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mjwebster comments on Money pumping: the axiomatic approach - Less Wrong
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Comments (93)
This claim is wrong for two reasons.
The usual money pump argument deals only with the difficulty of making a choice when your preferences are cyclic. The VM axioms have nothing to do with it.
Even in the presence of cyclic choices, there could be a maximal element among the convex combinations of the choices, see Peter Fishburn on SSB Utility, or Skew Symmetric Bilinear Utility.
Cyclic choices are what I termed a strict strong money pump. The VM axiom of transitivity forbids this.
I don't really see the relevance of the maximal element. Nowhere did I assume that there were no maximal elements.
@Stuart, you have misunderstood.
There may be a maximal element among the convex combinations of cyclic preferences, when the VM axioms fail to hold. SSB utility axioms have to hold in this case.
This maximal is a good candidate for choice from the cyclic preferences. So the claim that a violation of VM axioms leads to a money pump is false, even in the presence of cyclic preferences.
Read Fishburn's Nonlinear Preference and Utility Theory (1988) or the very recent Essays in Honor of Fishburn, edited by S. Brams et al.
You should probably start your discussion from Merrill Flood's 1952 article on preference cycles, available from Rand. http://www.rand.org/pubs/authors/f/flood_merrill_m.html
If I understand this correctly, you're not saying that you can't be money pumped with cyclic preferences; you're saying that if you start with the maximal, or choose to go to the maximal, then you can no longer be money pumped.
Is this what you are saying?
Page 42-44, Non Linear Preference, "The money-pump concept also reveals a narrow perspective on how choice might be based on preferences, and perhaps a lack of imagination in dealing with cyclic patterns. Although there is no transparent way to make a sensible choice from {p,q,r} when p>q>r>p, nothing prevents a person from considering preferences over the set of convex combinations of p, q, and r. And, if there is a combination in the set, then that persons has an ex ante maximally preferred alternative. As first shown in Kreweras (1961), this indeed can be the case, and we shall consider it later as part of the SSB theory."
Here is a modern paper addressing some of these issues: http://hal.archives-ouvertes.fr/docs/00/08/43/90/PDF/B06008.pdf
Indeed; but I can still money pump you for cyclic preferences if I'm the only trader around, and I only offer you the pure lotteries p, q and r rather than convex combinations. And if you never change your preferences after you realise what I'm doing...
Restrictive conditions, to be sure, but mathematically you can't escape. The fact that you generally do escape implies that something else is going on than your simple preferences.