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.

andmarginal utility of money (which you should probably call utility of money in this context), you do not have a question. – Benoît Kloeckner Dec 11 '13 at 8:54