JOHN FORBES NASH Meryem D İ LEKCAN

Outline  Who is John Forbes Nash?  The Contributions of Nash  Nash’s Theorem  References

Life of John Nash  born 13 June 1928 in West Virginia, USA  showed a keen interest in mathematics and chemistry  attended Carnegie Institute of Technology with a full scholarship andCarnegie Institute of Technology initially majored in Chemical Engineering John F. Nash,

 switched to Chemistry, and eventually to Mathematics  his talents were recognised while at Carnegie Institute “This man is a genius.” R.J. Duffin  Entered Princeton in 1948 for his doctorate(the equilibrium) at the age of 21  showed interest in mathematics; topology, geometry, game theory and logic.

 worked for the RAND Corporation on the Cold War  taught at Massachusetts Institute  married Alicia Lardé and had a son  endured long term mental problems and periods of treatment (schizophrenia)  Von Neumann Theory Prize in1978  Nobel Memorial Prize in Economic Sciences (Nash equilibrium) in 1994Nash equilibrium  in 2012 he became a fellow of the American Mathematical Society.American Mathematical Society

The Studies and Contributions of Nash n Nash Equilibrium  Equilibrium Points in N-person Games  The Bargaining Problem  Non-cooperative Games  Two person cooperative Games

Nash equilibrium An important concept in game theory, a solution concept of a game involving two or more players, in which no player has anything to gain by changing his own strategy unilaterally

More specifically…  GAME = (P,A,U)  Players (P 1 ; … ; P N ): Finite number (N≥2) of decision makers.  Action sets (A 1 ; … ;A N ): player P i has a nonempty set A i of actions.  Payoff functions u i : A 1 x … xA N :  R ; i = 1;….;N - materialize players’ preference, - take a possible action profile and assign to it areal number (von Neumann-Morgenstern).

Prisoner’s Dilemma An illustration of Nash Equilibrium Art’s Strategies Bob’s Strategies Confess Deny Confess Deny 10 yrs. 1 yr. 3 yrs. 1 yr. 10 yrs. 2 yrs. Consider Art’s options… 1. If Bob denies and Art denies, then Art will get two years. Art is better off confessing and getting one year. 2. If Bob confesses and Art denies, then Art will get ten years, so Art is much better off confessing and taking three years. Consider Bob’s options… 1. If Art denies and Bob denies, then Bob will get two years. Bob is better off confessing and getting one year. 2. If Art confesses and Bob denies, then Bob will get ten years, so Bob is much better to confess and take three years. Thus, both parties will rationally choose to confess, and take three years – even though they could have been better off denying. Each party does this because, considering the possible options of the other party, they always found the better option was to confess. When neither party has an incentive to change their strategy, they are in “Nash Equilibrium.” Art and Bob are both suspects in a crime, and they are both offered the following deal if they confess…

is used  Economics,  Network Economics,  Political Sciences,  Computer Sciences,  Biology  …

Also… n Nash imbedding theorem  shows that any abstract Riemannian manifold can be isometrically realized as a submanifold of Euclidean space.Riemannian manifoldisometricallyEuclidean space

Nash–Kuiper theorem (C 1 embedding theorem) n Theorem. Let (M,g) be a Riemannian manifold and ƒ: M m → R n a short C ∞ -embedding (or immersion) into Euclidean space R n, where n ≥ m+1. Then for arbitraryshortimmersion n ε > 0 there is an embedding (or immersion) ƒ ε : M m → R n which is  (i) in class C 1,  (ii) isometric: for any two vectors v,w ∈ T x (M) in the tangent space at x ∈ M,tangent space  (iii) ε-close to ƒ:|ƒ(x) − ƒ ε (x)| < ε for all x ∈ M.

References  bes_Nash,_Jr. bes_Nash,_Jr.  The movie “A Beautiful Mind”  les/PMC /?page=1 les/PMC /?page=1 

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