A fundamental problem in high-dimensional regression is to understand the trade-off between type I and type II errors or, equivalently, false discovery rate (FDR) and power in variable selection. To address this important problem, we offer the first complete diagram that distinguishes all pairs of FDR and power that can be asymptotically realized by the Lasso from the remaining pairs, in a regime of linear sparsity under random designs. The trade-off between the FDR and power characterized by our diagram holds no matter how strong the signals are. In particular, our results complete the earlier Lasso trade-off diagram of Su et al. (2017) by recognizing two simple constraints on the pairs of FDR and power. The improvement is more substantial when the regression problem is above the Donoho–Tanner phase transition. Finally, we present extensive simulation studies to confirm the sharpness of the complete Lasso trade-off diagram.