I suspect that I misunderstand the question, but my inclination is to answer the question of what day it is [nearly ]experimentally, by simulating a large number of Sleeping Beauties and figuring out the odds this way. In this case you have basically done the simulation in your first picture, and the odds are 2:1 that it is Monday/Room 1, and 2:1 for tails. This matches your “Self-Indicating Assumption” of being “randomly selected from the set of all possible observers”.
The other approach has the assumption that “Sleeping Beauty, before being put to sleep, expects that she will be awakened in future,” which makes no sense to me, as it is manifestly false on Monday+Heads and unconditionally on Tuesday, and she knows that full well.
Additionally, I do not understand how her utility function (and any amount of money or chocolate) can change the odds in any way. I also do not understand what this has to do with any decision theory, given that her fate is predetermined and there is nothing she can do to avoid being or not being awoken, so her decision doesn’t matter in the slightest.
My suspicion is that the Sleeping Beauty problem is a poor illustration for whatever concept you are advancing, but it is entirely possible that I simply missed your point.
by simulating a large number of Sleeping Beauties...
What criteria do you use to count up the results? Each incubator experiment produces either one, or two SBs. If we follow the criteria “in each experiment, we take the total number of people who were correct”, then SIA odds are the way to go. If instead, we follow “in each experiment, we take the average number of people who were correct”, then SSA is the way to go.
“Sleeping Beauty, before being put to sleep, expects that she will be awakened in future,”
Changed to clarify: “”Sleeping Beauty, before being put to sleep on Sunday, expects that she will be awakened in future,”
Additionally, I do not understand how her utility function (and any amount of money or chocolate) can change the odds in any way.
They do not change her odds, but her decisions, and I argue her decisions are the only important factors here, as her belief in her odds in not directly observable. You can believe in different odds, but still come to the same decision in any circumstance; I would argue that this is makes the odds irrelevant.
I suspect that I misunderstand the question, but my inclination is to answer the question of what day it is [nearly ]experimentally, by simulating a large number of Sleeping Beauties and figuring out the odds this way. In this case you have basically done the simulation in your first picture, and the odds are 2:1 that it is Monday/Room 1, and 2:1 for tails. This matches your “Self-Indicating Assumption” of being “randomly selected from the set of all possible observers”.
The other approach has the assumption that “Sleeping Beauty, before being put to sleep, expects that she will be awakened in future,” which makes no sense to me, as it is manifestly false on Monday+Heads and unconditionally on Tuesday, and she knows that full well.
Additionally, I do not understand how her utility function (and any amount of money or chocolate) can change the odds in any way. I also do not understand what this has to do with any decision theory, given that her fate is predetermined and there is nothing she can do to avoid being or not being awoken, so her decision doesn’t matter in the slightest.
My suspicion is that the Sleeping Beauty problem is a poor illustration for whatever concept you are advancing, but it is entirely possible that I simply missed your point.
What criteria do you use to count up the results? Each incubator experiment produces either one, or two SBs. If we follow the criteria “in each experiment, we take the total number of people who were correct”, then SIA odds are the way to go. If instead, we follow “in each experiment, we take the average number of people who were correct”, then SSA is the way to go.
Changed to clarify: “”Sleeping Beauty, before being put to sleep on Sunday, expects that she will be awakened in future,”
They do not change her odds, but her decisions, and I argue her decisions are the only important factors here, as her belief in her odds in not directly observable. You can believe in different odds, but still come to the same decision in any circumstance; I would argue that this is makes the odds irrelevant.