  We study outer billiards with contraction outside regular polygons. For
regular $n$-gons with $n = 3, 4, 5, 6, 8$, and $12$, we show that as the
contraction rate approaches $1$, dynamics of the system converges, in a certain
sense, to that of the usual outer billiards map. These are precisely the values
of $n \geq 3$ with $[\mathbb{Q}(e^{2\pi i/n}):\mathbb{Q}] \leq 2$. Then we
discuss how such convergence may fail in the case of $n=7$.
