The Validity of Self-Locating Probabilities

First of all, strong upvote. The points you raised have made me thought hard as well.

I don't think the probability about which room I am in is the same as the self-locating probability. Coincidentally I made my argument use color coding as well ( the probability of I am red or blue). The difference being which color I get labeled is determined by a particular process, the uncertainty is due to the randomness of that process or my lack of knowledge about it. Whereas for self-locating probability, there is nothing random/unknown about the experiment. The uncertainty, i.e. which physical person I am, is not determined by anything. If I ask myself why am I this particular human being? Why am I not Bill Gates? Then the only answer seems to be "Because the available subjective is connected to this person. Because I am experiencing the world from this person's perspective, not of Bill Gates'." It is not analyzable in terms of logic, only be regarded as a reasoning starting point. Something primitive.

Whether or not the questioner knows which person is being referred to by "I" is another interesting matter. Say the universe is infinite, and/or there are countless universes. So there could be many instances of human beings that are physically indistinguishable from me. But does that mean I don't know which one I am? It can be said that I do not know because I cannot provide any discernable details to distinguish myself from all of them. But on the other hand, it can be said I inherently know which is me. I can point to myself and say "I am this person" and call it a day. The physical similarities and differences are not even in the concern. This identification is nothing physical, it is inherently understandable to me because of my perspective. It is because of this primitive nature people consider "the probability of I am the Orginal" as a valid question instead of asking who is this "I" before answering.

My way of rejecting the self-locating probability is incompatible with the Many-Words interpretation. Sean Carroll calls this idea the "simple-mined" objection for the source of probability in Many-Worlds. Yet he admits that's a valid objection. I think treating perspectives as primitives would naturally lead to Copenhagen interpretation. It should also be noted that for Many-Worlds, "I" or "this branch" are still used as primitive notions when self-locating probabilities are derived.

Finally, the self-locating probability is not useful to decision-making. So even as tools, they are not justifiable. Goals such as maximizing the total or average benefit of a group can be determined by using probabilities of random samples from said group. e.g. probability of a randomly selected copy being Original. If the goal is strictly about the primitively identified "I" as in self-locating probability then there exists no valid strategy. As shown by the frequentist analysis in the post.

Yes, rejecting probability and refusing to make predictions about the future is just wrong here, no matter how many fancy primitives you put together.

I disagree that standard LW rejects that, though.

Of course, whether I am the Orignal or the Clone is a matter of fact. There is a definitive answer to it. I also see no problem saying "the probability I am the Original" is essentially the same as " What is the probability, given that he answers truthfully, that the scientist will say I am the original when I ask? ".

But does being a matter of fact imply there is a probability to it? Subsequently, what is the justification for 50% being the correct answer?

The Orignal/Clone is referring to the two physical persons in the experiment. One is a physical copy that existed before, the other created by mad scientists during the experiment. You can change the Original to "the causal descendant of the Original, that in a world without the mad scientist I would still exist?". But I don't think that's significant. Because the question does not depend on that. 

To illustrate this we can change the experiment. Instead of a direct cloning process, now the mad scientist will split you through the middle into two halves: the left part (L), and the right (R). Then he will complete the two by cloning the missing half onto them. So we still end up with two indiscernible copies. L and R. Now after waking up the second day, you can ask yourself  "what is the probability that I am L?". It is still a self-locating probability. And I thought about using this example in the post since it is more symmetrical. I ended up against it because it seems too exotic. 

Still the same. All I can say is I am either the Orignal or the Clone. For the credence of each is still "I don't know".

And this number-crunching goes both ways. Say if the Mad scientist is only producing valid Clone in 1% of the experiments. However if he successes, he will produce 1 Million of them. Then what is the probability of me being the Original? I assume people would say close to 0. 

This logic could lead to some strange actions such as Brain-Race as described by Adam Elga here. You could force someone to act to your liking by make 100s of his Clones with the same memory. For if he doesn't do so, you will torture all these clones. Then the best strategy for him is to play ball because he is most likely a clone. However, he could counter that by making 1000s of Clones of himself that will be tortured if they act to your liking. But you could make 100000s of Clones, and he could make 10000000s, etc. 

Our behavior should be different in many cases. However, base on my past experience, people who accept self-locating probabilities would often find various explanations so our decisions would still be the same. 

For example, in "Repeating the Experiment" the relative frequency of Me being the Original won't converge on any particular value. If we bet on that, I will say there is no strategy to maximize My personal gain. (There is a strategy to max the combined gain of all copies if everyone abides by it. As reflected by the probability of a randomly sampled copy being Original is 1/2)

On the other hand, you would say if I repeat the experiment long enough the relative frequency of me being the Original would converge on 50%, and the best strategy to max my personal gain is to bet accordingly. 

The problem of this example is that personal gain can only be verified by the first-person perspective of the subject. A verifiable example would be this: change the original experiment slightly. The Mad scientist would only perform the cloning if a fair coin toss landed on Tails. Then after waking up how should you guess the probability of Heads? What's the probability of Heads if you learn you are the Original? (Essentially the sleeping beauty problem).

If you endorse self-locating probability, then there are two options. First, the thirder. After waking up the probability of I am the Original is 2/3. The probability of Heads is 1/3. After learning I am the Original the probability of Heads updates to 1/2. 

The other option is to say after waking the probability of Heads is 1/2, the probability of I am the Original is 3/4. After learning I am the Orignal the probability of Heads needs to be updated. (How to do this update is very problematic, but let's skip it for now. The main point is the probability for Heads would have to be smaller than 1/2. And this is a very weak camp compare to the thirders)

Because I reject self-locating probability, I would say the probability of Heads is 1/2. And it is still 1/2 after learning I am the Original. No update because there is no probability in the first place. 

This should result in different betting strategies. Say you have just experienced 100 iterations of this toss and cloning and haven't learned whether you were the Orignal or the Clone in any of those iterations. Now you are offered to enter a bet for 2 dollars that will pay 5 dollars if the coin landed on Heads for each of those 100 iterations. Assuming you are a thirder, then you should not enter these bets, since you believe the probability of Heads is only 1/3. Whereas I would enter all these bets. But again, base on past experience thirders would come up with some explanation as to why they would also enter these bets. So our decisions would still be the same. 

If you think 1/2 is a valid probability in its own model. I would assume you are also interested in the probability update rule of this model, i.e. how can Beauty justify the probability of Heads to be 1/2 after learning it is Monday.