John Nash Jr. is a legendary mathematician known for his groundbreaking work in math and game theory and his battle with mental illness. He was born in West Virginia in 1928 and is regarded as a trailblazer for the study of differential geometry and partial differential equations. John Nash’s 1950 paper is a monumental contribution to the field of game theory and our basic grasp of strategic decision-making. The paper “Equilibrium points in N-person games” introduced a fundamental concept known as Nash equilibrium.^{1,2,3 }

John Nash is best known for his work on game theory, the math of decision-making and strategic planning, which earned him the Nobel Prize in economics in 1994. His fame was also boosted by Russell Crowe’s portrayal of him in the 2001 Oscar-winning film “*A Beautiful Mind.*”^{ }^{2} The film, which made him possibly one of the most well-known mathematicians in the world, is based on Sylvia Nasar’s excellent biography and follows his early career as well as his hardships with schizophrenia, which influenced most of his adult years.^{ }^{4 }

**Notable Accomplishments**

Nash is credited with several critical mathematical theories that helped him establish a reputation in the field, including:^{2,5 6,7,8,9}

**Nash Equilibrium:**According to the concept of Nash equilibrium from game theory, the best result in a game occurs when there is no reason to change from the initial plan of attack. In game theory, the Nash equilibrium is a notion that states that the best result of a game is one in which no player has the incentive to change their selected strategy after taking into account the choice of an opponent. It offers a method of anticipating what will happen when multiple individuals or institutions are making decisions concurrently, and the outcome is dependent on those of the others. Overall, assuming that other players maintain their current strategy, individuals cannot incrementally benefit from changing their actions. A game may have one, several, or no Nash equilibria.

**Game Theory:**While Nash didn’t invent game theory, Oskar Morganstern, an economist, and John Von Neumann, among one of the greatest mathematicians of the 20th century, share credit for its invention. They introduced tools for solving the main classes of games in their 1944 work “*Theory of Games and Economic Behavior,*” which was the first comprehensive formulation of the subject of game theory. The work of von Neumann and Morgenstern was crucial in establishing the groundwork for game theory and other branches of mathematical economics. However, the majority of their approach was restricted to resolving games in zero-sum games. A different way of thinking was required for non-zero-sum games. That’s where Nash’s brilliance came in handy; he invented what we now refer to as the Nash equilibrium. Additionally, he demonstrated that such a solution might be applied to various games, zero-sum or not. Nearly all subsequent game theory research makes use of Nash equilibrium. Finding the Nash equilibrium is now the aim of game analysis.

**Nash and Moser’s Inverse Function Theorem:**The theorem was discovered by Nash along with Jürgen Kurt Moser, a mathematician of German and American descent, who received recognition for contributions made over a four-decade period in the field of partial differential equations and Hamiltonian dynamical systems. When the required solution mapping for the linearized problem is not limited, the Nash-Moser theorem in analysis extends the inverse function theorem on Banach spaces.

- In 1994, Nash received the Nobel Memorial Prize in Economic Sciences. He was awarded the prize “
*for their pioneering analysis of equilibria in the theory of non-cooperative games*,” along with Reinhard Selten and John C. Harsanyi.^{10}

- In 2015, Nash was also awarded the Abel Prize. It is given by the Norwegian Academy of Science and Letters as a representative of the nation’s education ministry to those who reach significant milestones in math.

Nash made some astoundingly significant contributions to math. The Nash embedding theorems and the De Giorgi-Nash-Moser theorem are among the mathematical theories he created.

Nash started experiencing delusions and severe paranoia in 1959. He could only conduct meaningful mathematical research during brief windows of clarity for the following 40 years.^{ 9} Interestingly, though, he gradually improved, and his mental health returned by the time he received the Nobel Prize in 1994.^{ 9} Nash showed such tenacity and perseverance in his mathematical work and in overcoming his mental condition, making him a true inspiration to all.

Learn more about other eminent mathematicians by reading the articles listed below.

The Astounding Brilliance and Greatest Contributions of John von Neumann

What makes Isaac Newton one of the Most Influential Figures in Scientific History?

Who is Al-Khwārizmī, and What Contributions did He Make to the World of Math?

Albert Einstein’s Remarkable Contributions to the World of Mathematics

Archimedes Contributions to the World of Mathematical Physics

You can also visit BYJU’S FutureSchool Blog to read more inspiring articles on math and coding.

**References:**

*John Nash | Biography, Game Theory, Nobel Prize, & Facts | Britannica*. (n.d.). Retrieved November 2, 2022, from https://www.britannica.com/biography/John-Nash*John F. Nash Jr., Math Genius Defined by a ‘Beautiful Mind,’ Dies at 86 – The New York Times*. (n.d.). Retrieved November 2, 2022, from https://www.nytimes.com/2015/05/25/science/john-nash-a-beautiful-mind-subject-and-nobel-winner-dies-at-86.html*John F. Nash, Jr. (1950). Equilibrium points in n-person games. Proceedings of the National Academy of Sciences, 36(1), 48–49. https://doi.org/10.1073/PNAS.36.1.48**A Beautiful Mind – The story of John Forbes Nash Jr*. (n.d.). Retrieved November 2, 2022, from https://www.abeautifulmind.com/*John Nash’s ground-breaking contributions to maths – BBC News*. (n.d.). Retrieved November 2, 2022, from https://www.bbc.com/news/world-us-canada-32870646*John F. Nash Jr. – Biographical – NobelPrize.org*. (n.d.). Retrieved November 2, 2022, from https://www.nobelprize.org/prizes/economic-sciences/1994/nash/biographical/- Seminar, N., Arold, H., Kuhn, W., Harsanyi, J. C., Selten, R. E., Weibull, J. W., Damme, E. van, Nash, J. F., & Hammerstein, P. (1994).
*THE WORK OF JOHN NASH IN GAME THEORY*. Retrieved November 2, 2022, from https://www.nobelprize.org/uploads/2017/05/nash-lecture.pdf - Hamilton, R. S. (1982). THE INVERSE FUNCTION THEOREM OF NASH AND MOSER.
*BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY*,*7*(1). Retrieved November 2, 2022, from https://www.brainmaster.com/software/pubs/math/Hamilton%20Nash%20Moser.pdf *John Nash’s unique approach produced huge leaps in economics and maths | Mathematics | The Guardian*. (n.d.). Retrieved November 2, 2022, from https://www.theguardian.com/science/alexs-adventures-in-numberland/2015/may/24/john-nashs-unique-approach-produced-quantum-leaps-in-economics-and-maths*The Prize in Economics 1994 – Press release – NobelPrize.org*. (n.d.). Retrieved November 3, 2022, from https://www.nobelprize.org/prizes/economic-sciences/1994/press-release/