Does their thinking you received information have anything at all to do with your receiving information, even slightly correlated? If it does, then you have a situation in which receiving information hurts you and the proof only goes through because it doesn't consider the other agents.

It explicitly considers only cases where the information does not change payoffs. This is not interesting. This is akin to saying 'assume getting extra information either results in a gain or no loss; obviously, extra information weakly dominates not getting the extra information since in no circumstance is one worse off, and in some circumstances one os better off.'

In the blackmail examples you should in general be worst off if they think you know they can blackmail you, but you don't know they can blackmail you.

Unfortunately, this is exactly what I am objecting to. I agree it is a good heuristic to receive information. This is not what the post is about; it is not about ceteris paribus. Emphasis added:

More information is never a bad thing. ...The second of these is always at least as large as the first.

In a post claiming to offer proofs, I take these universal qualifiers at face value. They may be true in the simplified model. They are not true in many other models, one of which I have linked.

Since I was downvoted so very severely, I'll add another link, an entire paper by Nick Bostrom on all the kinds of information which receiving can hurt you: http://www.nickbostrom.com/information-hazards.pdf

In which case, you might as well include the costs for actually figuring it out.