Not directly, but all probability is betting. Or at least the modeling part is the same, where you define what the prediction is that your probability assessment applies to.
Sleeping beauty problems are interesting because they mess with the number of agents making predictions, and this very much confuses our intuitions. The confusion is in how to aggregate the two wakings (which are framed as independent, but I haven’t seen anyone argue that they’ll ever be different).
I think we all agree that post-amnesia, on Wednesday, you should predict 50% that the experimenter will reveal heads, and you were awoken once, and 50% tails, twice. When woken and you don’t know if it’s Monday or Tuesday, you should acknowledge that on Wednesday you’ll predict 50%. If right now you bet 1⁄3, it’s because you’re predicting something different than you will on Wednesday.
Of course you’re predicting something different. In all cases you’re making a conditional prediction of a state of the world given your epistemic state at the time. Your epistemic state on Wednesday is different from that on Monday or Tuesday. On Tuesday you have a 50% chance of not being asked anything at all due to being asleep, which breaks the symmetry between heads and tails.
By Wednesday the symmetry may have been restored due to the amnesia drug—you may not know whether the awakening you remember was Monday (which would imply heads) or Tuesday (which would imply tails). However, there may be other clues such as feeling extra hungry due to sleeping 30+ hours without eating.
you’re making a conditional prediction of a state of the world given your epistemic state at the time.
I think this is a crux. IMO, you can’t predict the state of the world, since you have no access to that except via your perceptions/experiences. You’re making a prediction of a future epistemic state (aka experience), given (of course) your current epistemic state, conditional on which prediction you make (what will happen if you guess either way, and if you’re right/wrong).
It’s perfectly reasonable to bet 1⁄3 if the reveal/payout is instantaneous and multiple, and to bet 1⁄2 if the reveal/payout is post-merge and singular. Each is correct, for predicting different future experiences.
I didn’t make a betting argument.
Not directly, but all probability is betting. Or at least the modeling part is the same, where you define what the prediction is that your probability assessment applies to.
Sleeping beauty problems are interesting because they mess with the number of agents making predictions, and this very much confuses our intuitions. The confusion is in how to aggregate the two wakings (which are framed as independent, but I haven’t seen anyone argue that they’ll ever be different).
I think we all agree that post-amnesia, on Wednesday, you should predict 50% that the experimenter will reveal heads, and you were awoken once, and 50% tails, twice. When woken and you don’t know if it’s Monday or Tuesday, you should acknowledge that on Wednesday you’ll predict 50%. If right now you bet 1⁄3, it’s because you’re predicting something different than you will on Wednesday.
Of course you’re predicting something different. In all cases you’re making a conditional prediction of a state of the world given your epistemic state at the time. Your epistemic state on Wednesday is different from that on Monday or Tuesday. On Tuesday you have a 50% chance of not being asked anything at all due to being asleep, which breaks the symmetry between heads and tails.
By Wednesday the symmetry may have been restored due to the amnesia drug—you may not know whether the awakening you remember was Monday (which would imply heads) or Tuesday (which would imply tails). However, there may be other clues such as feeling extra hungry due to sleeping 30+ hours without eating.
I think this is a crux. IMO, you can’t predict the state of the world, since you have no access to that except via your perceptions/experiences. You’re making a prediction of a future epistemic state (aka experience), given (of course) your current epistemic state, conditional on which prediction you make (what will happen if you guess either way, and if you’re right/wrong).
It’s perfectly reasonable to bet 1⁄3 if the reveal/payout is instantaneous and multiple, and to bet 1⁄2 if the reveal/payout is post-merge and singular. Each is correct, for predicting different future experiences.