On positionality of trigger strategies Nash Equilibria in SCAR

TL;DR

This paper investigates the positionality of trigger strategies Nash equilibria in N-player SCAR games, revealing that most are nonpositional and highlighting the influence of graph paths and parameters like epsilon and gamma.

Contribution

Introduces the concept of the state cop number and provides an exhaustive analysis of when trigger strategies are positional in SCAR games.

Findings

Most trigger strategy Nash equilibria are nonpositional.

Paths and parameters epsilon, gamma influence positionality.

The state cop number is a key analytical tool.

Abstract

We study the positionality of \emph{trigger strategies} Nash equilibria $\overline{\sigma}$ for the $N$-player SCAR games $\Gamma_{N}(G|s_{0},\gamma,\varepsilon)$ (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.