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

ConfessConfess DenyC

onfe

ssD

eny

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