One of my ways of thinking about these sorts of issues is in terms of “fair bets”
Well, as you may see it’s also is not helpful. Halfers and thirders disagree on which bets they consider “fair” but still agree on which bets to make, whether they call them fair or not. The extra category of a “fair bet” just adds another semantic disagreement between halfers and thirders. Once we specify whether we are talking per experiment or per awakening bet and on which, odds both theories are supposed to agree.
I don’t actually know what the Thirder position is supposed to be from a standpoint from before the experiment, but I see no contradiction in assigning equal utilities per awakening from the before-experiment perspective as well.
Thirders tend to agree with halfers that P(Heads|Sunday) = P(Heads|Wednesday) = 1⁄2. Likewise, because they make the same bets as the halfers, they have to agree on utilities. So it means that thirders utilities go back and forth which is weird and confusing behavior.
A Halfer has to discount their utility based on how many of them there are, a Thirder doesn’t. It seems to me, on the contrary to your perspective, that Thirder utility is more stable
You mean how many awakenings? That if there was not two awakenings on tails, but, for instance, ten, halfers will have to think that U(Heads) has to be ten times as much as U(Tails) for a utility neutral per awakening bet?
Sure, but it’s a completely normal behavior. It’s fine to have different utility estimates for different problems and different payout schemes—such things always happen. Sleeping Beauty with ten awakenings on Tails is a different problem than Sleeping Beauty with only two so there is no reason to expect that utilities of the events has to be the same. The point is that as long as we specified the experiment and a betting scheme, then the utilities has to be stable.
And thirder utilities are modified duringthe experiment. They are not just specified by a betting scheme, they go back and forth based on the knowledge state of the participant—behave the way probabilities are supposed to behave. And that’s because they are partially probabilities—a result of incorrect factorization of E(X).
Speculation; have you actually asked Thirders and Halfers to solve the problem? (while making clear the reward structure?
I’m asking it right in the post, explicitly stating that the bet is per experiment and recommending to think about the question more. What did you yourself answer?
My initial state that thirders model confuses them about this per experiment bet is based on the fact that a pro-thirder paper which introduced the technicolor sleeping beauty problem totally fails to understand why halfers scoring rule updates in it. I may be putting to much weight on the views of Rachael Briggs in particular, but it apparently was peer reviewed and so on, so it seems to be decent evidence.
… and I in my hasty reading and response I misread the conditions of the experiment
Well, I guess that answers my question.
Thirders can adapt to different reward structures but need to actually notice what the reward structure is!
Probably, but I’ve yet to see one actually derive the correct answer on their own, not post hoc after it was already spoiled or after consulting the correct model. I suppose I should have asked the question beforehand, and then publish the answer, oh well. Maybe I can still do it and ask nicely not to look.
The criterion I mainly use to evaluate probability/utility splits is typical reward structure
Well, if every other thirder reason like this, that would indeed explain the issue.
You can’t base the definition of probability on your intuitions about fairness. Or, rather, you can, but then you are risking contradicting the math. Probability is a mathematical concept with very specific properties. In my previous post I talk about it specifically and show that thirder probabilities for Sleeping Beauty are ill-defined.
My reasoning explicitly puts instrumental rationality ahead of epistemic. I hold this view precisely to the degree which I do in fact think it is helpful.
The extra category of a “fair bet” just adds another semantic disagreement between halfers and thirders.
It’s just a criterion by which to assess disagreements, not adding something more complicated to a model.
Regarding your remarks on these particular experiments:
If someone thinks the typical reward structure is some reward structure, then they’ll by default guess that a proposed experiment has that reward structure.
This reasonably can be expected to apply to halfers or thirders.
If you convince me that halfer reward structure is typical, I go halfer. (As previously stated since I favour the typical reward structure). To the extent that it’s not what I would guess by default, that’s precisely because I don’t intuitively feel that it’s typical and feel more that you are presenting a weird, atypical reward structure!
And thirder utilities are modified duringthe experiment. They are not just specified by a betting scheme, they go back and forth based on the knowledge state of the participant—behave the way probabilities are supposed to behave. And that’s because they are partially probabilities—a result of incorrect factorization of E(X).
Probability is a mathematical concept with very specific properties. In my previous post I talk about it specifically and show that thirder probabilities for Sleeping Beauty are ill-defined.
I’ve previously shown that some of your previous posts incorrectly model the Thirder perspective, but I haven’t carefully reviewed and critiqued all of your posts. Can you specify exactly what model of the Thirder viewpoint you are referencing here? (which will not only help me critique it but also help me determine what exactly you mean by the utilities changing in the first place, i.e. do you count Thirders evaluating the total utility of a possibility branch more highly when there are more of them as a “modification” or not (I would not consider this a “modification”).
Well, as you may see it’s also is not helpful. Halfers and thirders disagree on which bets they consider “fair” but still agree on which bets to make, whether they call them fair or not. The extra category of a “fair bet” just adds another semantic disagreement between halfers and thirders. Once we specify whether we are talking per experiment or per awakening bet and on which, odds both theories are supposed to agree.
Thirders tend to agree with halfers that P(Heads|Sunday) = P(Heads|Wednesday) = 1⁄2. Likewise, because they make the same bets as the halfers, they have to agree on utilities. So it means that thirders utilities go back and forth which is weird and confusing behavior.
You mean how many awakenings? That if there was not two awakenings on tails, but, for instance, ten, halfers will have to think that U(Heads) has to be ten times as much as U(Tails) for a utility neutral per awakening bet?
Sure, but it’s a completely normal behavior. It’s fine to have different utility estimates for different problems and different payout schemes—such things always happen. Sleeping Beauty with ten awakenings on Tails is a different problem than Sleeping Beauty with only two so there is no reason to expect that utilities of the events has to be the same. The point is that as long as we specified the experiment and a betting scheme, then the utilities has to be stable.
And thirder utilities are modified during the experiment. They are not just specified by a betting scheme, they go back and forth based on the knowledge state of the participant—behave the way probabilities are supposed to behave. And that’s because they are partially probabilities—a result of incorrect factorization of E(X).
I’m asking it right in the post, explicitly stating that the bet is per experiment and recommending to think about the question more. What did you yourself answer?
My initial state that thirders model confuses them about this per experiment bet is based on the fact that a pro-thirder paper which introduced the technicolor sleeping beauty problem totally fails to understand why halfers scoring rule updates in it. I may be putting to much weight on the views of Rachael Briggs in particular, but it apparently was peer reviewed and so on, so it seems to be decent evidence.
Well, I guess that answers my question.
Probably, but I’ve yet to see one actually derive the correct answer on their own, not post hoc after it was already spoiled or after consulting the correct model. I suppose I should have asked the question beforehand, and then publish the answer, oh well. Maybe I can still do it and ask nicely not to look.
Well, if every other thirder reason like this, that would indeed explain the issue.
You can’t base the definition of probability on your intuitions about fairness. Or, rather, you can, but then you are risking contradicting the math. Probability is a mathematical concept with very specific properties. In my previous post I talk about it specifically and show that thirder probabilities for Sleeping Beauty are ill-defined.
My reasoning explicitly puts instrumental rationality ahead of epistemic. I hold this view precisely to the degree which I do in fact think it is helpful.
It’s just a criterion by which to assess disagreements, not adding something more complicated to a model.
Regarding your remarks on these particular experiments:
If someone thinks the typical reward structure is some reward structure, then they’ll by default guess that a proposed experiment has that reward structure.
This reasonably can be expected to apply to halfers or thirders.
If you convince me that halfer reward structure is typical, I go halfer. (As previously stated since I favour the typical reward structure). To the extent that it’s not what I would guess by default, that’s precisely because I don’t intuitively feel that it’s typical and feel more that you are presenting a weird, atypical reward structure!
I’ve previously shown that some of your previous posts incorrectly model the Thirder perspective, but I haven’t carefully reviewed and critiqued all of your posts. Can you specify exactly what model of the Thirder viewpoint you are referencing here? (which will not only help me critique it but also help me determine what exactly you mean by the utilities changing in the first place, i.e. do you count Thirders evaluating the total utility of a possibility branch more highly when there are more of them as a “modification” or not (I would not consider this a “modification”).