  We study a version of the lights out game played on directed graphs. For a
digraph $D$, we begin with a labeling of $V(D)$ with elements of $\mathbb{Z}_k$
for $k \ge 2$. When a vertex $v$ is toggled, the labels of $v$ and any vertex
that $v$ dominates are increased by 1 mod $k$. The game is won when each vertex
has label 0. We say that $D$ is $k$-Always Winnable (also written $k$-AW) if
the game can be won for every initial labeling with elements of $\mathbb{Z}_k$.
We prove that all acyclic digraphs are $k$-AW for all $k$, and we reduce the
problem of determining whether a graph is $k$-AW to the case of strongly
connected digraphs. We then determine winnability for tournaments with a
minimum feedback arc set that arc-induces a directed path or directed star
digraph.
