I guess my main problem with your approach is that I don’t see a clear rational of which probability to use, or when to interpret it as “I see green” and when to interpret it as “Anyone see green” when both of the statement is based on the fact that I drew a green ball.
For example, my argument is that after seeing the green ball, my probability is 0.9, and I shall make all my decisions based on that. Why not update the pre-game plan based on that probability? Because the pre-game plan is not my decision. It is an agreement reached by all participants, a coordination. That coordination is reached by everyone reasoning objectively, which does not accommodate any any first-person self identification like “I”. In short, when reasoning from my personal perspective, use “I see green”; when reasoning from an objective perspective, use “someone see green”. All my solution (PBR) for anthropic and related questions are based on the exact same supposition of the axiomatic status of the first-person perspective. It gives the same explanation, and one can predict what this theory says about a problem. Some results are greatly disliked by many, like the nonexistence of self-locating probability and perspective disagreement, but those are clearly the conclusion of PBR, and I am advocating it.
You are arguing the two interpretation of “I see green” and “Anyone sees green” are both valid, and which one to use depends on the specific question. But, to me, what exact logic dictates this assignment is unclear. You argue that the bets structured not depending on which exact person gets green, then “my decision” shall be based on “anyone sees green”, it seems to me, a way of simply selecting whichever interpretation that does not yield a problematic result. A practice of fitting theory to results.
To the example I brought up in the last reply, what would you do if you drew a green ball and were told that all participants said yes, you used the probability of 0.9. Rational being you are the only decider in this case. It puzzles me because in exactly what sense “I am the only decider?” Didn’t other people also decide to say “yes”? Didn’t their “yes” contributed to whether the bet would be taken the same way as your “yes”? If you are saying I am the only decider because whatever I say would determine whether the bet would be taken. How is that different from deriving other’s responses by using the assumption of “everyone in my position would have the same decision as I do”? But you used probability of 0.5 (“someone sees green”) in that situation. If you are referring you being the only decider in a causal—counterfactual sense, then you are still in the same position as all other green ball holders. What justifies the change regarding which interpretation—which probability (0.5 or 0.9)—to use?
And also the case of our discussion about perspective disagreement in the other post where you and cousin-it were having a discussion. I, by PBR, concluded there should be a perspective disagreement. You held that there won’t be a probability disagreement, because the correct way for Alice to interpret the meeting is “Bob has met Alice in the experiment overall” rather than “Bob has met Alice today”. I am not sure your rational for picking one interpretation over the other. It seems the correct interpretation is always the one that does not give the problematic outcome. And that to me, is a practice of avoiding the paradoxes but not a theory to resolve them.
This post highlights my problem with your approach: I just don’t see a clear logic dictating which interpretation to use in a given problem—whether it’s the specific first-person instance or any instance in some reference class.
When Alice meets Bob, you are saying she should construe it as “I meet Bob in the experiment (on any day)” instead of “I meet Bob today” because—”both awakening are happening to her, not another person”. This personhood continuity, in your opinion, is based on what? Given you have distinguished the memory erasure problem from the fission problem, I would venture to guess you identify personhood by the physical body. If that’s the case, would it be correct to say you regard anthropic problems utilizing memory erasures fundamentally different from problems with fissures or clones? Entertain me this, what if the exact procedural is not disclose to you, then what? E.g. there is a chance that the “memory erasure” is actually achieved by creating a clone of Alice and wake that clone on Monday, then destroy it. Then wake the original on Tuesday. What would Alice’s probability calculation then? Anything changes if the fissure is used instead of cloning? What would Alice’s probability of Tails when she sees Bob when she is unsure of the exact procedure?
Furthermore you are holding that if saw Bob, Alice should interpret “I have met Bob (on some day) in the experiment”. But if if she didn’t see Bob, she shall interpret “I haven’t met Bob specifically for Today”. In another word, whether to use “specifically today” or “someday” depends on whether or not she sees Bob or not. Does this not seem problematic at all to you?
I’m not sure about what you mean in your example, Beauty is awakened on Monday with 50% chance, if she is awaken then what happens? Nothing? The experiment just ends, perhaps with a non-consequential fair coin toss anyway? If she is not awakened then if the coin toss is Tails then she wakes on Tuesday? Is that the setup? I fail to see there is any anthropic elements in this question at all. Of course I would update the probability to favour Tails in this case upon awakening. Because that is new information for me. I wasn’t sure that I would find myself awake during the experiment at all.