The bet is obviously in the scientist’s favour: there is an objective 50% chance that the scientist pays 3 extra gold, and an equal chance that they recover 4 gold. This doesn’t depend upon any subjective probabilities. Given that it’s a sucker bet and you don’t know who the sucker is, avoid it. This is only intensified since the duplicate probably has a much greater future utility for gold than the original who probably owns other property, has income, etc. Even if the utility is linear in both cases (risk-neutral), the slopes are likely different.
In the P(Heads has been flipped) = 1⁄2 case, yes the Bayesian update is to P(Heads has been flipped | I am the original) = 2⁄3 assuming you make a bunch of other reasonable assumptions. But note that this is specific to the conditions of the problem as stated. If the original will always be woken before the coin flip, then the true value of P(Heads has been flipped | I am the original) is identically 0. The coin hasn’t been flipped, so it certainly hasn’t been flipped to heads! Their estimate of the probability is mistaken because the scientist lied, which is not that surprising for a mad scientist and shouldn’t have been assumed impossible in the first place.
If you change the scenario, then you should go through and change the appropriate probability models.
I don’t see why owning other property would change the objective of having more gold. You are using gold to bet gold, where do other properties come into play? Nonetheless, if it bothers you just let’s just assume the subject has no other wealth other than what’s no him. Does that mean you still would not enter the bet?
The mad scientist does not need to lie. The experiment is changed to: 1. Sleep, 2.Scan, 3.Wake up the Orignal, 4. The original tosses the coin, 5. If tails create the Clone. 6. Wake up the Clone so he has indiscernible experience as the Orignal in 3. This whole process is disclosed to you. Now after waking up the coin may or may not have been tossed, what is the probability of Heads? What is the probability if you are the original? If you say they are both 1⁄2 then what is the probability that you are the clone?
Perhaps what confuses me the most is that you are arguing for both thirders and Halfers at the same time. If you think halfers are correct to say the probability of Heads is 1⁄2 wouldn’t you have taken the bet? If you think thirders are correct won’t you say the probability should be updated according to the standard Bayes rule? Why are you arguing against both of these? What’s your position?
If I am the original and have a job, house, 10000 gold in the bank, etc then I don’t need another 2 gold to buy shelter, food, etc. Being given 2 gold is vaguely nice, I can buy myself a nice book on decision theory or something. With 5 gold I could buy a book on decision theory and a concert ticket.
A duplicate with no legal claim to any property might need 2 gold to have a good chance to even survive. While 5 gold would give an even better chance of survival, only the original can win the extra 3 gold so risk-neutrality in the duplicate case is irrelevant.
If I’m unsure on whether or not I’m the duplicate, I’m not going to stake a nice book against years of life expectancy even if the odds were 100:1.
Now after waking up the coin may or may not have been tossed, what is the probability of Heads?
We’re using a model here analogous to the Sleeping Beauty 1⁄2 model right?
P(Heads | Observations upon awakening) = P(Heads) = 1⁄2 since the observations are stipulated to be identical and therefore independent of Heads.
What is the probability if you are the original?
That’s not actually an event in this space, and therefore has no measure. The measurable events in this space are just the sigma algebra generated by “Heads” and some set of possible “Observations on awakening”.
The bet is obviously in the scientist’s favour: there is an objective 50% chance that the scientist pays 3 extra gold, and an equal chance that they recover 4 gold. This doesn’t depend upon any subjective probabilities. Given that it’s a sucker bet and you don’t know who the sucker is, avoid it. This is only intensified since the duplicate probably has a much greater future utility for gold than the original who probably owns other property, has income, etc. Even if the utility is linear in both cases (risk-neutral), the slopes are likely different.
In the P(Heads has been flipped) = 1⁄2 case, yes the Bayesian update is to P(Heads has been flipped | I am the original) = 2⁄3 assuming you make a bunch of other reasonable assumptions. But note that this is specific to the conditions of the problem as stated. If the original will always be woken before the coin flip, then the true value of P(Heads has been flipped | I am the original) is identically 0. The coin hasn’t been flipped, so it certainly hasn’t been flipped to heads! Their estimate of the probability is mistaken because the scientist lied, which is not that surprising for a mad scientist and shouldn’t have been assumed impossible in the first place.
If you change the scenario, then you should go through and change the appropriate probability models.
I don’t see why owning other property would change the objective of having more gold. You are using gold to bet gold, where do other properties come into play? Nonetheless, if it bothers you just let’s just assume the subject has no other wealth other than what’s no him. Does that mean you still would not enter the bet?
The mad scientist does not need to lie. The experiment is changed to: 1. Sleep, 2.Scan, 3.Wake up the Orignal, 4. The original tosses the coin, 5. If tails create the Clone. 6. Wake up the Clone so he has indiscernible experience as the Orignal in 3. This whole process is disclosed to you. Now after waking up the coin may or may not have been tossed, what is the probability of Heads? What is the probability if you are the original? If you say they are both 1⁄2 then what is the probability that you are the clone?
Perhaps what confuses me the most is that you are arguing for both thirders and Halfers at the same time. If you think halfers are correct to say the probability of Heads is 1⁄2 wouldn’t you have taken the bet? If you think thirders are correct won’t you say the probability should be updated according to the standard Bayes rule? Why are you arguing against both of these? What’s your position?
If I am the original and have a job, house, 10000 gold in the bank, etc then I don’t need another 2 gold to buy shelter, food, etc. Being given 2 gold is vaguely nice, I can buy myself a nice book on decision theory or something. With 5 gold I could buy a book on decision theory and a concert ticket.
A duplicate with no legal claim to any property might need 2 gold to have a good chance to even survive. While 5 gold would give an even better chance of survival, only the original can win the extra 3 gold so risk-neutrality in the duplicate case is irrelevant.
If I’m unsure on whether or not I’m the duplicate, I’m not going to stake a nice book against years of life expectancy even if the odds were 100:1.
We’re using a model here analogous to the Sleeping Beauty 1⁄2 model right?
P(Heads | Observations upon awakening) = P(Heads) = 1⁄2 since the observations are stipulated to be identical and therefore independent of Heads.
That’s not actually an event in this space, and therefore has no measure. The measurable events in this space are just the sigma algebra generated by “Heads” and some set of possible “Observations on awakening”.