Re: Doomsday Example

Robin Hanson (
Wed, 26 Aug 1998 11:55:06 -0700

Nick Bostrom writes:
>> >> >an A universe contains 10 humans and nothing else. A B universe contains
>> >> >10 humans and one trillion trillion stones. ...
>> >> >a small C universe ...spawns one thousand baby-univeses through a random
>> >> >process (like rolling fair dice) that has a 10% chance of yielding an
>> >> >A and 90% of a B.
>> >It seems clear that there will probably be about 100 A univeses and
>> >900 B universes.
>> Let N = "trillion trillion", and assume there are exactly 100 A
>> "universes" and 900 B "universes". ...
>> As described the only remaining uncertainty is where in this world I am.
>> If I treat stone slots and human slots equally, there are
>> 100*10 + 900(10+N) slots. If my prior is uniform across these slots, ...
>consider the example that I formulated. It seems you have to say that
>when you find that you are a human, you have to conclude that all the
>1000 universes almost certainly are of type A. That means, you have
>to infer that the coin landed heads a thousand times. But is that
>really what you would infer? It seems very wrong.

I think you just haven't done the math here. I also think you meant to switch A and B in your first description. So I'll assume A universes have 10 humans + N stones, and B universes have just 10 humans.

I don't want to work out the math for 1000 universes, but two should be enough to see what works. In that case there are four possible worlds:

-- Two A "universes", with 2*(10+N) space-time slots, and "prior" of 1%.
-- Two B "universes", with 2*10 slots, and "prior" 81%.
-- An A and a B "universe," with 2*10 + N slots, and "prior" 9%.
-- Another possible world that looks just like the last one.  
These "priors" are over worlds, but not necessarily over states. If we extend these descriptions to include which slot "I" occupy, we get 800 + 4N states. If I make the relative priors between states equal to the relative "priors" between associated worlds, then, yes, very little state prior is associated with the second world with two B "universes." But conditioning on observing that I'm a human, I'm back to estimating a 81% chance that there are two B "universes."

Robin Hanson RWJF Health Policy Scholar, Sch. of Public Health 510-643-1884 140 Warren Hall, UC Berkeley, CA 94720-7360 FAX: 510-643-8614