# Understanding the Nash equilibrium for quadratic utilities

I need help to understand some steps of the article "Who's Who in Networks. Wanted: The Key Player" 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.

• 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". Apr 6, 2023 at 7:34
• "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? Apr 6, 2023 at 7:35
• "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. Apr 6, 2023 at 7:36
• All in all, it is not quite clear what your precise, concise and clearly answerable question is. Apr 6, 2023 at 7:37
• 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. Apr 6, 2023 at 7:38

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.
• 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, 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. Apr 11, 2023 at 7:24