In games of Bayesian Persuasion, under what conditions is Receiver better off than under uninformative signals?

Question

I've been working through a few Bayesian Persuasion (BP) models à la Kamenica and Gentzkow (2011), and a feature that seems to arise often is that commitment to a persuasion mechanism is Pareto improving, with Receiver typically indifferent between the BP equilibrium and the uninformative signal equilibrium.

I am having some trouble getting the intuition of why Receiver cannot be worse off than under uninformative signals. Is this always the case? Can we characterize the conditions for Receiver to be strictly better off under BP?

I am particularly thinking about a simple BP game where Receiver can decide whether to allow Sender to observe the state of the world or not, and so Receiver will do so only if the BP equilibrium gives Receiver more utility than one in which the state of the world is unknown. But if Receiver is always indifferent, the assumption that Receiver can decide whether Sender observes the state is redundant.

Answer 1

A receiver can always ignore any additional information and do what they would have done in the absence of informative signals. If they react optimally, they can therefore never be worse off.

It is a general principle of decision theory that more information cannot hurt a standard subjective expected utility maximizer and, since only the receiver chooses a payoff-relevant action, this is basically a decision problem.

Answer 2

In the paper, Kamenica nad Gentzkow argue that the geometrical approach they use to solve the problem, refers to the sender preferred equilibria. In other words, the agent will always break ties in favour of the sender's preferred action based on the posterior beliefs that are shaped by the signal of the sender (additional information that revises the probabilities with respect to the prior beliefs-namely Bayesian updating). These posterior beliefs of both the sender and the receiver must coincide as well. In case the sender sends a signal that is somehow uninformative, then the receiver will act again as a maximizer, because her action $\hat{\alpha(\mu)}$ that solves the receivers problem is her best action she can choose. Nevertheless, the critique in this literature could be, that the receiver acts, myopycally as a machine and she does not have any control over the precision of the signals she receives. This means that a receiver does not get to decide the time that she will stop ``aggregate" signals and start acting. In the latter occasion the problem becames difficult to solve, because the receiver infact acts at a stochastic time, namely she will decide the timing of her action based on how much of the inforamtion she observed at that time has triggered her to act.