  The paintability of a graph is a coloring parameter defined in terms of an
online list coloring game. In this paper we ask, what is the paintability of a
graph $G$ of maximum degree $\Delta$ and chromatic number $r$? By considering
the Alon-Tarsi number of $G$, we prove that the paintability of $G$ is at most
$\left(1 - \frac{1}{4r+1} \right ) \Delta + 2$. We also consider the
DP-paintability of $G$, which is defined in terms of an online DP-coloring
game. By considering the strict type $3$ degeneracy parameter recently
introduced by Zhou, Zhu, and Zhu, we show that when $r$ is fixed and $\Delta$
is sufficiently large, the DP-paintability of $G$ is at most $\Delta - \Omega(
\sqrt{\Delta \log \Delta})$.
