Schizophrenia...

He began to show signs of extreme paranoia and his wife later described his behavior as becoming increasingly erratic, stating that he began speaking of characters who were putting him in danger. Nash seemed to believe that there was an organization chasing him, in which all men wore "red ties". Nash mailed letters to foreign embassies in Washington, D.C. declaring that he was establishing a world government.

He was involuntarily admitted into the McLean Hospital, April–May 1959, where he was diagnosed with paranoid schizophrenia and mild clinical depression.^[3] Upon his release, Nash resigned from MIT, withdrew his pension, and went to Europe, unsuccessfully seeking political asylum in France and Eastern Germany. He tried to renounce his U.S. citizenship. After a problematic stay in Paris and Geneva, Nash was arrested by the French police and deported back to the United States at the request of the U.S. government.

In 1961 he was involuntarily hospitalized into the Trenton State Hospital. He was in and out of mental hospitals until 1970, being given insulin shock therapy and antipsychotic medications, usually as a result of being involuntarily committed.^[3][7][8]

Although prescribed antipsychotic medication, Nash has said he never really took it. For some periods he was forced to or voluntarily complied under the pressure, but after 1970 he was never committed to the hospital again and never took antipsychotic medication again. The film A Beautiful Mind fabricated him later taking the then new atypical antipsychotics, which Nash attributes to the screenwriter (whose mother, he notes, was a psychiatrist) not wanting to incite people with the disorder to stop taking their medication. Others, however, have questioned whether the fabrication obscured a key question as to whether recovery from problems like Nash's can actually be hindered by such drugs and Nash has said they are over-rated and the adverse effects are not given enough consideration once someone is considered mentally ill According to his biographer Nasar, Nash recovered gradually with the passage of time. Encouraged by his then former wife, Alicia, Nash worked in a communitarian setting where his eccentricities were accepted. Alicia also said that for Nash "it's just a question of living a quiet life".

Nash dates the start of what he terms "mental disturbances" to the early months of 1959 when his wife was pregnant. He has described a process of change "from scientific rationality of thinking into the delusional thinking characteristic of persons who are psychiatrically diagnosed as 'schizophrenic' or 'paranoid schizophrenic'" including seeing himself as a messenger or having a special function in some way, and with supporters and opponents and hidden schemers, and a feeling of being persecuted, and looking for signs representing divine revelation. Nash has suggested his delusional thinking was related to his unhappiness, and his striving to feel important and be recognized, and to his characteristic way of thinking such that "I wouldn't have had good scientific ideas if I had thought more normally." He has said that "If I felt completely pressureless I don't think I would have gone in this pattern". He does not see a categorical distinction between terms such as schizophrenia and bipolar disorder. Nash reports that he did not hear voices at first, only some years later around 1964, until later engaging in a process of rejecting them. Nash reports that he was always taken to hospital against his will, and only temporarily renounced his "dream-like delusional hypotheses" after being in hospital long enough to decide to superficially conform and behave normally or experience "enforced rationality". Only gradually on his own did he "intellectually reject" some of the "delusionally influenced" and "politically-oriented" thinking as a waste of effort. However, by 1995 he felt that although he was "thinking rationally again in the style that is characteristic of scientists", he felt more limited.

Nash equilibrium

In game theory, defines the Nash equilibrium (by John Forbes Nash) as a way to get an optimal strategy for games involving two or more players. If there is a set of strategies such that no player benefits by changing its strategy while the other will not change theirs, then that set of strategies and corresponding gains constitute Nash equilibrium.

The concept of Nash equilibrium first appeared in his dissertation Non-cooperative games (1950). John Forbes Nash showed that the different solutions that had previously been proposed to offer the games to produce Nash equilibrium.

A game may have no Nash equilibrium, or have more than one. Nash was able to show that if we allow mixed strategies (in which players may choose strategies at random, with a default probability), then every game of n players in which each player can choose between a finite number of strategies have at least a Nash equilibrium with mixed strategies.

If a game has a unique Nash equilibrium and the players are fully rational players will choose strategies that make up the balance.

Examples

Competitive Gaming

Consider the following game for two players:

"The players simultaneously choose a whole number between zero (0) and ten (10). The two players earn less in dollar value, but also, if the numbers are different, which has chosen the more you must pay another $ 2 andalusia.

This game has a unique Nash equilibrium: both players must choose zero (0). Any other strategy may be a disadvantage if another player chooses a number lower.

. If you change the game so that the two players earn the number chosen if both are equal, and otherwise would not win anything, there are 11 different Nash equilibrium.

Coordination games

This game is a game of coordination when driving. The options are driving or driving on the right or the left: 100 means that there is a crash and 0 means yes. The first number in each cell indicates the gain of the first player (whose options are shown at left) and the second gain of the second player (whose options are shown above).

. In this case there are two Nash equilibrium in pure strategies, where both lead to the right or both on the left lead. There is also Nash equilibrium with mixed strategies, where each player chooses randomly with a probability of 50% which of the two strategies applied.

Prisoner's Dilemma

The prisoner’s dilemma is Nash equilibrium: occurs when both players confess. Despite this, both confess is worse than "both cooperating" in the sense that the total time in prison to be met is higher. However, the strategy "both cooperating" is unstable because a player can improve your score if your opponent keeps deserting the cooperation strategy. Thus, "both cooperating" is not a Nash equilibrium, but a Paretian optimum. One way to arrive at this result is achieved through collusion and the promise of each player to "punish" the other if he breaks the agreement. It could also lead to a solution outside the Nash equilibrium if the game is repeated countless times, is achieved when the "eye for an eye."