The reason you don’t usually accept a bet which you can only accept at most once is that you have a 2/3s credence the coin is tails, but given that the coin is tails there’s a 50% chance the bet is fake (since you’ve already accepted it the first time you woke up). This halves your expected earnings, so the bet is net negative.
But this gives you a way to break the symmetry. You only take the bet if the room is blue, which means your expected takings are now = 1⁄3 * -$200 + 2⁄3 * $100 = $0. So you still don’t take the bet, but would if slightly better odds are available.
Taking a halfer position:
When you wake up and see the wall is blue, your credence that the coin was tails must remain %50 (by conservation of expected evidence—you knew the wall would be either red or blue before you opened your eyes). So why should you take the bet?
I think this is much easier to analyze if you think about your plans before the experiment starts, like on Sunday. In fact, let’s pretend we are going to write down a game plan on Sunday, and we will simply consult that plan wherever we wake up and do what it says. This sidesteps the whole half vs third debate, since both sides agree about how things look better the experiment begins.
Furthermore, let’s say we’re going to participate in this experiment 100 times, just so I don’t have to deal with annoying fractions.
Now, consider the following tentative game plan: agree to the bet if and only if the room is red. Let’s see what happens.
Out of 100 experiments, 50 will result in just one awakening. In 25 of them you will refuse the bet (costing you zero dollars), and in the other twenty five you will accept, which costs you $7500. So far so good. (Actually pretty bad, since we just lose lots of money.)
The other 50 will result in two awakenings. Here, we don’t need to worry about probabilities anymore. It is guaranteed we will see a red room once and a blue room once. Thus, we will agree to the bet once, and thus the bet will be in effect. So we will win $200 fifty times, for a total of $10k. Once we subtract what we lost when the coin landed for one awakening, our net profit is $2500, or an average of $25 dollars per experiment.
Taking a thirder position:
The reason you don’t usually accept a bet which you can only accept at most once is that you have a 2/3s credence the coin is tails, but given that the coin is tails there’s a 50% chance the bet is fake (since you’ve already accepted it the first time you woke up). This halves your expected earnings, so the bet is net negative.
But this gives you a way to break the symmetry. You only take the bet if the room is blue, which means your expected takings are now = 1⁄3 * -$200 + 2⁄3 * $100 = $0. So you still don’t take the bet, but would if slightly better odds are available.
Taking a halfer position:
When you wake up and see the wall is blue, your credence that the coin was tails must remain %50 (by conservation of expected evidence—you knew the wall would be either red or blue before you opened your eyes). So why should you take the bet?
I think this is much easier to analyze if you think about your plans before the experiment starts, like on Sunday. In fact, let’s pretend we are going to write down a game plan on Sunday, and we will simply consult that plan wherever we wake up and do what it says. This sidesteps the whole half vs third debate, since both sides agree about how things look better the experiment begins.
Furthermore, let’s say we’re going to participate in this experiment 100 times, just so I don’t have to deal with annoying fractions. Now, consider the following tentative game plan: agree to the bet if and only if the room is red. Let’s see what happens. Out of 100 experiments, 50 will result in just one awakening. In 25 of them you will refuse the bet (costing you zero dollars), and in the other twenty five you will accept, which costs you $7500. So far so good. (Actually pretty bad, since we just lose lots of money.) The other 50 will result in two awakenings. Here, we don’t need to worry about probabilities anymore. It is guaranteed we will see a red room once and a blue room once. Thus, we will agree to the bet once, and thus the bet will be in effect. So we will win $200 fifty times, for a total of $10k. Once we subtract what we lost when the coin landed for one awakening, our net profit is $2500, or an average of $25 dollars per experiment.