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Notes -
Ah but in this construction, I have to pay my stake twice. That's not actually a 3:2 bet, but a 3:4 bet. I'm staking my $20,000 against your $15,000 after all.
Recall that the original formulation was:
If similarly, the bet only happens during the first awakening, then clearly it is in my favor to take the bet, since it's equally likely that the coin came up heads or tails.
If we're betting on every awakening, that means that if I lose on timeline B, I have to pay twice. So I'm not actually betting on 3:2 odds, but on 3:4 odds, which are clearly not in my favor if I believe the odds are 1:1, and I wouldn't take that bet.
I read up a bit on the older thread and I liked the analogy about putting balls in a bag. If the coin toss comes up head, then we put 1 red ball in the bag, if it comes up tail, we put 2 green balls in the bag. Clearly all the balls in the bag are either red or green.
If you ask me: “what is the probability that the balls in the bag are red?” I would say 50%.
If you ask me: “do you take a bet where I pay you $15 if the balls in the bag are red, and you pay me $10 if the balls in the bag are green?” I would take that bet since I know the odds are even and so the expected value is positive ($2.5).
If you ask me: “do you take a bet where I pay you $15 for each red ball in the bag, and you pay me $10 for each green ball in the bag?” I would not take that bet because I know that the probability of the balls being red/green are 1:1, but if they are green, there are 2 of them, so you'll pay me $15 in one case, and I pay you $20 in the other, which is a negative expected value (-$2.5).
So when first awakened Sleeping Beauty ought to believe that the coin came up heads with 50% probability. She also oughtn't take a bet that is resolved on every awakening, knowing that awakenings happen more frequently in timeline B. If she is guaranteed her bet is only resolved on the first awakening, she can take the bet.
You are fighting the hypothetical. This sentence is key:
In order to maintain the logic of the thought experiment, I have to offer you the bet each time you wake up. If I only offered it on the first day, that would give away that it is the first day.
You cannot decide to only take the bet on the first day, because you have no way of knowing when you wake up whether or not it is the first day. Any decision algorithm you implement that would accept the bet on the first day would also accept the bet on the second day, because there is no way within the experimental setup for you to distinguish them.
You can offer it every day, just use the result from the first day to settle the bet on Wednesday, just like the original question asks: what is the credence on the first day, not on all days. If my answer is deterministic it will be the same in all cases anyway. Or we could agree the deposit will be spread equally over all days: $10,000 per day if I'm awakened twice, $20,000 if I'm awakened once.
Of course you don't want this, because you need me to double my stake only if I lose to turn even odds in your favor. And the problematic part is that you then claim this says something about the outcome of the coin toss, rather than stakes of the bet.
The problem can be simplified into a simply $1 to $2 bet. Let's start with this:
You toss a fair coin. If it's heads, you pay me $1. If it's tails, I pay you $1. Except, you have an accomplice that whacks me on the head with a stick as soon as I pay $1, inducing memory loss brief enough that I remember the bet but forgot I already paid up, so that when I recover and notice the coin came up tails, I pay you $1 again. (The stick breaks on first use, so you can only do this once.)
This is the same as the Sleeping Beauty problem. It's rigged against me, not because the probability of the coin coming up tails is less than 50%, but because you've found a way to trick me into paying twice if I lose. In the split second after you flip the coin but before I notice which face is up, I have every reason to insist my credence is 50:50 heads:tails. This is true before and after my memory loss.
Now if you tell me ahead of time, that if I lose, I'll be whacked in the head and made to pay twice, I know that I'm really staking $2 to your $1. Then I propose we skip the head whacking altogether, and if I lose I'll just pay you the $2. Clearly this is exactly equivalent.
Now if you toss the coin, and ask me before revealing the outcome: what is your credence that the coin came up tails? I will still say: 50%, but if you want me to bet on it, you will have to raise your stake to equal mine.
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