of a unique Nash equilibrium in games with linear best response functions in terms of the eigenvalues of the network of influences. We enumerate below some sufficient conditions that are related with our work:

\[Kranton condition\]Consider the class of games with linear best response functions described in Section \[Game Description\]. For these games, we have the following sufficient conditions for the existence of a unique Nash equilibrium:

(*i*)

:   $\delta <-1/\lambda _{n}(A_{\mathcal{G}})$, (Bramoullé et al., [@BKA10]).

(*ii*)

:   $\delta <1/\rho \left( A_{\mathcal{G}}\right) $, (Ballester et al., [@BC07]).

We can compare the set of spectral conditions in Proposition \[Kranton condition\] using the following inequalities [@BKA10]:

For any simple graph $\mathcal{G}$, we have that $-1/\lambda _{n}(A_{%
\mathcal{G}})\geq 1/\rho \left( A_{\mathcal{G}}\right) $, where this inequality is strict when no component of $\mathcal{G}$ is bipartite.

\[Active Remark\]Hence, Condition (*i*) in Proposition \[Kranton condition\] provides the best sufficient condition for the uniqueness of Nash equilibrium in these games. Furthermore, one can also prove that under Condition (i) or (ii), all players are active at the equilibrium point, i.e., $S=\mathcal{N}$ [@BKA10].

In Section \[Optimal Bounds\], we shall derive upper bounds on $\lambda
_{n}\left( A_{\mathcal{G}}\right) $ in terms of structural properties of the network. These bounds, in 