I used to be a great believer in FNC, but I’ve found it’s flawed. The main problem is that it’s not time-consistent.
For instance, if you start with some identical copies, and they are each going to flip a coin twenty times. Now FNC says that before they flip a coin, they should not believe that they are in a large universe, because they are identical.
However, after they have flipped, they will be nearly certainly very different, and so will believe that they are in a large universe.
So they know that after they flip the coin, their probability of being in a large universe will have increased, no matter what they see.
The problem isn’t just restricted to when you start with identical copies; whenever you increase your memory size by one bit, say, then FNC will be slightly inconsistent (because (1+e)^-n is approximately 1-ne for small e, but not exactly).
Yes, that is definitely a problem! The variation of FNC which I described in the final section of my UDT post has each person being allowed to help themselves to uniform random number in [0,1] - i.e. infinitely many random “coin flips”, as long as they don’t try to actually use the outcomes.
This solves the problem you mention, but others arise:
It’s hard to see how to give an independent justification of this trick.
More importantly, Eliezer’s tale of the Ebborians demonstrates that we can go continuously from one copy to two copies.
Actually, using (2), and variations alpha to gamma, I think I can construct a continuum of variations on Sleeping Beauty which stretch from one where the answer is unambiguously 1⁄3 (or 1⁄11 as in the link) to one where it’s unambiguously 1⁄2.
OK, I recant and denounce myself—the idea that any sensible variation of the Sleeping Beauty problem must have a ‘canonical’ answer is wrong, and FNC is broken.
OK, I recant and denounce myself—the idea that any sensible variation of the Sleeping Beauty problem must have a ‘canonical’ answer is wrong, and FNC is broken.
Very admirable stance to take :-) I wish I could claim I found the problem and immediately renounced SIA and FNC, but it was a long process :-)
I used to be a great believer in FNC, but I’ve found it’s flawed. The main problem is that it’s not time-consistent.
For instance, if you start with some identical copies, and they are each going to flip a coin twenty times. Now FNC says that before they flip a coin, they should not believe that they are in a large universe, because they are identical.
However, after they have flipped, they will be nearly certainly very different, and so will believe that they are in a large universe.
So they know that after they flip the coin, their probability of being in a large universe will have increased, no matter what they see.
The problem isn’t just restricted to when you start with identical copies; whenever you increase your memory size by one bit, say, then FNC will be slightly inconsistent (because (1+e)^-n is approximately 1-ne for small e, but not exactly).
Yes, that is definitely a problem! The variation of FNC which I described in the final section of my UDT post has each person being allowed to help themselves to uniform random number in [0,1] - i.e. infinitely many random “coin flips”, as long as they don’t try to actually use the outcomes.
This solves the problem you mention, but others arise:
It’s hard to see how to give an independent justification of this trick.
More importantly, Eliezer’s tale of the Ebborians demonstrates that we can go continuously from one copy to two copies.
Actually, using (2), and variations alpha to gamma, I think I can construct a continuum of variations on Sleeping Beauty which stretch from one where the answer is unambiguously 1⁄3 (or 1⁄11 as in the link) to one where it’s unambiguously 1⁄2.
OK, I recant and denounce myself—the idea that any sensible variation of the Sleeping Beauty problem must have a ‘canonical’ answer is wrong, and FNC is broken.
Very admirable stance to take :-) I wish I could claim I found the problem and immediately renounced SIA and FNC, but it was a long process :-)
Btw, a variant similar to your alpha to gamma was presented in my post http://lesswrong.com/lw/18r/avoiding_doomsday_a_proof_of_the_selfindication ; I found the problem with that in http://lesswrong.com/lw/4fl/dead_men_tell_tales_falling_out_of_love_with_sia/