„Whether or not your probability model leads to optimal descision making is the test allowing to falsify it.“
Sure, I don‘t deny that. What I am saying is, that your probability model don‘t tell you which probability you have to base on a certain decision. If you can derive a probability from your model and provide a good reason to consider this probability relevant to your decision, your model is not falsified as long you arrive at the right decision. Suppose a simple experiment where the experimenter flips a fair coin and you have to guess if Tails or Heads, but you are only rewarded for the correct decision if the coin comes up Tails. Then, of course, you should still entertain unconditional probabilities P(Heads)=P(Tails)=1/2. But this uncertainty is completely irrelevant to your decision. What is relevant, however, is P(Tails/Tails)=1 and P(Heads/Tails)=0, concluding you should follow the strategy always guessing Tails. Another way to arrive at this strategy is to calculate expected utilities setting U(Heads)=0 as you would propose. But this is not the only reasonable solution. It’s just a different route of reasoning to take into account the experimental condition that your decision counts only if the coin lands Tails.
„The model says that P(Heads|Red) = 1⁄3 P(Heads|Blue) = 1⁄3 but P(Heads|Red or Blue) = 1⁄2 Which obviosly translates in a betting scheme: someone who bets on Tails only when the room is Red wins 2⁄3 of times and someone who bets on Tails only when the room is Blue wins 2⁄3 of times, while someone who always bet on Tails wins only 1⁄2 of time.“
A quick translation of the probabilities is:
P(Heads/Red)=1/3: If your total evidence is Red, then you should entertain probability 1⁄3 for Heads.
P(Heads/Blue)=1/3: If your total evidence is Blue, then you should entertain probability 1⁄3 for Heads.
P(Heads/Red or Blue)=1/2: If your total evidence is Red or Blue, which is the case if you know that either red or blue or both, but not which exactly, you should entertain probalitity 1⁄2 for Heads.
If the optimal betting sheme requires you to rely on P(Heads/Red or Blue)=1/2 when receiving evidence Blue, then the betting sheme demands you to ignore your total evidence. Ignoring total evidence does not necessarily invalidate the probability model, but it certainly needs justification. Otherwise, by strictly following total evidence your model will let you also run foul of the Reflection Principle, since you will arrive at probability 1⁄3 in every single experimental run.
Going one step back, with my translation of the conditional probabilities above I have made the implicit assumption that the way the agent learns evidence is not biased towards a certain hypothesis. But this is obviously not true for the Beauty: Due to the memory loss Beauty is unable to learn evidence „Red and Blue“ regardless of the coin toss. This in combination with her sleep on Tuesday if Heads, she is going to learn „Red“ and „Blue“ (but not „Red and Blue“) if Tails while she is only going to learn either „Red“ or „Blue“ if Heads, resulting in a bias towards the Tails-hypothesis.
I admit that P(Heads/Red)=P(Heads/Blue)=1/3, but P(Heads/Red or Blue)=1/2 hints you towards the existence of that information selection bias. However, this is just as little a feature of your model as a flat tire is a feature of your car because it hints you to fix it. It is not your probability model that guides you to adopt the proper betting strategy by ignoring total evidence. In fact, it is just the other way around that your knowledge about the bias guides you to partially dismiss your model. As mentioned above, this does not necessarily invalidate your model, but it shows that directly applying it in certain decision scenarios does not guarantee optimal decisions but can even lead to bad decisions and violating Reflection Principle.
Therefore, as a halfer, I would prefer an updating rule that takes into account the bias and telling me P(Heads/Red)=P(Heads/Blue)=P(Red or Blue)=1/2. While offering me the possibility of a workaround to arrive at your betting sheme. One possible workaround is that Beauty runs a simulation of another experiment within her original Technicolor Experiment in which she is only awoken in a Red room. She can easily simulate that and the same updating rule that tells her P(Heads/Red)=1/2 for the original experiment tells her P(Heads/Red)=1/3 for the simulated experiment.
„This leads to a conclusion that observing event “Red” instead of “Red or Blue” is possible only for someone who has been expecting to observe event “Red” in particular. Likewise, observing HTHHTTHT is possible for a person who was expecting this particular sequence of coin tosses, instead of any combination with length 8. See Another Non-Anthropic Paradox: The Unsurprising Rareness of Rare Events“
I have already refuted this way of reasoning in the comments of your post.
If this were true that the concept of „indexical sample space“ does not capture the thirder position, how do you explain that it produces exactly the same probabilities that thirders entertain? Operating with indexicals is a necessary condition (and motivation) for Thirdism, which means assuming indexical sample spaces when it comes to the mathematical formalization of arguments in terms of probability theory. To my knowledge no relevant thirder literature denies that. And within the thirder model, these probabilities indeed hold true. If we assume Monday and Tuesday to be mutually exclusive, then this is mathematically the case. Math is not a judge of our assumptions here, it is merely the executive organ which in this case produces thirder probabilities. The point at issue is whether the theoretical assumptions of the thirder model fit reality and probabilities could be transfered into the real world. Thirders say yes, speaking of regular probabilities, halfers say no speaking of irregular, „weighted“ probabilities.