Statistically, it would seem improbable that any mathematician or scientist, at the age of 66, would be able through continued research efforts, to add much to his or her previous achievements. However I am still making the effort.

— John Nash, autobiographical note for the Nobel Foundation (1994)

John Nash is the only person to have received both the Nobel Prize and the Abel Prize, the two highest honours in economics and in mathematics. The two halves of his work seem to have nothing in common: one is about human decisions, the other about surfaces and equations. What unites them is the style: attacking problems everyone thought impossible with ideas nobody had had.

Game theory everywhere

The Nash equilibrium became the common language of economics. It explains why two petrol stations on the same corner charge almost the same price, why companies advertise even when all would be better off if none did, why climate agreements are so hard. In the 1990s governments in several countries began selling mobile-phone frequency bands in auctions designed with game theory, raising tens of billions of dollars.

In biology, the Briton John Maynard Smith showed in the 1970s that evolution also produces Nash equilibria: the behavioural strategies that survive in a population are those no mutation can improve upon. In political science, the equilibrium helps to understand elections, wars and votes in parliament. In computing, it appears in internet routing and in the design of algorithms for digital markets. Since 1994, several Nobel Prizes in Economics have been awarded for work in game theory.

Pure mathematics

The embedding theorem and the Nash-Moser method transformed geometry and analysis. The technique Nash invented for the $C^1$ case, which the Russian Mikhail Gromov called convex integration, gave rise to a whole field, that of “rigidity and flexibility” phenomena, and was used in 2018 to prove a conjecture of Lars Onsager about turbulence in fluids. The De Giorgi-Nash theorem is the foundation of the modern theory of differential equations with irregular coefficients, which describe composite materials and porous media.

Illness and recovery

Nash also left a lesson about mental illness. For thirty years he was seen as a finished man. His recovery, partial and slow, showed that schizophrenia is not necessarily a final sentence. After the Nobel, he and Alicia became advocates of better treatment and less stigma; the couple’s son, John Charles, also a mathematician, had schizophrenia.

What remained

  • Game theory. The Nash equilibrium for games of any kind, existence in mixed strategies and the bargaining solution.
  • Board games. Hex, the strategy-stealing argument and the link with the fixed-point theorem.
  • Geometry. The isometric embedding theorems, $C^1$ and smooth, and convex integration.
  • Analysis. The De Giorgi-Nash theorem and the Nash-Moser method for non-linear equations.
  • Applied science. Economics, evolutionary biology, political science, auctions and computing.

To remember

  • John Nash lived from 1928 to 2015. He did his main work between the ages of 20 and 30, at Princeton and MIT.
  • In Hex there is never a draw, and the first player can always win: this is the strategy-stealing argument.
  • In a Nash equilibrium, nobody improves by changing alone. The equilibrium can be bad for everyone, as in the prisoner’s dilemma.
  • Every finite game has an equilibrium in mixed strategies. In odds and evens, each player randomises with a 50% chance.
  • In Nash bargaining, one maximises the product of the gains over the disagreement point, and each takes their disagreement payoff plus a share of the surplus.
  • Every Riemannian manifold can be drawn without distortion in a Euclidean space; with smoothness of first order only, even the flat torus fits in ordinary space.
  • Even in irregular materials, heat smooths: solutions are Hölder continuous.

Sources: Sylvia Nasar, A Beautiful Mind (Simon & Schuster, 1998) · John Nash’s autobiographical note, Nobel Foundation · Nash’s letters to the NSA (1955), released by the agency in 2012.

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