  We study the positionality of \emph{trigger strategies} Nash equilibria
$\overline{\sigma}$ for the $N$-player SCAR games
$\Gamma_{N}(G|s_{0},\gamma,\varepsilon)$ (with $N\geq3$). Our study is
exhaustive with respect to types of graphs $G$, initial states $s_{0}$ and
values of $N,\gamma,\varepsilon$. We conclude that in the majority of cases,
profiles $\overline{\sigma}$ are nonpositional. Whenever $\overline{\sigma}$
are positional a key role is played by paths and the $\varepsilon$, $\gamma$
values (especially whether $\varepsilon>0$ or not). A crucial concept in our
analysis is the \emph{state cop number}, which is first introduced in the
current paper.
