[...] Richard Jeffrey is often said to have defended a specific one, namely the ‘news value’ conception of benefit. It is true that news value is a type of value that unambiguously satisfies the desirability axioms.
but at the same time
News value tracks desirability but does not constitute it. Moreover, it does not always track it accurately. Sometimes getting the news that X tells us more than just that X is the case because of the conditions under which we get the news.
And I can see how starting from this you would get that U(⊤)=0. However, I think one of the remaining confusions is how you would go in the other direction. How can you go from the premise that we shift utilities to be 0 for tautologies to say that we value something to a large part from how unlikely it is.
And then we also have the desirability axiom
U(A∨B)=P(A)U(A)+P(B)U(B)P(A)+P(B)
for all A and B such that P(A∧B)=0 together with Bayesian probability theory.
What I was talking about in my previous comment goes against the desirability axiom in the sense that I meant that for X="Sun with probability p and rain with probability (1−p)" in the more general case there could be subjects that prefer certain outcomes proportionally more (or less) than usual such that U(X)≠pU(Sun)+(1−p)U(Rain) for some probabilities p. As the equality derives directly from the desirability axiom, it was wrong of me to generalise that far.
But, to get back to the confusion at hand we need to unpack the tautology axiom a bit. If we say that a proposition ⊤ is a tautology if and only if P(⊤)=1[1], then we can see that any proposition that is no news to us has zero utils as well.
And I think it might be well to keep in mind that learning that e.g. sun tomorrow is more probable than we once thought does not necessarily make us prefer sun tomorrow less, but the amount of utils for sun tomorrow has decreased (in an absolute sense). This comes in nicely with the money analogy because you wouldn’t buy something that you expect with certainty anyway[2], but this doesn’t mean that you prefer it any less compared to some other worse outcome that you expected some time earlier. It is just that we’ve updated from our observations such that the utility function now reflects our current beliefs. If you prefer A to B then this is a fact regardless of the probabilities of those outcomes. When the probabilities change, what is changing is the mapping from proposition to real number (the utility function) and it is only changing with an shift (and possibly scaling) by a real number.
At least that is the interpretation that I’ve done.
So we have that
but at the same time
And I can see how starting from this you would get that U(⊤)=0. However, I think one of the remaining confusions is how you would go in the other direction. How can you go from the premise that we shift utilities to be 0 for tautologies to say that we value something to a large part from how unlikely it is.
And then we also have the desirability axiom
U(A∨B)=P(A)U(A)+P(B)U(B)P(A)+P(B)
for all A and B such that P(A∧B)=0 together with Bayesian probability theory.
What I was talking about in my previous comment goes against the desirability axiom in the sense that I meant that for X="Sun with probability p and rain with probability (1−p)" in the more general case there could be subjects that prefer certain outcomes proportionally more (or less) than usual such that U(X)≠pU(Sun)+(1−p)U(Rain) for some probabilities p. As the equality derives directly from the desirability axiom, it was wrong of me to generalise that far.
But, to get back to the confusion at hand we need to unpack the tautology axiom a bit. If we say that a proposition ⊤ is a tautology if and only if P(⊤)=1 [1], then we can see that any proposition that is no news to us has zero utils as well.
And I think it might be well to keep in mind that learning that e.g. sun tomorrow is more probable than we once thought does not necessarily make us prefer sun tomorrow less, but the amount of utils for sun tomorrow has decreased (in an absolute sense). This comes in nicely with the money analogy because you wouldn’t buy something that you expect with certainty anyway[2], but this doesn’t mean that you prefer it any less compared to some other worse outcome that you expected some time earlier. It is just that we’ve updated from our observations such that the utility function now reflects our current beliefs. If you prefer A to B then this is a fact regardless of the probabilities of those outcomes. When the probabilities change, what is changing is the mapping from proposition to real number (the utility function) and it is only changing with an shift (and possibly scaling) by a real number.
At least that is the interpretation that I’ve done.
This seems reasonable but non-trivial to prove depending on how we translate between logic and probability.
If you do, you either don’t actually expect it or has a bad sense of business.