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Continuity is no longer needed for these results...
If we're using the Independence II as an axiom, you should be a little more precise, when you introduced it above, you referred to the base four axioms, including continuity.
Now, I only noticed consistency needed to convert between the two Independence formulations, which would make your statement correct. But on the face of things, it looks like you are trying to show a money pump theorem under discontinuous preferences by calling upon the continuity axiom.
The continuity hypothesis really is an unimportant "technical assumption." The only kind of thing it rules out are lexicographical preferences, like if you maximize X, but use Y as a tie-breaker.
Specifically, it follows from independence that if A<B<C, then there is some P so that pA+(1-p)C is better than B for p<P and worse than B for p>P; the only thing the continuity axiom requires is that at P there is no preference between B and the mixture; there is no tie-breaker. (Without the continuity axiom, it may well be that P is 0 or 1.)
This is still true if you only have preferences involving p a rational number: the above is a Dedekind cut. If you restrict p to some smaller set that isn't dense, it's probably bad, but then I'd say you aren't taking probability seriously.
Correct, by definition, if you have a dense set (which by default we treat the probability space as) and we map it into another space than either that space is also dense, in which case the converging sequences will have limits or it will not be dense (in which case continuity fails). In the former case, continuity reduces to point-wise continuity.
Note, setting the limit to "no preference" does not resolve the discontinuity. But by intermediate value, there will exist at least one such point in any continuous approximation of the discontinuous function.
Nice to see Europe catching up with, say India in this regard.
Does that answer your question?
You really, really, really don't want to be touching continuity without knowing exactly what you're doing. See the hyperreals for an example of the sort of thing that happens in this case. Also look at non-measurable functions to see the fun in store.
But most of the time, when people deny continuity, it's not on theoretical grounds but because they have a particular non-continuous preference theory in mind. That's perfectly fine. But generally, the non-continous theory can be approximated arbitrarily well by a continuous version that looks exactly the same in virtually all circumstances.
This has been helpful. I'm much more familiar with the mathematics than the economics. Presently, I'm more worried about the mathematical chicanery involved in approximating a consistent continuous utility function out of things.
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.)
c(A) < c(B) < c(pA + (1-p)B) for a subset of possible outcomes A and B, and choices of p.
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.
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.
In response to
Money pumping: the axiomatic approach
Of note, you don't explain why discontinuous preferences necessarilly cause vulnerability to money pumping.
I'm concerned about this largely because the von Neumann-Morgenstern continuity axiom is problematic for constructing a functional utility theory from "fun theory".
In response to
Open Thread: November 2009
At it's height this poll registered 66 upvotes. As it is meta, no longer useful and not interesting enough for the top comments page please down vote it. Upvote the attached karma dump to compensate.
(It looks like CannibalSmith hasn't been on lately so I'll post this) This post tests how much exposure comments to open threads posted "not late" get. If you are reading this then please either comment or upvote. Please don't do both and don't downvote. The exposure count to this comment will then be compared to that of previous comment made "late". I won't link to the other comment and please don't go finding it yourself.
If the difference is insignificant, a LW forum is not warranted, and open threads are entirely sufficient (unless there are reasons other than exposure for having a forum).
I will post another comment in reply to this one which you can downvote if you don't want to give me karma for the post.
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Mathematically:
Independence + other 3 axioms => Independence II
Independence II => Independence
Hence: ~Independence => ~Independence II
My theorem implies: ~Independence II => You can be money pumped
Hence: ~Independence => You can be money pumped
Note, Independence II does not imply Independence, without using at least the consistency axiom.