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MendelSchmiedekamp comments on Money pumping: the axiomatic approach - Less Wrong
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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.