I’ve been trying to make this comment a bunch of times, no quotation from the post in case that’s the issue:
No, a thirder would not treat those possibilities as equiprobable. A thirder would instead treat the coin toss outcome probabilities as a prior, and weight the possibilities accordingly. Thus H1 would be weighted twice as much as any of the individual TH or TT possibilities.
A thirder would instead treat the coin toss outcome probabilities as a prior, and weight the possibilities accordingly
But then they will “update on awakening” and therefore weight the probabilities of each event by the number of awakenings that happen in them.
Every next Tails outcome, decreases the probability two fold, but it’s immediately compensated by the fact that twice as many awakenings are happening when this outcome is Tails.
Hmm, you’re right. Your math is wrong for the reason in my above comment, but the general form of the conclusion would still hold with different, weaker numbers.
The actual, more important issue relates to the circumstances of the bet:
If each awakening has an equal probability of receiving the bet, then receiving it doesn’t provide any evidence to Sleeping Beauty, but the thirder conclusion is actually rational in expectation, because the bet occurs more times in the high-awakening cases.
If the bet would not be provided equally to all awakenings, then a thirder would update on receiving the bet.
Your math is wrong for the reason in my above comment
What exactly is wrong? Could you explicitly show my mistake?
If each awakening has an equal probability of receiving the bet, then receiving it doesn’t provide any evidence to Sleeping Beauty, but the thirder conclusion is actually rational in expectation, because the bet occurs more times in the high-awakening cases.
The bet is proposed on every actual awakening, so indeed no update upon its receiving. However this “rational in expectation” trick doesn’t work anymore as shown by the betting argument. The bet does occur more times in high-awakening cases but you win the bet only when the maximum possible awakening happened. Until then you lose, and the closer the number of awakenings to the maximum, the higher the loss.
Ah, I forgot. You use assumptions where you don’t accumulate the winnings between the different times Sleeping Beauty agrees to the bet.
Well, in that case, if the thirder has certain beliefs about how to handle the situation, you may actually be able to money pump them. And it seems that you expect those beliefs.
My point of view, if adopting the thirder perspective[1], would be for the thirder to treat this situation using different beliefs. Specifically, consider what counterfactually might happen if Sleeping Beauty gave different answers in different awakenings. Possible responses by the bet proposer might be:
a) average the results across the awakenings.
b) accept the bet agreement from one awakening at random.
Regardless of which case (a) or (b) occurs, instrumentally Sleeping Beauty’s betting EV for her bet decision, with non-accumulated bets, should be divided by the number of awakenings to take into account the reduced winnings or reduced chance of influencing whether the bet occurs.
Even if we assume that such disagreement between bet decisions in different awakenings is impossible, it seems strange to assume that a thirder should give different results in that case than the answer they would give where it is not impossible?
This adjustment can be conceptualized as compensating for an “unfair” bet where the bet is unequal between awakenings overall (where parity between awakenings in different scenarios is seen as “fair” by the thirder). I see this as no different in principle to a halfer upweighting trials with more awakenings in the converse scenario where bets are accumulated between trials and are thus “unfair” from the halfer perspective which sees parity between trials as fair, but not awakenings.
reminder: my point of view is that either thirderism or halferism is viable, but I am relatively thirder-adjacent precisely because I find the scenario where the winnings are accumulated between awakenings more natural than if the bet is proposed and agreed at each awakening but not accumulated.
I’ve been trying to make this comment a bunch of times, no quotation from the post in case that’s the issue:
No, a thirder would not treat those possibilities as equiprobable. A thirder would instead treat the coin toss outcome probabilities as a prior, and weight the possibilities accordingly. Thus H1 would be weighted twice as much as any of the individual TH or TT possibilities.
But then they will “update on awakening” and therefore weight the probabilities of each event by the number of awakenings that happen in them.
Every next Tails outcome, decreases the probability two fold, but it’s immediately compensated by the fact that twice as many awakenings are happening when this outcome is Tails.
Hmm, you’re right. Your math is wrong for the reason in my above comment, but the general form of the conclusion would still hold with different, weaker numbers.
The actual, more important issue relates to the circumstances of the bet:
If each awakening has an equal probability of receiving the bet, then receiving it doesn’t provide any evidence to Sleeping Beauty, but the thirder conclusion is actually rational in expectation, because the bet occurs more times in the high-awakening cases.
If the bet would not be provided equally to all awakenings, then a thirder would update on receiving the bet.
What exactly is wrong? Could you explicitly show my mistake?
The bet is proposed on every actual awakening, so indeed no update upon its receiving. However this “rational in expectation” trick doesn’t work anymore as shown by the betting argument. The bet does occur more times in high-awakening cases but you win the bet only when the maximum possible awakening happened. Until then you lose, and the closer the number of awakenings to the maximum, the higher the loss.
Ah, I forgot. You use assumptions where you don’t accumulate the winnings between the different times Sleeping Beauty agrees to the bet.
Well, in that case, if the thirder has certain beliefs about how to handle the situation, you may actually be able to money pump them. And it seems that you expect those beliefs.
My point of view, if adopting the thirder perspective[1], would be for the thirder to treat this situation using different beliefs. Specifically, consider what counterfactually might happen if Sleeping Beauty gave different answers in different awakenings. Possible responses by the bet proposer might be:
a) average the results across the awakenings.
b) accept the bet agreement from one awakening at random.
Regardless of which case (a) or (b) occurs, instrumentally Sleeping Beauty’s betting EV for her bet decision, with non-accumulated bets, should be divided by the number of awakenings to take into account the reduced winnings or reduced chance of influencing whether the bet occurs.
Even if we assume that such disagreement between bet decisions in different awakenings is impossible, it seems strange to assume that a thirder should give different results in that case than the answer they would give where it is not impossible?
This adjustment can be conceptualized as compensating for an “unfair” bet where the bet is unequal between awakenings overall (where parity between awakenings in different scenarios is seen as “fair” by the thirder). I see this as no different in principle to a halfer upweighting trials with more awakenings in the converse scenario where bets are accumulated between trials and are thus “unfair” from the halfer perspective which sees parity between trials as fair, but not awakenings.
reminder: my point of view is that either thirderism or halferism is viable, but I am relatively thirder-adjacent precisely because I find the scenario where the winnings are accumulated between awakenings more natural than if the bet is proposed and agreed at each awakening but not accumulated.
See my top-level comment.