I find the problem statement to be completely unambiguous:
Sleeping Beauty volunteers to undergo the following experiment. On Sunday she is given a drug that sends her to sleep. A fair coin is then tossed just once in the course of the experiment to determine which experimental procedure is undertaken. If the coin comes up heads, Beauty is awakened and interviewed on Monday, and then the experiment ends. If the coin comes up tails, she is awakened and interviewed on Monday, given a second dose of the sleeping drug, and awakened and interviewed again on Tuesday. The experiment then ends on Tuesday, without flipping the coin again. The sleeping drug induces a mild amnesia, so that she cannot remember any previous awakenings during the course of the experiment (if any). During the experiment, she has no access to anything that would give a clue as to the day of the week. However, she knows all the details of the experiment.
Each interview consists of one question, “What is your credence now for the proposition that our coin landed heads?”
...and then revised that to “1/3 is the better answer, but not obviously so”. I wish to agree with both statements, with additional explanation. “Credence” is ambiguous in the question, “What is your credence now for … heads?” Depending on additional context, that could be a request for Beauty’s P(heads | Beauty woken up at least once). Or, it could be a request for her P(heads | Beauty just now woken).
The latter case first: Beauty is offered a bet with some payoff odds, for a few dollars, and she is neither risk-averse nor risk-seeking with regard to such amounts of money. She is to be offered this bet upon each awakening. Nothing else of consequence hinges on the coin flip. In this scenario, Beauty likely interprets “credence” in a manner directly corresponding to betting odds, and her correct answer is 1⁄3.
Now for the 1⁄2 case: Beauty knows that her President has decided to launch all-out nuclear war if and only if the coin lands heads. Upon being woken up, her first thought, naturally, is a deep dread at this possibility. How much dread does she feel, in comparison to how she would feel if nuclear war were certain, and in comparison to how she’d feel if it were out of the question? About halfway between.
Since the “offered a bet” scenario is more natural than scenarios where something momentous hangs on the coin flip, 1⁄3 is probably a “better” answer to the unadorned Sleeping Beauty problem. But even then, your mileage may vary. If you are the kind of person more interested in objective events (the coin flip itself) than the track record of your guesses (“I’ve just been woken, so I’ll say probably tails”), well then, be a halfer. If the opposite, be a thirder.
I’m confused. Are you saying there’s room for debate over what “credence” means?
Maybe in discussing what credence someone ought to have, there’s some default analogy to optimizing odds under some betting/payoff/utility scheme, but I think there’s a single correct answer to the Beauty problem under that default, and it should be possible to justify it without recourse to the analogy.
I like to simplify: suppose Beauty wakes and guesses that the coin was tails. How often is she expected to be right? For 2⁄3 of her guesses (but 1⁄2 of the experiments). So clearly in a wager to be played each time she’s woken in the experiment as described, she would need to lose twice as much utility as when she’s wrong as when she’s right, in order to be indifferent about making the wager. I believe that’s the default analogy between credence and lotteries.
That’s all perfectly true, but compare her strategy in this experiment to, say, an ordinary bet at 2:1 odds. If Beauty bets $10 on heads, she will either win $20 or lose $20 with equal likelihood over the course of the experiment—but if she bets $10 on an ordinary one-in-three chance, she will either win $20 or lose $10, with losing $10 being twice as likely. Mere risk aversion would make these two options different.
I’ll concede that, of the two options, 1⁄3 probably makes more sense to describe her credence, but it’s not sufficient to describe the variables she must account for.
I agree but don’t think it’s necessary to talk about risk at all (except to say that we wish to ignore it) for the purpose of the hypothetical bets an agent should make given a certain credence. I also think you confused the direction of the odds; if I believe something is 2⁄3 likely, I should take the positive side if I can gain anything more than half of what I stand to lose if the negative occurs (with p=1/3). But of course that doesn’t change the interesting difference you point out (that the bet involves a $40 swing rather than a $30 one).
I find the problem statement to be completely unambiguous:
Robin wrote:
...and then revised that to “1/3 is the better answer, but not obviously so”. I wish to agree with both statements, with additional explanation. “Credence” is ambiguous in the question, “What is your credence now for … heads?” Depending on additional context, that could be a request for Beauty’s P(heads | Beauty woken up at least once). Or, it could be a request for her P(heads | Beauty just now woken).
The latter case first: Beauty is offered a bet with some payoff odds, for a few dollars, and she is neither risk-averse nor risk-seeking with regard to such amounts of money. She is to be offered this bet upon each awakening. Nothing else of consequence hinges on the coin flip. In this scenario, Beauty likely interprets “credence” in a manner directly corresponding to betting odds, and her correct answer is 1⁄3.
Now for the 1⁄2 case: Beauty knows that her President has decided to launch all-out nuclear war if and only if the coin lands heads. Upon being woken up, her first thought, naturally, is a deep dread at this possibility. How much dread does she feel, in comparison to how she would feel if nuclear war were certain, and in comparison to how she’d feel if it were out of the question? About halfway between.
Since the “offered a bet” scenario is more natural than scenarios where something momentous hangs on the coin flip, 1⁄3 is probably a “better” answer to the unadorned Sleeping Beauty problem. But even then, your mileage may vary. If you are the kind of person more interested in objective events (the coin flip itself) than the track record of your guesses (“I’ve just been woken, so I’ll say probably tails”), well then, be a halfer. If the opposite, be a thirder.
It would make sense to respond “1/3” to that question, but it would not make sense to use 1⁄3 to make decisions with. The payoff grid is different.
I’m confused. Are you saying there’s room for debate over what “credence” means?
Maybe in discussing what credence someone ought to have, there’s some default analogy to optimizing odds under some betting/payoff/utility scheme, but I think there’s a single correct answer to the Beauty problem under that default, and it should be possible to justify it without recourse to the analogy.
I like to simplify: suppose Beauty wakes and guesses that the coin was tails. How often is she expected to be right? For 2⁄3 of her guesses (but 1⁄2 of the experiments). So clearly in a wager to be played each time she’s woken in the experiment as described, she would need to lose twice as much utility as when she’s wrong as when she’s right, in order to be indifferent about making the wager. I believe that’s the default analogy between credence and lotteries.
That’s all perfectly true, but compare her strategy in this experiment to, say, an ordinary bet at 2:1 odds. If Beauty bets $10 on heads, she will either win $20 or lose $20 with equal likelihood over the course of the experiment—but if she bets $10 on an ordinary one-in-three chance, she will either win $20 or lose $10, with losing $10 being twice as likely. Mere risk aversion would make these two options different.
I’ll concede that, of the two options, 1⁄3 probably makes more sense to describe her credence, but it’s not sufficient to describe the variables she must account for.
I agree but don’t think it’s necessary to talk about risk at all (except to say that we wish to ignore it) for the purpose of the hypothetical bets an agent should make given a certain credence. I also think you confused the direction of the odds; if I believe something is 2⁄3 likely, I should take the positive side if I can gain anything more than half of what I stand to lose if the negative occurs (with p=1/3). But of course that doesn’t change the interesting difference you point out (that the bet involves a $40 swing rather than a $30 one).
Agreed. I have indicated a change of opinion at my original comment.