“But,” says the causal decision theorist, “to take only one box, you must somehow believe that your choice can affect whether box B is empty or full—and that’s unreasonable! Omega has already left! It’s physically impossible!”
Unreasonable? I am a rationalist: what do I care about being unreasonable? I don’t have to conform to a particular ritual of cognition. I don’t have to take only box B because I believe my choice affects the box, even though Omega has already left. I can just… take only box B.
Similarly, Sleeping Beauty doesn’t have to answer “tails” because she believes that the probability of tails is 1⁄3. She can just… answer “tails”. Beauty is not obligated to believe anything whatsoever.
And to quote the post linked in the parent comment:
But in the original problem, when she is asked “What is your credence now for the proposition that our coin landed heads?”, a much better answer than “.5” is “Why do you want to know?”. If she knows how she’s being graded, then there’s an easy correct answer, which isn’t always .5; if not, she will have to do her best to guess what type of answer the experimenters are looking for; and if she’s not being graded at all, then she can say whatever the hell she wants (acceptable answers would include “0.0001,” “3⁄2,” and “purple”).
The linked post by ata is simply wrong. It presents the scenario where
Each interview consists of Sleeping Beauty guessing whether the coin came up heads or tails. After the experiment, she will be given a dollar if she was correct on Monday.
In this case, she should clearly be indifferent (which you can call “.5 credence” if you’d like, but it seems a bit unnecessary).
But this is not correct. 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.
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. But probability and decision theory have proven to be useful in numerous applications, and the Sleeping Beauty problem is about probability, presumably with the goal of clarifying how probability works, so that we can use it in practice with even more confidence. Saying that she could just make a decision without considering probabilities rather misses the point.
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.
Here’s another reframing of my point—borrowing from one of my favorite essays in the Sequences, “Newcomb’s Problem and Regret of Rationality”, where Eliezer says:
Similarly, Sleeping Beauty doesn’t have to answer “tails” because she believes that the probability of tails is 1⁄3. She can just… answer “tails”. Beauty is not obligated to believe anything whatsoever.
And to quote the post linked in the parent comment:
The linked post by ata is simply wrong. It presents the scenario where
But this is not correct. 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.
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. But probability and decision theory have proven to be useful in numerous applications, and the Sleeping Beauty problem is about probability, presumably with the goal of clarifying how probability works, so that we can use it in practice with even more confidence. Saying that she could just make a decision without considering probabilities rather misses the point.
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.