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We look at both the \emph{group labeling lights out game} and the \emph{neighborhood lights out game}. Our main focus is to determine necessary and sufficient conditions for when the group labeling lights out game on path graphs, cycle graphs, and complete bipartite graphs can be won for every possible initial labeling. In the process of solving this problem, we demonstrate a new proof for when the neighborhood lights out game on complete bipartite graphs can be won for every possible initial labeling.

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.