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kilobug comments on A fungibility theorem - Less Wrong
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Hum, yes, indeed I got the P and V_i backwards, sorry.
The argument still holds, but with the other inversion between the \forall and the \exists :
Having an utility function means the weighting (the c_i) can vary between each individuals, but not between situations. If for each situation ("world history" more exactly) you chose a different set of coefficients, it's no longer an utility function - and you can make about anything with that, just choosing the coefficients you want.
That doesn't work, because
is defined as a mapping from 
to the reals; if you change 
, then you also change 
, and so you can't define them out of order.
I suspect you're confusing
, the individual policies that an agent could adopt, and 
, the complete collection of policies that the agent could adopt.
Another way to express the theorem is that there is a many-to-one mapping from choices of
to Pareto optimal policies that maximize that choice of 
.
[Edit] It's not strictly many-to-one, since you can choose
s that make you indifferent between multiple Pareto optimal basic policies, but you recapture the many-to-one behavior if you massage your definition of "policy," and it's many-to-one for most choices of 
.