Novel solution to classic decision-theory problem!
- *Novel solution to classic decision-theory problem!*
- Background: For decades, the Sleeping Beauty puzzle has been one of the most controversial issues in decision theory. Here is a description, courtesy of Wikipedia:
- "Sleeping Beauty volunteers to undergo the following experiment and is told all of the following details: On Sunday she will be put to sleep. Once or twice, during the experiment, Beauty will be awakened, interviewed, and put back to sleep with an amnesia-inducing drug that makes her forget that awakening. A fair coin will be tossed to determine which experimental procedure to undertake: if the coin comes up heads, Beauty will be awakened and interviewed on Monday only. If the coin comes up tails, she will be awakened and interviewed on Monday and Tuesday. In either case, she will be awakened on Wednesday without interview and the experiment ends.
- Any time Sleeping Beauty is awakened and interviewed she will not be able tell which day it is or whether she has been awakened before. During the interview Beauty is asked: 'What is your credence now for the proposition that the coin landed heads?'"
- David Lewis, arguably the most influential philosopher of the 20th century, insisted that the answer is 1/2. Here, I present a novel refutation of this "halfer" position.
- Introducing... ***The Equivalence Argument Against Halfers***
- I. The zero information prior
- First, we reformulate the problem.
- In our version, beauty is first told that the result of the coin flip will not affect how many times she is awoken. Regardless of whether the result is heads or tails, she will be awoken on both Monday and Tuesday and asked the probability of heads.
- No one can dispute that in this scenario, she should assign a 50% probability to the fair coin coming up heads.
- The prior probability of heads was 50%. And beauty has not received any new information about the result of the coin toss.
- Thus, the posterior probability of heads must remain identical to the prior probability of heads: 50%.
- II. The additional information
- In our modified scenario, after beauty makes her 50% estimate, the scientist offers her a hint about the result of the coin flip.
- He says, "if today is Tuesday, then the result of the coin flip was tails."
- This hint indisputably provides new information about the result of the coin flip: Beauty can now exclude the possibility that today is Tuesday and the result of the coin flip was heads.
- Because the hint provides beauty with new information about the result of the coin flip, it is illogical for the probability of heads to remain the same both before and after receiving the hint.
- We need not advance any argument about what the proper posterior probability is after receiving the hint. Without addressing that issue, we can simply rule out the possibility of beauty maintaining the same probability before and after receiving the hint.
- This precludes the halfer position in our scenario: if the posterior probability of heads is 50%, then receipt of the hint does not affect beauty's conclusion. Thus, in our modified scenario, the halfer position is precluded.
- III. The equivalence of the standard and modified scenarios
- We can use our modified scenario to rule out the halfer position in the standard scenario.
- We need only demonstrate that if beauty receives the information available to her in the standard scenario, she can logically deduce the "hint" provided in our scenario.
- Oxford's Nick Bostrom has already done this in his work, so we will not belabor the point. He has shown that knowledge of the standard experimental procedure allows beauty to exclude the possibility that "today is Tuesday and the result of the coin flip was heads."
- Heads and tails are mutually exclusive. Thus, based on this information, beauty can derive the "hint" that "if today is Tuesday, then result of the coin flip was tails."
- This equivalence demonstrates that the probability of heads in the standard formulation of the problem cannot be 50%. That would make the probability of heads identical both before and after receiving the hint, which is untenable.