<p>What you describe as being used to seeing are bimatrix games, i.e. 2-person games where each player has only a finite (and usually small) number of pure strategies. What you have here is instead an <span class="math-container">$n$</span>-person game where each strategy space is a continuum. These cannot be written in bimatrix form, and the matrix of cross effects <span class="math-container">$\Sigma$</span> is a very different thing than a payoff bimatrix.</p>
<p>However, given the specific structure of the payoff functions, the matrix of cross effects, together with the vector of <span class="math-container">$\alpha_i$</span>'s, uniquely identifies the payoff functions and therefore the game. Thus, to find the NE or to prove its existence, <span class="math-container">$\Sigma$</span> is everything you need.</p>

CC BY-SA 4.0
2023-04-06T14:52:22.887Z
Thank you very much for answering, as would be the basis for that last thing you say "given the specific structure of the payoff functions, the matrix of cross effects, together with the vector of $\alpha_i$'s, uniquely identifies the payoff functions and therefore the game" That is my question, why or how is it that from there we can identify balances? Do you have any reference that I can read or an example with which I can illustrate?
Haus
CC BY-SA 4.0
2023-04-07T17:13:56.413Z
@Haus I'm not sure what you mean with "balances"...?
VARulle
CC BY-SA 4.0
2023-04-07T17:42:04.723Z
Sorry for my mistake, I wanted to write equilibrium's
Haus
CC BY-SA 4.0
2023-04-11T07:24:04.500Z
@Haus, identifying equilibria works as usual: You look for mutual best replies. Here this means that you focus on a player $i$, treat all $x_j$'s for $j\ne i$ as constants and maximize $u_i(x_i)$ by setting $\frac{\partial u_i}{\partial x_i}=0$ and solving for $x_i$. Doing this for all $i$'s gives you a system of linear equations (containing the cross effects parameters and the $\alpha_i$'s). The solution(s) are the NE.
VARulle
CC BY-SA 4.0
2023-04-06T11:38:32.727Z
2023-04-06T11:38:32.727Z
VARulle
<p>I need help to understand some steps of the article &quot;<a href="https://www.jstor.org/stable/3805930" rel="nofollow noreferrer">Who's Who in Networks. Wanted: The Key Player</a>&quot; and I would greatly appreciate if someone can provide me with references or if they can answer my questions.</p>
<p>My problem is focused on understanding the game. I've just taken my first game theory course and I've never seen anything like it. Here we describe a game with n players where each player selects an effort x_i and gets a bilinear payoff given by the following function:</p>
<p><span class="math-container">$$
u_i(x_1,\dots,x_n)=\alpha_ix_i+\frac{1}{2}\sigma_{ii}x_i^2+\sum_{j\neq i}\sigma_{ij}x_ix_j,
$$</span></p>
<p>which is strictly concave in own effort, that is <span class="math-container">$\frac{\partial^2u_i}{\partial x_i^2}=\sigma_{ii}&lt;0.$</span> Then form the matrix <span class="math-container">$\Sigma=[\sigma_{ij}]$</span></p>
<p>Finally, state the following theorem (theorem 1):</p>
<blockquote>
<p><strong>Theorem 1.</strong> Under some conditions, the game <span class="math-container">$\Sigma$</span> has a unique Nash equilibrium.</p>
</blockquote>
<p>Here comes my question, why we can obtain the Nash equilibrium from the matrix of crossed effects, I am usually used to seeing the matrix of the game as a matrix where each entry is made up of a vector that has in each entry the payments corresponding to each player. However, I do not understand why it is enough to analyze the cross effects matrix and what is the advantage or need of doing so? Is this used in the standard bibliography? How do I identify a Nash equilibrium using this type of representation?</p>
<p>I appreciate any response, hoping you can help me.</p>

CC BY-SA 4.0
2023-04-06T07:34:18.673Z
As currently stated, Theorem 1 is not a theorem. It is akin to "under some conditions, something is true". Without adding details, it is impossible to tell "*why it is enough to analyze the cross effects matrix*".
Giskard
CC BY-SA 4.0
2023-04-06T07:35:38.580Z
"*what is the advantage or need of doing so*" Advantage of doing so compared to what...? What is it that you are trying to do?
Giskard
CC BY-SA 4.0
2023-04-06T07:36:45.963Z
"*How do I identify a Nash equilibrium using this type of representation?*" I am not sure what you mean here. Theorem 1 does not claim to identify an equilibrium, it merely claims that one exists and that it is unique. The exact values of the equilibrium likely depend on the $\sigma_{ii}$'s as well.
Giskard
CC BY-SA 4.0
2023-04-06T07:37:07.243Z
All in all, it is not quite clear what your precise, concise and clearly answerable question is.
Giskard
CC BY-SA 4.0
2023-04-06T07:38:19.497Z
**If** you want to find **a** set of conditions under which Theorem 1 is true, you should probably have a look at the other fixed point/NE-existence theorems that you have studied so far.
Giskard
CC BY-SA 4.0
2023-04-06T14:49:33.543Z
@Giskard Thank you very much for commenting. Very surely what I asked was not understood, I did not want to state the complete theorem, because my doubt is not about what the theorem says, but about how or why the cross-effects matrix manages to find or identify equilibria.
Haus
<microeconomics><game-theory><reference-request><utility><nash-equilibrium>
Understanding the Nash equilibrium for quadratic utilities
Haus
# Understanding the Nash equilibrium for quadratic utilities

(Asked by: username_0 on 2023-04-06)

I need help to understand some steps of the article "[Who's Who in Networks. Wanted: The Key Player](https://www.jstor.org/stable/3805930)" and I would greatly appreciate if someone can provide me with references or if they can answer my questions.

My problem is focused on understanding the game. I've just taken my first game theory course and I've never seen anything like it. Here we describe a game with n players where each player selects an effort x\_i and gets a bilinear payoff given by the following function:

$$
u\_i(x\_1,\dots,x\_n)=\alpha\_ix\_i+\frac{1}{2}\sigma\_{ii}x\_i^2+\sum\_{j\neq i}\sigma\_{ij}x\_ix\_j,
$$

which is strictly concave in own effort, that is $\frac{\partial^2u\_i}{\partial x\_i^2}=\sigma\_{ii}<0.$ Then form the matrix $\Sigma=[\sigma\_{ij}]$

Finally, state the following theorem (theorem 1):

> **Theorem 1.** Under some conditions, the game $\Sigma$ has a unique Nash equilibrium.

Here comes my question, why we can obtain the Nash equilibrium from the matrix of crossed effects, I am usually used to seeing the matrix of the game as a matrix where each entry is made up of a vector that has in each entry the payments corresponding to each player. However, I do not understand why it is enough to analyze the cross effects matrix and what is the advantage or need of doing so? Is this used in the standard bibliography? How do I identify a Nash equilibrium using this type of representation?

I appreciate any response, hoping you can help me.

**Tags:** microeconomics, game-theory, reference-request, utility, nash-equilibrium

> **Comments:**
> **username_1**: As currently stated, Theorem 1 is not a theorem. It is akin to "under some conditions, something is true". Without adding details, it is impossible to tell "*why it is enough to analyze the cross effects matrix*".  
> **username_1**: "*what is the advantage or need of doing so*" Advantage of doing so compared to what...? What is it that you are trying to do?  
> **username_1**: "*How do I identify a Nash equilibrium using this type of representation?*" I am not sure what you mean here. Theorem 1 does not claim to identify an equilibrium, it merely claims that one exists and that it is unique. The exact values of the equilibrium likely depend on the $\sigma_{ii}$'s as well.  
> **username_1**: All in all, it is not quite clear what your precise, concise and clearly answerable question is.  
> **username_1**: **If** you want to find **a** set of conditions under which Theorem 1 is true, you should probably have a look at the other fixed point/NE-existence theorems that you have studied so far.  
> **username_0**: @username_1 Thank you very much for commenting. Very surely what I asked was not understood, I did not want to state the complete theorem, because my doubt is not about what the theorem says, but about how or why the cross-effects matrix manages to find or identify equilibria.  

---

## 1 answers

### Answer by username_2 on 2023-04-06. Score: 2 | Accepted answer

What you describe as being used to seeing are bimatrix games, i.e. 2-person games where each player has only a finite (and usually small) number of pure strategies. What you have here is instead an $n$-person game where each strategy space is a continuum. These cannot be written in bimatrix form, and the matrix of cross effects $\Sigma$ is a very different thing than a payoff bimatrix.

However, given the specific structure of the payoff functions, the matrix of cross effects, together with the vector of $\alpha\_i$'s, uniquely identifies the payoff functions and therefore the game. Thus, to find the NE or to prove its existence, $\Sigma$ is everything you need.

> **Comments:**
> **username_0**: Thank you very much for answering, as would be the basis for that last thing you say "given the specific structure of the payoff functions, the matrix of cross effects, together with the vector of $\alpha_i$'s, uniquely identifies the payoff functions and therefore the game" That is my question, why or how is it that from there we can identify balances? Do you have any reference that I can read or an example with which I can illustrate?  
> **username_2**: @username_0 I'm not sure what you mean with "balances"...?  
> **username_0**: Sorry for my mistake, I wanted to write equilibrium's  
> **username_2**: @username_0, identifying equilibria works as usual: You look for mutual best replies. Here this means that you focus on a player $i$, treat all $x_j$'s for $j\ne i$ as constants and maximize $u_i(x_i)$ by setting $\frac{\partial u_i}{\partial x_i}=0$ and solving for $x_i$. Doing this for all $i$'s gives you a system of linear equations (containing the cross effects parameters and the $\alpha_i$'s). The solution(s) are the NE.  

---

