory with the development of the Nash bargaining solution.

Nash’s work has had a profound impact in economics. Knowledge of game theory is essential training for every professional economist and it is a common – and popular – subject for undergraduate students as well. Nash’s work not only revolutionised modern economics, it has also had an important impact in fields as diverse as computer science, political science, sociology and biology.

His work on game theory won him a Nobel Prize for economics in 1994 and he just recently received Norway’s Abel prize for mathematics.

John Nash remained active at scientific conferences around the world. He was happy to talk with students, many of whom wanted a picture with him too. He and his wife, Alicia were both killed in a car accident on the New Jersey Turnpike on Saturday, May 24 on their way home from receiving the prize.

Both were kind people and they will be missed.

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Year 91 – 1951: “Non-cooperative Games” by John Nash, in: Annals of Mathematics 54 (2)

Posted on April 7, 2011 in All years , Uncategorized

Over the past 60 years, game theory has been one of the most influential theories in the social sciences, pervasive in economics, political science, business administration, and military strategy – the disciplines most consulted by the powers-that-be for “real-world,” high-stakes decisions. But just as there would be no semiconductors or (God forbid) laser pointers if not for the abstruse mathematics of quantum theory, game theory can be traced back to theoretical work by academic mathematicians. In a set of papers in the 1950s, mathematician John Forbes Nash set forth breakthrough ideas that helped transform game theory from an ivory tower abstraction into an indispensable analytical tool used by strategists from Wall Street to the Pentagon.

The foundational game theory work of mathematician John von Neumann and economist Oskar Morgenstern, published in 1944, provided a framework for solutions to zero-sum games, where one player’s win was the other’s loss. Nash, in his dissertation research at Princeton (published in this and three other papers), extended game theory to n -person games in which more than one party can gain, a better reflection of practical situations. Nash demonstrated that “a finite non-cooperative game always has at least one equilibrium point” or stable solution. This result came to be called the “Nash equilibrium,” a situation where no one player can get a better payoff by changing strategies, so long as other players also keep their strategies. Using Nash’s framework, predictions can be made about the outcomes of strategic interactions.

Based on Nash’s advances, game theory developed into one of the pre-eminent tools of economics in the second half of the 20th century. In recognition of his breakthrough work, Nash was joint recipient of the Nobel Prize for Economics in 1994 for “pioneering analysis of equilibria in the theory of non-cooperative games.”

If visions of Russell Crowe have danced in your head while you’ve been reading this post, that’s probably because you remember that Crowe played John Forbes Nash in the 2001 film A Beautiful Mind (at least we hope that’s why). Part of the movie takes place at MIT, portraying Nash’s years as an instructor in mathematics at the Institute, where he worked from 1951 to 1959, until mental illness curtailed his mathematical career.

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John Nash , in full John Forbes Nash, Jr. , (born June 13, 1928, Bluefield , West Virginia , U.S.—died May 23, 2015, near Monroe Township, New Jersey), American mathematician who was awarded the 1994 Nobel Prize for Economics for his landmark work, first begun in the 1950s, on the mathematics of game theory . He shared the prize with John C. Harsanyi and Reinhard Selten . In 2015 Nash won (with Louis Nirenberg ) the Abel Prize for his contributions to the study of partial differential equations .

Nash enrolled in chemical engineering at the Carnegie Institute of Technology (later Carnegie Mellon University ) in Pittsburgh before he switched to chemistry and then to mathematics, in which he finally received both bachelor’s and master’s degrees in 1948. Two years later, at age 22, he completed a doctorate at Princeton University . In 1951 he joined the faculty of the Massachusetts Institute of Technology (MIT), where he pursued research into partial differential equations. He resigned in the late 1950s after bouts of mental illness . He then began an informal association with Princeton, where he became a senior research mathematician in 1995.

While he was still in graduate school, Nash published (April 1950) his first paper, “The Bargaining Problem,” in the journal Econometrica . He expanded on his mathematical model for bargaining in his influential doctoral thesis, “ Non-Cooperative Games,” which appeared in September 1951 in the journal Annals of Mathematics . Nash thus established the mathematical principles of game theory, a branch of mathematics that examines the rivalries between competitors with mixed interests. Nash showed that for any finite game, all the players can arrive at an optimal outcome, known as the Nash equilibrium or the Nash solution , when considering the possible actions of the other players. Despite its practical limitations, the Nash equilibrium was widely applied by business strategists.

Nash’s research into differential equations at MIT led to his seminal paper “Real Algebraic Manifolds,” which was published in Annals of Mathematics in November 1952. His other influential work in mathematics included the Nash-Moser inverse function theorem, the Nash–De Giorgi theorem (a solution to David Hilbert ’s 19th problem, which Nash undertook at the suggestion of Nirenberg), and the Nash embedding (or imbedding) theorems, which the Norwegian Academy of Science and Letters described as “among the most original results in geometric analysis of the twentieth century”; the academy awarded Nash the Abel Prize. His other honours included the John von Neumann Theory Prize (1978) and the American Mathematical Society’s Leroy P. Steele Prize for a Seminal Contribution to Research (1999).

Nash’s research into game theory and his long struggle with paranoid schizophrenia became well known to the general public because of the Academy Award -winning motion picture A Beautiful Mind (2001), which was based on Sylvia Nasar’s 1998 biography of the same name. A more factually accurate exploration of Nash’s struggle with mental illness was offered by the public television documentary A Brilliant Madness (2002).

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Read John Nash’s Super Short PhD Thesis with 26 Pages & 2 Citations: The Beauty of Inventing a Field

in Math | June 1st, 2015 1 Comment

Last week John Nash , the Nobel Prize-winning mathematician, and subject of the blockbuster film A Beautiful Mind , passed away at the age of 86. He died in a taxi cab accident in New Jersey.

Days later, Cliff Pickover highlighted a curious factoid: When Nash wrote his Ph.D. thesis in 1950, “Non Cooperative Games” at Princeton University, the dissertation (you can read it online here) was brief. It ran only 26 pages. And more particularly, it was light on citations. Nash’s diss cited two texts: One was written by John von Neumann & Oskar Morgenstern, whose book, Theory of Games and Economic Behavior (1944), essentially created game theory and revolutionized the field of economics; the other cited text, “Equilibrium Points in n-Person Games,” was an article written by Nash himself. And it laid the foundation for his dissertation, another seminal work in the development of game theory, for which Nash won the Nobel Prize in Economic Sciences in 1994 .

The reward of inventing a new field, I guess, is having a slim bibliography.

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John F. Nash, Jr.

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John Nash Jr., a legendary fixture of Princeton University’s Department of Mathematics renowned for his breakthrough work in mathematics and game theory as well as for his struggle with mental illness, died with his wife, Alicia, in an automobile accident May 23 in Monroe Township, New Jersey. He was 86, she was 82.

During the nearly 70 years that Nash was associated with the University, he was an ingenious doctoral student; a specter in Princeton’s Fine Hall whose brilliant academic career had been curtailed by his struggle with schizophrenia; then, finally, a quiet, courteous elder statesman of mathematics who still came to work every day and in the past 20 years had begun receiving the recognition many felt he long deserved. He had held the position of senior research mathematician at Princeton since 1995.

Nash was a private person who also had a strikingly public profile, especially for a mathematician. His life was dramatized in the 2001 film “A Beautiful Mind” in which he and Alicia Nash were portrayed by actors Russell Crowe and Jennifer Connelly. The film centered on his influential work in game theory, which was the subject of his 1950 Princeton doctoral thesis and the work for which he received the 1994 Nobel Prize in economics.

At heart, however, Nash was a devoted mathematician whose ability to see old problems from a new perspective resulted in some of his most astounding and influential work, friends and colleagues said.

At the time of their deaths, the Nashes were returning home from Oslo, Norway, where John had received the 2015 Abel Prize from the Norwegian Academy of Science and Letters, one of the most prestigious honors in mathematics. The prize recognized his seminal work in partial differential equations, which are used to describe the basic laws of scientific phenomena. For his fellow mathematicians, the Abel Prize was a long-overdue acknowledgment of his contributions to mathematics.

For Nash to receive his field’s highest honor only days before his death marked a final turn of the cycle of astounding achievement and jarring tragedy that seemed to characterize his life. “It was a tragic end to a very tragic life. Tragic, but at the same time a meaningful life,” said Sergiu Klainerman, Princeton’s Eugene Higgins Professor of Mathematics, who was close to John and Alicia Nash, and whose own work focuses on partial differential equation analysis.

“We all miss him,” Klainerman said. “It was not just the legend behind him. He was a very, very nice person to have around. He was very kind, very thoughtful, very considerate and humble. All that contributed to his legacy in the department. The fact that he was always present in the department, I think that by itself was very moving. It’s an example that stimulated people, especially students. He was an inspiring figure to have around, just being there and showing his dedication to mathematics.”

Princeton President Christopher L. Eisgruber said Sunday that the University community was “stunned and saddened by news of the untimely passing of John Nash and his wife and great champion, Alicia.”

“Both of them were very special members of the Princeton University community,” Eisgruber said. “John’s remarkable achievements inspired generations of mathematicians, economists and scientists who were influenced by his brilliant, groundbreaking work in game theory, and the story of his life with Alicia moved millions of readers and moviegoers who marveled at their courage in the face of daunting challenges.”

Although Nash did not teach or formally take on students, his continuous presence in the department over the past several decades, coupled with the almost epic triumphs and trials of his life, earned him respect and admiration, said David Gabai, the Hughes-Rogers Professor of Mathematics and department chair.

“John Nash, with his long history of achievements and his incredible battle with mental health problems, was hugely inspirational,” Gabai said. “It’s a huge loss not to have him around anymore.”

Gabai said the Nashes regularly attended department events such as receptions, special teas, and special dinners, and they also were very supportive of undergraduate education and regularly attended undergraduate events. Gabai, who was with the couple in Norway when John received the Abel Prize, likened their deaths to the department losing two family members.

Even in the 1970s when Nash, still struggling with mental illness, was an elusive presence known as the “Phantom of Fine Hall,” his reputation for bravely original thinking motivated aspiring mathematicians, said Gabai, who was a Princeton graduate student at the time. Nash’s creativity helped preserve the department’s emphasis on risk-taking and exploration, he said.

“In those days, he was very present, but rarely said anything and just wandered benignly through Fine Hall. Nevertheless, we all knew that the mathematics he did was really spectacular,” Gabai said. “It went beyond proving great results. He had a profound originality as if he somehow had insights into developing problems that no one had even thought about.

“I think he prided himself that he had his way of thinking about things,” Gabai continued. “He was such an extraordinary exemplar of the things that this department strives for. Beyond great originality, he demonstrated tremendous tenacity, courage and fearlessness.”

Since winning the Nobel Prize , Nash had entered a long period of renewed activity and confidence — which coincided with Nash’s greater control of his mental state — that allowed him to again put his creativity to work, Klainerman said. He met Nash upon joining the Princeton faculty in 1987, but his doctoral thesis had made use of a revolutionary method introduced by Nash in connection to the Nash embedding theorems, which the Norwegian Academy described as “among the most original results in geometric analysis of the twentieth century.”

“When he got the Nobel Prize, there was this incredible transformation,” Klainerman said. “Prior to that we didn’t realize he was becoming normal again. It was a very slow process. But after the prize he was like a different person. He was much more confident in himself.”

During their frequent talks in recent years, Nash would offer unique perspectives on numerous topics spanning mathematics and current events, Klainerman said. “Even though his mind wasn’t functioning as it did in his youth, you could tell that he had an interesting point of view on everything. He was always looking for a different angle than everybody else. He always had something interesting to say.”

Nash’s quick and distinctive mind still shone in his later years, said Michail Rassias, a visiting postdoctoral research associate in mathematics at Princeton who was working with Nash on the upcoming book, “Open Problems in Mathematics.” He and Nash had just finished the preface of their book before Nash left for Oslo. They agreed upon a quote from Albert Einstein that resonated with Nash (although Nash pointed out that Einstein was a physicist, not a mathematician, Rassias said): “Learn from yesterday, live for today, hope for tomorrow. The important thing is not to stop questioning.”

“Even at 86, his mind was still open,” Rassias said. “He still wanted to have new ideas. Of course, he couldn’t work like when he was 20, but he still had this spark, the soul of a young mathematician. The fact that he moved slowly and talked with a quiet voice had nothing to do with the enthusiasm with which he did mathematics. It was very inspirational.”

Sixty years younger than Nash, Rassias said his work with Nash began with a conversation in the Fine Hall commons room in September.

“I could tell there was mathematical chemistry between us and that led to this intense collaboration. He was very simple, very open to discussing ideas with new people if you said something that attracted his interest,” Rassias said. “Nash gave this impression that he was distant, but when you actually had the opportunity to talk to him he was not like that. He tended to walk alone, but if you got the courage to talk to him it would be very natural for him to talk to you.”

Rassias has been inspired by the enthusiasm and willingness with which a person of Nash’s stature dedicated months of his time to working with a young mathematician. It was an example Rassias hopes to emulate during his own career.

“Remembering what John Nash did for me, I will definitely try to give all my heart and soul to younger people in all steps of their careers,” Rassias said. “I also will try to keep my mind and enthusiasm for math alive to the end. That is something I will try to achieve like him.”

Born in Bluefield, West Virginia, in 1928, Nash received his doctorate in mathematics from Princeton in 1950 and his graduate and bachelor’s degrees from Carnegie Institute of Technology (now Carnegie Mellon University) in 1948.

His honors included the American Mathematical Society’s 1999 Leroy P. Steele Prize for Seminal Contribution to Research and the 1978 John von Neumann Theory Prize. Nash held membership in the National Academy of Sciences and in 2012 was an inaugural fellow of the American Mathematical Society.

Nash is survived by his sister, Martha Nash Legg, and sons John David Stier and John Charles Martin Nash. He had his younger son, John Nash, with Alicia shortly after their marriage in 1957, which ended in divorce in 1963. They remarried in 2001.

Despite their divorce, Alicia, who was born in El Salvador in 1933, endured the peaks and troughs of Nash’s life alongside him, Klainerman said. Their deaths at the same time after such a long life together of highs and lows seemed literary in its tragedy and romance, he said.

“They were a wonderful couple,” Klainerman said. “You could see that she cared very much about him, and she was protective of him. You could see that she cared a lot about his image and the way he felt. I felt it was very moving.

“Coming home from Oslo, he must have been extremely happy, and she must have been extremely happy for him,” he continued. “They went for the apotheosis of his career, and died in this terrible way on the way back. But they were together.”

-By Morgan Kelley, Princeton University Office of Communications

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Nash, John (1928-2015) - Princeton University Library
Noted mathematician John Nash, Jr. (1928-2015) received his Ph.D. from Princeton University in 1950. The impact of his 27 page dissertation on the fields of mathematics and economics was tremendous. In 1951 he joined the faculty of the Massachusetts Institute of Technology in Cambridge.
John Forbes Nash Jr. - Wikipedia
As a graduate student in the Mathematics Department at Princeton University, Nash introduced a number of concepts (including Nash equilibrium and the Nash bargaining solution) which are now considered central to game theory and its applications in various sciences.
Non-Cooperative Games - Home | Princeton University Library
Non-Cooperative Games - Home | Princeton University Library
John Nash and his contribution to Game Theory and Economics
A two-page paper published by John Nash in 1950 is a seminal contribution to the field of Game Theory and of our general understanding of strategic decision-making. That paper, “ Equilibrium...
John Nash’s Super Short PhD Thesis: 26 Pages & 2 Citations
When John Nash wrote “Non Cooperative Games,” his Ph.D. dissertation at Princeton in 1950, the text of his thesis ( read it online) was brief. It ran only 26 pages. And more particularly, it was light on citations.
Year 91 – 1951: “Non-cooperative Games” by John Nash, in ...
Nash, in his dissertation research at Princeton (published in this and three other papers), extended game theory to n-person games in which more than one party can gain, a better reflection of practical situations. Nash demonstrated that “a finite non-cooperative game always has at least one equilibrium point” or stable solution.
John Nash | Biography, Game Theory, Nobel Prize, & Facts
John Nash, in full John Forbes Nash, Jr., (born June 13, 1928, Bluefield, West Virginia, U.S.—died May 23, 2015, near Monroe Township, New Jersey), American mathematician who was awarded the 1994 Nobel Prize for Economics for his landmark work, first begun in the 1950s, on the mathematics of game theory.
Read John Nash's Super Short PhD Thesis with 26 Pages & 2 ...
And it laid the foundation for his dissertation, another seminal work in the development of game theory, for which Nash won the Nobel Prize in Economic Sciences in 1994. The reward of inventing a new field, I guess, is having a slim bibliography. Related Content: John Nash: A Brilliant Madness — 2002 Film on the Nobel Prize Winning Mathematician
John F. Nash, Jr. | Math - Princeton University
2015 John Nash Jr., a legendary fixture of Princeton University’s Department of Mathematics renowned for his breakthrough work in mathematics and game theory as well as for his struggle with mental illness, died with his wife, Alicia, in an automobile accident May 23 in Monroe Township, New Jersey. He was 86, she was 82.