JOHN FORBES NASH
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Transcript of JOHN FORBES NASH
JOHN FORBES NASH
Meryem DİLEKCAN200822016
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 and
initially majored in Chemical Engineering
John F. Nash, 1928 -
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 1994 in 2012 he became a fellow of the
American Mathematical Society.
The Studies and Contributions of
Nash 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 (P1; … ; PN): Finite number (N≥2) of
decision makers. Action sets (A1; … ;AN): player Pi has a nonempty
set Ai of actions. Payoff functions ui : A1x … xAN: R; i = 1;….;N
- materialize players’ preference, - take a possible action profile and assign to
it areal number (von Neumann-Morgenstern).
Prisoner’s DilemmaAn illustration of Nash Equilibrium
Art’s Strategies
Bob’s Strategies
Confess Confess Deny
Confess
Deny
10 yrs.
1 yr.
3 yrs.
3 yrs.
1 yr.
10 yrs.
2 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 …
www.youtube.com/watch?v=8YuJDxSvL8I
Also… Nash imbedding theorem
shows that any abstract Riemannian manifold can be isometrically realized as a submanifold of Euclidean space.
Nash–Kuiper theorem (C1 embedding theorem)
Theorem. Let (M,g) be a Riemannian manifold and ƒ: Mm → Rn a short C∞-embedding (or immersion) into Euclidean space Rn, where n ≥ m+1. Then for arbitrary
ε > 0 there is an embedding (or immersion) ƒε: Mm → Rn which is (i) in class C1, (ii) isometric: for any two vectors v,w T∈ x(M) in the
tangent space at x M,∈ (iii) ε-close to ƒ:|ƒ(x) − ƒε(x)| < ε for all x M.∈
References
http://en.wikipedia.org/wiki/John_Forbes_Nash,_Jr.
The movie “A Beautiful Mind” http://www.ncbi.nlm.nih.gov/
pmc/articles/PMC1063129/?page=1
http://inside.bilgi.edu.tr/read/656/
Thanks for your
listening