John Nash
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In the short span of ten years, John Forbes Nash, Jr. published an astounding fourteen papers relating to such diverse mathematical subjects as game theory, differential equations, parabolic equations, and fluid dynamics. Although Nash is best known for his works in game theory, for which he received the Nobel Prize in economics in 1994, his other mathematical works do deserve investigation. Many argue however, that Nashs theories relating to non-cooperative games are much less significant than those applicable to other mathematical subjects.
In 1950, Nash published Equilibrium points in n-person games. Previous works (by von Newmann and Morganstern) state in non-cooperative games, all results achieve a zero sum. In other words, the theory of von Newmann and Morganstern states that in every non-cooperative game there is a winner and a loser. Nashs theory adds to the previous theory of von Newmann and Morganstern by stating that there does not always need to be a winner and a loser. In fact, it states that in a game each side will attempt to win to the best of his ability. The other player will know this and will attempt to counter the strategy of the other player. Eventually, an equilibrium point (also known as a Nash Equilibrium) will occur in the game, where each player neither wins nor loses.
For example, pretend that there are two car dealers in a small town. Together, they have a monopoly on the car market in this town. When pricing cars, they can choose a high, medium, or low price. A high price will maximize their profits, whereas a low price is the best value for the consumer. Dealer “A” can price his cars high, but he knows that if he does that, Dealer “B” will price his cars at the medium level, which in turn will force him to price his cars at a low price level. Since price fixing with the other dealer is against the rules (i.e. there are laws against it), each dealer prices his cars at the low level, in order to avoid a price war. Thus an equilibrium point is reached, even though this equilibrium point is not at its most beneficial for either dealer.
Nashs theory states that at least one equilibrium point exists per game. It is also possible to have multiple equilibrium points per game, or even have an equilibrium point that is disastrous for all players. For instance, a nuclear war is one game where the only equilibrium point is the total destruction of both sides. (Bernstein, 1998).
This theory and his expanded portions of this theory (published in 1950, 1951, 1953, and 1954), can be applied to any cooperative or non-cooperative game, such as contract negotiations, antitrust negotiations, and even the auctioning of pieces of the FCC radio spectrum. (Nasar, 1998).
The remainder of Nashs published papers consists of advances in different disciplines in algebra such as fluid dynamics, parabolic equations, and the insertion of three-dimensional