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# gjm comments on A summary of Savage's foundations for probability and utility. - Less Wrong

33 22 May 2011 07:56PM

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Comment author: 22 May 2011 11:04:07PM 2 points [-]

This is very nice.

One point I find less than perfectly convincing: the motivation of the "total" part of P1 by saying that if our preorder were partial then we'd have two different kinds of indifference.

First off, I don't see anything bad about that in terms of mathematical elegance. Consider, e.g., Conway's beautiful theory of numbers and (two player perfect-information) games, in which the former turn out to be a special case of the latter. When you extend the <= relation on numbers to games, you get a partial preorder with, yes, two kinds of "indifference". One means "these two games are basically the same game; they are interchangeable in almost all contexts". The other means "these are quite different games, but neither is unambiguously a better game to find yourself playing".

This sort of thing also seems eminently plausible to me on (so to speak) psychological grounds. Real agents do have multiple kinds of indifference. Sometimes two situations just don't differ in any way we care about. Sometimes they differ a great deal but neither seems clearly preferable.

It would probably be much harder to extract von Neumann / Morgenstern from a version of the axioms with P1 weakened to permit non-totality. But I wonder whether what you would get (perhaps with some other strengthenings somewhere) might end up being a better match for real agents' real preferences.

(That would not necessarily be a good thing; perhaps our experience of internal conflicts from multiple incommensurable-feeling values merely indicates a suboptimality in our thinking. After all, agents do have to decide what to do in any given situation.)

Comment author: 23 May 2011 04:01:29PM 1 point [-]

I basically agree with you on this. Savage doesn't seem to actually justify totality much; that was my own thought as I was writing this. The real question, I suppose, is not "are there two flavors of indifference" but "is indifference transitive", since that's equivalent to totality. I didn't bother talking about totality any further because, while I'm not entirely comfortable with it myself, it seems to be a standard assumption here.

I'll add a note to the post about how totality can be considered as transitivity of indifference.

Comment author: 23 May 2011 07:12:23PM 1 point [-]

Yes, that's a good way of looking at it.

If we (1) look at the way our preferences actually are and (2) consider "aargh, conflict of incommensurable values, can't decide" to be a kind of indifference, then indifference certainly isn't transitive. But, again, maybe we'd do better to consider idealized agents that don't have such confusions.

Comment author: 24 May 2011 12:05:50AM 1 point [-]

In particular because agents which do have such confusions should leave money on the table -- they are incapable of dutch-booking people who can be dutch-booked.

Comment author: 24 May 2011 12:46:42AM 1 point [-]

How so? (It looks to me as though the ability to dutch-book someone dutch-book-able doesn't depend at all on one's value system. In particular, the individual transactions that go to make up the d.b. don't need to be of positive utility on their own, because the dutch-book-er knows that the dutch-book-ee is going to be willing to continue through to the end of the process. I think. What am I missing?)

Comment author: 24 May 2011 02:11:25AM 2 points [-]

Hmm. That's not quite the right description of the illogic but something very odd is going on:

Suppose I find A and B incomparable and B and C incomparable but A is preferable to C.

Joe is willing to trade C=>B and B=>A.

I trade C into B knowing that I will eventually get A.

Then, I refuse to trade B to A!

But if Joe had not been willing to trade B=>A, I would not have traded C=>B!