  We study the following game on a finite graph $G = (V, E)$. At the start,
each edge is assigned an integer $n_e \ge 0$, $n = \sum_{e \in E} n_e$. In
round $t$, $1 \le t \le n$, a uniformly random vertex $v \in V$ is chosen and
one of the edges $f$ incident with $v$ is selected by the player. The value
assigned to $f$ is then decreased by $1$. The player wins, if the configuration
$(0, \dots, 0)$ is reached; in other words, the edge values never go negative.
Our main result is that there is a phase transition: as $n \to \infty$, the
probability that the player wins approaches a constant $c_G > 0$ when $(n_e/n :
e \in E)$ converges to a point in the interior of a certain convex set
$\mathcal{R}_G$, and goes to $0$ exponentially when $(n_e/n : e \in E)$ is
bounded away from $\mathcal{R}_G$. We also obtain upper bounds in the
near-critical region, that is when $(n_e/n : e \in E)$ lies close to $\partial
\mathcal{R}_G$. We supply quantitative error bounds in our arguments.
