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.
