You're given the option to torture everyone in the universe, or inflict a dust speck on everyone in the universe. Either you are the only one in the universe, or there are 3^^^3 perfect copies of you (far enough apart that you will never meet.) In the latter case, all copies of you are chosen, and all make the same choice. (Edit: if they choose specks, each person gets one dust speck. This was not meant to be ambiguous.)
As it happens, a perfect and truthful predictor has declared that you will choose torture iff you are alone.
What do you do?
How does your answer change if the predictor made the copies of you conditional on their prediction?
How does your answer change if, in addition to that, you're told you are the original?
Trial problem:
Omega appears before you, and gives you a pencil. He tells you that, in universes in which you break this pencil in half in the next twenty seconds, the universe ends immediately. Not as a result of your breaking the pencil - it's pure coincidence that all universes in which you break the pencil, the universe ends, and in all universes in which you don't, it doesn't.
Do you break the pencil in half? It's not like you're changing anything by doing so, after all; some set of universes will end, some set won't, and you aren't going to change that.
You're just deciding which set of universes you happen to occupy. Which implies something.
I don't break the pencil. But I already pointed out in Newcomb and in the Smoking Lesion that I don't care if I can change anything or not. So I don't care here either.