When describing John Forbes Nash, Jr (13 June 1928 – 23 May 2015), it’s hard to be more succinct than Richard Duffin, a professor at the Carnegie Institute of Technology, who wrote, in his letter of recommendation to Princeton, that ‘this man is a genius’. It was 1948: Nash, having abandoned a degree in Chemical Engineering for one in Mathematics, was only just embarking on a journey that would ultimately make him one of the most famous mathematicians of the 20th Century. Despite the interest of Harvard University, Nash eventually decided to pursue his graduate studies at Princeton and it was there that he published the 317 word paper, *Equilibrium points in N-person games*, that introduced the Nash Equilibrium and won him the Nobel Prize for Economics (jointly with Reinhard Selten and John Harsanyi) in 1994. As a result of this work in game theory, Nash was appointed to the RAND Corporation, which applied this relatively young field to the pressing policy issues of the time: nuclear weapons, the space race, the Cold War.

This man is a genius.

However, Nash’s contributions to mathematics go far beyond game theory. Nash was the archetypal problem solver: if there was an important open problem that mathematicians had struggled for years to solve and failed, it warranted his attention – regardless of his possible lack of knowledge in the subject. Indeed, it was often this absence of expertise, coupled with his genius, that would allow him to discover the revolutionary approach required and astound the mathematical community. The papers that became known as the Nash Embedding Theorems, published in 1954 and 1956 and which proved that every Riemannian manifold could be embedded into some Euclidean space, followed the challenge of a fellow faculty member at MIT, Warren Ambrose, who asked ‘if you’re so good, why don’t you solve the embedding problem for manifolds?’. An unsolved problem in differential geometry that had first been posed by Ludwig Schläfli in 1873, Nash’s proof involved the invention of a new technique to solve a system of partial differential equations that had previously been considered unsolvable; a technique now applied in the field of celestial mechanics. The proof is, in the opinion of John Conway, ‘one of the most important pieces of mathematical analysis in this century’; led Mikhail Gromov to state that ‘what he has done in geometry is… incomparably greater than what he has done in economics’; and, ultimately, was partly responsible for Nash’s death: on his way back from collecting the Abel Prize for this ‘striking and seminal contribution to the theory of nonlinear partial differential equations and its applications to geometric analysis’, the taxi he was in crashed, killing both him and his wife, Alicia.

Of course, a discussion of Nash’s life cannot be complete without a mention of the schizophrenia that robbed the world of his genius for the best part of three decades; and yet, paradoxically, also brought that genius to the attention of the world (in part through Sylvia Nasar’s powerful biography, *A Beautiful Mind*, and its Hollywood dramatisation). First hospitalised in 1959, the disease resulted in Nash resigning from his post at MIT that same year, caused his divorce from Alicia in 1962 (they remarried in 2001) and saw him spend much of those lost years wandering the Princeton campus, a phantom. He slowly began to recover and in the 1980s began communicating again with fellow mathematicians, including Harold Kuhn to whom, in a 1996 email, he ascribed his emergence ‘from irrational thinking’ to no medicine ‘other than the natural hormonal changes of ageing’.

In an age where mathematicians became ever more specialised, Nash stood out: his interests and successes ranged from game theory to analysis, geometry to fluid dynamics, nonlinear partial differential equations to cosmology (as a student in Princeton he once approached Einstein to discuss an idea he had had about gravity, friction and radiation). It’s pointless to speculate what contributions Nash might have made in these fields were it not for his illness; better to celebrate what Richard Duffin had spotted right at the beginning: his genius.

### Nash equilibrium

The mathematical study of decision making can be traced back to Antoine Cournot, a French mathematician, who was the first to publish a theoretical analysis of games in 1838; but it really took off in 1944 with the publication of *Theory of Games and Economic Behaviour* by John von Neumann and Oskar Morgenstern. It became known as game theory, where a game is defined as consisting of players, a set of actions (or strategies) that they can choose from and a pay-off function, which determines what each player receives or loses based on the actions of all the players.

Initially game theory dealt only with zero-sum games: those in which the gains of the players are exactly matched by the losses (in rock-paper-scissors, for example, one person will win and the other will lose). However, although most board games you will play are zero-sum, they have little relevance to real-world issues in economics or, especially important at the time, war. People were beginning to realise that two-person zero-sum games could only be applied to wars of complete extermination; in reality, two opponents will always have some common interest and would have something to gain from cooperation (during the Second World War, for example, the Allies did not destroy the Ruhr’s coal mines, knowing that to do so would be counter-productive in the long term).

Nash’s 1950 paper (and a later one published in 1951) introduced the Nash equilibrium and revolutionised the approach to game theory, moving it away from zero-sum games. Suppose that there are *N* players and each player has a strategy, with everyone knowing the strategies of everybody else. Nash equilibrium occurs when nobody can increase their reward by changing strategy. The prisoner’s dilemma is often given as an example when explaining Nash equilibrium (see **our next article**); another is the stag hunt, where two players can choose whether to hunt a stag or a rabbit. A stag gives more food than a rabbit, but both people need to hunt the stag in order to successfully catch it: if only one does so, then that person will fail and eat nothing. In this game, there are two Nash equilibria: both choose to hunt rabbit and both choose to hunt stag; since if you change your strategy then you get, respectively, either no food or less food. Nash showed that all games would have at least one equilibrium point.

Despite not always being a completely realistic representation of human interactions – we usually don’t know what choice everyone else will make, humans often make irrational decisions or mistakes and we might not always trust others to follow their stated strategy (if someone says that they will hunt the stag, are you sure that at the last moment they won’t change their mind and go for the rabbit, leaving you to starve?) – game theory and Nash’s contribution to it is often used to define policy in war and arms races (as was done when Nash was publishing his papers on the subject), explain social interactions, come up with marketing strategies and develop theories in economics.