  We introduce a new notion of stability for periodic orbits in polygonal
billiards. We say that a periodic orbit of a polygonal billiard is
$\lambda$-stable if there is a periodic orbit for the corresponding pinball
billiard which converges to it as $\lambda$ $\rightarrow$ 1. This notion of
stability is unrelated to the notion introduced by Galperin, Stepin and
Vorobets. We give sufficient and necessary conditions for a periodic orbit to
be $\lambda$-stable and prove that the set of d-gons having at most finite
number of $\lambda$-stable periodic orbits is dense is the space of d-gons.
Moreover, we also determine completely the $\lambda$-stable periodic orbits in
integrable polygons.
