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.