<p>It seems unlikely that an elementary proof exists, because the Kakutani fixed point theorem <a href="https://link.springer.com/article/10.1186/s13663-016-0562-z" rel="nofollow noreferrer">follows</a> from the theorem proving the existence of Nash equilibria. So this is a strong theorem.</p>

<hr>

<p>The rest of this answer has been made before the question has been <a href="https://economics.stackexchange.com/posts/24809/revisions">edited</a> to specifically exclude it. (After the answer was already posted and the OP read it.)</p>

<hr>

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

<p><a href="https://en.wikipedia.org/wiki/Nash_equilibrium#Alternate_proof_using_the_Brouwer_fixed-point_theorem" rel="nofollow noreferrer">One can use Brouwer's fixed point theorem</a>.</p>

<p>Or, one can do so using Sperner's lemma, stopping by at Brouwer's as an intermediate step. <a href="https://web.stanford.edu/~saberi/lecture1.pdf" rel="nofollow noreferrer">Sperner, Brouwer and Nash</a></p>

CC BY-SA 4.0
2018-10-03T07:54:05.620Z
But isn't Kakutani's FPT a generalization of Brouwer's FPT?
superhulk
CC BY-SA 4.0
2018-10-03T07:55:06.367Z
Does my answer contradict this in any way?
Giskard
CC BY-SA 4.0
2018-10-03T07:59:04.490Z
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?
superhulk
CC BY-SA 4.0
2018-10-03T08:01:07.220Z
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.
Giskard
CC BY-SA 4.0
2018-10-03T08:05:29.893Z
I've rephrased my question. I'm specifically looking for approaches other than Sperner's Lemma/Brouwer's FPT/Kakutani's FPT.
superhulk
CC BY-SA 4.0
2018-10-03T08:08:23.073Z
That is actually a new question. Rephrasing your question after it has been answered is ungraceful in my opinion. And you wasted my time by making me write my answer to your original question, which you did not mean to ask anyway. So I hope you understand that I don't feel like spending more time on this. Have a nice day.
Giskard
CC BY-SA 4.0
2018-10-03T08:15:53.567Z
Sorry for editing the question after your answer. I did it unintentionally, I don't mean to offend anyone.
superhulk
CC BY-SA 4.0
2018-10-03T07:52:29.063Z
2018-10-03T08:10:46.050Z
2018-10-03T08:10:46.050Z
Giskard
<p>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.</p>

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

<p>I would really appreciate any kind of help.</p>

CC BY-SA 4.0
2018-10-03T07:43:55.860Z
I have edited my question.
superhulk
CC BY-SA 4.0
2018-10-03T07:45:05.347Z
@denesp So finding NE essentially means solving an optimization problem, right?
superhulk
CC BY-SA 4.0
2018-10-03T07:47:32.633Z
In somewhat mathematical terms, is finding a 'fixed point' some sort an optimization problem?
superhulk
CC BY-SA 4.0
2018-10-03T08:05:09.443Z
[On editing answered questions](https://economics.meta.stackexchange.com/questions/1460/editing-answered-questions)
Giskard
CC BY-SA 4.0
2018-10-03T14:57:40.623Z
[This question](https://mathoverflow.net/q/121542/105233) on MO might be of interest. See, in particular, Anthony Mendes’ answer.
Theoretical Economist
<reference-request><mathematical-economics><game-theory><nash-equilibrium>
Proving the existence of Nash Equilibrium using alternate approaches
superhulk
# Proving the existence of Nash Equilibrium using alternate approaches

(Asked by: username_0 on 2018-10-03)

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.

**Tags:** reference-request, mathematical-economics, game-theory, nash-equilibrium
> **Comments:**
> **username_0**: I have edited my question.  
> **username_0**: @denesp So finding NE essentially means solving an optimization problem, right?  
> **username_0**: In somewhat mathematical terms, is finding a 'fixed point' some sort an optimization problem?  
> **username_1**: [On editing answered questions](https://economics.meta.stackexchange.com/questions/1460/editing-answered-questions)  
> **username_2**: [This question](https://mathoverflow.net/q/121542/105233) on MO might be of interest. See, in particular, Anthony Mendes’ answer.  

---

## 1 answers

### Answer by username_1 on 2018-10-03. Score: 3

It seems unlikely that an elementary proof exists, because the Kakutani fixed point theorem [follows](https://link.springer.com/article/10.1186/s13663-016-0562-z) 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](https://economics.stackexchange.com/posts/24809/revisions) 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](https://en.wikipedia.org/wiki/Nash_equilibrium#Alternate_proof_using_the_Brouwer_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](https://web.stanford.edu/~saberi/lecture1.pdf)

> **Comments:**
> **username_0**: But isn't Kakutani's FPT a generalization of Brouwer's FPT?  
> **username_1**: Does my answer contradict this in any way?  
> **username_0**: 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?  
> **username_1**: 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.  
> **username_0**: I've rephrased my question. I'm specifically looking for approaches other than Sperner's Lemma/Brouwer's FPT/Kakutani's FPT.  
> **username_1**: That is actually a new question. Rephrasing your question after it has been answered is ungraceful in my opinion. And you wasted my time by making me write my answer to your original question, which you did not mean to ask anyway. So I hope you understand that I don't feel like spending more time on this. Have a nice day.  
> **username_0**: Sorry for editing the question after your answer. I did it unintentionally, I don't mean to offend anyone.  

---

