Nash's Theorem proof in 2by2 games

[Bipolarity]

According to Nash's Theorem, every game has at least one Nash Equilibrium, whether that be a pure strategy or a mixed strategy Nash equilibrium. However, I have not been able to find a proof for the theorem.

I am looking for a proof of the theorem in 2by2 games involving simultaneous strategies. Perhaps someone here knows good places where these proofs can be found? I googled but most seem to explain the theorem rather superficially without a rigorous mathematical approach.

Thanks!

BiP

 

[mathman]

http://en.wikipedia.org/wiki/Nash_equilibrium#Proof_using_the_Kakutani_fixed_point_theorem

Did you google "nash's theorem proof"?

 

[Bipolarity]

Yep! The proofs that show up on google generalize it to 'n' players each having many strategies.

I was looking for a short proof on the simple case of 2by2 games with only 2 players. It should proof the existence of at least one Nash equilibrium.

BiP

 

[Bacle2]

Have you tried a search of the Minimax Theorem?

http://en.wikipedia.org/wiki/Minimax_theorem#Minimax_theorem

 

[CellCoree]

olarBear, as a scientist, I can understand your frustration in not being able to find a rigorous mathematical proof for Nash's Theorem in 2by2 games. However, it is important to note that Nash's Theorem is a fundamental result in game theory and has been extensively studied and proven by various mathematicians and economists over the years.

One of the earliest proofs of Nash's Theorem can be found in John Nash's original paper, "Equilibrium Points in N-Person Games," published in 1950. In this paper, Nash provides a proof for the existence of a mixed strategy Nash equilibrium in any finite game.

Since then, many other mathematicians and economists have also provided their own proofs for Nash's Theorem in 2by2 games. These include the proofs by John Harsanyi and Reinhard Selten in their paper "A General Theory of Equilibrium Selection in Games" (1988), as well as the proof by Ariel Rubinstein in his paper "Equilibrium in Supergames with the Same Number of Players and a Continuum of Strategies" (1989).

Furthermore, there are various textbooks and online resources that provide detailed and rigorous proofs for Nash's Theorem in 2by2 games. Some recommended resources include "Game Theory: An Introduction" by Steven Tadelis, "Game Theory for Applied Economists" by Robert Gibbons, and the lecture notes on game theory by Professor Kevin Leyton-Brown from the University of British Columbia.

In summary, while it may be challenging to find a single comprehensive proof for Nash's Theorem in 2by2 games, there are many reliable sources available that provide rigorous and detailed proofs for this fundamental result in game theory. I encourage you to continue your search and consult these resources for a better understanding of Nash's Theorem.