  We construct two planar homeomorphisms $f$ and $g$ for which the origin is a
globally asymptotically stable fixed point whereas for $f \circ g$ and $g \circ
f$ the origin is a global repeller. Furthermore, the origin remains a global
repeller for the iterated function system generated by $f$ and $g$ where each
of the maps appears with a certain probability. This planar construction is
also extended to any dimension greater than 2 and proves for first time the
appearance of the Parrondo's dynamical paradox in odd dimensions.
