# Proving the existence of Nash Equilibrium using alternate approaches

Most of the standard books/papers/reading materials prove/state the existence of a Nash Equilibrium by appealing to Sperner's Lemma, or to Brouwer's/Kakutani's FPT. However, I've recently come to know that the existence can be proved in other ways, although I wasn't able to find any relevant material regarding the same.

My question is, that is it really possible to prove the existence of a Nash Equilibrium using some other result(s)?

I would really appreciate any kind of help.

• I have edited my question. Oct 3 '18 at 7:43
• @denesp So finding NE essentially means solving an optimization problem, right? Oct 3 '18 at 7:45
• In somewhat mathematical terms, is finding a 'fixed point' some sort an optimization problem? Oct 3 '18 at 7:47
• On editing answered questions Oct 3 '18 at 8:05
• This question on MO might be of interest. See, in particular, Anthony Mendes’ answer. Oct 3 '18 at 14:57

It seems unlikely that an elementary proof exists, because the Kakutani fixed point theorem follows from the theorem proving the existence of Nash equilibria. So this is a strong theorem.

The rest of this answer has been made before the question has been edited to specifically exclude it. (After the answer was already posted and the OP read it.)

There are several ways the existence of NE's in finite games can be proven.

Or, one can do so using Sperner's lemma, stopping by at Brouwer's as an intermediate step. Sperner, Brouwer and Nash

• But isn't Kakutani's FPT a generalization of Brouwer's FPT? Oct 3 '18 at 7:54
• Does my answer contradict this in any way? Oct 3 '18 at 7:55
• No, but Brouwer's FPT and Kakutani's FPT are the standard results discussed in classrooms, and standard textbooks like MWG, A Course in Game Theory by Rubinstein, etc. My question is, are there any other theorems/results one can use to prove the existence of NE? Oct 3 '18 at 7:59
• No, your question is whether there are theorems other than Kakutani that can be used. I gave you two. I also answered why a more elementary proof seems unlikely. Oct 3 '18 at 8:01
• Sorry for editing the question after your answer. I did it unintentionally, I don't mean to offend anyone. Oct 3 '18 at 8:15