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.
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.

One can use Brouwer's fixed point theorem.

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