I’m very busy at the moment, but the short version is that one of my good candidates for a utility component function, c, has, c(A) < c(B) < c(pA + (1-p)B) for a subset of possible outcomes A and B, and choices of p.
This is only a piece of the puzzle, but if continuity in the von Neumann-Morgenstern sense falls out of it, I’ll be surprised. Some other bounds are possible I suspect.
But does doesn’t the money pump result for non-independence rely on continuity? Perhaps I missed something there.
(Of note, this is what happens when I try to pull out a few details which are easy to relate and don’t send entirely the wrong intuition—can’t vouch for accuracy, but at least it seems we can talk about it.)
Actually, I realised you didn’t need continuity at all. See the addendum; if you violate independence, you can be weakly money-pumped even without continuity (though the converse may be false).
Perhaps I’m confused, but I thought that the inequality you described simply refers to a utility function with convex preferences (i.e. diminishing returns).
I agree in general that discontinuity does not by itself entail the ability to be money-pumped—this should be trivially true from utility functions over strictly complementary goods.
Can you elaborate? Maybe there is another solution to your problem than abandoning continuity.
I’m very busy at the moment, but the short version is that one of my good candidates for a utility component function, c, has, c(A) < c(B) < c(pA + (1-p)B) for a subset of possible outcomes A and B, and choices of p.
This is only a piece of the puzzle, but if continuity in the von Neumann-Morgenstern sense falls out of it, I’ll be surprised. Some other bounds are possible I suspect.
Independence fails here. We have B > A, yet there is a p such that (pA + (1-p)B) > B = (pB + (1-p)B). This violates independence for C = B.
As this is an existence result (“for a subset of possible A, B and p...”), it doesn’t say anything about continuity.
Sorry I left this out. It’s a huge simplification, but treat the set of p as a discrete subset set in the standard topology.
And that is discontinuous; but you can model it by a narrow spike around the value of p, making it continuous.
Hum, this seems to imply that the set of p is a finite set...
Still doesn’t change anything about the independence violation, though.
But does doesn’t the money pump result for non-independence rely on continuity? Perhaps I missed something there.
(Of note, this is what happens when I try to pull out a few details which are easy to relate and don’t send entirely the wrong intuition—can’t vouch for accuracy, but at least it seems we can talk about it.)
Actually, I realised you didn’t need continuity at all. See the addendum; if you violate independence, you can be weakly money-pumped even without continuity (though the converse may be false).
Perhaps I’m confused, but I thought that the inequality you described simply refers to a utility function with convex preferences (i.e. diminishing returns).
I agree in general that discontinuity does not by itself entail the ability to be money-pumped—this should be trivially true from utility functions over strictly complementary goods.