The Game theory of Von Neumann, Morgenstern and Nash

The Game Theory of Von Neumann, Morgenstern and Nash

The game theory is a mathematical method of analyzing strategic behaviour of people when placed in competitive situation. The scientists behind it are John Von Neumann, Oskar Morgenstern and John Nash, Jr.

The Game Theory in a Nutshell
According to the game theory, all games have three things in common:

• rules,
• payoffs, and
• strategies.

Games include zero-sum games (when each player benefits at the expense of others), non-zero-sum games, co-operative games (when people can make bargains), and games of complete information.
The game has an equilibrium referred to as the “Nash equilibrium”, coined after the American mathematician, John Nash, Jr. It is a solution concept that maximizes everyone’s benefit.


Nash Equilibrium

The "Nash Equilibrium" is a solution concept of a non-cooperative game involving two or more players, in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only their own strategy.
For example, a group of players are in Nash equilibrium if each one is making the best decision possible, taking into account the decisions of the others in the game as long the other party's decision remains unchanged. Or simply stated: players A (she) and B (he) are in Nash equilibrium if A is making the best decision she can, taking into account B's decision while his decision remains unchanged, and he is making the best decision he can, taking into account her decision while her decision remains unchanged.

The Scientists

The theory was born when John von Neumann realized that poker, a game he played occasionally was not guided by probability theory alone (conceived by famous French mathematician, Blaise Pascal), and that “bluffing”, a strategy to hide information from other players, was also crucial. It was further extended by scientists John von Neumann with Oskar Morgenstern in 1944, and with Nash in 1949.

John von Neumann (1903-1957), was a Hungarian-born American mathematician who made major contributions in many mathematics and statistics fields. He is generally regarded as one of the greatest mathematicians in the modern era.

Oskar Morgenstern (1902-1977), was a German-born Austrian economist who helped found the mathematical field of game theory with John von Neumann.

John Nash, Jr. (1928-2015), was a mathematician who wrote his seminal paper “No-cooperative Games” while studying for his PhD at Princeton University. Aside from his work in game theory, other works of Nash include differential geometry and partial differential equations. He served as Senior Research Mathematician at Princeton University later in his life and shared the 1994 Nobel Prize in Economic Sciences. A few years later Nash was diagnosed with paranoid schizophrenia. In the early 1990s he overcame his disease and resumed his work. Sylvia Nasser’s book about Nash, A Beautiful Mind (1998), was made into a movie of the same name (2001), starring Russell Crowe as Nash.

Von Neumann and Morgenstern both died before the Economics Nobel Prize was introduced.
Game theory as a branch of applied mathematics is used in areas of discipline such as economics, biology, engineering, political science, international relations, computer science, social psychology, philosophy and management.

Resources:
GameTheory200.net. Image accessed Dec 22, 2013.
Moore, Pete. E=MC2: The Great Ideas That Shaped Our World. London: New Burlington Books. 2002