It seems to me that Rule 1 is a direct translation of the Sleeping Beauty problem into a betting strategy question, while the other rules correspond to different questions where a single outcome depends on some function of the two guesses in the case of tails. Doing the experiment 100 times under that rule, Beauty will have around 150 identical awakening experiences. The payout for each correct guess is the same, $1, and the correct guess would be tails 2⁄3 of the time. So surely the probability that the coin had landed tails prior to these events is 2/3? Not because it’s an unfair coin or there was an information update (neither is true), but because the SB problem asks the probability from the perspective of someone being awakened, and 2⁄3 of these experiences happen after flipping tails. It seems a stretch to say the bet is 50⁄50 but the 2nd 50% happens twice as often.
There is no such thing as direct translation of a problem into a betting strategy question. A model for a problem should be able to deal with any betting schemes, no matter how extravagant.
And the scheme where the Beauty can bet on every awakening is quite extravagant. It’s an asymetric bet on a coin toss, where Tails outcome is rewarded twice as Heads outcome.
So surely the probability that the coin had landed tails prior to these events is 2/3? Not because it’s an unfair coin or there was an information update (neither is true)
If there is no information update then the probability of the coin to be Tails can’t change from 1⁄2 to 2⁄3. It would contradict the law of conservation of expected evidence.
but because the SB problem asks the probability from the perspective of someone being awakened, and 2⁄3 of these experiences happen after flipping tails.
As I’ve written in the Effects of Amnesia section, from Beauty’s perspective Tails&Monday and Tails&Tuesday awakening are still part of the same elementary outcome because she remembers the setting of the experiment. If she didn’t know that Tails&Monday and Tails&Tuesday necessary follow each other, if all she knew is that there are three states in which she can awaken, then yes, she should’ve reasoned that P(Tails)=2/3.
Alternatively if the question was about a random awakening of the Beauty among multiple possible experiments, then, once again, P(Heads) would be 1⁄3. But in the experiment as stated, the Beauty isn’t experiencing a random awakening, she is experiencing ordered awakening, determined by a coin toss.
It seems to me that Rule 1 is a direct translation of the Sleeping Beauty problem into a betting strategy question, while the other rules correspond to different questions where a single outcome depends on some function of the two guesses in the case of tails. Doing the experiment 100 times under that rule, Beauty will have around 150 identical awakening experiences. The payout for each correct guess is the same, $1, and the correct guess would be tails 2⁄3 of the time. So surely the probability that the coin had landed tails prior to these events is 2/3? Not because it’s an unfair coin or there was an information update (neither is true), but because the SB problem asks the probability from the perspective of someone being awakened, and 2⁄3 of these experiences happen after flipping tails. It seems a stretch to say the bet is 50⁄50 but the 2nd 50% happens twice as often.
There is no such thing as direct translation of a problem into a betting strategy question. A model for a problem should be able to deal with any betting schemes, no matter how extravagant.
And the scheme where the Beauty can bet on every awakening is quite extravagant. It’s an asymetric bet on a coin toss, where Tails outcome is rewarded twice as Heads outcome.
If there is no information update then the probability of the coin to be Tails can’t change from 1⁄2 to 2⁄3. It would contradict the law of conservation of expected evidence.
As I’ve written in the Effects of Amnesia section, from Beauty’s perspective Tails&Monday and Tails&Tuesday awakening are still part of the same elementary outcome because she remembers the setting of the experiment. If she didn’t know that Tails&Monday and Tails&Tuesday necessary follow each other, if all she knew is that there are three states in which she can awaken, then yes, she should’ve reasoned that P(Tails)=2/3.
Alternatively if the question was about a random awakening of the Beauty among multiple possible experiments, then, once again, P(Heads) would be 1⁄3. But in the experiment as stated, the Beauty isn’t experiencing a random awakening, she is experiencing ordered awakening, determined by a coin toss.