If you work out the result with standard decision theory, you get indifference between guessing Heads or Tails only if Beauty’s subjective probability of Heads is 1⁄3, not 1⁄2.
I don’t know about “standard decision theory”, but it seems to me that—in the described case (only Monday’s answer matters)—guessing Heads yields an average of 50¢ at the end, and guessing Tails also yields an average of 50¢ at the end. I don’t see that Beauty has to assign any credences or subjective probabilities to anything in order to deduce this.
As the linked post says, you can call this “0.5 credence”. But, if you don’t want to, you can also not call it “0.5 credence”. You don’t have to call it anything. You can just be indifferent.
You are of course right that anyone can just decide to act, without thinking about probabilities, or decision theory, or moral philosophy, or anything else.
My point is that “just deciding to act” in this case actually gets us the result that we want. Saying that probability and decision theory are “useful” is beside the point, since we already have the answer we actually care about, which is: “what do I [Sleeping Beauty] say to the experimenters in order to maximize my profit?”
But the thing is you can’t call it “0.5 credence” and have your credence be anything like a normal probability. The Halfer will assign probability 1⁄2 for Heads and Monday, 1⁄4 for Tails and Monday, and 1⁄4 for Tails and Tuesday. Since only the guess on Monday is relevant to the payoff, we can ignore the Tuesday possibility (in which the action taken has no effect on the payoff), and see that a halfer would have a 2:1 preference for Heads. In contrast, a Thirder would give 1⁄3 probability to Heads and Monday, 1⁄3 to Tails and Monday, and 1⁄3 to Tails and Tuesday. Ignoring Tuesday, they’re indifferent between guessing Heads or Tails.
With a slight tweak to payoffs so that Tails are slightly more rewarding, the Halfer will make a definitely wrong decision, while the Thirder will make the right decision.
I don’t know about “standard decision theory”, but it seems to me that—in the described case (only Monday’s answer matters)—guessing Heads yields an average of 50¢ at the end, and guessing Tails also yields an average of 50¢ at the end. I don’t see that Beauty has to assign any credences or subjective probabilities to anything in order to deduce this.
As the linked post says, you can call this “0.5 credence”. But, if you don’t want to, you can also not call it “0.5 credence”. You don’t have to call it anything. You can just be indifferent.
My point is that “just deciding to act” in this case actually gets us the result that we want. Saying that probability and decision theory are “useful” is beside the point, since we already have the answer we actually care about, which is: “what do I [Sleeping Beauty] say to the experimenters in order to maximize my profit?”
But the thing is you can’t call it “0.5 credence” and have your credence be anything like a normal probability. The Halfer will assign probability 1⁄2 for Heads and Monday, 1⁄4 for Tails and Monday, and 1⁄4 for Tails and Tuesday. Since only the guess on Monday is relevant to the payoff, we can ignore the Tuesday possibility (in which the action taken has no effect on the payoff), and see that a halfer would have a 2:1 preference for Heads. In contrast, a Thirder would give 1⁄3 probability to Heads and Monday, 1⁄3 to Tails and Monday, and 1⁄3 to Tails and Tuesday. Ignoring Tuesday, they’re indifferent between guessing Heads or Tails.
With a slight tweak to payoffs so that Tails are slightly more rewarding, the Halfer will make a definitely wrong decision, while the Thirder will make the right decision.