Theorem 3 asserts that, for ordered target rates, there are structured prices under which an optimal policy accepts every surge trip and all non-surge trips below a cutoff. In its higher-ratio regime it further gives the accept-all conclusion.
The printed Lemma 9 fixes a non-surge policy before it selects a feasible surge price ratio. Its conclusion therefore has the form “for each policy, there is a feasible ratio.” The subsequent Theorem 3 argument needs the stronger conclusion that one chosen ratio works against every policy. The clarification concerns the quantifier order required by that proof route.
A direct proof selects one structured price before the optimality comparison and splits the target-rate range into two cases. Above an explicit threshold, the fixed-price Bellman comparison gives accept-all. At or below that threshold, the non-surge cutoff is zero (the empty non-surge policy) and the surge policy is accept-all; the resulting statewise inequalities give optimality. The paper’s cutoff domain includes zero and permits the required sign of the non-surge adjustment.
Thus the stated Theorem 3 conclusion remains valid on its source policy domain. The direct proof changes the route used to establish that conclusion.