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Your point is that in the case of the low entropy universe you have much possibilities for the time to consider for its random formation compared to the single brain?

Can we estimate the probability of this 3rd hypothesis or even compare it with the probability of the other two?

Itt seems to me that it is actually easy to define a function $u'(...)>=0$ such that the preferences are represented by $E(u'^2)$ and not by $E(u')$: just take u'=sqrt(u), and you can do the same for any value of the exponent, so the expectation does not play a special role in the theorem, you can replace it with any $L^p$ norm.

There are infinitely many ways to find utility functions that represents preferences on outcomes, for example if outcomes are monetary than any increasing function is equivalent  on outcomes but not when you try to extend it to distributions and lotteries with the expected value.
I wander if given a specific function u(...) on every outcome you can also chose "rational" preferences (as in the theorem) according to some other operator on the distributions that is not the average, for example what about the L^p norm or the sup of the distribution (if they are continuous)?
Or is the expected value the special unique operator that have the propety stated by the VN-M theorem?

You don't necessarily need to start from the preference and use the theorem to define the function, you can also start from the utility function and try to produce an intuitive explanation of why you should prefer to have the best expected value

Thank you for your insight. The problem with this view of utility "just as a language" is that sometimes I feel that the conclusion of utility maximization are not "rational" and I cannot figure out why they should be indeed rational if the language is not saying anything that is meaningful to my intuition.

Very interesting observations. I woudln't say the theorem is used to support his assumption because the assumptions don't speak about utils but only about preference over possible outcomes and lotteries, but I see your point.

Actually the assumptions are implicitly saying that you are not rational if you don't want to risk to get a 1'000'000'000'000$ debt with a small enough probability rather than losing 1 cent (this is strightforward from the archimedean property).

Ok we have a theorem that says that if we are not maximizing the expected value of some function "u" then our preference are apparently "irrational" (violating some of the axioms). But assume we already know our utility function before applying the theorem, is there an argument that shows how and why the preference of B over A (or maybe indifference) is irrational if E(U(A))>E(U(B))?

Apparently the axioms can be considered to talk about preferences, not necessarily about probabilistic expectations. Am I wrong in seeing them in this way?

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It seems indeed quite reasonable to maximize utility if you can choose an option that makes it possible, my point is why you should maximize expected utility when the choice is under uncertainty

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