Proposition 1, p. 12 — informative A-Learning equilibrium iff θH/θL ≥ Δ

Consider Guo's two-type A-Learning customization game with $\theta_H>\theta_L>0$, $\alpha,\gamma\in(0,1)$, and $k,s>0$. Buyers send costless messages, the seller chooses a nonnegative quality that maximizes its expected payoff given the message-induced belief, and each type chooses its best message. Informative communication means that the two types send different messages in a perfect Bayesian equilibrium.

Such an equilibrium exists if and only if

This characterizes when cheap talk can credibly reveal buyer preferences and support type-dependent customization.

Formalization Note The two-message game has on-path type-consistent beliefs, attained seller optimization over $q\geq0$, and buyer optimality against both messages. The price-setting stage is represented by equations (4)–(5). The weak boundary is included, as in Proposition 1.