Proof of Proposition 7, p. 29 — Δ_H < Λ_{H2}

Let $(\theta_H,\theta_L,\alpha,\gamma,k,s)$ satisfy the standing assumptions $\theta_H>\theta_L>0$, $\alpha,\gamma\in(0,1)$, $k>0$, $s>0$. Then the high type's A-Learning threshold is strictly smaller than the second high-type O-Learning threshold:

Together with $\Delta_H<\Lambda_{H1}$ this gives $\Delta_H<\min\{\Lambda_{H1},\Lambda_{H2}\}$, the paper's statement that the high-type buyers' motivation for truthful revelation is higher under A-Learning than under O-Learning.