Honestly, I do not see any unlawful reasoning going on here. First of all, it‘s certainly important to distinguish between a probability model and a strategy. The job of a probability model is simply to suggest the probability of certain events and to describe how probabilities are affected by the realization of other events. A strategy on the other hand is to guide decision making to arrive at certain predefined goals.
My point is, that the probabilities a model suggests you to have based on the currently available evidence do NOT neccessarily have to match the probabilities that are relevant to your strategy and decisions. If Beauty is awake and doesn‘t know if it is the day her bet counts, it is in fact a rational strategy to behave and decide as if her bet counts today. If she knows that her bet only counts on Monday and her probability model suggests that „Today is Monday“ is relevant for H, then ideal rationality requires her to base her decision on P(H/Monday) cause she knows that Monday is realized when her decision counts. This guarantees that on her Monday awakening when her decision counts, she is calculating the probability for heads based on all relevant evidence that is realized on that day.
It is true that the thirder model does not suggest such a strategy, but suggesting strategies and therefore suggesting which probabilities are relevant for decisions is not the job of a probability model anyway. Similar is the case of the Technicolor Beauty: The strategy „only updating if Red“ is neither suggested nor hinted by your model. All your model suggests are probabilities conditional on the realization of certain events. It can’t tell you to treat the observation „Red room“ as a realization of the event „There is an awakening in a red room“ while treating the observation „Blue room“ merely as a realization of the event „There is an awakening in a red or a blue room“ instead of „There is an awakening in a blue room“. The observation of a blue room is always a realization of both of these events, and it is your strategy „tracking red“ and not your probability model that suggests to prefer one over the other as the relevant evidence to calculate your probabilities. I had been thinking over this for a while after I recently discovered this „Updating only if Red“-strategy for myself and how this strategy could be directly derived from the halfer model. But I honestly see no better justification to apply it than the plain fact that it proves to be more successful in the long run.
„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 doesn‘t tell you which probability is relevant for a certain decision. If you can derive a probability from your model and provide a good reason to consider this probability as relevant to your decision, your model is not falsified as long you arrive at the right decision this way.
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 answer 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 deduce this strategy is to calculate expected utilities setting U(Heads)=0 as you would propose. But this is not the only permissible solution. It’s just a different route of reasoning to take into account the experimental condition that your answer „counts“ only if the coin lands Tails.
Technicolor Beauty:
„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.“
I don‘t agree that these probabilities obviously translate into the betting sheme you are proposing. A plausible 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.
This is all your model tells you and if you strictly follow it you will arrive at the final conclusion that the probability of Heads is 1⁄3 in every single experimental run of the Technicolor Beauty version, thus violating the Reflection Principle.
Why is this? Notice that the strategy „update probalitity only if observing Red“ is the best strategy only for an agent who is suffering from memory loss and that an agent whose memory is not erased would not violate Reflection Principle. What‘s the difference between the agent with and without memory loss when applying your model? Well, the agent without memory loss can (when awoken in the course of the experiment) only have total evidence „Red“ (first awakening) or „Red and Blue“ (if there is a second awakening). Or he can have total evidence „Blue“ or „Blue and Red“. But it is impossible for him to have total evidence of both „Red“ and „Blue“ alone within same experimental run, which is only possible for an agent who is affected by memory loss. This makes the agent with memory loss vulnerable for violation of Reflection which he can avoid following your strategy that is basically ignoring total evidence in order to eliminate the effects of memory loss. You can think of your strategy as if Beauty aims to simulate another experiment within the setting of the Technicolor Experiment. In this simulated experiment, she is awoken only if the room is red and so no memory loss occurs. This „simulation approach“ is one way your model can be used to deal with memory loss effects but it does not capture them. It cannot even tell you that they exist. Without knowing that she is affected by memory loss Beauty cannot deduce such a strategy.
„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 argument in the comments.