On positionality of trigger strategies Nash Equilibria in SCAR

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.