<p style='text-indent:20px;'>We consider the logistic family <inline-formula><tex-math id="M2">\begin{document}$ f_{a} $\end{document}</tex-math></inline-formula> and a family of homeomorphisms <inline-formula><tex-math id="M3">\begin{document}$ \phi _{q} $\end{document}</tex-math></inline-formula>. The <inline-formula><tex-math id="M4">\begin{document}$ q $\end{document}</tex-math></inline-formula>-deformed system is given by the composition map <inline-formula><tex-math id="M5">\begin{document}$ f_{a}\circ \phi _{q} $\end{document}</tex-math></inline-formula>. We study when this system has non zero fixed points which are LAS and GAS. We also give an alternative approach to study the dynamics of the <inline-formula><tex-math id="M6">\begin{document}$ q $\end{document}</tex-math></inline-formula>-deformed system with special emphasis on the so-called Parrondo's paradox finding parameter values <inline-formula><tex-math id="M7">\begin{document}$ a $\end{document}</tex-math></inline-formula> for which <inline-formula><tex-math id="M8">\begin{document}$ f_{a} $\end{document}</tex-math></inline-formula> is simple while <inline-formula><tex-math id="M9">\begin{document}$ f_{a}\circ \phi _{q} $\end{document}</tex-math></inline-formula> is dynamically complicated. We explore the dynamics when several <inline-formula><tex-math id="M10">\begin{document}$ q $\end{document}</tex-math></inline-formula>-deformations are applied.</p>