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Phase transition in a sequential assignment problem on graphs

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.