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army1987 comments on Secrets of the eliminati - Less Wrong
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<nitpick>
You can average but not add elements of an affine space. The average between the position of the tip of my nose and the point two metres west of it is the point one metre west of it, but their sum is not a well-defined concept (you'd have to pick an origin first, and the answer will depend on it).
(More generally, you can only take linear combinations whose coefficients sum to 1 (to get another element of the affine space) or to 0 (to get a vector). Anyway, the values of two different utility functions aren't even elements of the same affine space, so you still can't average them. The values of the same utility function are, and the average between U1 and U2 is U3 such that you'd be indifferent between 100% probability of U3, and 50% probability of each of U1 and U2.)
</nitpick>
Correct but irrelevant. Utility functions are families of mappings from futures to reals, which don't live in an affine space, as you mention.
This looks more like a mention of an unrelated but cool mathematical concept than a nitpick.
My point is that “If you can't add, how can you average?” is not a valid argument, even though in this particular case both the premise and the conclusion happen to be correct.
If I ask "If you can't add, how can you average?" and TimFreeman responds with "by using utilities that live in affine spaces," I then respond with "great, those utilities are useless for doing what you want to do." When a rhetorical question has an answer, the answer needs to be material to invalidate its rhetorical function; where's the invalidity?
I took the rhetorical question to implicitly be the syllogism 'you can't sum different people's utilities, you can't average what you can' t sum, therefore you can' average different people's utilities'. I just pointed out that the second premise isn't generally true. (Both the first premise and the conclusion are true, which is why it's a nitpick.) Did I over-interpret the rhetorical question?
The direction I took the rhetorical question was "utilities aren't numbers, they're mappings," which does not require the second premise. I agree with you that the syllogism you presented is flawed.
Are you sure? The only thing one really wants from a utility function is ranking, which is even weaker a requirement than affine spaces. All monotonic remappings are in the same equivalency class.
It's practically useful to have reals rather than rankings, because that lets one determine how the function will behave for different probabilistic combinations of futures. If you already have the function fully specified over uncertain futures, then only providing a ranking is sufficient for the output.
The reason why I mentioned that it was a mapping, though, is because the output of a single utility function can be seen as an affine space. The point I was making in the ancestral posts was that while it looks like the outputs of two different utility functions play nicely, careful consideration shows that their combination destroys the mapping, which is what makes utility functions useful.
Hence the 'families' comment.
Huh, no. If army1987.U($1000) = shminux.U($1000) = 1, army1987.U($10,000) = 1.9, shminux.U($10,000) = 2.1, and army1987.U($100,000) = shminux.U($100,000) = 3, then then I would prefer 50% probability of $1000 and 50% probability of $100,000 rather than 100% probability of $10,000, and you wouldn't.
I'm hearing an echo of praxeology here; specifically the notion that humans use something like stack-ranking rather than comparison of real-valued utilities to make decisions. This seems like it could be investigated neurologically ....