The Mad Scientist Decision Problem

Yes, of course! The only requirement is that FAI must know the source code of hypothetical Clippy and vice versa. Assuming that, here's one way it could work:

Humans are risk-averse - a 100% chance for humanity to get 40% of the universe is strictly better than 50% chance to get the whole universe. While Clippy is presumably risk-neutral about paperclips. So we can come up with a deal that's profitable for both parties, e.g. FAI converts 60% of the universe to paperclips in exchange for Clippy giving 40% of the universe to humanity. Let's handwave away the difficulties of bargaining and assume that a single optimal deal is known to both parties.

Now both FAI and Clippy would benefit if the following sentence S about computer programs was true: "FAI holds up its end of the deal if and only if Clippy holds up its end". The key part is that S can be made true, if both sides implement a certain decision procedure that has no effect unless the other side does the same. Namely, FAI should implement the decision procedure "hold up my end of the deal if and only if S is provable in less than a million steps", and Clippy should do the same. That will make S provable by a bounded variant of Löb's theorem, so both sides will find the proof and hold up their ends of the deal.

Note that the proof of S relies on both sides implementing the decision procedure. If one side tries to cheat, S simply becomes unprovable and both sides know the deal isn't happening.

References: Rolf Nelson came up with AI deterrence in 2007, I came up with the proof-based mechanism in 2010, then MIRI took it further in Modal Combat.

This is an interesting reformulation of Counterfactual Mugging. In the case where the cooperation of the paperclip maximiser is provable I don't see it any different from a Counterfactual Mugging taking place before the AI comes into existence. The only way I see this as becoming more complicated is when the AI tries to blackmail you in the counterfactual world.

Depends on what value the FAI places on human flourishing in hypothetical alternate realities I guess. If it's focused on the universe it's in then there's no reason to waste half of it on paperclips. If it's trying to help out the people living in a universe where the paperclip maximizer got activated then it should cooperate. I guess a large part of that is also about whether it determines there really are parallel universes or not to be concerned about.

I'm not an expert on decision theory, but my understanding (of FDT) is that there is no reason for the AI to cooperate with the paperclip maximizer (cooperate how?) because there is no scenario in which the paperclip maximizer treats the friendly AI differently based on it cooperating in counter-factual worlds. For it to be a question at all, it would require that

1) the paperclip maximizer is not a paperclip maximizer but a different kind of unfriendly AI

2) this unfriendly AI is actually launched (but may be in an inferior position)

I think there could be situations where it should cooperate. As I understand it, updateless/functional may say yes, causal and evidental would say no.

Just to be clear, i'm imagining counterfactual cooperation to mean the FAI building vaults full of paperclips in every region where there is a surplus of aluminium (or a similar metal). In the other possibility branch, the paperclip maximizer (which thinks identically) reciprocates by preserving semi-autonomous cities of humans among the mountains of paperclips.

If my understanding above is correct, then yes, i think these two would cooperate IF this type of software agent shares my perspective on acausal game theory and branching timelines.