In this paper, we consider some open problems related to \(\textbf{Q}\) -matrix and quitting games raised by Solan and Solan (Math Oper Res 45(2):434–454 , 2020). We revisit the connection between \(\textbf{Q}\) -matrix, \(\epsilon \) -equilibrium, and sunspot \(\epsilon \) -equilibrium; addressing these open problems and presenting some important characterization of the \(\textbf{Q}\) -matrix. Further, we discuss how the concept of principal pivot transform is useful for identifying a \(\textbf{Q}\) -matrix and related characterization of \(\epsilon \) -equilibrium/ sunspot \(\epsilon \) -equilibrium. We obtain a new characterization for sunspot \(\epsilon \) -equilibrium for the class of matrices that belongs to \(\textbf{Q} {\setminus } \textbf{M}.\)
