“A Beautiful Mind”

The movie that you are about to see deals with the life of John Nash. John Nash was a brilliant mathematician, cryptologist and extremely intelligent. He entered Carnegie Tech in Pittsburgh with a major of chemical engineering, shifting to chemistry and then to math. Nash received a BA and an MA in mathematics in 1948. His graduate studies, beginning in September 1948, were in Princeton. In 1949, while studying for his doctorate, he wrote a paper which 45 years later was to win a Nobel Prize for economics. In 1950, Nash received his doctorate from Princeton with a thesis entitled “Non-cooperative Games”. He was on the math faculty at M.I.T. from 1951 until he resigned in the spring of 1959, interspaced with assignments in the private sector. A brilliant mathematician, he is a schizophrenic who happened to win the Nobel Prize in Economics in 1994. This was the same year that his illness went into remission. The illness has not disappeared, as hallucinations still occur, but he is able to deal with them on a rational basis.

1. List the symptoms, social ramifications, and possible dangerous actions that John Nash displays.

2. When does the onset of his schizophrenia occur?

3. Is mental illness in any way correlated with some level of intelligence? Do you think that schizophrenia occurs more in extremely intelligent people or with people of a lesser intelligence?

4. Describe his hallucinations.

5. What were some techniques used to help him?

6. Should he have been given the important tasks that he was? Was he responsible enough? Does his illness affect that responsibility?

7. When Nash left his infant son unattended in the bathtub, could he be charged with neglect or even murder if the child had drowned? Would an insanity plea serve as a reasonable defense?

8. What are the chances, do you think, of his child becoming schizophrenic?

9. How did seeing this movie inform you about the condition of schizophrenia? Did it help in understanding the tendencies, symptoms and the condition itself? Explain.

Theory:

When everybody is playing his best moves to everyone else’s best moves, everyone loses. When everybody is not moving, the winner is determined by statistical chance. When one is using his best moves while everyone else is not moving, than he wins. In competitive games, there is a sticking point where no players are benefited by changing their moves- this is called a Nash equilibrium.

A theorem in game theory which guarantees the existence of a set of mixed strategies for finite, noncooperative games of two or more players in which no player can improve his payoff by unilaterally changing strategy.

A Nash equilibrium of a strategic game is a profile of strategies , where ( is the strategy set of player i), such that for each player i, , , where and .

Another way to state the Nash equilibrium condition is that solves for each i. In words, in a Nash equilibrium, no player has an incentive to deviate from the strategy chosen, since no player can choose a better strategy given the choices of the other players.