I’ve been thinking 1⁄2 as well (though I’m also definitely in the “problem is underdefined” camp).
Here is how describe the appropriate payoff scheme. Prior to the experiment (but after learning the details) Beauty makes a wager with the Prince. If the coin comes up heads the Prince will pay Beauty $10. If it comes up tails Beauty will pay $10. Even odds. This wager represents Beauty’s prior belief that the coin is fair and head/tails have equal probability: her credence that heads will or did come up. At any point before Beauty learns what day of the week it is she is free alter the bet such that she takes tails but must pay $10 more dollars to do so (making the odds 2:1).
Beauty should at no point (before learning what day of the week it is) alter the wager. Which means when she is asked what her credence is that the coin came up heads she should continue to say 1⁄2.
This seems at least as good an payoff interpretation as a new bet every time Beauty is asked about her credence.
You don’t measure an agent’s subjective probability like that, though—not least because in many cases it would be bad experimental methodology. Bets made which are intended to represent the subject’s probability at a particular moment should pay out—and not be totally ignored. Otherwise there may not be any motivation for the subject making the bet to give an answer that represents what they really think. If the subject knows that they won’t get paid on a particular bet, that can easily defeat the purpose of offering them a bet in the first place.
I’ve been thinking 1⁄2 as well (though I’m also definitely in the “problem is underdefined” camp).
Here is how describe the appropriate payoff scheme. Prior to the experiment (but after learning the details) Beauty makes a wager with the Prince. If the coin comes up heads the Prince will pay Beauty $10. If it comes up tails Beauty will pay $10. Even odds. This wager represents Beauty’s prior belief that the coin is fair and head/tails have equal probability: her credence that heads will or did come up. At any point before Beauty learns what day of the week it is she is free alter the bet such that she takes tails but must pay $10 more dollars to do so (making the odds 2:1).
Beauty should at no point (before learning what day of the week it is) alter the wager. Which means when she is asked what her credence is that the coin came up heads she should continue to say 1⁄2.
This seems at least as good an payoff interpretation as a new bet every time Beauty is asked about her credence.
You don’t measure an agent’s subjective probability like that, though—not least because in many cases it would be bad experimental methodology. Bets made which are intended to represent the subject’s probability at a particular moment should pay out—and not be totally ignored. Otherwise there may not be any motivation for the subject making the bet to give an answer that represents what they really think. If the subject knows that they won’t get paid on a particular bet, that can easily defeat the purpose of offering them a bet in the first place.
This doesn’t make any sense to me. Or at least the sense it does make doesn’t sound like sufficient reason to reject the interpretation.