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PaperclipPerfector  5d ago (text post)   17728 thread views

Culture War Roundup for the week of November 10, 2025

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Alice has a computer terminal in her room, and the only thing she can do with this computer terminal3 is input into it a single number, her "credence that the coin flip came up heads".

..

I contend that it is obvious that in Variant 1, Alice should still tell Bob that the probability of the coin flip is p, even though she is going to personally bet on heads with probability (1-p)/(p+1). That is, if p=1/2, Alice should bet heads with probability 1/3, but tell Bob that the probability of the coin flip is 1/2.

Forgive me for not being initiated in the lit behind this question, but I'm not following why in variant 1, if Bob is expecting the message to denote Alice's credence for heads and they have mutually consistent methods for deriving it, i.e. (1-p)/(p+1), why Alice would provide anything other than her true credence (which is acknowledged to be invariant based on which wake/day they are in).

i.e.

  1. Alice wakes, knows p_tails, derives her P(H|wake) as (1-p)/(p+1), sends it to the computer
  2. Bob wakes, sees Alice's P(H|wake) on the computer. He knows how he'd derive it from p if he were Alice, so he reverses the calculation to get the coin weighting p_tails
  3. Bob uses the p_tails to derive his own credence for heads (2 wakes per head, heads results at 1-p), i.e. 2(1-p)/(2-p)

For Bob to benefit from being told p_tails instead of Alice's P(H|wake), then Bob must either not be aware that Alice's exposure setup is an inversion of his own, or otherwise believe that Alice will communicate 'true' p_tails instead of her P(H|wake), neither of which seems apparent from the set-up. If Bob expects Alice to input her actual credence and he knows the experiment setup, there's no need for Alice to strategically misreport.

You're perfectly correct. I semi-strategically left this possibility open for Variant 1. That you are able to realize that either can be done means that you adeptly realize that all of these probabilities can be 'things' at the same time. The only thing that matters is that Alice and Bob both know whether Alice is going to put p_tails or P(O_A(H|wake)) (probability of Alice's observation function) into the computer.

...I waited until Variant 3 to add the constraint that Bob doesn't really have a clue what's going on with Alice's observation function, just so that by that point, it became really really clear that we can do whatever it takes to force Alice to give a 'true' (or whatever you want to call it) estimate of p_tails apart from her estimate of what she's going to observe.

EDIT: This is extra important for actually driving home the Wiki description of it being an "ambiguous question". In Variant 1, it's ambiguous which one they're gonna communicate, right!? They have to specify in order to be able to communicate properly!

I think if Alice was specifically directed to input her "credence that the coin flip came up heads" then it's not really ambiguous if everyone is on the same page, as it were. I agree that it's not correct to characterise Alice (or Bob) waking as 'gaining' information, perhaps that's just some Bayesian baggage from Monty Hall or the way the notation is typically used. Alice is fully able to preregister her bets before she falls asleep the first time.

I think if Alice was specifically directed to input her "credence that the coin flip came up heads" then it's not really ambiguous if everyone is on the same page, as it were.

This is actually kind of the core of the problem! The original problem statement, long ago, used this phrase like "credence that the coin flip came up heads". But what does that mean? Obviously, if they all get on the same page and say, "It specifically means this and not that," then there's no ambiguity. But the "ambiguous question" position is saying, "Actually, maybe you need to specify, because maybe there are just multiple different things?"

Is there a definition here of "credence the coin flip came up heads" that is not equivalent to "what is the p*, such that you would bet the coin flip came up heads if given odds (1-p*)/p* or greater"?

Which number in the above examples do you think that is? The one Alice bets, or the one she tells Bob, for him to use to make his bet? Or maybe the one she tells Bob to bet in Variant 2? Which bet? Which version of "came up heads"? The one that you observe some variable number of times? Or, like, "the one true one"?

EDIT: Or even just in your comment. You gave two possibilities. Which one do you think that is?

Alice's bets are neutral EV at odds of 2:1, corresponding to p*=1/3 for a fair coin, yes. Unless I am missing something this is directly analogous to stating that Alice has a credence that the coin flip came up heads of 1/3. Therefore if Alice is directed to communicate their credence to Bob, they would communicate 1/3 (which Bob would understand to be subject to Alice's predetermined pattern of exposure and handle appropriately to derive their credence of heads at 2/3).

The ambiguity only arises if "credence" is allowed to mean something other than Alice's implied probability from her 0-EV betting odds. As I said, not across the formal literature here but that doesn't seem to be the case to me.

edit: perhaps the different probabilities can be better compared if the (fair) coin is flipped on the Sunday before either are put to sleep, and they provide their credence at that point as well. Alice would say: Today I have a credence that the coin came up heads of 1/2. Tomorrow, on waking, I will have a credence that it came up heads of 1/3. Bob will have a credence it came up heads of 2/3. This is no more unusual, mathematically, than if we were to flip the coin today, and ask me tomorrow by mail, if a result of tails today meant you opted for a mail service that was exactly twice as reliable than the service you'd have chosen if the result was heads. Equivalently, my response on receiving the question would be 1/3 and I could pre-register that response with you now.

Alice's bets are neutral EV at odds of 2:1, corresponding to p=1/3 for a fair coin, yes.

What do you mean p=1/3? See, you're back to not specifying what you mean anymore. We already had a value for p. It was 1/2. You had called it p_tails, which was clear. We used it to compute a different value ((1-p)/(p+1) = 1/3), which was being used to make Alice's bets. You had called this latter thing P(H|wake), and I slightly quibbled that I thought it was P(O_A(H|wake)), but in either event, it was clear that it was a different thing from p_tails or "p". It seems like you're using the same mathematical symbol to mean two different things.

Prior comment is assuming a fair coin, so p_tails=0.5, but I've clarified to specify p* as Alice's credence upon waking that the coin result was heads.

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