The Nash equilibrium

Defined in a Princeton doctoral thesis, 1950

Economics — game theory and strategic behaviour

The central solution concept of game theory: a combination of strategies, one for each player, such that no player can do better by changing their own strategy alone. John Forbes Nash Jr. defined it in a Princeton doctoral thesis of 1950 barely thirty pages long, and proved that every finite game has at least one such point if players are allowed to mix strategies at random. The result turned game theory from a set of results about two-player zero-sum contests into a general language for analysing strategic behaviour — in economics, evolutionary biology, political science and the design of auctions. It is not a claim that outcomes are good: the prisoner's dilemma has exactly one, and both players would prefer another.

How to spot them

▸A twenty-seven-page Princeton doctoral thesis of 1950 titled 'Non-Cooperative Games'
▸No player can gain by deviating unilaterally
▸Its existence proof rests on a fixed-point theorem
▸Shared the 1994 economics prize with John Harsanyi and Reinhard Selten

Key facts

A twenty-seven-page Princeton doctoral thesis of 1950 entitled 'Non-Cooperative Games' (1950)

One of the shortest theses ever to produce a Nobel Memorial Prize

The proof that every finite game has at least one solution in mixed strategies, using a fixed-point theorem (1950)

Via Kakutani's, and in the later version Brouwer's, fixed-point theorem

Two arrested suspects, each better off confessing whatever the other does, and both worse off for it (1950)

The prisoner's dilemma, devised at RAND by Merrill Flood and Melvin Dresher; Albert Tucker supplied the prison story

John von Neumann is said to have dismissed it as 'trivial — it's just a fixed point theorem'

Von Neumann and Oskar Morgenstern had published Theory of Games and Economic Behavior in 1944

Sylvia Nasar's 1998 biography, filmed in 2001 with Russell Crowe and winning Best Picture (1998)

A Beautiful Mind; the film's bar scene is a famously poor illustration of the concept

The Abel Prize in 2015, shared with Louis Nirenberg for work on partial differential equations (2015)

Awarded days before its recipient was killed in a taxi crash on the New Jersey Turnpike

An embedding theorem showing that any Riemannian manifold can be isometrically embedded in Euclidean space (1956)

Work its author himself considered more important than the game theory

Key ideas

The definition is about unilateral deviation only: it says nothing about what two players could achieve by colluding, which is why co-operative outcomes can be unreachable.
The concept guarantees existence, not uniqueness. Many games have several such points, which is why later work added refinements — Reinhard Selten's subgame perfection, for instance — to pick among them.
It is not the same as a Pareto-efficient outcome. In the prisoner's dilemma the unique stable point is the one both players like least of the symmetric options.
Mixed strategies — randomising between options with fixed probabilities — are what make the existence theorem work. In a game like rock-paper-scissors there is no stable pure strategy at all.

Quick-fire facts

•John Forbes Nash Jr. lived from 1928 to 2015 and was born in Bluefield, West Virginia
•His doctoral thesis, 'Non-Cooperative Games', was submitted at Princeton in 1950; a summary appeared in the Proceedings of the National Academy of Sciences that year and the full paper in the Annals of Mathematics in 1951
•He shared the 1994 Nobel Memorial Prize in Economic Sciences with John Harsanyi and Reinhard Selten
•Theory of Games and Economic Behavior, by John von Neumann and Oskar Morgenstern, had appeared in 1944
•Nash was diagnosed with schizophrenia in the late 1950s and was largely absent from mathematics for some thirty years
•Sylvia Nasar's biography A Beautiful Mind (1998) was filmed in 2001 by Ron Howard, with Russell Crowe as Nash; it won the Academy Award for Best Picture
•Nash received the Abel Prize jointly with Louis Nirenberg in May 2015
•He and his wife Alicia were killed in a taxi crash on the New Jersey Turnpike on 23 May 2015, returning from the Abel Prize ceremony in Norway
•The Nash embedding theorems of the mid-1950s are considered by many mathematicians his deepest work
•The prisoner's dilemma was formulated at the RAND Corporation in 1950 by Merrill Flood and Melvin Dresher; Albert W. Tucker gave it its narrative and its name

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Sources

John F. Nash — Equilibrium Points in N-Person Games Proceedings of the National Academy of Sciences, 1950. Cited without URL.
Sylvia Nasar — A Beautiful Mind 1998