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John Forbes Nash (John Forbes Nash Jr, Jr.), an American mathematician, made fundamental contributions to game theory, real algebraic geometry, differential geometry and partial differential equations. Nash and his colleagues were awarded the 1994 Nobel Prize in economics. In 2015, he was awarded the Abel Prize for his contributions to the field of partial differential equations.

As a graduate student in mathematics at Princeton University, Nash put forward a number of concepts (including Nash equilibrium and Nash bargaining solutions), which are now considered to be at the core of game theory and its applications in various sciences. In the 1950s, Nash discovered and proved the Nash embedding theorem by solving the system of nonlinear partial differential equations in Riemannian geometry. This work also introduced the preliminary form of the Nash-Moser theorem, which was later recognized by the American Mathematical Society and won the Leroy P. Steele Award for his pioneering contribution to research. In addition, Nash found a series of results in different methods, which paved the way for a systematic understanding of elliptic and parabolic partial differential equations.

His life was perfectly interpreted in the award-winning biographical film Beautiful mind in 2001. The film focuses on Nash's life, from his early years as a smart but eccentric student at Princeton, to his groundbreaking work in math and game theory, to his struggle with schizophrenia.

Maybe it's a good thing to have a beautiful mind, but the greater gift is to find a beautiful heart-- John Nash

Early Life and Education according to his biographer Sylvia Nassa, Nash is not a traditional good student and he does not like to take classes. He is regarded as a "loner" who doesn't like to socialize with his peers. He has a tendency to be absent-minded in class and often spends time thinking about math problems or doing his own research.

Nash was very fond of reading, and Eric Temple Bell's mathematician initially influenced him and stimulated his interest in mathematics. In 1948, he began studying at the Carnegie Institute of Technology (later named Carnegie Mellon University), where he majored in chemical engineering. However, after studying several courses such as tensor calculus and the theory of relativity, he soon became interested in mathematics, transferred to mathematics and graduated in 1949.

Nash then went on to pursue graduate studies in mathematics at Princeton University. Duffin wrote a short but very powerful letter of recommendation, which became famous as part of his application to Princeton, saying:

Mr. Nash is nineteen years old and a math genius!

Nash made rapid progress in his studies, and in 1950, at the age of 21, he received his doctorate. His paper, "non-cooperative Game", directed by Albert W. Tucker, laid the foundation for the field of game theory. Game theory is one of the most important fields in economics and mathematics.

Doctoral thesis

Nash's paper is relatively short, with a total of 26 pages. However, it is famous for its simplicity and the powerful ideas it contains. One noteworthy aspect of his paper is that it is cited only twice, one of which is John von Neumann's book Game Theory and Economic behavior, which was published in 1944. Nash was deeply influenced by Von Neumann's revolutionary work, so his paper was based on the concepts introduced in that book. Another citation is his own paper, "equilibrium in the n-player Game."

Although Nash's paper is groundbreaking, it has only been cited twice, which undoubtedly proves his originality and independent thinking ability.

In his paper, John Nash put forward the concept of "Nash equilibrium" as a way to analyze a non-cooperative game in which players cannot reach binding agreements. Nash equilibrium means that each player in the game chooses a strategy, that is, no one will benefit from changing the strategy, while the other players keep the strategy the same.

He also introduced the concept of "mixed strategy" in his paper, that is, players choose randomly between different strategies with a certain probability. This allows us to analyze situations in which players may respond best to the actions of other players. His paper also explores the concept of a "cooperative game" in which players can reach binding agreements and form alliances. He put forward the concept of cooperative negotiation as a way to distribute the benefits of cooperative game among the participants in the alliance. The concept of Nash bargaining solution is widely used in labor economics, political science, international relations and other fields to analyze the formation and dissolution of alliances and the allocation of resources among different groups.

Nash was diagnosed with schizophrenia in the late 1950s, characterized by hallucinations, delusions, thinking disorders and decreased functions of daily life. He struggled with his illness for many years and was unable to have a stable job or maintain interpersonal relationships. Decades later, however, in the 1990s, with the help of drugs and treatment, he was able to control his symptoms and eventually return to work in math.

He said,

If I can't create good things at work, I won't think I've recovered.

Nash's paper is not only famous for its concept of Nash equilibrium, but also attracted wide attention in the field of mathematics because of its rigor and complexity. His use of advanced mathematical concepts such as topology and fixed point theorems is advanced, setting a high standard for future game theory.

As a result, he won many famous academic positions and research opportunities, including a Rand scholarship and a professorship at the Massachusetts Institute of Technology.

The concept of n-tuple of Nash equilibrium is probably the most important concept in non-cooperative game theory. Whether we are analyzing the election strategy of the candidates, the causes of the war, the manipulation of the agenda by the legislature, or the actions of interest groups, the prediction of events boils down to the search and description of equilibrium. In short, the equilibrium strategy is our prediction of people.

Unfortunately, his research received little attention when it was published. He said,

Although my research has been successful, in the end, I feel that if I think and act like a 'normal person', my work will be better respected.

It was not until the late 1960s and 1970s that his works began to be widely recognized and adopted by the scientific community, and were used for analysis in various fields.

John Nash died in a car accident in Norway on May 23, 2015.

This article comes from the official account of Wechat: Lao Hu Shuo Science (ID:LaohuSci). Author: I am Lao Hu.

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