A troubled mind that found some equilibrium


Under the Microscope/Prof William Reville: John Forbes Nash Jnr, along with two others, shared the Nobel Prize in Economics in 1994, in recognition of his contribution to game theory. After a brilliant early career in mathematics, Nash suffered a protracted period of mental instability from which he eventually recovered.

His story is the subject of the book A Beautiful Mind by Sylvia Nasar (Simon and Shuster, 1998), and also of a recent movie starring Russell Crowe.

John Forbes Nash Jnr was born on June 13th 1928 in Bloomfield West Virginia. A sister, Martha, was born in 1930. His father was an electrical engineer and his mother was a teacher. Although he was brought up in a loving household, he did not relate emotionally to other boys, seeming remote and confrontational. Nash first showed an interest in mathematics at the age of 14, inspired by the book Men of Mathematics by ET Bell. He was also very interested in chemistry.

Nash entered the Carnegie Institute of Technology in 1945 to study chemical engineering. His teachers recognised his mathematical skills and persuaded him to change to mathematics. He continued to have poor relations with his fellow students, many of whom sensed he had a mental problem. Nash behaved at times in a very odd manner, for example, playing a single piano cord over and over.

Nash was awarded a BA and an MA in mathematics in 1948. He was accepted into the postgraduate programme at Princeton in September 1948. Nash took an interest in a broad range of mathematics - topology, algebraic geometry, game theory and logic. The point in game theory is to find the best rational strategy to use when your opponents make no mistakes and always pursue their own best outcome. The simplest games have two players and a zero-sum payoff rule, which means that one player's gain is the other player's loss.

John von Neumann had shown in the 1920s that optimal strategies exist for such players and that such games reach equilibrium where none of the players can improve their position. Von Neumann's argument couldn't be extended to multiplayer games or to non-zero-sum games.

Nash developed game theory to include multiple players in non-cooperative situations where each player must play his best game in response to his opponent's best strategy. Nash showed there is also a point of equilibrium for those games for which the rational strategies of all players are in balance. This is called the Nash Equilibrium Theory.

Equilibrium Theory has wide applicability in many fields. The economist P. Ordeshook has written: "The concept of the Nash Equilibrium is perhaps the most important idea in non-cooperative game theory. Whether we are analysing candidates' election strategies, the causes of war, agenda manipulation in legislatures, or the actions of interest groups, predictions about events reduce to a search for a description of equilibria. Put simply, equilibrium strategies are the things that we predict about people."

Nash received his doctorate from Princeton in 1950. He moved to the Massachusetts Institute of Technology (MIT) in 1952. Between 1950 and 1958 Nash published many significant mathematical contributions. Many believe that game theory was not his best work, pointing to his major contributions in topology and other areas.

In 1952 he established a romantic relationship with a nurse, Eleanor Stier. However, he concealed her existence from his family and most colleagues at MIT. When she became pregnant, Nash apparently made no offer of marriage and after his son was born, Eleanor had difficulty supporting the boy. Eventually she was forced to enrol her son in an orphanage.

The second woman in Nash's life is Alicia Larde who was a student in his calculus class. They married in 1957. Shortly afterwards their son John was born and around this time also Nash's mental state became very disturbed.

He appeared at a New Year's party dressed only in a nappy and spent the evening curled up on his wife's lap. He declared that the New York Times contained encrypted messages from outer space that were meant only for him. He was diagnosed as schizophrenic and Alicia had him involuntarily committed to a hospital. On his release, Nash resigned from MIT and went to Europe where he intended to renounce his US citizenship. Alicia had him deported back to the US.

Alicia settled with Nash in Princeton, but his illness continued and he spent most of his time hanging around the campus behaving strangely. He made temporary recoveries followed by further treatment. Alicia divorced Nash in 1962 but in 1970 she took him into her house again because no one else would have him. They are together still but have not re-married.

Ever so slowly Nash regained his lucidity. He claims that his recovery was partly an act of will, saying: "I began to intellectually reject some of the delusional lines of thinking which had been characteristic of my orientation."

Nash resumed his research at Princeton. A colleague once asked Nash how, as a man devoted to reason and logic, he could believe that extraterrestrials were sending him messages. "Because the ideas I had about supernatural beings came to me the same way that my mathematical ideas did. So I took them seriously," he replied.

William Reville is associate professor of biochemistry and director of microscopy at University College Cork