all Nash equilibria, one can check the conditions in Proposition \[Nash Conditions\] for each one of the $2^{n}$ possibilities of $S$. For each possible $S$, these conditions can be checked by computing $\mathbf{x}_{S}=(I_{\left\vert S\right\vert
}+\delta A_{S})^{-1}\mathbf{1}_{\left\vert S\right\vert }$, and checking whether $\delta A_{\mathcal{N}\backslash S,S}~\mathbf{x}%
_{S}\geq \mathbf{1}_{n-\left\vert S\right\vert }$. If the last inequality holds, then $\mathbf{x}_{S}$ is an equilibrium outcome. Note that using this approach to compute the set of equilibria runs in exponential time. However, we can analyze some properties of the Nash equilibria, such as uniqueness and stability, by looking into the eigenvalues of the adjacency matrix.

The Shape of Nash Equilibria
----------------------------

In order to relate the equilibrium outcomes of the game to the network structure, the authors in [@BKA10] defined the following potential function:$$\varphi (\mathbf{x};\delta ,A_{\mathcal{G}})\triangleq \sum_{i=1}^{n}\left(
x_{i}-\frac{1}{2}x_{i}^{2}\right) -\frac{1}{2}\delta
\sum_{i=1}^{n}\sum_{j=1}^{n}a_{ij}x_{i}x_{j}.$$Then, they proved, using Kuhn-Tucker conditions, that the set of Nash equilibria coincides with the critical points of the following optimization problem:$$\begin{array}{lll}
(\text{P}) & \max_{\mathbf{x}} & \varphi (\mathbf{x};\delta ,A_{\mathcal{G}})
\\ 
& s.t. & x_{i}\geq 0,\text{ for all }i,%
\end{array}%$$for a given network structure $A_{\mathcal{G}}$ and a parameter $\delta $.

Eigenvalues and Nash Equilibria
-------------------------------

We can find several results in the literature providing sufficient conditions for the existence 