There is a simple way to answer the question without resorting anthropic reasoning. Then you can try to make the anthropic reasoning fit the (correct) answer.
Flip two coins on Sunday Night, a Dime and a Quarter. Lock them in a glass box showing the results. On Monday morning, perform this procedure: Look at the two coins. If either is showing Tails, wake SB, interview her, and put her back to sleep with the amnesia drug. If both are showing Heads, leave her asleep.
On Monday night, open the box, turn the Dime over, and re-lock it. On Tuesday morning, repeat the same procedure described for Monday.
In the interview(s), ask SB for her credence that the Quarter is currently showing Heads. Since the Quarter is never changed, it is always showing the same face that was the result of the flip. SB knows that when the locked box was examined, there are four equally probably possibilities for {Dime,Quarter}. They are HH, HT, TH, and TT. Since SB is now awake, she knows that HH is eliminated as a possibility. Her credence for each of TH, HT, and TT are each 1⁄3.
The only difference between this version, and the original, is that we don’t need to say anything about what day it is.
I fail to see how this variation is going to settle the debate. Thirders will agree with your solution but halfers would disagree with it the same way as in the original sleeping beauty problem.
Halfers will ask why should beauty regard the four outcomes (HH, HT, TH, TT) equal probable? Yes they are equal probables if this is a simple tossing of two coins. Yet the experiment is far from that simple: my awakenings depend on it, the dime is being manipulated half way, my memory is erased after the first awakening....Halfers will say HH is never in the sample space to begin with, and there is no good reason to believe HT, TH and TT are equal probables. Beauty should just examine the information she has once waken up. She knew that she would definitely find herself awake in the experiment since the dime is manipulated, so right now being awake gives no new information about the Quarter, the probability ought to remain at 1⁄2. The same old dispute as in the original sleeping beauty.
Without trying to figure out the correct way to interpret today or this awakening the debate is not going to be settled. Some halfers (SSA) think this awakening shall be interpreted as a random awakening. Some thirders (SIA) think this awakening should be regarded as a random sample from all potential awakenings (thus being actually awake gives new information as the case of your argument). There are others (FNC) who think we should ignore indexicals such as today or this awakening all together but consider all objective information available. And I’m suggesting treating indexicals like fundamentals: they are primitively understood from the first-person perspective and irreducible. These are all attempts to solve the anthropic mystery. I don’t think this debate can magically go away just by using a different experiment setup.
Let’s say Beauty is paid (once, at the end) if & only if she guesses the Quarter correctly on every wake-up. The reward will be £111 if she correctly guessed Heads, £100 if she correctly guessed Tails. So she should guess Heads.
You would still say that her credence for Heads was 13, but you’d argue that she adapts her bet to take account of the experimental protocol, betting in defiance of her greater credence for Tails. Right?
Now if the single non-HH Monday morning wake-up was replaced with some undisclosed number of wake-ups, I imagine you’d say that her credence was undefined. Yet she’s still able to take the bet, absent of any credence. How is this?
The answer is that she does not need to resort to credence in making the decision. So it is vacuous to argue that Halfing or Thirding is the “correct” approach.
She gets paid once because that’s how I choose to demonstrate my point, supported by your reply, that arguments concerning the “correctness” of halving/thirding are impotent.
There is a simple way to answer the question without resorting anthropic reasoning. Then you can try to make the anthropic reasoning fit the (correct) answer.
Flip two coins on Sunday Night, a Dime and a Quarter. Lock them in a glass box showing the results. On Monday morning, perform this procedure: Look at the two coins. If either is showing Tails, wake SB, interview her, and put her back to sleep with the amnesia drug. If both are showing Heads, leave her asleep.
On Monday night, open the box, turn the Dime over, and re-lock it. On Tuesday morning, repeat the same procedure described for Monday.
In the interview(s), ask SB for her credence that the Quarter is currently showing Heads. Since the Quarter is never changed, it is always showing the same face that was the result of the flip. SB knows that when the locked box was examined, there are four equally probably possibilities for {Dime,Quarter}. They are HH, HT, TH, and TT. Since SB is now awake, she knows that HH is eliminated as a possibility. Her credence for each of TH, HT, and TT are each 1⁄3.
The only difference between this version, and the original, is that we don’t need to say anything about what day it is.
I fail to see how this variation is going to settle the debate. Thirders will agree with your solution but halfers would disagree with it the same way as in the original sleeping beauty problem.
Halfers will ask why should beauty regard the four outcomes (HH, HT, TH, TT) equal probable? Yes they are equal probables if this is a simple tossing of two coins. Yet the experiment is far from that simple: my awakenings depend on it, the dime is being manipulated half way, my memory is erased after the first awakening....Halfers will say HH is never in the sample space to begin with, and there is no good reason to believe HT, TH and TT are equal probables. Beauty should just examine the information she has once waken up. She knew that she would definitely find herself awake in the experiment since the dime is manipulated, so right now being awake gives no new information about the Quarter, the probability ought to remain at 1⁄2. The same old dispute as in the original sleeping beauty.
Without trying to figure out the correct way to interpret today or this awakening the debate is not going to be settled. Some halfers (SSA) think this awakening shall be interpreted as a random awakening. Some thirders (SIA) think this awakening should be regarded as a random sample from all potential awakenings (thus being actually awake gives new information as the case of your argument). There are others (FNC) who think we should ignore indexicals such as today or this awakening all together but consider all objective information available. And I’m suggesting treating indexicals like fundamentals: they are primitively understood from the first-person perspective and irreducible. These are all attempts to solve the anthropic mystery. I don’t think this debate can magically go away just by using a different experiment setup.
Let’s say Beauty is paid (once, at the end) if & only if she guesses the Quarter correctly on every wake-up. The reward will be £111 if she correctly guessed Heads, £100 if she correctly guessed Tails. So she should guess Heads.
You would still say that her credence for Heads was 13, but you’d argue that she adapts her bet to take account of the experimental protocol, betting in defiance of her greater credence for Tails. Right?
Now if the single non-HH Monday morning wake-up was replaced with some undisclosed number of wake-ups, I imagine you’d say that her credence was undefined. Yet she’s still able to take the bet, absent of any credence. How is this?
The answer is that she does not need to resort to credence in making the decision. So it is vacuous to argue that Halfing or Thirding is the “correct” approach.
Why does she get paid only once, at the end? Why not once for each waking?
This is the problem with all betting arguments. They incorporate an answer to the anthropic question by providing one, or #wakings, payoffs.
She gets paid once because that’s how I choose to demonstrate my point, supported by your reply, that arguments concerning the “correctness” of halving/thirding are impotent.