On positionality of trigger strategies Nash Equilibria in SCAR

Abstract

We study the positionality of \emph{trigger strategies} Nash equilibria $\overline{\sigma }$ for the $N$-player SCAR games ${\Gamma }_N(G|s_0,\gamma ,\epsilon )$ (with $N\ge 3$). Our study is exhaustive with respect to types of graphs $G$, initial states $s_0$ and values of $N,\gamma ,\epsilon$. 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 $\epsilon$, $\gamma$ values (especially whether $\epsilon >0$ or not). A crucial concept in our analysis is the \emph{state cop number}, which is first introduced in the current paper.