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.