Down with Solomonoff Induction, up with the Presumptuous Philosopher

by Charlie Steiner2 min read12th Jun 20209 comments


RationalityWorld Modeling

Followup to "The Presumptuous Philosopher, self-locating information, and Solomonoff induction"

In the comments last time, everyone seemed pretty content to ditch the Presumptuous Philosopher (who said that the universe with a trillion times the copies of you was a trillion times as likely), and go with Solomonoff induction (which says that the universe with a trillion times the copies of you is only a few times as likely). To rally some support for the Presumptuous Philosopher, here's a decision game for you.

The Game:

If you start out with a universe that has only one of you, and then Omega comes along and copies you, you now need an extra bit to say which you you are. If Omega has flipped a coin (or consulted a high-entropy bit string) to decide whether to copy you, you now think there is a 50% chance you're un-copied, and a 25% chance you're each of the copies (if you're not a Solomonoff inductor, pretend to be one for now).

So far, this seems, although not obvious, at least familiar. It's the "halfer" position in Sleeping Beauty, which you might also call Bostrom's "SSA" position. It's not widely accepted because of frequentist / pragmatic arguments like "if you did this experiment a bunch of times, each person would converge to a frequency of 2/3 copied and 1/3 non-copied" and abstract arguments about symmetry of information. But of course Solomonoff induction would reply that there's no symmetry between the different hypotheses about the universe, and so it's okay to make an asymmetrical assumption under which the more you're copied, the less your universe matters to the average.

But suppose now that Omega, in addition to maybe copying you at t_0, comes along and shoots the copy (or a deterministic one of the two yous) at t_1. After this happens, you now no longer need that extra bit to describe which you is you, and so the probabilities go back to being the same for the two hypotheses about the universe and Omega's coin.

So here's what Omega does. He has all of you play for chocolate bars, or some other short-term satisfying experience. After t_0, you get offered this deal: If you accept, then the no-copy you will get 2 chocolate bars and both copy-yous will lose 1.9 chocolate bars. Since you're so much more likely to be in the no-copy universe, you accept.

Next, still before t_1, Omega gives a bonus for the two copies transferring chocolate. You all get offered a deal where the copy that will happen to survive loses 0.5 chocolate bars, and the copy that will get shot gains 1 chocolate bar. Since you're not sure which you are but think they're equally likely, and this doubles the chocolate, you accept.

Then, after t_1, Omega deterministically shoots one copy but still doesn't tell the survivor that they're in the copy-universe. The two of you are offered a deal to lose 2 chocolate bars if you're in the no-copy universe, but gain 2.1 if you're in the copy universe. Since this is more chocolate among equally likely possibilities, you accept a third time.

You are now neutral on chocolate in the no-copy universe, the you that died in the copy universe paid 0.9 chocolate bars, and the you that survived in the copy universe paid 0.3 chocolate bars. You have now been Dutch-process booked.

If we treat the hypotheses of Solomonoff induction as if they are descriptions of physical universes (specifically, things with a property of continuity), then Solomonoff induction seems to be doing something wrong. And if we don't treat Turing machines as specifications of physical universes, then this removes much of the bite to applying Solomonoff induction in the Presumptuous Philosopher problem.