  Given a simple graph $G$, one can define a hyperplane arrangement called the
$G$-Shi arrangement. The Pak-Stanley algorithm labels the regions of this
arrangement with $G_\bullet$-parking functions. When $G$ is a complete graph,
we recover the Shi arrangement, and the Pak-Stanley labels give a bijection
with ordinary parking functions. However, for proper subgraphs $G \subset K_n$,
while the Pak-Stanley labels still include every $G_{\bullet}$-parking
function, some appear more than once. These repetitions of Pak-Stanley labels
are a topic of interest in the study of $G$-Shi arrangements and
$G_{\bullet}$-parking functions. Furthermore, $G_{\bullet}$-parking functions
are connected to many other combinatorial objects (for example, superstable
configurations in chip-firing). In studying these repetitions, we can draw on
existing results about these objects such as Dhar's Burning Algorithm.
Conversely, our results have implications for the study of these objects as
well. The key insight of our work is the introduction of a combinatorial model
called the Three Rows Game. Analyzing the histories of this game and how they
induce identical outcomes lets us characterize the multiplicities of the
Pak-Stanley labels. Using this model, we develop a classification theorem for
the multiplicities of the Pak-Stanley labels of the regions in the $P_n$-Shi
arrangement, where $P_n$ is the path graph on $n$ vertices. Then, we generalize
the Three Rows Game into the $T$-Three Rows Game. This allows us to study the
multiplicities of the Pak-Stanley labels of the regions in $T$-Shi
arrangements, where $T$ is any tree. Finally, we discuss the possibilities and
difficulties in applying our method to arbitrary graphs. In particular, we
analyze multiplicities in the case when $G$ is a cycle graph, and prove a
uniqueness result for maximal $G_{\bullet}$-parking functions for all graphs
using the Three Rows Game.
