All of these problems are variants on the Monty Hall problem, and the wrong answers (thirder in this case) reflect the same failure to take appropriate account of information
Marilyn Vos Savant, who popularised the Monty Hall problem back in the 1990s, used much the same argument to show you should switch (which corresponds to the halfer position in Sleeping Beauty).
It's related to the argument about causal decision theory vs evidential decision theory. I've had a couple of goes at writing about this, but never finished anything This would be a good thing to collaborate on, I think.
I'm going to change my mind about Sleeping Beauty though not my general point about information. Information should always change your beliefs, so if priors are symmetical (halfer) posteriors should be asymmetrical (third)
Suppose, instead of the fanciful story, there are two coins tossed in order. If the coins are HH, TH or TT, the other party offers you the chance to bet on whether the first coin is H or T. If it is HT, they don't offer you a bet. It's clear now, that, conditional on being offered the bet you should go for T unless odds are greater than 2:1 against.
"It is the year 2100 and physicists have narrowed down the search for a theory of everything to only two remaining plausible candidate theories, T1 and T2 (using considerations from super-duper symmetry). According to T1 the world is very, very big but finite, and there are a total of a trillion trillion observers in the cosmos. According to T2, the world is very, very, very big but finite, and there are a trillion trillion trillion observers. The super-duper symmetry considerations seem to be roughly indifferent between these two theories. The physicists are planning on carrying out a simple experiment that will falsify one of the theories. Enter the presumptuous philosopher: "Hey guys, it is completely unnecessary for you to do the experiment, because I can already show to you that T2 is about a trillion times more likely to be true than T1 (whereupon the philosopher runs the God’s Coin Toss thought experiment and explains Model 3)!"
One suspects the Nobel Prize committee to be a bit hesitant about awarding the presumptuous philosopher the big one for this contribution."
Ahh, perfect, yes, this is exactly what I have in mind. Do you know if anyone has explicitly given the presumptuous philosopher as a reason not to be a thirder in the sleeping beauty case?
The many-worlds piece of this seems to top out at simulation, if you buy in to the (at this point totally philosophical) idea that this is possible. In any world where it’s possible to simulate more than one world, it’s more likely than not that you’re in a simulation. So, “one of many simulations inside one of many non-simulated worlds” is a higher cardinality, and so forth.
Where this gets weird(est) is that it’s not even really possible to disprove it as it dissolves into tautology: it’s infinitely more likely that we’re in the last possible tier of simulations (that is, one where running another world-simulation would require more resources than the world simulating our world is able to allocate) than that we’re in any parent tier. And if so, then we won’t be able to simulate a world, but can’t prove with that negative the impossibility of a “bigger” world. We should in fact expect to be unable to simulate a world, and if we can, then it throws off an otherwise infinitely reasonable prediction(?!)
On the other hand, this is one of those things where I can’t imagine that knowing the truth of whether or not there’s another world besides ours can possibly help with any problem or decision if it’s already guaranteed that we can never otherwise observe or interact with it (beyond “knowing” it exists).
It seems to me that in many worlds the observations are of others so should have no relevance to my credences, whereas in Sleeping Beauty the observations are my own. But I’m more confident that there’s some key difference in probability-counting between the two that breaks their symmetry.
All of these problems are variants on the Monty Hall problem, and the wrong answers (thirder in this case) reflect the same failure to take appropriate account of information
Marilyn Vos Savant, who popularised the Monty Hall problem back in the 1990s, used much the same argument to show you should switch (which corresponds to the halfer position in Sleeping Beauty).
Has anyone written on this?
It's related to the argument about causal decision theory vs evidential decision theory. I've had a couple of goes at writing about this, but never finished anything This would be a good thing to collaborate on, I think.
I'm going to change my mind about Sleeping Beauty though not my general point about information. Information should always change your beliefs, so if priors are symmetical (halfer) posteriors should be asymmetrical (third)
Suppose, instead of the fanciful story, there are two coins tossed in order. If the coins are HH, TH or TT, the other party offers you the chance to bet on whether the first coin is H or T. If it is HT, they don't offer you a bet. It's clear now, that, conditional on being offered the bet you should go for T unless odds are greater than 2:1 against.
I need to come back to the counterexample
Bostrom had an argument called just "The presumptuous philosopher" along these lines.
"It is the year 2100 and physicists have narrowed down the search for a theory of everything to only two remaining plausible candidate theories, T1 and T2 (using considerations from super-duper symmetry). According to T1 the world is very, very big but finite, and there are a total of a trillion trillion observers in the cosmos. According to T2, the world is very, very, very big but finite, and there are a trillion trillion trillion observers. The super-duper symmetry considerations seem to be roughly indifferent between these two theories. The physicists are planning on carrying out a simple experiment that will falsify one of the theories. Enter the presumptuous philosopher: "Hey guys, it is completely unnecessary for you to do the experiment, because I can already show to you that T2 is about a trillion times more likely to be true than T1 (whereupon the philosopher runs the God’s Coin Toss thought experiment and explains Model 3)!"
One suspects the Nobel Prize committee to be a bit hesitant about awarding the presumptuous philosopher the big one for this contribution."
Ahh, perfect, yes, this is exactly what I have in mind. Do you know if anyone has explicitly given the presumptuous philosopher as a reason not to be a thirder in the sleeping beauty case?
Yes, with the thought experiment "Extreme Sleeping Beauty" in his paper "Sleeping Beauty and Self-Location: A Hybrid Model".
The many-worlds piece of this seems to top out at simulation, if you buy in to the (at this point totally philosophical) idea that this is possible. In any world where it’s possible to simulate more than one world, it’s more likely than not that you’re in a simulation. So, “one of many simulations inside one of many non-simulated worlds” is a higher cardinality, and so forth.
Where this gets weird(est) is that it’s not even really possible to disprove it as it dissolves into tautology: it’s infinitely more likely that we’re in the last possible tier of simulations (that is, one where running another world-simulation would require more resources than the world simulating our world is able to allocate) than that we’re in any parent tier. And if so, then we won’t be able to simulate a world, but can’t prove with that negative the impossibility of a “bigger” world. We should in fact expect to be unable to simulate a world, and if we can, then it throws off an otherwise infinitely reasonable prediction(?!)
On the other hand, this is one of those things where I can’t imagine that knowing the truth of whether or not there’s another world besides ours can possibly help with any problem or decision if it’s already guaranteed that we can never otherwise observe or interact with it (beyond “knowing” it exists).
It seems to me that in many worlds the observations are of others so should have no relevance to my credences, whereas in Sleeping Beauty the observations are my own. But I’m more confident that there’s some key difference in probability-counting between the two that breaks their symmetry.