Initially, I had a strong feeling/intuition that the answer was 1⁄3, but felt that because you can also construct a betting situation for 1⁄2, the question was not decided. In general, I’ve always found betting arguments the strongest forms of arguments: I don’t much care how philosophers feel about what the right way to assign probabilities is, I want to make good decisions in uncertain situations for which betting arguments are a good abstraction. “Rationality is systematized winning” and all that.
Then, I’ve read this comment, which showed me that I made a mistake by accepting the halfer betting situation as an argument for 1⁄2. In retrospect, I could have avoided this by actually doing the math, but it’s an understandable mistake, people have finite time. In particular, this sentence on the Sleeping Beauty Paradox tag page also makes the mistake: “If Beauty’s bets about the coin get paid out once per experiment, she will do best by acting as if the probability is one half.” No, as the linked comment shows, it is advantageous to bet 1:1 in some interpretations, but that’s exactly because the actual probability is 1⁄3. Note: there is no rule/axiom that a bet’s odds should always correspond with the event’s probability, that is something that can be derived in non-anthropic situations assuming rational expected money-maximizing agents. It’s more accurate to call what the above situation points to a scoring rule, you can make up situations with other scoring rules too: “Sleeping Beauty, but Omega will kick you in nuts/vulva if you don’t say your probability is 7⁄93.” In this case it is also advantageous “to behave as if” the probability is 7⁄93 in some respect, but the probability in your mind should still be the correct one.
Thank you for bringing this to my attention. As a matter of fact in the linked comment Radford Neal is dealing with a weak-man, while conveniently assuming that other alternatives “are beyond the bounds of rational discussion”, which is very much not the case.
But it is indeed a decent argument that deserves a detailed rebuttal. And I’ll make sure to provide it in the future.
Initially, I had a strong feeling/intuition that the answer was 1⁄3, but felt that because you can also construct a betting situation for 1⁄2, the question was not decided. In general, I’ve always found betting arguments the strongest forms of arguments: I don’t much care how philosophers feel about what the right way to assign probabilities is, I want to make good decisions in uncertain situations for which betting arguments are a good abstraction. “Rationality is systematized winning” and all that.
Then, I’ve read this comment, which showed me that I made a mistake by accepting the halfer betting situation as an argument for 1⁄2. In retrospect, I could have avoided this by actually doing the math, but it’s an understandable mistake, people have finite time. In particular, this sentence on the Sleeping Beauty Paradox tag page also makes the mistake: “If Beauty’s bets about the coin get paid out once per experiment, she will do best by acting as if the probability is one half.” No, as the linked comment shows, it is advantageous to bet 1:1 in some interpretations, but that’s exactly because the actual probability is 1⁄3. Note: there is no rule/axiom that a bet’s odds should always correspond with the event’s probability, that is something that can be derived in non-anthropic situations assuming rational expected money-maximizing agents. It’s more accurate to call what the above situation points to a scoring rule, you can make up situations with other scoring rules too: “Sleeping Beauty, but Omega will kick you in nuts/vulva if you don’t say your probability is 7⁄93.” In this case it is also advantageous “to behave as if” the probability is 7⁄93 in some respect, but the probability in your mind should still be the correct one.
Thank you for bringing this to my attention. As a matter of fact in the linked comment Radford Neal is dealing with a weak-man, while conveniently assuming that other alternatives “are beyond the bounds of rational discussion”, which is very much not the case.
But it is indeed a decent argument that deserves a detailed rebuttal. And I’ll make sure to provide it in the future.
Please do so in a post, I subscribed to those