<p>To prove that that some given combination of strategies is a Nash equilibrium you don't <em>need</em> to use a fixed-point theorem (such as Brouwer's or the fixed-point theorem for contractions on Banach spaces).
What you do have to do, is check that they are best responses to one another.
This true for mixed and pure strategies.</p>
<p>You also seem to be asking how you can use the Banach fixed-point theorem to prove existence of a pure strategy equilibrium.
As @Giskard pointed out, this is doomed to failure without restrictions on the game, as not every game admits a pure strategy equilibrium.</p>
<p>If you wanted to try anyway:
Let <span class="math-container">$S_{i}$</span> denote <span class="math-container">$i$</span>'s set of pure strategies. Let <span class="math-container">$X = \times_{i} S_{i}$</span> and define <span class="math-container">$T \colon X \to X$</span> as follows: <span class="math-container">$T(x) = (T_{i}(x))_{i}$</span> where <span class="math-container">$T_{i}(x)$</span> is some pure strategy of the best response correspondence of agent <span class="math-container">$i$</span> to the strategy profile <span class="math-container">$x_{-i}$</span> of the others.
(If <span class="math-container">$T$</span> is supposed to be a contraction on <span class="math-container">$X$</span>, then it has to map to <span class="math-container">$X$</span>, not the power set. Note, however, that this selection is irrelevant if the game happens to be such that best responses are always unique.)
A profile of pure strategies is a Nash equilibrium if it is a fixed-point of <span class="math-container">$T$</span>.
Conversely, for every Nash equilibrium <span class="math-container">$x$</span>, there is a way to define the selection from the correspondence such that <span class="math-container">$x$</span> is a fixed-point of the thusly defined mapping <span class="math-container">$T$</span>.</p>
<p>The difficult in this approach thus lies in finding a selection from the best response correspondence together with a metric such that <span class="math-container">$X$</span> is a complete metric space on which <span class="math-container">$T$</span> is a contraction.</p>
<p>But this is not possible in general: Not every game admits a pure strategy Nash equilibirum, as pointed out by @Giskard. Not only that: Not every game which does admit such an equilibrium has a unique one (consider the standard coordination game, where best responses are always unique but which has multiple pure strategy equilibria).
There is simply no good reason (at least one that I can see) why, without restrictions on the game, best responses would allow you to define a contraction.
(But perhaps someone knows a class of games where this is actually the case.)</p>
<p>All of this is to say that there is no existence result for pure strategy Nash equilibria in the generality that you seem to describe.
Results for games in which such pure equilibria can be proven to existence abstractly (e.g. via Kakutani's fixed-point theorem) rely on more special features of the game (e.g. in Athey (2001), certain single-crossing properties are key).
What tools might be useful to you will depend, to a degree, on the details of your problem.</p>

CC BY-SA 4.0
2020-08-02T07:35:38.563Z
thank you for your answer. Well, I have a problem, where the game is played in the demand scedules, where I parametrie a part of the intercept point in particoular. Within this class of games, that is common way of introducing some strategic behavior in demands, I have not seen to much work of trying to prove the existence, maybe beacause they do not have restrictions? What type of restrictions? On the other hand, I refer to Banach fixed point theorem since I work in $\mathbb{L}^2$. However, I have cited a link of a simplified version of my problem in the commets above
Nav89
CC BY-SA 4.0
2020-08-02T07:37:19.347Z
I do not know if you are in mood to take a look...
Nav89
CC BY-SA 4.0
2020-08-03T06:29:41.447Z
@Tomcat you can check again the link
https://math.stackexchange.com/questions/3773106/how-can-i-build-the-banach-fixed-point-argument-when-the-set-of-strategic-choice
I have written down the answer
Nav89
CC BY-SA 4.0
2020-08-02T07:15:04.690Z
2020-08-02T07:15:04.690Z
induction_is_a_laddah
<p>To begin with, I am recalling the Banach Fixed Point Theorem.</p>
<p>Let <span class="math-container">$(X,d)$</span> be a non-empty complete metric space with a contraction mapping <span class="math-container">$T:X\to X$</span>. Then <span class="math-container">$T$</span> admits a unique fixed-point <span class="math-container">$x^*$</span> in <span class="math-container">$X$</span> i.e. <span class="math-container">$T(x^*) = x^*$</span>.</p>
<p>In order to prove that the combinantion of the best response strategies of <span class="math-container">$N \geqslant 2$</span> agents consitutes a Nash Equilibrium, we need to use a fixed point theorem argument. Is this necessary only in cases where the agents employ mixed strategies or this holds in the case where they also follow pure strategies? I am searcing for some guidance to use the Banch fixed point theorem in guadratic utility functions when agents act by submitting pure strategies. How can I build this argument? What confuses me the most is that the mapping <span class="math-container">$T:X\to X$</span> is about the strategies isn't it?</p>
<p>I can provide more details of my problem if you wish so. I would also be glad if you could answer with great details, since I do not have any idea about the topic</p>

CC BY-SA 4.0
2020-07-31T17:32:44.840Z
From reading your question I indeed do not understand what you are and aren't familiar with. Have you ever seen a fixed point theorem based proof for the existence of a Nash-equilibrium? If yes, that should tell you a lot about what the space $X$ is. If not, read one? Perhaps also clarify what is that you are trying to do, what is your goal?
Giskard
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2020-07-31T18:23:50.993Z
Well, this $X$ is a complete metric space as the definition says, but in our case it is the set of the cartesian product of the individual sets of strategies, i.e. if we have three agents that compete, that is $X=S_1\times S_2 \times S_3$, where $S_i$ is the strategy set of agent $i$, isnt it?
Nav89
CC BY-SA 4.0
2020-07-31T20:02:35.307Z
I don't really understand whether you want to prove existence of NE in pure strategies for some general case or whether you want to calculate a NE in pure strategies for some class of games? (Or something completely different...)
VARulle
CC BY-SA 4.0
2020-07-31T20:21:04.517Z
I want to prove the existence of NE in pure strategies for a general case
Nav89
CC BY-SA 4.0
2020-07-31T21:26:15.533Z
What is the pure NE for [matching pennies](https://en.wikipedia.org/wiki/Matching_pennies) or rock-paper-scissors?
Giskard
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2020-07-31T21:54:11.607Z
(Or what exactly do you mean by general case, if these games are not included.)
Giskard
CC BY-SA 4.0
2020-07-31T21:54:16.623Z
It is pure NE for rock-paper-scissors
Nav89
CC BY-SA 4.0
2020-07-31T21:55:07.553Z
...what is the pure NE? Rock-rock?
Giskard
CC BY-SA 4.0
2020-07-31T21:56:27.037Z
@Giskard yes this is the case
Nav89
CC BY-SA 4.0
2020-07-31T21:56:40.620Z
Why would player 1 not change to paper?
Giskard
CC BY-SA 4.0
2020-07-31T23:19:41.463Z
Let us [continue this discussion in chat](https://chat.stackexchange.com/rooms/111285/discussion-between-nav89-and-giskard).
Nav89
CC BY-SA 4.0
2020-07-31T23:22:00.397Z
Well @Giskard what about this specific problem https://math.stackexchange.com/questions/3773106/how-can-i-build-the-banach-fixed-point-argument-when-the-set-of-strategic-choice
Have you ever employed something close to this?
Nav89
CC BY-SA 4.0
2020-08-02T06:30:29.857Z
Well @Tomcat, i think (but I am not a specialist) that this comes form the utility faunction that you have. Namely, the contraction mappling, If I am not mistaken is in the following way.

Say $G_i(\cdot)$ is the utility function of the individual agent $i$, $b_i$ is her best response while $b_{-i}$ is the best response strategy of the agent that she competes. Then 
$$b_i^*=\underbrace{Argmax_{b_i}G(b_i;b_{-i})}_{\text{the contraction mapping}\quad T:S \to S}$$
where $S=S_1\times S_2$ is the space (cartesian product) that contains the strategies of every agent.
Nav89
CC BY-SA 4.0
2020-08-02T06:38:01.403Z
@Tomcat but I am not sure about my argument, and also I can not be sure about using the fixed point theorem. I hope someone can confirm my thoughts and writings...You can see that problem here also. This is something that I try to solve...
https://math.stackexchange.com/questions/3773106/how-can-i-build-the-banach-fixed-point-argument-when-the-set-of-strategic-choice
Nav89
CC BY-SA 4.0
2020-08-02T15:13:54.677Z
@Giskard I am more confused than ever! Well, could you please tell me where should I search for a fixed point theorem based proof for the existence of a Nash-equilibrium?
Nav89
<microeconomics><game-theory><mathematical-economics><nash-equilibrium>
How can I build a fixed point theorem argument in pure strategies?
Nav89
# How can I build a fixed point theorem argument in pure strategies?

(Asked by: username_0 on 2020-07-31)

To begin with, I am recalling the Banach Fixed Point Theorem.

Let $(X,d)$ be a non-empty complete metric space with a contraction mapping $T:X\to X$. Then $T$ admits a unique fixed-point $x^\*$ in $X$ i.e. $T(x^\*) = x^\*$.

In order to prove that the combinantion of the best response strategies of $N \geqslant 2$ agents consitutes a Nash Equilibrium, we need to use a fixed point theorem argument. Is this necessary only in cases where the agents employ mixed strategies or this holds in the case where they also follow pure strategies? I am searcing for some guidance to use the Banch fixed point theorem in guadratic utility functions when agents act by submitting pure strategies. How can I build this argument? What confuses me the most is that the mapping $T:X\to X$ is about the strategies isn't it?

I can provide more details of my problem if you wish so. I would also be glad if you could answer with great details, since I do not have any idea about the topic

**Tags:** microeconomics, game-theory, mathematical-economics, nash-equilibrium

> **Comments:**
> **username_1**: From reading your question I indeed do not understand what you are and aren't familiar with. Have you ever seen a fixed point theorem based proof for the existence of a Nash-equilibrium? If yes, that should tell you a lot about what the space $X$ is. If not, read one? Perhaps also clarify what is that you are trying to do, what is your goal?  
> **username_0**: Well, this $X$ is a complete metric space as the definition says, but in our case it is the set of the cartesian product of the individual sets of strategies, i.e. if we have three agents that compete, that is $X=S_1\times S_2 \times S_3$, where $S_i$ is the strategy set of agent $i$, isnt it?  
> **username_2**: I don't really understand whether you want to prove existence of NE in pure strategies for some general case or whether you want to calculate a NE in pure strategies for some class of games? (Or something completely different...)  
> **username_0**: I want to prove the existence of NE in pure strategies for a general case  
> **username_1**: What is the pure NE for [matching pennies](https://en.wikipedia.org/wiki/Matching_pennies) or rock-paper-scissors?  
> **username_1**: (Or what exactly do you mean by general case, if these games are not included.)  
> **username_0**: It is pure NE for rock-paper-scissors  
> **username_1**: ...what is the pure NE? Rock-rock?  
> **username_0**: @username_1 yes this is the case  
> **username_1**: Why would player 1 not change to paper?  
> **username_0**: Let us [continue this discussion in chat](https://chat.stackexchange.com/rooms/111285/discussion-between-nav89-and-giskard).  
> **username_0**: Well @username_1 what about this specific problem https://math.stackexchange.com/questions/3773106/how-can-i-build-the-banach-fixed-point-argument-when-the-set-of-strategic-choice Have you ever employed something close to this?  
> **username_0**: Well @Tomcat, i think (but I am not a specialist) that this comes form the utility faunction that you have. Namely, the contraction mappling, If I am not mistaken is in the following way.  Say $G_i(\cdot)$ is the utility function of the individual agent $i$, $b_i$ is her best response while $b_{-i}$ is the best response strategy of the agent that she competes. Then  $$b_i^*=\underbrace{Argmax_{b_i}G(b_i;b_{-i})}_{\text{the contraction mapping}\quad T:S \to S}$$ where $S=S_1\times S_2$ is the space (cartesian product) that contains the strategies of every agent.  
> **username_0**: @Tomcat but I am not sure about my argument, and also I can not be sure about using the fixed point theorem. I hope someone can confirm my thoughts and writings...You can see that problem here also. This is something that I try to solve... https://math.stackexchange.com/questions/3773106/how-can-i-build-the-banach-fixed-point-argument-when-the-set-of-strategic-choice  
> **username_0**: @username_1 I am more confused than ever! Well, could you please tell me where should I search for a fixed point theorem based proof for the existence of a Nash-equilibrium?  

---

## 1 answers

### Answer by username_3 on 2020-08-02. Score: 1

To prove that that some given combination of strategies is a Nash equilibrium you don't *need* to use a fixed-point theorem (such as Brouwer's or the fixed-point theorem for contractions on Banach spaces).
What you do have to do, is check that they are best responses to one another.
This true for mixed and pure strategies.

You also seem to be asking how you can use the Banach fixed-point theorem to prove existence of a pure strategy equilibrium.
As @username_1 pointed out, this is doomed to failure without restrictions on the game, as not every game admits a pure strategy equilibrium.

If you wanted to try anyway:
Let $S\_{i}$ denote $i$'s set of pure strategies. Let $X = \times\_{i} S\_{i}$ and define $T \colon X \to X$ as follows: $T(x) = (T\_{i}(x))\_{i}$ where $T\_{i}(x)$ is some pure strategy of the best response correspondence of agent $i$ to the strategy profile $x\_{-i}$ of the others.
(If $T$ is supposed to be a contraction on $X$, then it has to map to $X$, not the power set. Note, however, that this selection is irrelevant if the game happens to be such that best responses are always unique.)
A profile of pure strategies is a Nash equilibrium if it is a fixed-point of $T$.
Conversely, for every Nash equilibrium $x$, there is a way to define the selection from the correspondence such that $x$ is a fixed-point of the thusly defined mapping $T$.

The difficult in this approach thus lies in finding a selection from the best response correspondence together with a metric such that $X$ is a complete metric space on which $T$ is a contraction.

But this is not possible in general: Not every game admits a pure strategy Nash equilibirum, as pointed out by @username_1. Not only that: Not every game which does admit such an equilibrium has a unique one (consider the standard coordination game, where best responses are always unique but which has multiple pure strategy equilibria).
There is simply no good reason (at least one that I can see) why, without restrictions on the game, best responses would allow you to define a contraction.
(But perhaps someone knows a class of games where this is actually the case.)

All of this is to say that there is no existence result for pure strategy Nash equilibria in the generality that you seem to describe.
Results for games in which such pure equilibria can be proven to existence abstractly (e.g. via Kakutani's fixed-point theorem) rely on more special features of the game (e.g. in Athey (2001), certain single-crossing properties are key).
What tools might be useful to you will depend, to a degree, on the details of your problem.

> **Comments:**
> **username_0**: thank you for your answer. Well, I have a problem, where the game is played in the demand scedules, where I parametrie a part of the intercept point in particoular. Within this class of games, that is common way of introducing some strategic behavior in demands, I have not seen to much work of trying to prove the existence, maybe beacause they do not have restrictions? What type of restrictions? On the other hand, I refer to Banach fixed point theorem since I work in $\mathbb{L}^2$. However, I have cited a link of a simplified version of my problem in the commets above  
> **username_0**: I do not know if you are in mood to take a look...  
> **username_0**: @Tomcat you can check again the link https://math.stackexchange.com/questions/3773106/how-can-i-build-the-banach-fixed-point-argument-when-the-set-of-strategic-choice I have written down the answer  

---

