John Nash: a beautiful mind and its exquisite mathematics

John Nash, May 2015. NTB/Scanpix

John Nash, mathematician and Nobel laureate in economics, died in a taxi accident on May 23; he was 86. His wife, Alicia, was with him and also did not survive the crash. The Nashes were on their way home to Princeton from Norway, where John was honored as a recipient (along with Louis Nirenberg) of this year’s Abel Prize in mathematics.

Thanks to A Beautiful Mind, Sylvia Nasar’s chronicle of Nash’s life, and its film adaptation starring Russell Crowe, Nash was one of the few mathematicians well known outside the halls of academia. The general public may remember the story of Nash’s mental illness and eventual recovery from paranoid schizophrenia. But Nash’s influence goes far beyond the Hollywood version of his biography. His colleagues count his mathematical innovations, particularly on noncooperative games (the work that would earn him his Nobel Prize), among the great economic ideas of the 20th century.

Noncooperative games

Nash is best known for his work in game theory. In mathematics, a game involves two or more “players” who earn rewards or penalties depending on the actions of all the participants. Some games are called zero-sum, which means that one player’s gain is another player’s loss. Nash’s work applied to noncooperative games. In these situations, players may unilaterally change strategy to improve (or worsen) their own outcome without affecting the other players.

The prototypical example of such a game is the basic Prisoner’s Dilemma. Two criminals have been captured and detained in separate cells, unable to communicate with each other. The prosecutors do not have sufficient evidence to convict them on the primary charge, but they can convict them on a lesser charge which comes with a one-year sentence. The prisoners are offered a deal: Testify against the other defendant (ie, defect) and go free while he serves three years. However, if both defendants betray each other, both will serve two years. If neither betrays the other (ie, they cooperate), then they will both be convicted of the lesser charge and serve the one year. The outcomes may be summarized in a payoff matrix.

The payoff matrix for the Prisoner’s Dilemma. In parentheses, the sentences are ordered (Player A, Player B). generated by the author

What Nash discovered is that any such game has a strategy, now called a Nash equilibrium, where any unilateral change in strategy by a player results in a worse outcome for that player. In the case of the Prisoner’s Dilemma, there are two such equilibria, the upper-left and lower-right squares in the payoff matrix. Indeed, in the situation in the lower-right corner, if either player changes his strategy unilaterally and decides not to defect, he will increase his sentence and thereby end up with a worse outcome. This example is particularly vexing because the upper-left strategy is clearly the best way to go for the prisoners (they should remain silent), but purely rational players will end up at the lower-right position.

A Beautiful Mind famously dramatized Nash’s discovery of what’s now called the Nash equilibrium.

Game theory has applications in many fields, including economics and political science. Many scenarios in international relations may be modeled as noncooperative games. For example, the development of nuclear programs during World War II can be modeled as a sort of Prisoner’s Dilemma, in which the two sides each decide to pursue a bomb for fear the other side will do so. This, of course, led to the less-desirable outcome of nuclear proliferation, analogous to both prisoners defecting.

Embedding theorems

While Nash is best known globally for his work on game theory, most mathematicians think of his results on embeddings of Riemannian manifolds as his most innovative and important. In this subspecialty of geometry, an n-manifold is a space which locally looks like n-dimensional Euclidean space (the typical three spatial dimensions we’re used to form a 3-dimensional Euclidean space). For example, a surface, such as a sphere or hollow donut, is a 2-manifold since any point on the surface has a small disc around it; to a small bug standing at the point, the surface looks like a flat 2-dimensional plane (hence the ancient belief that the Earth is flat).

Given two vectors in Euclidean space, the angle (theta) between them may be defined via the formulas on the right. generated by the author

A manifold is Riemannian if there is a globally consistent way to define angles between vectors tangent to the manifold at a point. In particular, this allows us to define distances between points on the manifold and to find the lengths of curves embedded in the manifold. Euclidean space with its usual notion of angle and distance is the simplest example.

Now imagine trying to put an abstract Riemannian manifold inside Euclidean space. You might twist it up and do all sorts of strange things that end up distorting the angles between tangent vectors on your manifold. The Nash-Kuiper Embedding Theorem asserts that we can always fix this problem; that is, we can find a realization of a Riemannian manifold of dimension n into a Euclidean space of dimension n+1 such that the angles are preserved. Then you can compute distances between points on the manifold more easily using the Riemannian structure inherited from Euclidean space.

This may not sound earth-shattering, but the problem had vexed mathematicians for more than a century. That the dimension of the Euclidean space cannot be made smaller than n+1 is familiar to anyone who has studied a map – the surface of a sphere cannot be flattened onto a plane without distorting angles.

There are many counterintuitive implications of Nash’s theorem. For example, it implies that any closed surface may be realized inside an arbitrarily small ball in 3-dimensional space.

The board in Hex; blue wins.

An actual game

Nash is also credited with inventing a game, eventually marketed by Parker Brothers as a board game called Hex. This game, played on a parallelogram-shaped field of hexagonal cells, was discovered independently in Denmark around the same time. In Princeton it was called Nash, after its creator, or John, a double entendre involving the fact that it was played on the tiles in the mathematics department’s men’s room floor. There are two players, each of whom has tokens of a single color (red and blue, say). The object is to form an unbroken path from one side of the board to the other before one’s opponent does the same in the opposite direction.

There are online versions of the game. The first player always has a winning strategy; that is, the player who makes the first move can always win, provided he executes the proper sequence of moves.

Nash’s beautiful mind focused on mathematics throughout his long life. Bobby Yip/Reuters

A life’s work remembered

In any given century there are a handful of mathematicians who stand out, whose work is so original and groundbreaking that it becomes a part of the language. As journalist Erica Klarreich pointed out, no one cites Nash’s papers any longer because “Nash equilibrium” is standard vocabulary; every mathematician knows what it means. While he published only a small handful of papers, John Nash will be remembered as one of the most original and influential mathematicians of the 20th century, whose work continues to inspire new results and research directions.