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AlexMennen comments on Harsanyi's Social Aggregation Theorem and what it means for CEV - Less Wrong Discussion
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Unlikely, unless there are at least as many agents as outcomes.
Yes. In fact, I think something like that will be necessary. For example, suppose there is a population of two agents, each of which has a "hedon function" which specifies their agent-centric preferences. One of the agents is an egoist, so his utility function is his hedon function. The other agent is an altruist, so his utility function is the average of his and the egoist's hedon functions. If you add up the two utility functions, you find that the egoist's hedon function gets three times the weight of the altruist's hedon function, which seems unfair. So we would want to give extra weight to the altruist's utility function (you could argue that in this example you should use only the altruist's utility function).
It may contain negative elements.
It's unlikely that the weights of existing agents would change under either of those cases, or that the multiplication could be expressed as a weighted sum, or that the multiplication would have axiom 2-ness?
Indeed. The problem is more general- I would classify the parts as "internal" and "external," rather than agent-centric and other, because that makes it clearer that agents don't have to positively weight each other's utilities. If you have a 'maltruist' whose utility is his internal utility minus the egoist's utility (divided by two to normalize), we might want to balance their weight and the egoist's weight so that the agents' internal utilities are equally represented in the aggregator.
Such meta-weight arguments, though, exist in an entirely different realm from this result, and so this result has little bearing on those arguments (which is what people are interested in when they resist the claim that social welfare functions are linear combinations of individual utility).
Ah! Of course.
Unlikely that the multiplication could be expressed as a weighted sum (and hence by extension, also unlikely it would obey axiom 2).
I agree in general, because we would need the left inverse of the combined linearly independent individual utilities and e, and that won't exist. We do have freedom to affinely transform the individual utilities before taking their element-wise product, though, and that gives us an extra degree of freedom per agent. I suspect we can do it so long as the number of agents is at least half the number of outcomes.
Oh, I see what you mean. It should be possible to find some affinely transformed product that is also a linear combination if the number of agents is at least half the number of outcomes, but some arbitrary affinely transformed product is only likely to also be a linear combination if the number of agents is at least the number of outcomes.