Keep in mind: Controlling Constant Programs, Notion of Preference in Ambient Control.
There is a reasonable game-theoretic heuristic, "don't respond to blackmail" or "don't negotiate with terrorists". But what is actually meant by the word "blackmail" here? Does it have a place as a fundamental decision-theoretic concept, or is it merely an affective category, a class of situations activating a certain psychological adaptation that expresses disapproval of certain decisions and on the net protects (benefits) you, like those adaptation that respond to "being rude" or "offense"?
We, as humans, have a concept of "default", "do nothing strategy". The other plans can be compared to the moral value of the default. Doing harm would be something worse than the default, doing good something better than the default.
Blackmail is then a situation where by decision of another agent ("blackmailer"), you are presented with two options, both of which are harmful to you (worse than the default), and one of which is better for the blackmailer. The alternative (if the blackmailer decides not to blackmail) is the default.
Compare this with the same scenario, but with the "default" action of the other agent being worse for you than the given options. This would be called normal bargaining, as in trade, where both parties benefit from exchange of goods, but to a different extent depending on which cost is set.
Why is the "default" special here? If bargaining or blackmail did happen, we know that "default" is impossible. How can we tell two situations apart then, from their payoffs (or models of uncertainty about the outcomes) alone? It's necessary to tell these situations apart to manage not responding to threats, but at the same time cooperating in trade (instead of making things as bad as you can for the trade partner, no matter what it costs you). Otherwise, abstaining from doing harm looks exactly like doing good. A charitable gift of not blowing up your car and so on.
My hypothesis is that "blackmail" is what the suggestion of your mind to not cooperate feels like from the inside, the answer to a difficult problem computed by cognitive algorithms you don't understand, and not a simple property of the decision problem itself. By saying "don't respond to blackmail", you are pushing most of the hard work into intuitive categorization of decision problems into "blackmail" and "trade", with only correct interpretation of the results of that categorization left as an explicit exercise.
(A possible direction for formalizing these concepts involves introducing some kind of notion of resources, maybe amount of control, and instrumental vs. terminal spending, so that the "default" corresponds to less instrumental spending of controlled resources, but I don't see it clearly.)
(Let's keep on topic and not refer to powerful AIs or FAI in this thread, only discuss the concept of blackmail in itself, in decision-theoretic context.)
My take: what we call "extortion" or "blackmail" is where agent A1 offers A2 a choice between X and Y, both of which are harmful to A2, and where A1 has selected X to be less harmful to A2 than Y with the intention of causing A2 to choose X.
"Not responding to blackmail" comprises A2 choosing Y over X whenever A2 suspects this is going on.
A1 can still get A2 to choose X over Y, even if A2 has a policy of not responding to blackmail, by not appearing to have selected X... that is, by not appearing to be blackmailing A2.
For example, if instead of "I will hurt you if you don't give me money" A1 says "I've just discovered that A3 is planning to hurt you! I can prevent it by taking certain steps on your behalf, but those steps are expensive, and I have other commitments for my money that are more important to me than averting your pain. But if you give me the money, I can take those steps, and you won't get hurt," A2 may not recognize this as blackmail, in which case A1 can finesse A2's policy.
Of course, any reasonably sophisticated human will recognize that as likely blackmail, so a kind of social arms race ensues. Real-world blackmail attempts can be very subtle. (ETA: That extortion is illegal also contributes to this, of course... subtle extortion attempts can reduce A1's legal liability, even when they don't actually fool anyone.)
(Indeed, in some cases A1 can fool themselves, which brings into question whether it's still blackmail. IMHO, the best way to think about cases like that is to stop treating people fooling themselves as unified agents, but that's way off-topic.)