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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 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.