Why is A+B Better Than B? A Simple Graph Perspective on Task Transfer

Joint training on an auxiliary task $A$ and a target task $B$ routinely outperforms training on $B$ alone, but the conditions and mechanism behind this gap are poorly understood. We ask *when*, and *by how much*, an auxiliary task boosts target learning. We model the two tasks as goal-conditioned navigation on a shared directed graph and identify a *joint-dominance regime*, where neither task alone covers the full target trajectory but their union does. In this regime joint training preserves teacher-level accuracy along the entire target path while single-task training incurs an exponential penalty in the length of its uncovered region. We verify the theory in a controlled GPT-2 study on two algorithmic tasks sharing a depth-first-search backbone, a Qwen2.5 GRPO study on two related math tasks, and a cross-representation procedural-reasoning setting. The findings translate into three checkable criteria for selecting auxiliary tasks: shared sub-skills, joint horizon coverage, and base-policy solvability. Our code is available at [https://anonymous.4open.science/r/task_transfer_fogen26/](https://anonymous.4open.science/r/task_transfer_fogen26/).