The Validity of Self-Locating Probabilities (Pt. 2)

I think we've had this discussion before, but let me try one more time...

You say, "And the "probability I am the Orignal" is not a valid concept. "I" is an identification not based on anything but my first-person perspective. Whether "I" am the Orignal or the Clone is something primitive, not analyzable. Any attempt to justify this probability requires additional postulates such as equating "I" to a random sample of some sort."

But to me, this means throwing out the whole notion of probability as a subjective measure of uncertainty.  Perhaps you're fine with that, but it also means throwing out all use of probability in scientific applications, such as  evaluating the probability that a drug will work and/or have side effects - because the practical use of such evaluations is to conclude that "I" will probably be cured by that drug, which is a statement you have declared meaningless.  Maybe you're assuming that some "additional postulate" will fix that, but if so, I don't see why something similar wouldn't also render the probability that "I" am the Original in your problem meaningful.

I think an underlying problem here is an insistence on overly abstract thought experiments.  You're assuming that the subject of the experiment cannot simply walk out the door of the room they're in and see whether they're in the same place where they went to sleep (in which case they're the Original), or in a different place.  They can also do all sorts of other things, whose effects for good or bad may depend on whether or not they are the Original (before they figure this out). They will in general need some measure of uncertainty in making decisions of this sort - they can't simply say that self-locating probabilities are meaningless, when implicitly they will be using them to decide. This is all true even if they in fact decide to cooperate with the experimenter and do none of this.

The assumption that the experiment must proceed in the manner as it is abstractly described severs all connection between the answers being proposed and the real world.  There is then nothing stopping anyone from proposing that the probability of Heads is 1/2, or 3/4, or 2/7 - since none of these have consequences - and similarly the probability of "I am the Original" can be anything you like, or be meaningless, if you treat the person making the judgement as an abstract entity constrained to do nothing but what they're supposed to do in the problem statement, rather than as a real person.

Probability is the measure of your uncertainty.

If the procedure is to flip a coin, clone me on tails but not on heads, separate the original and the clone if needed, then let me (or us) wake up, then when I wake up I will think the probability that I am the original is 3/4 and that the probability the coin landed on heads is 1/2. If I am then informed that I am the original, I will think that the probability the coin landed on heads is 2/3.

But that can't be right. For this experiment, the Original and the Clone do not have to be waked up at the same time. The mad scientist could wake up the Original first. In fact, the coin can be tossed after it. For dramatic effect, after telling you you are the original, the mad scientist can give you the coin and let you toss it. It seems absurd to say the probability for Heads is anything but 1/2. Why is does the probability for Heads remains unchanged after learning you are the Original?

I don't understand the objection. Yes, I say 1/2, and yes, anything but that seems absurd. This coinflip isn't correlated with whether I was cloned, why should its probability depend on whether I am the original or the clone? In the first situation I believe the pre-experiment coinflip has 50% probability of having landed heads, then after learning some information positively correlated with the actual result being heads, I update to 67% probability of the coin having landed heads. In the dramatic situation I believe the post-experiment coinflip has 50% chance of heads and never learn any information correlated with the result. Zero contradiction here.

P(original) = 2/3
P(heads) = 1/3
P(heads|original) = 1/2
Value of bet is (1/3)*5 = 1.667 < 2, so don't take the bet.

The policy of not betting maximizes the total payoff to all clones (6 vs. 5). The policy of betting maximizes the average payoff-per-clone over worlds (2.5 vs 2). If I don't care at all about my (possible) future clones, I should pre-commit to taking the bet, since this maximizes the payoff of the original. However, after waking up, I have to consider the possibility (with its associated probability) that I am the clone, so that changes my answer.

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.