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Xodarap comments on A Pure Math Argument for Total Utilitarianism - Less Wrong
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The only part that makes this total utilitarianism is the ranking you match the embedding to. So what, mathematically, goes wrong if you embed the average of your individual numbers into a directed graph like (Very Good) > (Good, Good, Good, Good) ~~ (Good) > (Medium).
I think this is a great question, as people who accept the premises of this article are likely to accept some sort of utilitarianism, so a major result is that average utilitarianism doesn't work.
If we are average utilitarians, then we believe that (2) ~~ (1,2,3). But this must mean that (2,6) ~~ (1,2,3,6) to be order preserving, which is not true. (The former's average utility is 4, the latter's 3.)
Ah, great, I understand more now - the linchpin is the premise that what we really want, is to preserve order when we add another person. So what sort of premise would lead to average utilitarianism?
How about - order should be preserved if we shift the zero-point of our happiness measurement. That seems pretty common-sense. And yet it rules out total utilitarianism. (2,2,2) > (5), but (1,1,1) < (4).
Or maybe we could allow average utilitarianism just by weakening the premise - so that we want to preserve the ordering only if we add an average member.
The usual definition of "zero-point" is "it doesn't matter whether that person exists or not". By that definition, there is no (universal) zero-point in average utilitarianism. (2,2,0) != (2,2) etc.
By the way, it's true you can't shift by a constant in total utilitarianism, but you can scale by a constant/
...Or you could notice that requiring that order be preserved when you add another member is outright assuming that you care about the total and not about the average. You assume the conclusion as one of your premises, making the argument trivial.