What does Arrow's theorem say about Kaldor-Hicks social welfare functions with von Neumann-Morgenstern utility? Let $A$ be the set of all possible states of the world, let $G(A)$ be the set of all "lotteries" or "gambles", i.e. the set of all probability distributions over $A$.  Now consider an individual with a preference ordering of the various lotteries in $G(A)$.  Then the von Neumann-Morgenstern theorem states that, assuming the individual's preferences obeys certain rationality conbditions, there exists a function $u: A \rightarrow \mathbb{R}$, such that the individual's preference ordering maximizes the expected value of $u$.  Moreover, the function $u$ is unique up to linear transformations, i.e. maximizing the expected value of $u$ and maximizing the expected value of $a + bu$ yield equivalent results.
Now consider a society with N individuals, where each individual's preferences obey the von Neumann Morgenstern axioms.  Then we can define a social welfare function $W = a_1u_1 + a_2u_2 + ... + a_Nu_N$, where $u_i$ is the von Neumann-Morgenstern utility function for the $i^{\textrm{th}}$ individual, and $a_i$ is the reciprocal of the marginal utility of money for the $i^{\textrm{th}}$ individual.  As shown in this thread, $W$ is well-defined, because it's invariant under linear transformations of the $u_i$'s.  More importantly for our purposes, it is my understanding that maximizing $W$ will achieve a Kaldor-Hicks optimal result.  (Can someone back me up on this, and preferably tell me where I can find a proof?)
My question is, how does Arrow's impossibility theorem apply to a social preference ordering based on Kaldor-Hicks efficiency?  Specifically, given two outcomes in $A$, what would happen if we let the social ordering prefer the outcome that has a greater value of W?  Arrow's theorem, as usually stated, is about rules that are maps from $L(A)^N$ to $L(A)$, i.e. rules that take each individual's preference ordering on $A$, and then spit out a social preference ordering on $A$.  ($L(A)$ is the set of linear orders on the set $A$.)
But the rule I'm describing is not just based on each individual's preference ordering on $A$ (their preferences for certain outcomes), but on their von Neumannn-Morgenstern utility function $u$, i.e. on their preference ordering on $G(A)$ as well (their preferences under uncertainty).  So are there generalizations of Arrow's theorem that deal with maps from $L(G(A))^N$ to either $L(G(A))$ or failing that, maps from $L(G(A))^N$ to $L(A)$, as is the case with the rule I'm describing?  If an extension of Arrow's theorem does apply, what does it say about this rule?  What conditions does the rule obey or not obey? 
Any help would be greatly appreciated.
Thank You in Advance.
 A: There exists adaptions of Arrow's theorem to von Neumann-Morgenstern preferences. see for example Theorem 4.3 here.
the weighted utilitarianism you propose violates the independence axiom, and one can multiply each of the utility functions by some positive number. This changes the SWF, but not the preferences over lotterie represented.
There is an extensive literature on informational requirements and utility comparisons in social choice developed by Gevers, Sen, and others. A fairly comprehensive survey is Social welfare functionals and interpersonal comparability by d'Aspremont and Gevers. For a mor leisurely overview, see chapter 1 of Theories of Distributive Justice by Roemer.  
A: First, Arrow's Theorem says that no map 
$$L(A)^n\rightarrow L(A)$$
can simultaneously satisfy a certain list of properties.
You are trying, more or less, to construct a counterexample where $A$ is replaced by $G(A)$.
There are (at least) two problems with your idea.
1)  You are not constructing a map
$$L(G(A))^n\rightarrow L(G(A))$$
Instead you are constructing a map
$$V^n\rightarrow L(G(A))$$
where $V\subset L(G(A))$ consists of those linear orders that satisfy the vonNeumann-Morgenstern axioms.   Therefore there's no reason Arrow's Theorem should apply.
2)  Your proposal is ill-defined because it gives me no clue how to normalize the functions $u_i$.  (I have no idea what "marginal utility of money" means in the general context of lotteries over states of the world.)  You can set the weights $a_i$ arbitrarily, but you still need a map from preferences to utility functions (which are, a priori, well-defined only up to affine transformations).  What makes you think you can construct that map in such a way that your proposed mechanism will satisfy all of the Arrow conditions?  In particular, what makes you think that any such map will satisfy the independence of irrelevant alternatives?  
