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 Nash's 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. Nash's 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...

...John Forbes Nash, Jr. (born June 13, 1928) is an American mathematician whose works in game theory, differential geometry, and partial differential equations have provided insight into the forces that govern chance and events inside complex systems in daily life. His theories are used in market economics, computing, evolutionary biology, artificial intelligence, accounting, politics and military theory. Serving as a Senior Research Mathematician at Princeton University during the later part of his life, he shared the 1994 Nobel Memorial Prize in Economic Sciences with game theorists Reinhard Selten and John Harsanyi.
Nash is the subject of the Hollywood movie A Beautiful Mind. The film, loosely based on the biography of the same name, focuses on Nash's mathematical genius and struggle with paranoid schizophrenia.[1][2]
Contents
Early life
Nash was born on June 13, 1928 in Bluefield, West Virginia. His father, after whom he is named, was an electrical engineer for the Appalachian Electric Power Company. His mother, Margaret, had been a school teacher prior to marriage. Nash's parents pursued opportunities to supplement their son's education with encyclopedias and even allowed him to take advanced mathematics courses at a local college while still in high school. Nash accepted a scholarship to Carnegie Institute of Technology (now Carnegie Mellon University) and graduated with a...

...John Forbes Nash Jr. (born June 13, 1928) is a mathematician who worked in game theory and differential geometry. He shared the 1994 Nobel Prize for economics with two other game theorists, Reinhard Selten and John Harsanyi.
After a promising start to his mathematical career, Nash began to suffer from schizophrenia around his 30th year, an illness from which he has only recovered some 25 years later.
JohnNash was born in Bluefield, West Virginia as son of JohnNash Sr. and Virginia Martin. His father was an electrotechnician; his mother a language teacher. As a young boy he spent much time reading books and experimenting in his room, which he had converted into a laboratory.
From June 1945-June 1948 Nash studied at the Carnegie Institute of Technology in Pittsburgh, intending to become a technical engineer like his father. Instead, he developed a deep love for mathematics and a lifelong interest in subjects such as number theory, Diophantine equations, quantum mechanics and relativity theory. He loved solving problems.
At Carnegie he became interested in the 'negotiation problem', which John von Neumann had left unsolved in his book The Theory of Games and Economic Behavior (1928). He participated in the game theory group there.
From Pittsburgh he went to Princeton University where he worked on his equilibrium theory....

...beautiful mind is a great way to describe JohnNash because he was a brilliant person who suffered and fought through Schizophrenia. Nash was born on June 13, 1928, in Bluefield, West Virginia. His father was an electrical engineer for the Appalachian Electric Power Company. His mother, name was Virginia Martin and she had been a schoolteacher before she married. Nash had a younger sister, Martha, born November 16, 1930.Nash attended kindergarten and public school. Nash's parents worked hard to create a challenging learning program for their son's education, and arranged for him to take advanced mathematics courses at a local community college during his final year of high school. Nash attended Carnegie Mellon University with a full scholarship, and the George Westinghouse, which was a scientific/mathematic competition that helped students earn scholarship money. He initially majored in Chemical Engineering. He switched to Chemistry, and eventually to Mathematics. After graduating in 1948 with Bachelor of Science and Master of Science degrees in mathematics, he received a scholarship to Princeton University where he pursued his graduate studies in Mathematics. At Princeton he worked on his equilibrium theory. JohnNash had a bright future ahead of him at Princeton but he did go through some devastating problem which created obstacles, like his mental illness,...

...JohnNash Biography
JohnNash (June 13, 1928 – present) is a brilliant mathematician, specializing in economics. He was born n Westfield, West Virginia, into a family of three, he, his father – an electrical engineer, and his mother – a school teacher who pushed him to do many great things that led to his superb education and extraordinary mind. As a child, he had a quiet and withdrawn personality, but was very intelligent. He started reading at four, skipped a grade, and even learned Latin, all of which his mother pressed on him. As he grew up, he became aware of how smart he was and could be seen by some as being arrogant and an introvert. In his eyes, extracurricular activity such as music and sports were a waste of time and distracted him from his math and science studies. Starting in the fourth grade, his aptitude for mathematics became evident when he would solve complicated problems easily in front of the teachers’ eyes. Nash went to Carnegie Institute of Technology under a George Westinghouse Scholarship, with a George Westinghouse Award, which was only given to ten people. He studied chemical engineering, but disliked it and switched to chemistry. From chemistry, he switched again to mathematics to where he found his passion. He ended up with a Masters Degree and a Bachelor Degree in mathematics. Harvard University accepted him, but the chairman of Princeton University wrote him and persuaded...

...Nash Equilibrium and Dominant Strategies
Nash Equilibrium is a term used in game theory to describe an equilibrium where each player's strategy is optimal given the strategies of all other players. A Nash Equilibrium exists when there is no unilateral profitable deviation from any of the players involved. In other words, no player in the game would take a different action as long as every other player remains the same. Nash Equilibria are self-enforcing; when players are at a Nash Equilibrium they have no desire to move because they will be worse off.
Necessary Conditions
The following game doesn't have payoffs defined:
L
R
T
a,b
c,d
B
e,f
g,h
In order for (T,L) to be an equilibrium in dominant strategies (which is also a Nash Equilibrium), the following must be true:
a > e
c > g
b > d
f > h
In order for (T,L) to be a Nash Equilibrium, only the following must be true:
a > or = e
b > or = d
Prisoners' Dilemma (Again)
If every player in a game plays his dominant pure strategy (assuming every player has a dominant pure strategy), then the outcome will be a Nash equilibrium. The Prisoners' Dilemma is an excellent example of this. It was reviewed in the introduction, but is worth reviewing again. Here's the game (remember that in the Prisoners' Dilemma, the numbers represent years in prison):
Jack
C
NC
Tom
C
-10,-10
0,-20
NC...

...Arjun Pahwa Math Research Paper The Application of the Nash Equilibrium in Game Theory to Microeconomics ! One of the most challenging problems a business owner comes across is the
amount of a certain item he or she should stock and the price at which to sell it. Many factors play into ﬁnding this appropriate price. These include the cost of stocking the item, the projected demand, and what the competition is pricing the same item at. The latter of the three factors is considered to be the most challenging to consider. When attempting to tackle this problem we must consider three factors. First, we must be able to accurately predict the outcome of our decisions. Second, we must be able to accurately predict our competitionʼs decisions. Finally, we must be able to predict the outcome of our competitionʼs decisions. If all of these three requisites can be met our solution will be able to provide us the necessary information needed to price our item. ! The solution to this problem is game theory. Game theory is a branch of
mathematics that is used to predict the actions of an opponent or competitor in a certain “game.” Game theory has many applications including war, macro and microeconomics, and even biology. Game theory is a relatively new concept created by John von Neumann and Oskar Morgenstern in 1944. Over time it has evolved into a very complex ﬁeld of mathematics that has applications in many other ﬁelds. John von...

...Security Dilemma the Collective Action Problem and the Nash Equilibrium.
Criticism of the United Nations highlight the lack of power it has and its reliance on superpowers for legitimacy. The use by states of the UN is conditional on whether it serves state self-interest and whether the value of participating outweighs the cost (Abbott and Snidal 2005: 27). This brings into question why states would allow the UN to impose International laws and Norms that erode state sovereignty and how this increases international peace and security. It is seemingly irrational that despite the issue of national sovereignty and individual grievances states are extremely hesitant to leave the United Nations (Diehl 2005:4). The importance of the UN in international peace and security can be explained by the dominance of the ‘security dilemma’ and the connection between realism, rational choice theory and the Nash equilibrium.
The security dilemma is the international predicament that can best be categorized as aiming to reduce the uncertainty of an anarchistic world order (Booth and Wheeler 2009:132). The uncertainty of states actions has led some realist theorists to attempt to find the optimum strategy for mitigating external threats and thus secure its own interests.There are two levels to the security dilemma; the dilemma of interpretation which is attempting to discover what other nations are doing behind closed doors, and the dilemma of response; how...