Many papers on Bayesian persuasion adopt SPE as a solution, by arguing that the sender has no private information. This does not make sense to me: Regardless of whether the sender has private information, we need to restrict the belief of the receiver about the state of the world after the receive observes an off-path disclosure policy, which SPE has no bite. Should we understand SPE in Bayesian persuasion literature as something that does not really mean subgame perfect equilibrium?
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$\begingroup$ No off-path messages can be sent due to the commitment of the sender. $\endgroup$– user25110Jun 20 '20 at 2:02
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Here are my thoughts. Because the sender commits to a disclosure policy, how you stipulate off-path beliefs doesn't matter (phrased loosely). That is, the modification to PBE is trivial: any optimal persuasion policy corresponds to a class of optimal persuasion policies in which any belief is assigned to each off-path message.
Moreover, you could also resolve your off-path message beliefs issue by assuming that the cardinality of the message set is so small that no off-path messages are sent. This would correspond to, e.g., there only being two messages available in the canonical prosecutor-judge example from KG.