Much reasoning in anthropics, like the Sleeping Beauty problem or Eliezer’s lottery, relies on the notion of ‘copies’. However, to my (limited) knowledge, no one has seriously investigated what counts as a ‘copy’.

Consider a normal human undergoing an anthropic experiment. I instantly cool him/her to near absolute zero, so the atoms stop wiggling. I make one perfect copy, but then move one of the atoms a tiny amount, say a nanometer. I then make another copy, moving a second atom, make a third copy, and so on, moving one atom each step of the way, until I wind up at (his father/her mother) as they existed at (his/her) age. At every step, I maintain a physically healthy, functioning human being. The number of copies needed should be on the order of 10^30. Large, but probably doable for a galactic superintelligence.

But then I repeat the process, creating another series of copies going back to (his grandfather/her grandmother). I then go back another generation, and so on, all the way back to the origin of Earth-based life. As the organisms become smaller, the generations grow shorter, but the number of copies per generation also becomes less. Let’s wave our hands, and say the total number of copies is about 10^40.

Now, before putting you to sleep, I entered you into a deterministic lottery drawing. I then warm up all the modified copies, and walk into their rooms with the lottery results. For the N copies closest to you, I tell them they have won, and hand them the pile of prize money. For the others, I apologize and say they have lost (and most of them don’t understand me). All copies exist in all universes, so SIA vs. SSA shouldn’t matter here.

Before you go to sleep, what should your subjective probability of winning the lottery be after waking up, as a function of N? When doing your calculations, it seems there are only two possibilities:

1. The cutoff between ‘copy’ and ‘not a copy’ is sharp, and you assign each organism a weighting of zero or one. That is, it’s possible to move an atom one nanometer, and thereby make an organism go from “not you” to “you”.

2. There exist ‘partial copies’ out in mind space, and you assign some organisms partial weightings. That is, there exist hypothetical entities which are two-thirds you.

Both seem problematic, for different reasons. Is there a third option?